M. Verbitsky D. Kaledin

NEW EXAMPLES OF HYPERKÄHLER MANIFOLDS Preface

Hyperkähler manifolds have appeared at first within the framework of differential geometry as an example of Riemannian manifolds with holonomy of a special restricted type. However, they have soon exhibited such diverse and unexpected links with various branches of mathematics that now hyperkähler geometry by itself forms a separate research subject. Among the traditional areas fused within this new subject are differential and algebraic geometry of complex manifolds, holomorphic symplectic geometry, geometric representation theory, Hodge theory and many others. The most recent addition to the list is the link between hyperkähler geometry and theoretical physics: it turns out that hyperkähler manifolds play a critical part in the modern version of the string theory, which is in itself the basis for the future unified field theory and quantum gravity.

Perhaps because the structure of a hyperkähler manifold is so rich, such manifolds are quite rare and hard to construct. Thus every new example or a class of examples of hyperkähler manifolds is of considerable interest. The main goal of this book is to describe two recent developments in this area, one dealing with compact hyperkähler manifolds, the other - with a rather general class of non-compact ones. In order to make the presentation as self-contained as possible, we have included much preliminary material and gave an exposition of most of the basic facts of the theory. We believe that this makes it possible to read the book without any prior knowledge of hyperkähler geometry and to use it as an introduction to the subject. On the other hand, it is our hope that the new examples of hyperkähler manifolds constructed here would be of interest to a specialist in the field.

For the detailed description of the new results proved in the book the reader is referred to the introductions to the individual chapters. In this general introduction we restrict ourselves to giving a brief historical overview of the theory of hyperkähler manifolds and indicating the place of our results in the general framework of hyperkähler geometry.

Historical overview

Recall that one can define a Kähler manifold as a Riemannian manifold M equipped with an almost complex structure parallel with respect to the Levi-Civita connection. It is well-known that such an almost complex structure is automatically integrable, thus every Kähler M is a complex manifold. Moreover, the Riemannian metric and the complex structure together define a non-degenerate closed 2-form ω on M, thus making M a symplectic manifold.

The notion of a hyperkähler manifold is obtained from this definition by replacing the field of complex numbers with the algebra of quaternions. A hyperkähler manifold is by definition a Riemannian manifold M equipped with two anticommuting almost complex structures parallel with respect to the Levi-Civita connection. These two almost complex structures generate an action of the quaternion algebra in the tangent bundle to M, which is also parallel. Every quaternion h with h^{2} = −1 defines an almost complex struture on M. This almost complex structure is parallel, hence integrable. Thus every hyperkähler manifold is canonically complex, and in many different ways.

It is convenient to fix once and for all a quaternion I with I^{2} = −1 and to consider a hyperkähler manifold M as complex by means of the corresponding complex structure. It is canonically Kähler. Moreover, one can combine the other complex structures on M with the Riemannian metric and obtain, apart from the Kähler 2-form ω, a canonical closed non-degenerate holomorphic 2-form Ω on M. Thus every hyperkähler manifold carries canonical Kähler and holomorphically symplectic structures.

A Riemannian manifold of dimension 4n is hyperkähler if and only if its holonomy group is contained in the symplectic group Sp(n). As such, hyperkähler manifolds first appeared in the classification of all possible holonomy groups given by M. Berger [Ber]. The term “hyperkähler manifold” was introduced by E, Calabi in his paper [C], where he also constructed several non-trivial examples of hyperkähler metrics. All of Calabi’s examples were non-compact. In fact, all these manifolds were total spaces of cotangent bundles to Kähler manifolds.

At the time of the original paper of Calabi’s, it seemed that hyperkähler manifolds are a rather unusual phenomenon, not unlike, for example, sporadic simple groups. However, starting with the beginning of the eighties, there has been a wave of discoveries in the area, and we now know a lot of examples of hyperkähler metrics which occur “in the nature”. These examples split naturally into two groups, depending on whether the underlying complex manifold is compact.

A powerful tool for constructing compact hyperkähler manifolds is the famous Calabi-Yau Theorem, which provides a Ricci-flat Kähler metric on every compact manifold of Kähler type with trivial canonial bundle. Its usefulness for the hyperkähler geometry lies in the fact that every hyperkähler manifold is canonically holomorphically symplectic. The converse statement is far from being true: a holomorphically symplectic Kähler manifold does not have to be hyperkähler. However, the converse is true if we require in addition that the hololomorphic symplectic form is parallel with respect to the Levi-Civita connection. It is easy to see that every Kähler manifold equipped with a parallel holomorphic symplectic form is hyperkähler.

In general it is very hard to check whether a holomorphic symplectic form on a compact Kähler manifold is parallel. However, there exists a theorem of S. Bochner’s [Boch] which claims that this is always the case when the Kähler metric is Ricci-flat. Since the canonical bundle of a holomorphically symplectic manifold is trivial, indeed, trivialized by the power of the symplectic from, the Calabi-Yau Theorem shows that every compact holomorphically symplectic manifold of Kähler type carries a Ricci-flat Kähler metric. This metric must be hyperkähler by the Bochner Theorem. Thus every compact holomorphically symplectic manifold of Kähler type is hyperkähler.

Well-known examples of compact holomorphically symplectic manifolds of dimension 2 are abelian complex surfaces and K3 surfaces. In higher dimensions non-trivial examples of such manifolds have been given by A. Beauville [Beau], extending earlier results of A. Fujiki [F]. Beauville’s examples are the Hilbert schemes of points on an abelian or a K3 surface. All these manifolds are of Kähler type, hence hyperkähler.

Given a compact hyperkähler manifold M, one can consider moduli spaces M of stable holomorphic vector bundles on M with fixed Chern classes. When M is 4-dimensional, hence either an abelian surface or a K3 surface, the moduli space M is known to be smooth and hyperkähler (see [Kob] for an excellent exposition of these results). When the Chern classes are such that M is compact, we obtain in this way a new compact hyperkähler manifold. This situation was studied in detail by S. Mukai in [M1], [M2]. It is now known that some of the compact moduli spaces of stable bundles on a K3 surface are deformationally equivalent to hyperkähler manifolds of the type constructed by Beauville. Conjecturally all these moduli spaces lie in the Beauville’s deformation class. We refer the reader to [Huy] for an overview of this subject.

These results were partially generalized to higher dimensions in [V1].

The moduli space of stable bundles on a higher-dimensional hyperkähler manifold is no longer automatically smooth. However, it is a singular hyperkähler variety in the sense of [V2]. This implies, in particular, that it is hyperkähler near every smooth point. Moreover, a singular hyperkähler veriety can be canonically desingularized to a smooth hyperkähler manifold.

An outstanding problem in the theory of compact hyperkähler manifolds is to find an example of such a manifold which would be simply connected and not equivalent deformationally to a product of the ones constructed by Beauville. Several important results ([OGr]) on this subject have appeared recently, but the problem is still not completely closed. The first chapter of the present book describes a different approach to this subject. The results of Mukai and Verbitsky are extended and strengthened in a way that conjecturally leads to the hoped-for examples of compact hyperkähler manifolds belonging to a new deformation class.

Both the Calabi-Yau Theorem and the Bochner Theorem are results of a global nature and cannot be used to construct non-compact hyperkähler manifolds. Two general methods are known which can be used for this purpose. The first one uses the link between the theory of hyperkähler manifolds and the holomorphic geometry provided by the notion of the twistor space. The twistor space construction, introduced by R. Penrose, has a long and glorious history. The reader is referred to [HKLR] for a detailed exposition of this subject. Here we only mention that the twistor space X for a 4n-dimensional hyperkähler manifold M is a holomorphic manifold of complex dimension 2n+2 canonically associated to M, and that the holomorphic structure on the twistor space X embodies most of the differential-geometric properties of the hyperkähler manifold M.

There exists a theorem which allows one to reconstruct a hyperkähler manifold M from its twistor space X equipped with some additional structures. Therefore, if it is impossible to construct M explicitly, one can construct X instead. In this way one can, for example, construct an infinitedimensional family of hyperkähler metrics on the vector space ℂ^{2n} (see [HKLR]).

Another general method of constructing non-compact hyperkähler manifolds is the famous hyperkähler reduction technique introduced by Hitchin [Hi1], Hitchin et al. [HKLR]. It is this technique that led to recent discoveries of hyperkähler structures on many interesting manifolds. Fortunately, the hyperkähler reduction is well-covered in the literature (see, for example, [Hi3]). Therefore we will only list the most important examples of hyperkähler manifolds obtained by this method.

• One of the original examples of hyperkähler manifolds given by Calabi was the total space of the cotangent bundle to a complex projective space. This manifold admits an elementary construction via the hyperkähler reduction.

• Let Γ ∈ SL(2, ℂ) be a finite subgroup. The quotient ℂ^{2}/Γ has an isolated singularity at 0 which can be blown up to a non-singular complex manifold. This manifold can be equipped with a hyperkähler metric by means of the hyperkähler reduction. The resulting metric is asymptotically locally Euclidean (ALE) at infinity. It was discovered by P.B. Kronheimer [Kr1] and generalized by Kronheimer and H. Nakajima [KN]. Nakajima [N] has recently generalized this example even further and obtained a whole family of hyperkähler manifolds called the quiver varieties.

• Let G be a compact Lie group, and let LG be the (infinite-dimensional) Lie group of maps from the unit cricle S^{1} to G. A hyperkähler metric on the quotient LG/G has been constructed by S. Donaldson [D].

• Let S be a compact complex curve, and let M be the moduli space of bundles on S equipped with a flat connection. Hitchin [Hi2] has constructed a hyperkähler structure on the space M. This construction has been recently generalized by C. Simpson [Sim] to the case when S is an arbitrary projective complex manifold.

A related group of examples is obtained by considering the solutions to a system of ordinary differential equations called the Nahm equations. These equations first appeared in the work of Schmid [Sch] on the variations of Hodge structures. They have been used by Kronheimer [Kr2],[Kr3] to obtain a hyperkähler metric on an orbit in the coadjoint representstion of an arbitrary semisimple complex ie group G. More recently Kronheimer’s method was used in papers [BG],[DS] to obtain new examples of hyperkähler manifolds. This method is not directly related to the hyperkähler reduction but shares many features with it. Some of the metrics obtained by reduction can be also constructed via the Nahm equations, and vice versa.

Unfortunately, while hyperkähler reduction is a generous source of new hyperkähler metrics, it can only be pushed so far. One of the problems which seems to lie outside of the scope of this approach is that of constructing a hyperkähler metric on the total space of the cotangent bundle to a non-homogeneous Kähler manifold. The second chapter of the present book describes such a construction. This construction is local and works for an arbitrary Kähler manifold. The methods used are necessarily different from the ones already exploited in the literature and consist of explicit but cumbersome application of the deformation theory in the spirit of Kodaira and Spencer [Kod].

Bibliography

[Beau] A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Diff. Geom. 19 (1983), 755–782.

[Ber] M. Berger, Sur les groupes d’holonomie des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France, 83 (1955), 279–330.

[BG] O. Biquard and P. Gauduchon, Hyperkähler metrics on cotangent bundles of Hermitian symmetric spaces, in ”Proc. of the Special Session an Geometry and Physics held at Aarhus University and of the Summer School at Odense University, Denmark, 1995”, Lecture Notes in Pure and Applied Mathematics, volume 184, Marcel Dekker Inc. 1996, 768 pages.

[Boch] S. Bochner, Curvature in Hermitian metric, Bull. Amer. Math. Soc. 53 (1947), 179–195.

[C] E. Calabi, Métriques kähleriennes et fibrés holomorphes, Ann. Ecol. Norm. Sup. 12 (1979), 269–294.

[DS] A. Dancer and A. Swann, The structure of quaternionic kähler quotients, in ”Proc. of the Special Session an Geometry and Physics held at Aarhus University and of the Summer School at Odense University, Denmark, 1995”, Lecture Notes in Pure and Applied Mathematics, volume 184, Marcel Dekker Inc. 1996, 768 pages.

[D] S.K. Donaldson, Boundary value problems for Yang-Mills fields, J. Geom. Phys. 8 (1992), 89–122.

[F] A. Fujiki, On primitively symplectic compact Kähler V-manifolds of dimension four, in Classification of algebraic and analytic manifolds, Progr. Math. 39 (1983), 71–250.

[Hi1] N.J. Hitchin, Metrics on moduli spaces, Proc. Lefschetz Centennial Conference (Mexico City 1984), Contemp. Math. 58, Part I, AMS, Providence, RI, 1986.

[Hi2] N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59–126.

[Hi3] N.J. Hitchin, Hyperkähler manifolds, Séminaire Bourbaki, Exposé 748, Astérisque, 206 (1992), 137–166.

[HKLR] N.J. Hitchin, A. Karlhede, U. Lindström, M. Roček, Hyperkähler metrics and supersymmetry, Comm. Math. Phys (1987).

[Huy] D. Huybrechts, Birational symplectic manifolds and their deformations, alg-geom/9601015.

[Kod] K. Kodaira and D. Spencer, On deformations of complex analytic structures I, II, Ann. Math. 67 (1958), 328–466.

[Kob] S. Kobayashi, Differential geometry of complex vector bundles, Princeton University Press, 1987.

[Kr1] P.B. Kronheimer, The construction of ALE spaces as hyper-Kaehler quotients, J. Diff. Geom. 29 (1989), 665–683.

[Kr2] P.B. Kronheimer, A hyper-Kählerian structure on coadjoint orbits of a semisimple complex group, J. of LMS, 42 (1990), 193–208.

[Kr3] P.B. Kronheimer, Instantons and the nilpotent variety, J. Diff. Geom. 32 (1990), 473–490.

[KN] P.B. Kronheimer and H. Nakajima, Yang-Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990), 263–307.

[M1] S. Mukai, Symplectic structure of the moduli space of sheaves on abelian or K3 surfaces, Invent. Math. 77 (1984), 101–116.

[M2] S. Mukai, Moduli of vector bundles on K3 surfaces, and symplectic manifolds, Sugaku Expositions 1 (1988), 138–174.

[N] H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), 365–416.

[OGr] Kieran G. O’Grady, Desingularized moduli spaces of sheaves on a K3, alg-geom/9708009.

[Sim] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety, I: Publ. Math. I.H.E.S. 79 (1994), 47–129; II: Publ. Math. I.H.E.S. 80 (1994), 5–79.

[Sch] W. Schmid, Variations of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319.

[V1] M. Verbitsky, Hyperholomorphic bundles over a hyperkähler manifold, alg-geom/9307008 (1993), also published in: Jour. of Alg. Geom. 5 (1996), 633–669.

[V2] M. Verbitsky, Hypercomplex Varieties, alg-geom/9703016 (1997) (to appear in Comm. in Anal. and Geom.)

12

Contents

Part I. Hyperholomorphic sheaves and new examples of hyperkähler manifolds M. Verbitsky 17 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.1 An overview . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 C-restricted complex structures: an introduction … 24 1.3 Quaternionic-Kähler manifolds and Swann’s formalism 25 1.4 Contents . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Hyperkähler manifolds . . . . . . . . . . . . . . . . . . . . . . 30 2.1 Hyperkähler manifolds . . . . . . . . . . . . . . . . . . 30 2.2 Simple hyperkähler manifolds . . . . . . . . . . . . . . 32 2.3 Trianalytic subvarieties in hyperkähler manifolds. … 32 2.4 Hyperholomorphic bundles . . . . . . . . . . . . . . . 35 2.5 Stable bundles and Yang–Mills connections. . . . . . . 36 2.6 Twistor spaces . . . . . . . . . . . . . . . . . . . . . . 38

3 Hyperholomorphic sheaves . . . . . . . . . . . . . . . . . . . . 39 3.1 Stable sheaves and Yang-Mills connections . . . . . . . 39 3.2 Stable and semistable sheaves over hyperkähler man-

ifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Hyperholomorphic connection in torsion-free sheaves . 43 3.4 Existence of hyperholomorphic connections . . . . . . 46 3.5 Tensor category of hyperholomorphic sheaves . . . . . 49

4 Cohomology of hyperkähler manifolds . . . . . . . . . . . . . 51 4.1 Algebraic induced complex structures . . . . . . . . . 51 4.2 The action of 𝔰𝔬(5) on the cohomology of a hyperkähler

manifold . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Structure of the cohomology ring . . . . . . . . . . . . 54 4.4 Cohomology classes of CA-type . . . . . . . . . . . . . 58

5 C-restricted complex structures on hyperkähler manifolds . . 59 5.1 Existence of C-restricted complex structures . . . . . 59 5.2 Hyperkähler structures admitting C-restricted com-

plex structures . . . . . . . . . . . . . . . . . . . . . . 66 5.3 Deformations of coherent sheaves over manifolds with

C-restricted complex structures . . . . . . . . . . . . . 69

6 Desingularization of hyperholomorphic sheaves . . . . . . . . 70 6.1 Twistor lines and complexification . . . . . . . . . . . 71 6.2 The automorphism Ψ_{I,J} acting on hyperholomorphic

sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.3 A ℂ^{∗}-action on a local ring of a hyperkähler manifold 76 6.4 Desingularization of ℂ^{∗}-equivariant sheaves . . . . . . 79

7 Twistor transform and quaternionic-Kähler geometry . . . . . 80 7.1 Direct and inverse twistor transform . . . . . . . . . . 81 7.2 Twistor transform and Hermitian structures on vector

bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.3 B_{2}-bundles on quaternionic-Kähler manifolds . . . . . 85 7.4 Hyperkähler manifolds with special ℍ^{∗}-action and qua-

ternionic-Kähler manifolds of positive scalar curvature 88

8 ℂ^{∗}-equivariant twistor spaces . . . . . . . . . . . . . . . . . . 92

8.1 B_{2}-bundles on quaternionic-Kähler manifolds and ℂ^{∗}-

equivariant holomorphic bundles over twistor spaces . 93 8.2 ℂ^{∗}-equivariant bundles and twistor transform . . . . . 95 8.3 Twistor transform and the ℍ^{∗}-action . . . . . . . . . . 98 8.4 Hyperholomorphic sheaves and ℂ^{∗}-equivariant bun-

dles over M_{fl} . . . . . . . . . . . . . . . . . . . . . . . 100 8.5 Hyperholomorphic sheaves and stable bundles on ℂP^{2n+1}103

9 Moduli spaces of hyperholomorphic sheaves and bundles … 104

9.1 Deformation of hyperholomorphic sheaves with iso-

lated singularities . . . . . . . . . . . . . . . . . . . . . 104 9.2 The Maruyama moduli space of coherent sheaves … 109 9.3 Moduli of hyperholomorphic sheaves and C-restricted

comples structures . . . . . . . . . . . . . . . . . . . . 111

10 New examples of hyperkähler manifolds . . . . . . . . . . . . 113 10.1 Twistor paths . . . . . . . . . . . . . . . . . . . . . . . 113 10.2 New examples of hyperkähler manifolds . . . . . . . . 118 10.3 How to check that we obtained new examples of hy-

perkähler manifolds? . . . . . . . . . . . . . . . . . . . 120

Part II. Hyperkähler structures on total spaces of holomorphic cotangent bundles. D. Kaledin 129 1 Preliminary facts from linear algebra . . . . . . . . . . . . . . 135 1.1 Quaternionic vector spaces . . . . . . . . . . . . . . . 135 1.2 The complementary complex structure . . . . . . . . . 137 1.3 ℝ-Hodge structures . . . . . . . . . . . . . . . . . . . . 138 1.4 Weakly Hodge maps . . . . . . . . . . . . . . . . . . . 140 1.5 Polarizations . . . . . . . . . . . . . . . . . . . . . . . 143

2 Hodge bundles and quaternionic manifolds . . . . . . . . . . . 144 2.1 Hodge bundles . . . . . . . . . . . . . . . . . . . . . . 144 2.2 Equivariant quaternionic manifolds . . . . . . . . . . . 147 2.3 Quaternionic manifolds and Hodge bundles . . . . . . 148 2.4 Holonomic derivations . . . . . . . . . . . . . . . . . . 149

3 Hodge manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 151 3.1 Integrability . . . . . . . . . . . . . . . . . . . . . . . . 151 3.2 The de Rham complex of a Hodge manifold . . . . . . 153 3.3 Polarized Hodge manifolds . . . . . . . . . . . . . . . 154

4 Regular Hodge manifolds . . . . . . . . . . . . . . . . . . . . 156 4.1 Regular stable points . . . . . . . . . . . . . . . . . . . 156 4.2 Linearization of regular Hodge manifolds . . . . . . . 157 4.3 Linear Hodge manifold structures . . . . . . . . . . . . 160

5 Tangent bundles as Hodge manifolds . . . . . . . . . . . . . . 163 5.1 Hodge connections . . . . . . . . . . . . . . . . . . . . 163 5.2 The relative de Rham complex of U over M . . . . . . 167 5.3 Holonomic Hodge connections . . . . . . . . . . . . . . 171 5.4 Hodge connections and linearity . . . . . . . . . . . . 173

6 Formal completions . . . . . . . . . . . . . . . . . . . . . . . . 177 6.1 Formal Hodge manifolds . . . . . . . . . . . . . . . . . 177 6.2 Formal Hodge manifold structures on tangent bundles 179 6.3 The Weil algebra . . . . . . . . . . . . . . . . . . . . . 180 6.4 Extended connections . . . . . . . . . . . . . . . . . . 183

7 Preliminaries on the Weil algebra . . . . . . . . . . . . . . . . 187 7.1 The total de Rham complex . . . . . . . . . . . . . . . 187 7.2 The total Weil algebra . . . . . . . . . . . . . . . . . . 192 7.3 Derivations of the Weil algebra . . . . . . . . . . . . . 195

8 Classification of flat extended connections . . . . . . . . . . . 199 8.1 Kählerian connections . . . . . . . . . . . . . . . . . . 199 8.2 Linearity and the total Weil algebra . . . . . . . . . . 203 8.3 The reduced Weil algebra . . . . . . . . . . . . . . . . 204 8.4 Reduction of extended connections . . . . . . . . . . . 208

9 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 9.1 Hyperkähler metrics on Hodge manifolds . . . . . . . 209 9.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 211 9.3 The Dolbeult differential on \overline{T} M_{J} . . . . . . . . . . . 214 9.4 The proof of Theorem 9.1 . . . . . . . . . . . . . . . . 215 9.5 The cotangent bundle . . . . . . . . . . . . . . . . . . 217

10 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 10.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 220 10.2 Combinatorics . . . . . . . . . . . . . . . . . . . . . . 225 10.3 The main estimate . . . . . . . . . . . . . . . . . . . . 229 10.4 The proof of Theorem 10.1 . . . . . . . . . . . . . . . 235

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Index of terms and notation . . . . . . . . . . . . . . . . . . . . . . 247 Authors index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

Part I. Hyperholomorphic sheaves and new examples of hyperkähler manifolds

Misha Verbitsky

Given a compact hyperkähler manifold M and a holomorphic bundle B over M, we consider a Hermitian connection ∇ on B which is compatible with all complex structures on M induced by the hyperkähler structure. Such a connection is unique, because it is Yang-Mills. We call the bundles admitting such connections hyperholomorphic bundles. A bundle is hyperholomorphic if and only if its Chern classes c_{1}, c_{2} are SU(2)-invariant, with respect to the natural SU(2)-action on the cohomology. For several years, it was known that the moduli space of stable hyperholomorphic bundles is singular hyperkähler. More recently, it was proven that singular hyperkähler varieties admit a canonical hyperkähler desingularization. In the present paper, we show that a moduli space of stable hyperholomorphic bundles is compact, given some assumptions on Chern classes of B and hyperkähler geometry of M (we also require dim_{ℂ} M > 2). Conjecturally, this leads to new examples of hyperkähler manifolds. We develop the theory of hyperholomorphic sheaves, which are (intuitively speaking) coherent sheaves compatible with hyperkähler structure. We show that hyperholomorphic sheaves with isolated singularities can be canonically desingularized by a blow-up. This theory is used to study degenerations of hyperholomorphic bundles.

1 Introduction

For an introduction to basic results and the history of hyperkähler geometry, see [Bes].

This Introduction is independent from the rest of this paper.

1.1 An overview

Examples of hyperkähler manifolds

A Riemannian manifold M is called hyperkähler if the tangent bundle of M is equipped with an action of quaternian algebra, and its metric is Kähler with respect to the complex structures I_{ι}, for all embeddings ℂ ↪^{ι} ℍ. The complex structures I_{ι} are called induced complex structures; the corresponding Kähler manifold is denoted by (M, I_{ι}).

For a more formal definition of a hyperkähler manifold, see Definition 2.1. The notion of a hyperkähler manifold was introduced by E. Calabi ([C]).

Clearly, the real dimension of M is divisible by 4. For dim_{ℝ} M = 4, there are only two classes of compact hyperkähler manifolds: compact tori and K3 surfaces.

Let M be a complex surface and M^{(n)} be its n-th symmetric power, M^{(n)} = M^{n}/S_{n}. The variety M^{(n)} admits a natural desingularization M^{[n]}, called the Hilbert scheme of points.

The manifold M^{[n]} admits a hyperkähler metrics whenever the surface M is compact and hyperkähler ([Bea]). This way, Beauville constructed two series of examples of hyperkähler manifolds, associated with a torus (so-called “higher Kummer variety”) and a K3 surface. It was conjectured that all compact hyperkähler manifolds M with H^{1}(M) = 0, H^{2,0}(M) = ℂ are deformationally equivalent to one of these examples. In this paper, we study the deformations of coherent sheaves over higher-dimensional hyperkähler manifolds in order to construct counterexamples to this conjecture. A different approach to the construction of new examples of hyperkähler manifolds is found in the recent paper of K. O’Grady, who studies the moduli of semistable bundles over a K3 surface and resolves the singularities using methods of symplectic geometry ([O’G]).

Hyperholomorphic bundles

Let M be a compact hyperkähler manifold, and I an induced complex structure. It is well known that the differential forms and cohomology of M are equipped with a natural SU(2)-action (Lemma 2.5). In [V1], we studied the holomorphic vector bundles F on (M, I) which are compatible with a hyperkähler structure, in the sense that any of the following conditions hold:

(i) The bundle F admits a Hermitian connection ∇ with a curvature Θ ∈ Λ^{2}(M, End(F)) which is of Hodge type (1,1) with respect to any of induced complex structures.

(ii) The bundle F is a direct sum of stable bundles, and its Chern classes c_{1}(F), c_{2}(F) are SU(2)-invariant.

(1.1)

These conditions are equivalent (Theorem 2.27). Moreover, the connection ∇ of (1.1) (i) is Yang-Mills (Proposition 2.25), and by Uhlenbeck–Yau theorem (Theorem 2.24), it is unique.

A holomorphic vector bundle satisfying any of the conditions of (1.1) is called hyperholomorphic ([V1]).

Clearly, a stable deformation of a hyperholomorphic bundle is again a hyperholomorphic bundle. In [V1], we proved that a deformation space of hyperholomorphic bundles is a singular hyperkähler variety. A recent development in the theory of singular hyperkähler varieties ([V-d], [V-d2], [V-d3]) gave a way to desingularize singular hyperkähler manifolds, in a canonical way. It was proven (Theorem 2.16) that a normalization of a singular hyperkähler variety (taken with respect to any induced complex structure I) is a smooth hyperkähler manifold.

This suggested a possibility of constructing new examples of compact hyperkähler manifolds, obtained as deformations of hyperholomorphic bundles. Two problems arise.

Problem 1. The deformation space of hyperholomorphic bundles is a priori non-compact and must be compactified.

Problem 2. The geometry of deformation spaces is notoriously hard to study. Even the dimension of a deformation space is difficult to compute, in simplest examples. How to find, for example, the dimension of the deformation space of a tangent bundle, on a Hilbert scheme of points on a K3 surface? The Betti numbers are even more difficult to compute. Therefore, there is no easy way to distinguish a deformation space of hyperholomorphic bundles from already known examples of hyperkähler manifolds.

In this paper, we address Problem 1. Problem 2 can be solved by studying the algebraic geometry of moduli spaces. It turns out that, for a generic deformation of a complex structure, the Hilbert scheme of points on a K3 surface has no closed complex subvarieties ([V5]; see also Theorem 2.17). It is possible to find a 21-dimensional family of deformations of the moduli space Def(B) of hyperholomorphic bundles, with all fibers having complex subvarieties (Lemma 10.28). Using this observation, it is possible to show that Def(B) is a new example of a hyperkähler manifold. Details of this approach are given in Subsection 10.3, and the complete proofs will be given in a forthcoming paper.

It was proven that a Hilbert scheme of a generic K3 surface has no trianalytic subvarieties.^{1} Given a hyperkähler manifold M and an appropriate hyperholomorphic bundle B, denote the deformation space of hyperholomorphic connections on B by Def(B). Then the moduli of complex structures on M are locally embedded to a moduli of complex structures on Def(B) (Claim 10.26). Since the dimension of the moduli of complex structures on Def(B) is equal to its second Betti number minus 2 (Theorem 5.9), the second Betti number of Def(B) is no less than the second Betti number of M. The Betti numbers of Beauville’s examples of simple hyperkähler manifolds are 23 (Hilbert scheme of points on a K3 surface) and 7 (generalized Kummer variety). Therefore, for M a generic deformation of a Hilbert scheme of points on K3, Def(B) is either a new manifold or a generic deformation of a Hilbert scheme of points on K3. It is easy to construct trianalytic subvarieties of the varieties Def(B), for hyperholomorphic B (see [V2], Appendix for details). This was the motivation of our work on trianalytic subvarieties of the Hilbert scheme of points on a K3 surface ([V5]).

For a generic complex structure on a hyperkähler manifold, all stable bundles are hyperholomorphic ([V2]). Nevertheless, hyperholomorphic bundles over higher-dimensional hyperkähler manifolds are in short supply. In fact, the only example to work with is the tangent bundle and its tensor powers, and their Chern classes are not prime. Therefore, there is no way to insure that a deformation of a stable bundle will remain stable (like it happens, for instance, in the case of deformations of stable bundles of rank 2 with odd first Chern class over a K3 surface). Even worse, a new kind of singularities may appear which never appears for 2-dimensional base manifolds: a deformation of a stable bundle can have a singular reflexization.

1Trianalytic subvariety (Definition 2.9) is a closed subset which is complex analytic with respect to any of induced complex structures.

We study the singularities of stable coherent sheaves over hyperkähler manifolds, using Yang-Mills theory for reflexive sheaves developed by S. Bando and Y.-T. Siu ([BS]).

Hyperholomorphic sheaves

A compactification of the moduli of hyperholomorphic bundles is the main purpose of this paper. We require the compactification to be singular hyperkähler. A natural approach to this problem requires one to study the coherent sheaves which are compatible with a hyperkähler structure, in the same sense as hyperholomorphic bundles are holomorphic bundles compatible with a hyperkähler structure. Such sheaves are called hyperholomorphic sheaves (Definition 3.11). Our approach to the theory of hyperholomorphic sheaves uses the notion of admissible Yang-Mills connection on a coherent sheaf ([BS]).

The equivalence of conditions (1.1) (i) and (1.1) (ii) is based on Uhlenbeck–Yau theorem (Theorem 2.24), which states that every stable bundle F with deg c_{1}(F) = 0 admits a unique Yang–Mills connection, that is, a connection ∇ satisfying Λ∇^{2} = 0 (see Subsection 2.5 for details). S. Bando and Y.-T. Siu developed a similar approach to the Yang–Mills theory on (possibly singular) coherent sheaves. Consider a coherent sheaf F and a Hermitian metric h on a locally trivial part of F |_{U} . Then h is called admis-

sible (Definition 3.5) if the curvature ∇^{2} of the Hermitian connection on F |_{U}

is square-integrable, and the section Λ∇^{2} ∈ End(F |_{U} ) is uniformly bounded. The admissible metric is called Yang-Mills if Λ∇^{2} = 0 (see Definition 3.6 for details). There exists an analogue of Uhlenbeck–Yau theorem for coherent sheaves (Theorem 3.8): a stable sheaf admits a unique admissible Yang–Mills metric, and conversely, a sheaf admitting a Yang–Mills metric is a direct sum of stable sheaves with the first Chern class of zero degree.

A coherent sheaf F is called reflexive if it is isomorphic to its second dual sheaf F^{∗∗}. The sheaf F^{∗∗} is always reflexive, and it is called a reflexization of F (Definition 3.1).

Applying the arguments of Bando and Siu to a reflexive coherent sheaf F over a hyperkähler manifold (M, I), we show that the following conditions are equivalent (Theorem 3.19).

(i) The sheaf F is stable and its Chern classes c_{1}(F), c_{2}(F) are SU(2)- invariant (ii) F admits an admissible Yang–Mills connection, and its curvature is of

type (1,1) with respect to all induced complex structures.

A reflexive sheaf satisfying any of the these conditions is called reflexive stable hyperholomorphic. An arbitrary torsion-free coherent sheaf is called stable hyperholomorphic if its reflexization is hyperholomorphic, and its second Chern class is SU(2)-invariant, and semistable hyperholomorphic if it is a successive extension of stable hyperholomorphic sheaves (see Definition 3.11 for details).

This paper is dedicated to the study of hyperholomorphic sheaves.

Deformations of hyperholomorphic sheaves

By Proposition 2.14, for an induced complex structure I of general type, all coherent sheaves are hyperholomorphic. However, the complex structures of general type are never algebraic, and in complex analytic situation, the moduli of coherent sheaves are, generaly speaking, non-compact. We study the flat deformations of hyperholomorphic sheaves over (M, I), where I is an algebraic complex structure.

A priori, a flat deformation of a hyperholomorphic sheaf will be no longer hyperholomorphic. We show that for some algebraic complex structures, called C-restricted complex structures, a flat deformation of a hyperholomorphic sheaf remains hyperholomorphic (Theorem 5.14). This argument is quite convoluted, and takes two sections (Sections 4 and 5).

Further on, we study the local structure of stable reflexive hyperholomorphic sheaves with isolated singularities. We prove the Desingularization Theorem for such hyperholomorphic sheaves (Theorem 6.1). It turns out that such a sheaf can be desingularized by a single blow-up. The proof of this result is parallel to the proof of Desingularization Theorem for singular hyperkähler varieties (Theorem 2.16).

The main idea of the desingularization of singular hyperkähler varieties ([V-d2]) is the following. Given a point x on a singular hyperkähler variety M and an induced complex structure I, the complex variety (M, I) admits a local ℂ^{∗}-action which preserves x and acts as a dilatation on the Zariski tangent space of x. Here we show that any stable hyperholomorphic sheaf F is equivariant with respect to this ℂ^{∗}-action (Theorem 6.6, Definition 6.11). Then an elementary algebro-geometric argument (Proposition 6.12) implies that F is desingularized by a blow-up.

Using the desingularization of hyperholomorphic sheaves, we prove that a hyperholomorphic deformation of a hyperholomorphic bundle is again a bundle (Theorem 9.3), assuming that it has isolated singularities. The proof of this result is conceptual but quite difficult, it takes 3 sections (Sections 7–9), and uses arguments of quaternionic-Kähler geometry ([Sw], [N2]) and twistor transform ([KV]).

In our study of deformations of hyperholomorphic sheaves, we usually assume that a deformation of a hyperholomorphic sheaf over (M, I) is again hyperholomorphic, i. e. that an induced complex structure I is C-restricted, for C sufficiently big (Definition 5.1). Since C-restrictness is a tricky condition, it is preferable to get rid of it. For this purpose, we use the theory of twistor paths, developed in [V3-bis], to show that the moduli spaces of hyperholomorphic sheaves are real analytic equivalent for different complex structures I on M (Theorem 10.14). This is done as follows.

A hyperkähler structure on M admits a 2-dimensional sphere of induced complex structures. This gives a rational curve in the moduli space Comp of complex structures on M, so-called twistor curve. A sequence of such rational curves connect any two points of Comp (Theorem 10.4). A sequence of connected twistor curves is called a twistor path. If the intersection points of these curves are generic, the twistor path is called admissible (Definition 10.6). It is known (Theorem 10.8) that an admissible twistor path induces a real analytic isomorphism of the moduli spaces of hyperholomorphic bundles. There exist admissible twistor paths connecting any two complex structures (Claim 10.13). Thus, if we prove that the moduli of deformations of hyperholomorphic bundles are compact for one generic hyperkähler structure, we prove a similar result for all generic hyperkähler structures (Theorem 10.14). Applying this argument to the moduli of deformations of a tangent bundle, we obtain the following theorem.

Theorem 1.1: Let M be a Hilbert scheme of points on a K3 surface, dim_{ℍ}(M) > 1 and H a generic hyperkähler structure on M. Assume that for all induced complex structures I, except at most a finite many of, all semistable bundle deformations of the tangent bundle T (M, I) are stable. Then, for all complex structures J on M and all generic polarizations ω on (M, J), the deformation space M_{J,ω}(T (M, J)) is singular hyperkähler and compact, and admits a smooth compact hyperkähler desingularization.

Proof: This is Theorem 10.20.

In the course of this paper, we develop the theory of C-restricted complex structures (Sections 4 and 5) and another theory, which we called the Swann’s formalism for vector bundles (Sections 7 and 8). These themes are of independent interest. We give a separate introduction to Crestricted complex structures (Subsection 1.2) and Swann’s formalism (Subsection 1.3).

1.2 C-restricted complex structures: an introduction

This part of the Introduction is highly non-precise. Our purpose is to clarify the intuitive meaning of C-restricted complex structure.

Consider a compact hyperkähler manifold M, which is simple (Definition 2.7), that is, satisfies H^{1}(M) = 0, H^{2,0}(M) = ℂ.

A reflexive hyperholomorphic sheaf is by definition a semistable sheaf which has a filtration of stable sheaves with SU(2)-invariant c_{1} and c_{2}. A hyperholomorphic sheaf is a torsion-free sheaf which has hyperholomorphic reflexization and has SU(2)-invariant c_{2} (Definition 3.11). If the complex structure I is of general type, all coherent sheaves are hyperholomorphic (Definition 2.13, Proposition 2.14), because all integer (p, p)- classes are SU(2)-invariant. However, for generic complex structures I, the corresponding complex manifold (M, I) is never algebraic. If we wish to compactify the moduli of holomorphic bundles, we need to consider algebraic complex structures, and if we want to stay in hyperholomorphic category, the complex structures must be generic. This paradox is reconciled by considering the C-restricted complex structures (Definition 5.1).

Given a generic hyperkähler structure H, consider an algebraic complex structure I with P ic(M, I) = ℤ. The group of rational (p, p)-cycles has form

\begin{matrix}H_{I}^{p,p}(M, ℚ) = & H^{2p}(M, ℚ)^{SU(2)} ⊕ a · H^{2p}(M, ℚ)^{SU(2)} \\ a^{2} · ⊕H^{2p}(M, ℚ)^{SU(2)} ⊕ ... \end{matrix}  (1.2)

where a is a generator of P ic(M, I) ⊂ H_{I}^{p,p}(M, ℤ) and H^{2p}(M, ℚ)^{SU(2)} is the group of rational SU(2)-invariant cycles. This decomposition follows from an explicit description of the algebra of cohomology given by Theorem 4.6. Let

Π : H_{I}^{p,p}(M, ℚ) ⟶ a · H^{2p}(M, ℚ)^{SU(2)} ⊕ a^{2} · H^{2p}(M, ℚ)^{SU(2)} ⊕ ...

be the projection onto non-SU(2)-invariant part. Using Wirtinger’s equality, we prove that a fundamental class [X] of a complex subvariety X ⊂ (M, I) is SU(2)-invariant unless deg Π([X]) ≠ 0 (Proposition 2.11). A similar result holds for the second Chern class of a stable bundle (Corollary 3.24,).

A C-restricted complex structure is, heuristically, a structure for which the decomposition (1.2) folds, and deg a > C. For a C-restricted complex structure I, and a fundamental class [X] of a complex subvariety X ⊂ (M, I) of complex codimension 2, we have deg[X] > C or X is trianalytic. A version of Wirtinger’s inequality for vector bundles (Corollary 3.24) implies that a stable vector bundle B over (M, I) is hyperholomorphic, unless | deg c_{2}(B)| > C. Therefore, over a C-restricted (M, I), all torsion-free semistable coherent sheaves with bounded degree of the second Chern class are hyperholomorphic (Theorem 5.14).

The utility of C-restricted induced complex structures is that they are algebraic, but behave like generic induced complex structures with respect to the sheaves F with low | deg c_{2}(F)| and | deg c_{1}(F)|.

We prove that a generic hyperkähler structure admits C-restricted induced complex structures for all C, and the set of C-restricted induced complex structures is dense in the set of all induced complex structures (Theorem 5.13). We prove this by studying the algebro-geometric properties of the moduli of hyperkähler structures on a given hyperkähler manifold (Subsection 5.2).

1.3 Quaternionic-Kähler manifolds and Swann’s formalism

Quaternionic-Kähler manifolds (Subsection 7.3) are a beautiful subject of Riemannian geometry. We are interested in these manifolds because they are intimately connected with singularities of hyperholomorphic sheaves. A stable hyperholomorphic sheaf is equipped with a natural connection, which is called hyperholomorphic connection. By definition, a hyperholomorphic connection on a torsion-free coherent sheaf is a connection ∇ defined outside of singularities of F, with square-integrable curvature ∇^{2} which is an SU(2)- invariant 2-form (Definition 3.15). We have shown that a stable hyperholomorphic sheaf admits a hyperholomorphic connection, and conversely, a reflexive sheaf admitting a hyperholomorphic connection is a direct sum of stable hyperholomorphic sheaves (Theorem 3.19).

Consider a reflexive sheaf F over (M, I) with an isolated singularity in x ∈ M. Let ∇ be a hyperholomorphic connection on F. We prove that F can be desingularized by a blow-up of its singular set. In other words, for π : M̃ ⟶ (M, I) a blow-up of x ∈ M, the pull-back π^{∗}F is a bundle over M̃.

Consider the restriction π^{∗}F |_{C} of π^{∗}F to the blow-up divisor

C = ℙT_{x}M ≅ ℂP^{2n−1}.

To be able to deal with the singularities of F effectively, we need to prove that the bundle π^{∗}F |_{C} is semistable and satisfies c_{1} (π^{∗}F |_{C} ) = 0.

The following intuitive picture motivated our work with bundles over quaternionic-Kähler manifolds. The manifold C = ℙT_{x}M is has a quaternionic structure, which comes from the SU(2)-action on T_{x}M. We know that bundles which are compatible with a hyperkähler structure ( hyperholomorphic bundles) are (semi-)stable. If we were able to prove that the bundle π^{∗}F |_{C} is in some way compatible with quaternionic structure on C, we could hope to prove that it is (semi-)stable.

To give a precise formulation of these heuristic arguments, we need to work with the theory of quaternionic-Kähler manifolds, developed by Berard Bergery and Salamon ([Sal]). A quaternionic-Kähler manifold (Definition 7.8) is a Riemannian manifold Q equipped with a bundle W of algebras acting on its tangent bundle, and satisfying the following conditions. The fibers of W are (non-canonically) isomorphic to the quaternion algebra, the map W ↪ End(T Q) is compatible with the Levi-Civita connection, and the unit quaternions h ∈ W act as orthogonal automorphisms on T Q. For each quaternionic-Kähler manifold Q, one has a twistor space Tw(Q) (Definition 7.10), which is a total space of a spherical fibration consisting of all h ∈ W satisfying h^{2} = −1. The twistor space is a complex manifold ([Sal]), and it is Kähler unless W is flat, in which case Q is hyperkähler. Further on, we shall use the term “quaternionic-Kähler” for manifolds with non-trivial W.

Consider the twistor space Tw(M) of a hyperkähler manifold M, equipped with a natural map

σ : Tw(M) ⟶ M.

Let (B, ∇) be a bundle over M equipped with a hyperholomorphic connection. A pullback (σ^{∗}B, σ^{∗}∇) is a holomorphic bundle on Tw(M) (Lemma 7.2), that is, the operator σ^{∗}∇^{0,1} is a holomorphic structure operator on σ^{∗}B. This correspondence is called the direct twistor transform. It is invertible: from a holomorphic bundle (σ^{∗}B, σ^{∗}∇^{0,1}) on Tw(M) it is possible to reconstruct (B, ∇), which is unique ([KV]; see also Theorem 7.3).

A similar construction exists on quaternionic-Kähler manifolds, due to T. Nitta ([N1], [N2]). A bundle (B, ∇) on a quaternionic-Kähler manifold Q is called a B_{2}-bundle if its curvature ∇^{2} is invariant with respect to the adjoint action of ℍ^{∗} on Λ^{2}(M, End(B)) (Definition 7.12). An analogue of direct and inverse transform exists for B_{2}-bundles (Theorem 7.14). Most importantly, T. Nitta proved that on a quaternionic-Kähler manifold of positive scalar curvature a twistor transform of a B_{2}-bundle is a Yang-Mills bundle on Tw(Q) (Theorem 7.17). This implies that a twistor transform of a Hermitian B_{2}-bundle is a direct sum of stable bundles with deg c_{1} = 0.

In the situation described in the beginning of this Subsection, we have a manifold C = ℙT_{x}M ≅ ℂP^{2n−1} which is a twistor space of a quaternionic projective space

ℙ_{ℍ}T_{x}M = (T_{x}M\0)/ℍ^{∗} ≅ ℍP^{n}.

To prove that π^{∗}F |_{C} is stable, we need to show that π^{∗}F |_{C} is obtained as twistor transform of some Hermitian B_{2}-bundle on ℙ_{ℍ}T_{x}M.

This is done using an equivalence between the category of quaternionic-Kähler manifolds of positive scalar curvature and the category of hyperkähler manifolds equipped with a special type of ℍ^{∗}-action, constructed by A. Swann ([Sw]). Given a quaternionic-Kähler manifold Q, we consider a principal bundle U(Q) consisting of all quaternion frames on Q (7.4). Then U(Q) is fibered over Q with a fiber ℍ/{±1}. It is easy to show that U(Q) is equipped with an action of quaternion algebra in its tangent bundle. A. Swann proved that if Q has with positive scalar curvature, then this action of quaternion algebra comes from a hyperkähler structure on U(M) (Theorem 7.24).

The correspondence Q ⟶ U(Q) induces an equivalence of appropriately defined categories (Theorem 7.25). We call this construction Swann’s formalism.

The twistor space Tw(U(Q)) of the hyperkähler manifold U(Q) is equipped with a holomorphic action of ℂ^{∗}. Every B_{2}-bundle corresponds to a ℂ^{∗}-invariant holomorphic bundle on Tw(U(Q)) and this correspondence induces an equivalence of appropriately defined categories, called Swann’s formalism for budnles (Theorem 8.5). Applying this equivalence to the ℂ^{∗}-equivariant sheaf obtained as an associate graded sheaf of a hyperholomorphic sheaf, we obtain a B_{2} bundle on ℙ_{ℍ}T_{x}M, and π^{∗}F |_{C} is obtained from this B_{2}-bundle by a twistor transform.

The correspondence between B_{2}-bundles on Q and ℂ^{∗}-invariant holomorphic bundles on Tw(U(Q)) is an interesting geometric phenomenon which is of independent interest. We construct it by reduction to dim Q = 0, where it follows from an explicit calculation involving 2-forms over a flat manifold of real dimension 4.

1.4 Contents

The paper is organized as follows.

• Section 1 is an introduction. It is independent from the rest of this apper.

• Section 2 is an introduction to the theory of hyperkähler manifolds. We give a compenduum of results from hyperkähler geometry which are due to F. Bogomolov ([Bo]) and A. Beauville ([Bea]), and give an introduction to the results of [V1], [V-d3], [V2(II)].

• Section 3 contains a definition and basic properties of hyperholomorphic sheaves. We prove that a stable hyperholomorphic sheaf admits a hyperholomorphic connection, and conversely, a reflexive sheaf admitting a hyperholomorphic connection is stable hyperholomorphic (Theorem 3.19). This equivalence is constructed using Bando-Siu theory of Yang–Mills connections on coherent sheaves.

We prove an analogue of Wirtinger’s inequality for stable sheaves (Corollary 3.24), which states that for any induced complex structure J ≠ ±I, and any stable reflexive sheaf F on (M, I), we have

deg_{I} (2c_{2}(F) − \frac{r − 1}{r}c_{1}(F)^{2}) ⩾ |deg_{J} (2c_{2}(F) − \frac{r − 1}{r}c_{1}(F)^{2})| ,

and the equality holds if and only if F is hyperholomorphic.

• Section 4 contains the preliminary material used for the study of Crestricted complex structures in Section 5. We give an exposition of various algebraic structures on the cohomology of a hyperkähler manifold, which were discovered in [V0] and [V3]. In the last Subsection, we apply the Wirtinger’s inequality to prove that the fundamental classes of complex subvarieties and c_{2} of stable reflexive sheaves satisfy a certain set of axioms. Cohomology classes satisfying these axioms are called classes of CA-type. This definition simplifies the work on Crestricted complex structures in Section 5.

• In Section 5 we define C-restricted complex structures and prove the following. Consider a compact hyperkähler manifold and an SU(2)- invariant class a ∈ H^{4}(M). Then for all C-restricted complex structures I, with C > deg a, and all semistable sheaves I on (M, I) with c_{2}(F) = a, the sheaf F is hyperholomorphic (Theorem 5.14). This is used to show that a deformation of a hyperholomorphic sheaf is again hyperholomorphic, over (M, I) with I a C-restricted complex structure, c > deg c_{2}(F).

We define the moduli space of hyperkähler structures, and show that for a dense set C of hyperkähler structures, all H ∈ C admit a dense set of C-induced complex structures, for all C ∈ ℝ (Theorem 5.13).

• In Section 6 we give a proof of Desingularization Theorem for stable reflexive hyperholomorphic sheaves with isolated singularities (Theorem 6.1). We study the natural ℂ^{∗}-action on a local ring of a hyperkähler manifold defined in [V-d2]. We show that a sheaf F admitting a hyperholomorphic connection is equivariant with respect to this ℂ^{∗}-action. Then F can be desingularized by a blow-up, because any ℂ^{∗}-equivariant sheaf with an isolated singularity can be desingularized by a blow-up (Proposition 6.12).

• Section 7 is a primer on twistor transform and quaternionic-Kähler geometry. We give an exposition of the works of A. Swann ([Sw]), T. Nitta ([N1], [N2]) on quaternionic-Kähler manifolds and explain the direct and inverse twistor transform over hyperkähler and quaternionic--Kähler manifolds.

• Section 8 gives a correspondence between B_{2}-bundles on a quaternionic--Kähler manifold, and ℂ^{∗}-equivariant holomorphic bundles on the twistor space of the corresponding hyperkähler manifold constructed by A. Swann. This is called “Swann’s formalism for vector bundles”. We use this correspondence to prove that an associate graded sheaf of a hyperholomorphic sheaf is equipped with a natural connection which is compatible with quaternions. This implies polystability of the bundle π^{∗}F |_{C} (see Subsection 1.3).

• In Section 9, we use the polystability of the bundle π^{∗}F |_{C} to show that a hyperholomorphic deformation of a hyperholomorphic bundle is again a bundle. Together with results on C-restricted complex structures and Maruyama’s compactification ([Ma2]), this implies that the moduli of semistable bundles are compact, under conditions of C-restrictness and non-existence of trianalytic subvarieties (Theorem 9.11).

• In Section 10, we apply these results to the hyperkähler geometry. Using the desingularization theorem for singular hyperkähler manifolds (Theorem 2.16), we prove that the moduli of stable deformations of a hyperholomorphic bundle has a compact hyperkähler desingularization (Theorem 10.17). We give an exposition of the theory of twistor paths, which allows one to identify the categories of stable bundles for different Kähler structures on the same hyperkähler manifold (Theorem 10.8). These results allow one to weaken the conditions necessary for compactness of the moduli spaces of vector bundles. Finally, we give a conjectural exposition of how these results can be used to obtain new examples of compact hyperkähler manifolds.

2 Hyperkähler manifolds

2.1 Hyperkähler manifolds

This subsection contains a compression of the basic and best known results and definitions from hyperkähler geometry, found, for instance, in [Bes] or in [Bea].

Definition 2.1: ([Bes]) A hyperkähler manifold is a Riemannian manifold M endowed with three complex structures I, J and K, such that the following holds.

(i) the metric on M is Kähler with respect to these complex structures and

(ii) I, J and K, considered as endomorphisms of a real tangent bundle, satisfy the relation I ∘ J = −J ∘ I = K.

The notion of a hyperkähler manifold was introduced by E. Calabi ([C]).

Clearly, a hyperkähler manifold has a natural action of the quaternion algebra ℍ in its real tangent bundle T M. Therefore its complex dimension is even. For each quaternion L ∈ ℍ, L^{2} = −1, the corresponding automorphism of T M is an almost complex structure. It is easy to check that this almost complex structure is integrable ([Bes]).

Definition 2.2: Let M be a hyperkähler manifold, and L a quaternion satisfying L^{2} = −1. The corresponding complex structure on M is called an induced complex structure. The M, considered as a Kähler manifold, is denoted by (M, L). In this case, the hyperkähler structure is called compatible with the complex structure L.

Let M be a compact complex manifold. We say that M is of hyperkähler type if M admits a hyperkähler structure compatible with the complex structure.

Definition 2.3: Let M be a complex manifold and Θ a closed holomorphic 2-form over M such that Θ^{n} = Θ ∧ Θ ∧ ..., is a nowhere degenerate section of a canonical class of M (2n = dim_{ℂ}(M)). Then M is called holomorphically symplectic.

Let M be a hyperkähler manifold; denote the Riemannian form on M by < ·, · >. Let the form ω_{I} :=< I(·), · > be the usual Kähler form which is closed and parallel (with respect to the Levi-Civita connection). Analogously defined forms ω_{J} and ω_{K} are also closed and parallel.

A simple linear algebraic consideration ([Bes]) shows that the form Θ := ω_{J} + \sqrt{−1}ω_{K} is of type (2, 0) and, being closed, this form is also holomorphic. Also, the form Θ is nowhere degenerate, as another linear algebraic argument shows. It is called the canonical holomorphic symplectic form of a manifold M. Thus, for each hyperkähler manifold M, and an induced complex structure L, the underlying complex manifold (M, L) is holomorphically symplectic. The converse assertion is also true:

Theorem 2.4: ([Bea], [Bes]) Let M be a compact holomorphically symplectic Kähler manifold with the holomorphic symplectic form Θ, a Kähler class [ω] ∈ H^{1,1}(M) and a complex structure I. Let n = dim_{ℂ} M. Assume that ∫_{M} ω^{n} = ∫_{M}(ReΘ)^{n}. Then there is a unique hyperkähler structure (I, J, K, (·, ·)) over M such that the cohomology class of the symplectic form ω_{I} = (·, I·) is equal to [ω] and the canonical symplectic form ω_{J} + \sqrt{−1} ω_{K} is equal to Θ.

Theorem 2.4 follows from the conjecture of Calabi, proven by Yau ([Y]).

Let M be a hyperkähler manifold. We identify the group SU(2) with the group of unitary quaternions. This gives a canonical action of SU(2) on the tangent bundle, and all its tensor powers. In particular, we obtain a natural action of SU(2) on the bundle of differential forms.

Lemma 2.5: The action of SU(2) on differential forms commutes with the Laplacian.

Proof: This is Proposition 1.1 of [V2(II)].

Thus, for compact M, we may speak of the natural action of SU(2) in cohomology.

Further in this article, we use the following statement.

Lemma 2.6: Let ω be a differential form over a hyperkähler manifold M. The form ω is SU(2)-invariant if and only if it is of Hodge type (p, p) with respect to all induced complex structures on M.

Proof: This is [V1], Proposition 1.2.

2.2 Simple hyperkähler manifolds

Definition 2.7: ([Bea]) A connected simply connected compact hyperkähler manifold M is called simple if M cannot be represented as a product of two hyperkähler manifolds:

M ≠ M_{1} × M_{2}, where dim M_{1} > 0 and dim M_{2} > 0

Bogomolov proved that every compact hyperkähler manifold has a finite covering which is a product of a compact torus and several simple hyperkähler manifolds. Bogomolov’s theorem implies the following result ([Bea]):

Theorem 2.8: Let M be a compact hyperkähler manifold. Then the following conditions are equivalent.

(i) M is simple

(ii) M satisfies H^{1}(M, ℝ) = 0, H^{2,0}(M) = ℂ, where H^{2,0}(M) is the space of (2, 0)-classes taken with respect to any of induced complex structures.

2.3 Trianalytic subvarieties in hyperkähler manifolds.

In this subsection, we give a definition and basic properties of trianalytic subvarieties of hyperkähler manifolds. We follow [V2(II)], [V-d2].

Let M be a compact hyperkähler manifold, dim_{ℝ} M = 2m.

Definition 2.9: Let N ⊂ M be a closed subset of M. Then N is called trianalytic if N is a complex analytic subset of (M, L) for any induced complex structure L.

Let I be an induced complex structure on M, and N ⊂ (M, I) be a closed analytic subvariety of (M, I), dim_{ℂ}N = n. Consider the homology class represented by N. Let [N] ∈ H^{2m−2n}(M) denote the Poincare dual cohomology class, so called fundamental class of N. Recall that the hyperkähler structure induces the action of the group SU(2) on the space H^{2m−2n}(M).

Theorem 2.10: Assume that [N] ∈ H^{2m−2n}(M) is invariant with respect to the action of SU(2) on H^{2m−2n}(M). Then N is trianalytic.

Proof: This is Theorem 4.1 of [V2(II)].

The following assertion is the key to the proof of Theorem 2.10 (see [V2(II)] for details).

Proposition 2.11: (Wirtinger’s inequality) Let M be a compact hyperkähler manifold, I an induced complex structure and X ⊂ (M, I) a closed complex subvariety for complex dimension k. Let J be an induced complex structure, J ≠ ±I, and ω_{I}, ω_{J} the associated Kähler forms. Consider the numbers

deg_{I} X := ∫_{X} ω_{I}^{k}, deg_{J} X := ∫_{X} ω_{J}^{k}

(these numbers are called degree of a subvariety X with respect to I, J. Then deg_{I} X ⩾ | deg_{J} X|, and the inequality is strict unless X is trianalytic.

Remark 2.12: Trianalytic subvarieties have an action of quaternion algebra in the tangent bundle. In particular, the real dimension of such subvarieties is divisible by 4.

Definition 2.13: Let M be a complex manifold admitting a hyperkähler structure H. We say that M is of general type or generic with respect to H if all elements of the group

⨁_{p} H^{p,p}(M) ∩ H^{2p}(M, ℤ) ⊂ H^{∗}(M)

are SU(2)-invariant. We say that M is Mumford–Tate generic if for all n ∈ ℤ^{>0}, all the cohomology classes

α ∈ ⨁_{p} H^{p,p}(M^{n}) ∩ H^{2p}(M^{n}, ℤ) ⊂ H^{∗}(M^{n})

are SU(2)-invariant. In other words, M is Mumford–Tate generic if for all n ∈ ℤ^{>0}, the n-th power M^{n} is generic. Clearly, Mumford–Tate generic implies generic.

Proposition 2.14: Let M be a compact manifold, H a hyperkähler structure on M and S be the set of induced complex structures over M. Denote by S_{0} ⊂ S the set of L ∈ S such that (M, L) is Mumford-Tate generic with respect to H. Then S_{0} is dense in S. Moreover, the complement S\S_{0} is countable.

Proof: This is Proposition 2.2 from [V2(II)]

Theorem 2.10 has the following immediate corollary:

Corollary 2.15: Let M be a compact holomorphically symplectic manifold. Assume that M is of general type with respect to a hyperkähler structure H. Let S ⊂ M be closed complex analytic subvariety. Then S is trianalytic with respect to H.

In [V-d3], [V-d], [V-d2], we gave a number of equivalent definitions of a singular hyperkähler and hypercomplex variety. We refer the reader to [V-d2] for the precise definition; for our present purposes it suffices to say that all trianalytic subvarieties are hyperkähler varieties. The following Desingularization Theorem is very useful in the study of trianalytic subvarieties.

Theorem 2.16: ([V-d2]) Let M be a hyperkähler variety, and I an induced complex structure. Consider the normalization

{ \atop (M, I)} ⟶^{n} (M, I)

of (M, I). Then { \atop (M, I)} is smooth and has a natural hyperkähler structure H, such that the associated map n : { \atop (M, I)} ⟶ (M, I) agrees with H. Moreover, the hyperkähler manifold M̃ := { \atop (M, I)} is independent from the choice of induced complex structure I.

Let M be a K3 surface, and M^{[n]} be a Hilbert scheme of points on M. Then M^{[n]} admits a hyperkähler structure ([Bea]). In [V5], we proved the following theorem.

Theorem 2.17: Let M be a complex K3 surface without automorphisms. Assume that M is Mumford-Tate generic with respect to some hyperkaḧler structure. Consider the Hilbert scheme M^{[n]} of points on M. Pick a hyperkähler structure on M^{[n]} which is compatible with the complex structure. Then M^{[n]} has no proper trianalytic subvarieties.

2.4 Hyperholomorphic bundles

This subsection contains several versions of a definition of hyperholomorphic connection in a complex vector bundle over a hyperkähler manifold. We follow [V1].

Let B be a holomorphic vector bundle over a complex manifold M, ∇ a connection in B and Θ ∈ Λ^{2} ⊗ End(B) be its curvature. This connection is called compatible with a holomorphic structure if ∇_{X}(ζ) = 0 for any holomorphic section ζ and any antiholomorphic tangent vector field X ∈ T^{0,1}(M). If there exists a holomorphic structure compatible with the given Hermitian connection then this connection is called integrable.

One can define a Hodge decomposition in the space of differential forms with coefficients in any complex bundle, in particular, End(B).

Theorem 2.18: Let ∇ be a Hermitian connection in a complex vector bundle B over a complex manifold. Then ∇ is integrable if and only if Θ ∈ Λ^{1,1}(M, End(B)), where Λ^{1,1}(M, End(B)) denotes the forms of Hodge type (1,1). Also, the holomorphic structure compatible with ∇ is unique.

Proof: This is Proposition 4.17 of [Ko], Chapter I.

This result has the following more general version:

Proposition 2.19: Let ∇ be an arbitrary (not necessarily Hermitian) connection in a complex vector bundle B. Then ∇ is integrable if and only its (0, 1)-part has square zero.

This proposition is a version of Newlander-Nirenberg theorem. For vector bundles, it was proven by Atiyah and Bott.

Definition 2.20: Let B be a Hermitian vector bundle with a connection ∇ over a hyperkähler manifold M. Then ∇ is called hyperholomorphic if ∇ is integrable with respect to each of the complex structures induced by the hyperkähler structure.

As follows from Theorem 2.18, ∇ is hyperholomorphic if and only if its curvature Θ is of Hodge type (1,1) with respect to any of complex structures induced by a hyperkähler structure.

As follows from Lemma 2.6, ∇ is hyperholomorphic if and only if Θ is a SU(2)-invariant differential form.

Example 2.21: (Examples of hyperholomorphic bundles)

(i) Let M be a hyperkähler manifold, and T M be its tangent bundle equipped with the Levi–Civita connection ∇. Consider a complex structure on T M induced from the quaternion action. Then ∇ is a Hermitian connection which is integrable with respect to each induced complex structure, and hence, is hyperholomorphic.

(ii) For B a hyperholomorphic bundle, all its tensor powers are also hyperholomorphic.

(iii) Thus, the bundles of differential forms on a hyperkähler manifold are also hyperholomorphic.

2.5 Stable bundles and Yang–Mills connections.

This subsection is a compendium of the most basic results and definitions from the Yang–Mills theory over Kähler manifolds, concluding in the fundamental theorem of Uhlenbeck–Yau [UY].

Definition 2.22: Let F be a coherent sheaf over an n-dimensional compact Kähler manifold M. We define the degree deg(F) (sometimes the degree is also denoted by deg c_{1}(F)) as

deg(F) = ∫_{M} \frac{c_{1}(F) ∧ ω^{n−1}}{vol(M)}

and slope(F) as

slope(F) = \frac{1}{rank(F)} · deg(F).

The number slope(F) depends only on a cohomology class of c_{1}(F).

Let F be a coherent sheaf on M and F′ ⊂ F its proper subsheaf. Then F′ is called destabilizing subsheaf if slope(F′) ⩾ slope(F)

A coherent sheaf F is called stable^{1} if it has no destabilizing subsheaves. A coherent sheaf F is called semistable if for all destabilizing subsheaves F′ ⊂ F, we have slope(F′) = slope(F).

Later on, we usually consider the bundles B with deg(B) = 0.

Let M be a Kähler manifold with a Kähler form ω. For differential forms with coefficients in any vector bundle there is a Hodge operator L : η ⟶ ω ∧ η. There is also a fiberwise-adjoint Hodge operator Λ (see [GH]).

Definition 2.23: Let B be a holomorphic bundle over a Kähler manifold M with a holomorphic Hermitian connection ∇ and a curvature Θ ∈ Λ^{1,1} ⊗ End(B). The Hermitian metric on B and the connection ∇ defined by this metric are called Yang-Mills if

Λ(Θ) = constant · Id |_{B} ,

where Λ is a Hodge operator and Id |_{B} is the identity endomorphism which is a section of End(B).

Further on, we consider only these Yang–Mills connections for which this constant is zero.

A holomorphic bundle is called indecomposable if it cannot be decomposed onto a direct sum of two or more holomorphic bundles.

The following fundamental theorem provides examples of Yang--Mills bundles.

Theorem 2.24: ( Uhlenbeck-Yau) Let B be an indecomposable holomorphic bundle over a compact Kähler manifold. Then B admits a Hermitian Yang-Mills connection if and only if it is stable, and this connection is unique.

Proof: [UY].

Proposition 2.25: Let M be a hyperkähler manifold, L an induced complex structure and B be a complex vector bundle over (M, L). Then every hyperholomorphic connection ∇ in B is Yang-Mills and satisfies Λ(Θ) = 0 where Θ is a curvature of ∇.

1In the sense of Mumford-Takemoto

Proof: We use the definition of a hyperholomorphic connection as one with SU(2)-invariant curvature. Then Proposition 2.25 follows from the

Lemma 2.26: Let Θ ∈ Λ^{2}(M) be a SU(2)-invariant differential 2-form on M. Then Λ_{L}(Θ) = 0 for each induced complex structure L.^{2}

Proof: This is Lemma 2.1 of [V1].

Let M be a compact hyperkähler manifold, I an induced complex structure. For any stable holomorphic bundle on (M, I) there exists a unique Hermitian Yang-Mills connection which, for some bundles, turns out to be hyperholomorphic. It is possible to tell when this happens.

Theorem 2.27: Let B be a stable holomorphic bundle over (M, I), where M is a hyperkähler manifold and I is an induced complex structure over M. Then B admits a compatible hyperholomorphic connection if and only if the first two Chern classes c_{1}(B) and c_{2}(B) are SU(2)-invariant.^{3}

Proof: This is Theorem 2.5 of [V1].

2.6 Twistor spaces

Let M be a hyperkähler manifold. Consider the product manifold X = M × S^{2}. Embed the sphere S^{2} ⊂ ℍ into the quaternion algebra ℍ as the subset of all quaternions J with J^{2} = −1. For every point x = m × J ∈ X = M × S^{2} the tangent space T_{x}X is canonically decomposed T_{x}X = T_{m}M ⊕ T_{J}S^{2}. Identify S^{2} = ℂP^{1} and let I_{J} : T_{J}S^{2} → T_{J}S^{2} be the complex structure operator. Let I_{m} : T_{m}M → T_{m}M be the complex structure on M induced by J ∈ S^{2} ⊂ ℍ.

The operator I_{x} = I_{m} ⊕ I_{J} : T_{x}X → T_{x}X satisfies I_{x} ∘ I_{x} = −1. It depends smoothly on the point x, hence defines an almost complex structure on X. This almost complex structure is known to be integrable (see [Sal]).

Definition 2.28: The complex manifold ⟨X, I_{x}⟩ is called the twistor space for the hyperkähler manifold M, denoted by Tw(M). This manifold is equipped with a real analytic projection σ : Tw(M) ⟶ M and a complex analytic projection π : Tw(M) ⟶ ℂP^{1}.

2By Λ_{L} we understand the Hodge operator Λ associated with the Kähler complex structure L.

3We use Lemma 2.5 to speak of action of SU(2) in cohomology of M.

The twistor space Tw(M) is not, generally speaking, a Kähler manifold. For M compact, it is easy to show that Tw(M) does not admit a Kähler metric. We consider Tw(M) as a Hermitian manifold with the product metric.

Definition 2.29: Let X be an n-dimensional Hermitian manifold and let \sqrt{−1}ω be the imaginary part of the metric on X. Thus ω is a real (1, 1)- form. Assume that the form ω satisfies the following condition of Li and Yau ([LY]).

ω^{n−2} ∧ dω = 0.  (2.1)

Such Hermitian metrics are called metrics satisfying the condition of Li–Yau.

For a closed real 2-form η let

deg η = ∫_{X} ω^{n−1} ∧ η.

The condition (2.1) ensures that deg η depends only on the cohomology class of η. Thus it defines a degree functional deg : H^{2}(X, ℝ) → ℝ. This functional allows one to repeat verbatim the Mumford-Takemoto definitions of stable and semistable bundles in this more general situation. Moreover, the Hermitian Yang-Mills equations also carry over word-by-word. Li and Yau proved a version of Uhlenbeck–Yau theorem in this situation ([LY]; see also Theorem 3.8).

Proposition 2.30: Let M be a hyperkähler manifold and Tw(M) its twistor space, considered as a Hermitian manifold. Then Tw(M) satisfies the conditions of Li–Yau.

Proof: [KV], Proposition 4.5.

3 Hyperholomorphic sheaves

3.1 Stable sheaves and Yang-Mills connections

In [BS], S. Bando and Y.-T. Siu developed the machinery allowing one to apply the methods of Yang-Mills theory to torsion-free coherent sheaves. In the course of this paper, we apply their work to generalise the results of [V1]. In this Subsection, we give a short exposition of their results.

Definition 3.1: Let X be a complex manifold, and F a coherent sheaf on X. Consider the sheaf F^{∗} := Hom_{O_{X}} (F, O_{X}). There is a natural functorial map ρ_{F} : F ⟶ F^{∗∗}. The sheaf F^{∗∗} is called a reflexive hull, or reflexization of F. The sheaf F is called reflexive if the map ρ_{F} : F ⟶ F^{∗∗} is an isomorphism.

Remark 3.2: For all coherent sheaves F, the map ρ_{F^{∗}} : F^{∗} ⟶ F^{∗∗∗} is an isomorphism ([OSS], Ch. II, the proof of Lemma 1.1.12). Therefore, a reflexive hull of a sheaf is always reflexive.

Claim 3.3: Let X be a Kähler manifold, and F a torsion-free coherent sheaf over X. Then F (semi)stable if and only if F^{∗∗} is (semi)stable.

Proof: This is [OSS], Ch. II, Lemma 1.2.4.

Definition 3.4: Let X be a Kähler manifold, and F a coherent sheaf over X. The sheaf F is called polystable if F is a direct sum of stable sheaves.

The admissible Hermitian metrics, introduced by Bando and Siu in [BS], play the role of the ordinary Hermitian metrics for vector bundles.

Let X be a Kähler manifold. In Hodge theory, one considers the operator Λ : Λ^{p,q}(X) ⟶ Λ^{p−1,q−1}(X) acting on differential forms on X, which is adjoint to the multiplication by the Kähler form. This operator is defined on differential forms with coefficient in every bundle. Considering a curvature Θ of a bundle B as a 2-form with coefficients in End(B), we define the expression ΛΘ which is a section of End(B).

Definition 3.5: Let X be a Kähler manifold, and F a reflexive coherent sheaf over X. Let U ⊂ X be the set of all points at which F is locally trivial. By definition, the restriction F |_{U} of F to U is a bundle. An admissible

metric on F is a Hermitian metric h on the bundle F |_{U} which satisfies the following assumptions

(i) the curvature Θ of (F, h) is square integrable, and

(ii) the corresponding section ΛΘ ∈ End(F |_{U} ) is uniformly bounded.

Definition 3.6: Let X be a Kähler manifold, F a reflexive coherent sheaf over X, and h an admissible metric on F. Consider the corresponding Hermitian connection ∇ on F |_{U} . The metric h and the connection ∇ are called Yang-Mills if its curvature satisfies

ΛΘ ∈ End(F |_{U} ) = c · id

where c is a constant and id the unit section id ∈ End(F |_{U} ).

Further in this paper, we shall only consider Yang-Mills connections with ΛΘ = 0.

Remark 3.7: By Gauss-Bonnet formule, the constant c is equal to deg(F), where deg(F) is the degree of F (Definition 2.22).

One of the main results of [BS] is the following analogue of Uhlenbeck– Yau theorem (Theorem 2.24).

Theorem 3.8: Let M be a compact Kähler manifold, or a compact Hermitian manifold satisfying conditions of Li-Yau (Definition 2.29), and F a coherent sheaf without torsion. Then F admits an admissible Yang– Mills metric is and only if F is polystable. Moreover, if F is stable, then this metric is unique, up to a constant multiplier.

Proof: In [BS], Theorem 3.8 is proved for Kähler M ([BS], Theorem 3). It is easy to adapt this proof for Hermitian manifolds satisfying conditions of Li–Yau.

Remark 3.9: Clearly, the connection, corresponding to a metric on F, does not change when the metric is multiplied by a scalar. The Yang–Mills metric on a polystable sheaf is unique up to a componentwise multiplication by scalar multipliers. Thus, the Yang–Mills connection of Theorem 3.8 is unique.

Another important theorem of [BS] is the following.

Theorem 3.10: Let (F, h) be a holomorphic vector bundle with a Hermitian metric h defined on a Kähler manifold X (not necessary compact nor complete) outside a closed subset S with locally finite Hausdorff measure of real co-dimension 4. Assume that the curvature tensor of F is locally square integrable on X. Then F extends to the whole space X as a reflexive sheaf F. Moreover, if the metric h is Yang-Mills, then h can be smoothly extended as a Yang-Mills metric over the place where F is locally free.

Proof: This is [BS], Theorem 2.

3.2 Stable and semistable sheaves over hyperkähler manifolds

Let M be a compact hyperkähler manifold, I an induced complex structure, F a torsion-free coherent sheaf over (M, I) and F^{∗∗} its reflexization. Recall that the cohomology of M are equipped with a natural SU(2)-action (Lemma 2.5). The motivation for the following definition is Theorem 2.27 and Theorem 3.8.

Definition 3.11: Assume that the first two Chern classes of the sheaves F, F^{∗∗} are SU(2)-invariant. Then F is called stable hyperholomorphic if F is stable, and semistable hyperholomorphic if F can be obtained as a successive extension of stable hyperholomorphic sheaves.

Remark 3.12: The slope of a hyperholomorphic sheaf is zero, because a degree of an SU(2)-invariant 2-form is zero (Lemma 2.26).

Claim 3.13: Let F be a semistable coherent sheaf over (M, I). Then the following conditions are equivalent.

(i) F is stable hyperholomorphic

(ii) Consider the support S of the sheaf F^{∗∗}/F as a complex subvariety of (M, I). Let X_{1}, ... , X_{n} be the set of irreducible components of S of codimension 2. Then X_{i} is trianalytic for all i, and the sheaf F^{∗∗} is stable hyperholomorphic.

Proof: Consider an exact sequence

0 ⟶ F ⟶ F^{∗∗} ⟶ F^{∗∗}/F ⟶ 0.

Let [F/F^{∗∗}] ∈ H^{4}(M) be the fundamental class of the union of all codimension-2 components of support of the sheaf F/F^{∗∗}, taken with appropriate multiplicities. Then, c_{2}(F^{∗∗}/F) = −[F/F^{∗∗}]. From the product formula for Chern classes, it follows that

c_{2}(F) = c_{2}(F_{i}^{∗∗}) + c_{2}(F^{∗∗}/F) = c_{2}(F_{i}^{∗∗}) − [F/F^{∗∗}].  (3.1)

Clearly, if all X_{i} are trianalytic then the class [F/F^{∗∗}] is SU(2)-invariant. Thus, if the sheaf F^{∗∗} is hyperholomorphic and all X_{i} are trianalytic, then the second Chern class of F is SU(2)-invariant, and F is hyperholomorphic. Conversely, assume that F is hyperholomorphic. We need to show that all X_{i} are trianalytic. By definition,

[F/F^{∗∗}] = ∑_{i} λ_{i}[X_{i}]

where [X_{i}] denotes the fundamental class of X_{i}, and λ_{i} is the multiplicity of F/F^{∗∗} at X_{i}. By (3.1), (F hyperholomorphic) implies that the class [F/F^{∗∗}] is SU(2)-invariant. Since [F/F^{∗∗}] is SU(2)-invariant, we have

∑_{i} λ_{i} deg_{J}(X_{i}) = ∑_{i} λ_{i} deg_{I}(X_{i}).

By Wirtinger’s inequality (Proposition 2.11),

deg_{J}(X_{i}) ⩽ deg_{I}(X_{i}),

and the equality is reached only if X_{i} is trianalytic. By definition, all the numbers λ_{i} are positive. Therefore,

∑_{i} λ_{i} deg_{J}(X_{i}) ⩽ ∑_{i} λ_{i} deg_{I}(X_{i}).

and the equality is reached only if all the subvarieties X_{i} are trianalytic. This finishes the proof of Claim 3.13.

Claim 3.14: Let M be a compact hyperkähler manifold, and I an induced complex structure of general type. Then all torsion-free coherent sheaves over (M, I) are semistable hyperholomorphic.

Proof: Let F be a torsion-free coherent sheaf over (M, I). Clearly from the definition of induced complex structure of general type, the sheaves F and F^{∗∗} have SU(2)-invariant Chern classes. Now, all SU(2)-invariant 2forms have degree zero (Lemma 2.26), and thus, F is semistable.

3.3 Hyperholomorphic connection in torsion-free sheaves

Let M be a hyperkähler manifold, I an induced complex structure, and F a torsion-free sheaf over (M, I). Consider the natural SU(2)-action in the bundle Λ^{i}(M, B) of the differential i-forms with coefficients in a vector bundle B. Let Λ_{inv}^{i}(M, B) ⊂ Λ^{i}(M, B) be the bundle of SU(2)-invariant i-forms.

Definition 3.15: Let X ⊂ (M, I) be a complex subvariety of codimension at least 2, such that F |_{M\X} is a bundle, h be an admissible metric on

F |_{M\X} and ∇ the associated connection. Then ∇ is called hyperholomorphic if its curvature

Θ_{∇} = ∇^{2} ∈ Λ^{2} (M, End (F |_{M\X} ))

is SU(2)-invariant, i. e. belongs to Λ_{inv}^{2} (M, End (F |_{M\X} )).

Claim 3.16: The singularities of a hyperholomorphic connection form a trianalytic subvariety in M.

Proof: Let J be an induced complex structure on M, and U the set of all points of (M, I) where F is non-singular. Clearly, (F, ∇) is a bundle with admissible connection on (U, J). Therefore, the holomorphic structure on F |_{(U,J)} can be extended to (M, J). Thus, the singular set of F is holomorphic with respect to J. This proves Claim 3.16.

Proposition 3.17: Let M be a compact hyperkähler manifold, I an induced complex structure and F a reflexive sheaf admitting a hyperholomorphic connection. Then F is a polystable hyperholomorphic sheaf.

Proof: By Remark 3.20 and Theorem 3.8, F is polystable. We need only to show that the Chern classes c_{1}(F) and c_{2}(F) are SU(2)-invariant. Let U ⊂ M be the maximal open subset of M such that F |_{U} is locally trivial. By Theorem 3.10, the metric h and the connection ∇ can be extended to U. Let Tw U ⊂ Tw M be the corresponding twistor space, and σ : Tw U ⟶ U the standard map. Consider the bundle σ^{∗}F |_{U} , equipped with

a connection σ^{∗}∇. It is well known ^{1} that σ^{∗}F |_{U} is a bundle with an admissible Yang-Mills metric (we use Yang-Mills in the sense of Li-Yau, see Definition 2.29). By Theorem 3.10, σ^{∗}F |_{U} can be extended to a reflexive sheaf F on Tw M. Clearly, this extension coincides with the push-forward of σ^{∗}F |_{U} . The singular set S̃ of F is a pull-back of the singular set S of F. Thus, S is trianalytic. By desingularization theorem (Theorem 2.16), S can be desingularized to a hyperkähler manifold in such a way that its twistors form a desingularization of S. From the exact description of the singularities of S, provided by the desingularization theorem, we obtain that the standard projection π : S ⟶ ℂP^{1} is flat. By the following lemma, the restriction of F to the fiber (M, I) = π^{−1}({I}) of π coincides with F.

1See for instance the section “Direct and inverse twistor transform” in [KV].

Lemma 3.18: Let π : X ⟶ Y be a map of complex varieties, and S ↪ X a subvariety of X of codimension at least 2, which is flat over Y. Denote by U ↪^{j} X the complement U = (X\S). Let F be a vector bundle over U, and j_{∗}F its push-forward. Then the restriction of j_{∗}F to the fibers of π is reflexive.

Proof: Let Z = π^{−1}({y}) be a fiber of π. Since S is flat over Y and of codimension at least 2, we have j_{∗}(O_{Z∩U}) = O_{Z}. Clearly, for an open embedding γ : T_{1} ⟶ T_{2} and coherent sheaves A, B on T_{1}, we have γ_{∗}(A ⊗ B) = γ_{∗}A ⊗ γ_{∗}B. Thus, for all coherent sheaves A on U, we have

j_{∗}A ⊗ O_{Z} = j_{∗}(A ⊗ O_{Z∩U}).  (3.2)

This implies that j_{∗}(F |_{Z}) = j_{∗}F |_{Z}. It is well known ([OSS], Ch. II, 1.1.12; see also Lemma 9.2) that a push-forward of a reflexive sheaf under an open embedding γ is reflexive, provided that the complement of the image of γ has codimension at least 2. Therefore, j_{∗}F |_{Z} is a reflexive sheaf over Z. This proves Lemma 3.18.

Return to the proof of Proposition 3.17. Consider the sheaf F on the twistor space constructed above. Since F is reflexive, its singularities have codimension at least 3 ([OSS], Ch. II, 1.1.10). Therefore, F is flat in codimension 2, and the first two Chern classes of F = F|_{π^{−1}(I)} can be obtained by restricting the first two Chern classes of F to the subvariety (M, I) = π^{−1}(I) ⊂ Tw(M). It remains to show that such restriction is SU(2)-invariant. Clearly, H^{2}((M, I)) = H^{2}((M, I)\S), and H^{4}((M, I)) = H^{4}((M, I)\S). Therefore,

c_{1} (F|_{(M,I)} ) = c_{1} (F|_{(M,I)\S} )

and

c_{2} (F|_{(M,I)} ) = c_{2} (F|_{(M,I)\S} ) .

On the other hand, the restriction F|_{Tw(M)\S} is a bundle. Therefore, the classes

c_{1} (F|_{(M,I)\S} ) , c_{2} (F|_{(M,I)\S} )

are independent from I ∈ ℂP^{1}. On the other hand, these classes are of type (p, p) with respect to all induced complex structures I ∈ ℂP^{1}. By Lemma 2.6, this implies that the classes c_{1}(F|_{(M,I)} ), c_{1}(F|_{(M,I)} ) are SU(2)- invariant. As we have shown above, these two classes are equal to the first Chern classes of F. Proposition 3.17 is proven.

3.4 Existence of hyperholomorphic connections

The following theorem is the main result of this section.

Theorem 3.19: Let M be a compact hyperkähler manifold, I an induced complex structure and F a reflexive sheaf on (M, I). Then F admits a hyperholomorphic connection if and only if F is polystable hyperholomorphic in the sense of Definition 3.11.

Remark 3.20: From Lemma 2.26, it is clear that a hyperholomorphic connection is always Yang-Mills. Therefore, such a connection is unique (Theorem 3.8).

The “only if” part of Theorem 3.19 is Proposition 3.17. The proof of “if” part of Theorem 3.19 takes the rest of this subsection.

Let I be an induced complex structure. We denote the corresponding Hodge decomposition on differential forms by Λ^{∗}(M) = ⊕Λ_{I}^{p,q}(M), and the standard Hodge operator by Λ_{I} : Λ_{I}^{p,q}(M) ⟶ Λ_{I}^{p−1,q−1}(M). All these structures are defined on the differential forms with coefficients in a bundle. Let deg_{I} η := ∫_{M} T r(Λ_{I})^{k}(η), for η ∈ Λ^{k}(M, End B). The following claim is implied by an elementary linear-algebraic computation.

Claim 3.21: Let M be a hyperkähler manifold, B a Hermitian vector bundle over M, and Θ a 2-form on M with coefficients in 𝔰𝔲(B). Assume that

Λ_{I}Θ = 0, Θ ∈ Λ_{I}^{1,1}(M, End B)

for some induced complex structure I. Assume, moreover, that Θ is squareintegrable. Let J be another induced complex structure, J ≠ ±I. Then

deg_{I} Θ^{2} ⩾ | deg_{J} Θ^{2}|,

and the equality is reached only if Θ is SU(2)-invariant.

Proof: The following general argument is used.

Sublemma 3.22: Let M be a Kähler manifold, B a Hermitian vector bundle over M, and Ξ a square-integrable 2-form on M with coefficients in 𝔰𝔲(B). Then:

(i) For

Λ_{I}Ξ = 0, Ξ ∈ Λ_{I}^{1,1}(M, End B)

we have

deg_{I} Ξ^{2} = C ∫_{M} |Ξ|^{2} Vol M,

where C = (4π^{2}n(n − 1))^{−1} M.

(ii) For

Ξ ∈ Λ_{I}^{2,0}(M, End B) ⊕ Λ_{I}^{0,2}(M, End B)

we have

deg_{I} Ξ^{2} = −C ∫_{M} |Ξ|^{2} Vol M,

where C is the same constant as appeared in (i).

Proof: The proof is based on a linear-algebraic computation (so-called Lübcke-type argument). The same computation is used to prove Hodge-Riemann bilinear relations.

Return to the proof of Claim 3.21. Let Θ = Θ_{J}^{1,1} + Θ_{J}^{2,0} + Θ_{J}^{0,2} be the Hodge decomposition associated with J. The following Claim shows that Θ_{J}^{1,1} satisfies conditions of Sublemma 3.22 (i).

Claim 3.23: Let M be a hyperkähler manifold, I, L induced complex structures and Θ a 2-form on M satisfying

Λ_{I}Θ = 0, Θ ∈ Λ_{I}^{1,1}(M).

Let Θ_{L}^{1,1} be the (1, 1)-component of Θ taken with respect to L. Then Λ_{L}Θ_{L}^{1,1} = 0.

Proof: Clearly, Λ_{L}Θ_{L}^{1,1} = Λ_{L}Θ. Consider the natural Hermitian structure on the space of 2-forms. Since Θ is of type (1, 1) with respect to I, Θ is fiberwise orthogonal to the holomorphic symplectic form Ω = ω_{J}+\sqrt{−1}Ω_{K} ∈ Λ_{I}^{2,0}(M). By the same reason, Θ is orthogonal to \overline{Ω}. Therefore, Θ is orthogonal to ω_{J} and ω_{K}. Since Λ_{I}Θ = 0, Θ is also orthogonal to ω_{I}. The map Λ_{L} is a projection to the form ω_{L} which is a linear combination of ω_{I}, ω_{J} and ω_{K}. Since Θ is fiberwise orthogonal to ω_{L}, we have Λ_{L}Θ = 0.

By Sublemma 3.22, we have

deg_{J} (Θ_{J}^{1,1})^{2} = C ∫_{M} |Θ_{J}^{1,1}|^{2}

and

deg_{J} (Θ_{J}^{2,0} + Θ_{J}^{0,2})^{2} = −C ∫_{M} |Θ_{J}^{2,0} + Θ_{J}^{0,2}|^{2}.

Thus,

deg_{J} Θ^{2} = C ∫_{M} |Θ_{J}^{1,1}| 2 − C ∫_{M} |Θ_{J}^{2,0} + Θ_{J}^{0,2}| 2.

On the other hand,

deg_{I} Θ^{2} = C ∫_{M} |Θ| 2 = C ∫_{M} |Θ_{J}^{1,1}| 2 + C ∫_{M} |Θ_{J}^{2,0} + Θ_{J}^{0,2}| 2.

Thus, deg_{I} Θ^{2} > | deg_{J} Θ^{2}| unless Θ_{J}^{2,0} + Θ_{J}^{0,2} = 0. On the other hand, Θ_{J}^{2,0} + Θ_{J}^{0,2} = 0 means that Θ is of type (1, 1) with respect to J. Consider the standard U(1)-action on differential forms associated with the complex structures I and J. These two U(1)-actions generate the whole Lie group SU(2) acting on Λ^{2}(M) (here we use that I ≠ ±J). Since Θ is of type (1, 1) with respect to I and J, this form is SU(2)-invariant. This proves Claim 3.21.

Return to the proof of Theorem 3.19. Let ∇ be the admissible Yang-Mills connection in F, and Θ its curvature. Recall that the form T rΘ^{2} represents the cohomology class 2c_{2}(F) − \frac{r−1}{r}c_{1}(F)^{2}, where c_{i} are Chern classes of F. Since the form T rΘ^{2} is square-integrable, the integral

deg_{J} Θ^{2} = ∫_{M} T rΘ^{2}ω_{J}^{n−2}

makes sense. In [BS], it was shown how to approximate the connection ∇ by smooth connections, via the heat equation. This argument, in particular, was used to show that the value of integrals like ∫_{M} T rΘ^{2}ω_{J}^{n−2} can be computed through cohomology classes and the Gauss–Bonnet formula

T rΘ^{2} = 2c_{2}(F) − \frac{r − 1}{r}c_{1}(F)^{2}.

Since the classes c_{2}(F), c_{1}(F) are SU(2)-invariant, we have

deg_{I} Θ^{2} = deg_{J} Θ^{2}

for all induced complex structures I, J. By Claim 3.21, this implies that Θ is SU(2)-invariant. Theorem 3.19 is proven.

The same argument implies the following corollary.

Corollary 3.24: Let M be a compact hyperkähler manifold, I an induced complex structure, F a stable reflexive sheaf on (M, I), and J be an induced complex structure, J ≠ ±I. Then

deg_{I} (2c_{2}(F) − \frac{r − 1}{r}c_{1}(F)^{2}) ⩾ |deg_{J} (2c_{2}(F) − \frac{r − 1}{r}c_{1}(F)^{2})| ,

and the equality holds if and only if F is hyperholomorphic.

3.5 Tensor category of hyperholomorphic sheaves

This subsection is extraneous. Further on, we do not use the tensor structure on the category of hyperholomorphic sheaves. However, we need the canonical identification of the categories of hyperholomorphic sheaves associated with different induced complex structures.

From Bando-Siu (Theorem 3.8) it follows that on a compact Kähler manifold a tensor product of stable reflexive sheaves is polystable. Similarly, Theorem 3.19 implies that a tensor product of polystable hyperholomorphic sheaves is polystable hyperholomorphic. We define the following category.

Definition 3.25: Let M be a compace hyperkähler manifold and I an induced complex structure. Let F_{st}(M, I) be a category with objects reflexive polystable hyperholomorphic sheaves and morphisms as in category of coherent sheaves. This category is obviously additive. The tensor product on F_{st}(M, I) is induced from the tensor product of coherent sheaves.

Claim 3.26: The category F_{st}(M, I) is abelian. Moreover, it is a Tannakian tensor category.

Proof: Let φ : F_{1} ⟶ F_{2} be a morphism of hyperholomorphi sheaves. In Definition 2.22 , we introduced a slope of a coherent sheaf. Clearly, sl(F_{1}) ⩽ sl(im φ) ⩽ sl(F_{2}). All hyperholomorphic sheaves have slope 0 by Remark 3.12. Thus, sl(im φ) = 0 and the subsheaf im φ ⊂ F_{2} is destabilizing. Since F_{2} is polystable, this sheaf is decomposed:

F_{2} = im φ ⊕ coker φ.

A similar argument proves that F_{1} = ker φ ⊕ coim φ, with all summands hyperholomorphic. This proves that F_{st}(M, I) is abelian. The Tannakian properties are clear.

The category F_{st}(M, I) does not depend from the choice of induced complex structure I:

Theorem 3.27: Let M be a compact hyperkähler manifold, I_{1}, I_{2} induced complex structures and F_{st}(M, I_{1}), F_{st}(M, I_{2}) the associated categories of polystable reflexive hyperholomorphic sheaves. Then, there exists a natural equivalence of tensor categories

Φ_{I_{1},I_{2}} : F_{st}(M, I_{1}) ⟶ F_{st}(M, I_{2}).

Proof: Let F ∈ F_{st}(M, I_{1}) be a reflexive polystable hyperholomorphic sheaf and ∇ the canonical admissible Yang-Mills connection. Consider the sheaf F on the twistor space Tw(M) constructed as in the proof of Proposition 3.17. Restricting F to π^{−1}(I_{2}) ⊂ Tw(M), we obtain a coherent sheaf F′ on (M, I_{2}). As we have shown in the proof of Proposition 3.17, the sheaf (F′)^{∗∗} is polystable hyperholomorphic. Let Φ_{I_{1},I_{2}}(F) := (F′)^{∗∗}. It is easy to check that thus constructed map of objects gives a functor

Φ_{I_{1},I_{2}} : F_{st}(M, I_{1}) ⟶ F_{st}(M, I_{2}),

and moreover, Φ_{I_{1},I_{2}} ∘ Φ_{I_{2},I_{1}} = Id. This shows that Φ_{I_{1},I_{2}} is an equivalence. Theorem 3.27 is proven.

Definition 3.28: By Theorem 3.27, the category F_{st}(M, I_{1}) is independent from the choice of induced complex structure. We call this category the category of polystable hyperholomorphic reflexive sheaves on M and denote it by F(M). The objects of F(M) are called hyperholomorphic sheaves on M. For a hyperholomorphic sheaf on M, we denote by F_{I} the corresponding sheaf from F_{st}(M, I_{1}).

Remark 3.29: Using the same argument as proves Theorem 10.8 (ii), it is easy to check that the category F(M) is a deformational invariant of M. That is, for two hyperkähler manifolds M_{1}, M_{2} which are deformationally equivalent, the categories F(M_{i}) are also equivalent, assuming that P ic(M_{1}) = P ic(M_{2}) = 0. The proof of this result is essentially contained in [V3-bis].

Remark 3.30: As Deligne proved ([D]), for a each Tannakian category C equipped with a fiber functor, there exists a natural pro-algebraic group G such that C is a group of representations of G. For F(M), there are several natural fiber functors. The simplest one is defined for each induced complex structure I such that (M, I) is algebraic (such complex structures always exist, as proven in [F]; see also [V-a] and Subsection 4.1). Let K(M, I) is the space of rational functions on (M, I). For F ∈ F_{st}(M, I), consider the functor F ⟶ η_{I}(F), where η_{I}(F) is the space of global sections of F ⊗ K(M, I). This is clearly a fiber functor, which associates to F(M) the group G_{I}. The corresponding pro-algebraic group G_{I} is a deformational, that is, topological, invariant of the hyperkähler manifold.

4 Cohomology of hyperkähler manifolds

This section contains a serie of preliminary results which are used further on to define and study the C-restricted complex structures.

4.1 Algebraic induced complex structures

This subsection contains a recapitulation of results of [V-a].

A more general version of the following theorem was proven by A. Fujiki ([F], Theorem 4.8 (2)).

Theorem 4.1: Let M be a compact simple hyperkähler manifold and R be the set of induced complex structures R ≅ ℂP^{1}. Let R_{alg} ⊂ R be the set of all algebraic induced complex structures. Then R_{alg} is countable and dense in R.

Proof: This is [V-a], Theorem 2.2.

In the proof of Theorem 4.1, the following important lemma was used.

Lemma 4.2:

(i) Let O ⊂ H^{2}(M, ℝ) be the set of all cohomology classes which are Kähler with respect to some induced comples structure. Then O is open in H^{2}(M, ℝ). Moreover, for all ω ∈ O, the class ω is not SU(2)-invariant.

(ii) Let η ∈ H^{2}(M, ℝ) be a cohomology class which is not SU(2)-invariant. Then there exists a unique up to a sign induced complex structure I ∈ R/{±1} such that η belongs to H_{I}^{1,1}(M).

Proof: This statement is a form of [V-a], Lemma 2.3.

4.2 The action of 𝔰𝔬(5) on the cohomology of a hyperkähler manifold

This subsection is a recollection of data from [V0] and [V2(II)].

Let M be a hyperkähler manifold. For an induced complex structure R over M, consider the Kähler form ω_{R} = (·, R·), where (·, ·) is the Riemannian form. As usually, L_{R} denotes the operator of exterior multiplication by ω_{R}, which is acting on the differential forms A^{∗}(M, ℂ) over M. Consider the adjoint operator to L_{R}, denoted by Λ_{R}.

One may ask oneself, what algebra is generated by L_{R} and Λ_{R} for all induced complex structures R? The answer was given in [V0], where the following theorem was proven.

Theorem 4.3: ([V0]) Let M, H be a hyperkähler manifold, and 𝔞_{H} be a Lie algebra generated by L_{R} and Λ_{R} for all induced complex structures R over M. Then the Lie algebra 𝔞_{H} is isomorphic to 𝔰𝔬(4, 1).

The following facts about a structure of 𝔞_{H} were also proven in [V0]. Let I, J and K be three induced complex structures on M, such that

I ∘ J = −J ∘ I = K. For an induced complex structure R, consider an operator adR on cohomology, acting on (p, q)-forms as a multiplication by (p − q)\sqrt{−1} . The operators adR generate a 3-dimensional Lie algebra 𝔤_{H}, which is isomorphic to 𝔰𝔲(2). This algebra coincides with the Lie algebra associated to the standard SU(2)-action on H^{∗}(M). The algebra 𝔞_{H} contains 𝔤_{H} as a subalgebra, as follows:

[Λ_{J}, L_{K}] = [L_{J}, Λ_{K}] = ad I (etc).  (4.1)

The algebra 𝔞_{H} is 10-dimensional. It has the following basis: L_{R}, Λ_{R}, ad R (R = I, J, K) and the element H = [L_{R}, Λ_{R}]. The operator H is a standard Hodge operator; it acts on r-forms over M as multiplication by a scalar n − r, where n = dim_{ℂ}M.

Definition 4.4: Let 𝔤 be a semisimple Lie algebra, V its representation and V = ⊕V_{α} a 𝔤-invariant decompostion of V, such that for all α, V_{α} is a direct sum of isomorphic finite-dimensional representations W_{α} of V, and all W_{α} are distinct. Then the decomposition V = ⊕V_{α} is called the isotypic decomposition of V.

It is clear that for all finite-dimensional representations, isotypic decomposition always exists and is unique.

Let M be a compact hyperkähler manifold. Consider the cohomology space H^{∗}(M) equipped with the natural action of 𝔞_{H} = 𝔰𝔬(5). Let H_{o}^{∗} ⊂ H^{∗}(M) be the isotypic component containing H^{0}(M) ⊂ H^{∗}(M). Using the root system written explicitly for 𝔞_{H} in [V0], [V3], it is easy to check that H_{o}^{∗}(M) is an irreducible representation of 𝔰𝔬(5). Let p : H^{∗}(M) ⟶ H_{o}^{∗}(M) be the unique 𝔰𝔬(5)-invariant projection, and i : H_{o}^{∗}(M) ↪ H^{∗}(M) the natural embedding.

Let M be a compact hyperkähler manifold, I an induced complex structure, and ω_{I} the corresponding Kähler form. Consider the degree map deg_{I} : H^{2p}(M) ⟶ ℂ, η ⟶ ∫_{M} η ∧ ω_{I}^{n−p}, where n = dim_{C} M.

Proposition 4.5: The space

H_{o}^{∗}(M) ⊂ H^{∗}(M)

is a subalgebra of H^{∗}(M), which is invariant under the SU(2)-action. Moreover, for all induced complex structures I, the degree map

deg_{I} : H^{∗}(M) ⟶ ℂ

satisfies

deg_{I}(η) = deg_{I}(i(p(η)),

where i : H_{o}^{∗}(M) ↪ H^{∗}(M), p : H^{∗}(M) ⟶ H_{o}^{∗}(M) are the 𝔰𝔬(5)-invariant maps defined above. And finally, the projection p : H^{∗}(M) ⟶ H_{o}^{∗}(M) is SU(2)-invariant.

Proof: The space H_{o}^{∗}(M) is generated from 111 ∈ H^{0}(M) by operators L_{R}, Λ_{R}. To prove that H_{o}^{∗}(M) is closed under multiplication, we have to show that H_{o}^{∗}(M) is generated (as a linear space) by expressions of type L_{r_{1}} ∘L_{R_{2}} ∘...∘111. By (4.1), the commutators of L_{R}, Λ_{R} map such expressions to linear combinations of such expressions. On the other hand, the operators Λ_{R} map 111 to zero. Thus, the operators Λ_{R} map expressions of type L_{r_{1}} ∘ L_{R_{2}} ∘ ... ∘ 111 to linear combinations of such expressions. This proves that H_{o}^{∗}(M) is closed under multiplication. The second statement of Proposition 4.5 is clear (see, e. g. [V2(II)], proof of Proposition 4.5). It remains to show that H_{o}^{∗}(M) ⊂ H^{∗}(M) is an SU(2)-invariant subspace and that p : H^{∗}(M) ⟶ H_{o}^{∗}(M) is compatible with the SU(2)-action. From (4.1), we obtain that the Lie group G_{A} associated with 𝔞_{H} ≅ 𝔰𝔬(1, 4) contains SU(2) acting in a standard way on H^{∗}(M). Since the map p : H^{∗}(M) ⟶ H_{o}^{∗}(M) commutes with G_{A}-action, p also commutes with SU(2)-action. We proved Proposition 4.5.

4.3 Structure of the cohomology ring

In [V3] (see also [V3-bis]), we have computed explicitly the subalgebra of cohomology of M generated by H^{2}(M). This computation can be summed up as follows.

Theorem 4.6: ([V3], Theorem 15.2) Let M be a compact hyperkähler manifold, H^{1}(M) = 0, dim_{ℂ} M = 2n, and H_{r}^{∗}(M) the subalgebra of cohomology of M generated by H^{2}(M). Then

\{ \begin{matrix}H_{r}^{2i}(M) ≅ S^{i}H^{2}(M) & for i ⩽ n, and \\ H_{r}^{2i}(M) ≅ S^{2n−i}H^{2}(M) & for i ⩾ n \end{matrix}

Theorem 4.7: Let M be a simple hyperkähler manifold. Consider the group G generated by a union of all SU(2) for all hyperkähler structures on M. Then the Lie algebra of G is isomorphic to 𝔰𝔬(H^{2}(M)), for a certain natural integer bilinear symmetric form on H^{2}(M), called Bogomolov-Beauville form.

Proof: [V3] (see also [V3-bis]).

The key element in the proof of Theorem 4.6 and Theorem 4.7 is the following algebraic computation.

Theorem 4.8: Let M be a simple hyperkähler manifold, and H a hyperkähler structure on M. Consider the Lie subalgebra

𝔞_{H} ⊂ End(H^{∗}(M)), 𝔞_{H} ≅ 𝔰𝔬(1, 4),

associated with the hyperkähler structure (Subsection 4.2). Let

𝔤 ⊂ End(H^{∗}(M))

be the Lie algebra generated by subalgebras 𝔞_{H} ⊂ End(H^{∗}(M)), for all hyperkäher structures H on M. Then

(i) The algebra 𝔤 is naturally isomorphic to the Lie algebra 𝔰𝔬(V ⊕ ℌ), where V is the linear space H^{2}(M, ℝ) equipped with the Bogomolov– Beauville pairing, and ℌ is a 2-dimensional vector space with a quadratic form of signature (1, −1).

(ii) The space H_{r}^{∗}(M) is invariant under the action of 𝔤, Moreover,

H_{r}^{∗}(M) ⊂ H^{∗}(M)

is an isotypic ^{1} component of the space H^{∗}(M) considered as a representation of 𝔤.

Proof: [V3] (see also [V3-bis]).

As one of the consequences of Theorem 4.6, we obtain the following lemma, which will be used further on in this paper.

Lemma 4.9: Let M be a simple hyperkähler manifold, dim_{ℍ} M = n, and p : H^{∗}(M) ⟶ H_{o}^{∗}(M) the map defined in Subsection 4.2. Then, for all x, y ∈ H_{r}^{∗}(M), we have

p(x)p(y) = p(xy), whenever xy ∈ ⨁_{i⩽2n} H^{i}(M).

Proof: Let ω_{I}, ω_{J}, ω_{K}, x_{1}, ..., x_{n} be an orthonormal basis in H^{2}(M). Clearly, the vectors x_{1}, ..., x_{k} are SU(2)-invariant. Therefore, these vectors are highest vectors of the corresponding 𝔞_{H}-representations, with respect to the root system and Cartan subalgebra for 𝔞_{H} which is written in [V0] or [V3]. We obtain that the monomials

1See Definition 4.4 for the definition of isotypic decomposition.

P_{k_{1},k_{2},k_{3},{n_{i}}} = ω_{I}^{k_{1}}ω_{J}^{k_{2}}ω_{K}^{k_{3}} ∏ x_{i}^{n_{i}}, ∑ n_{i} = N, P_{k_{1},k_{2},k_{3},{n_{i}}} ∈ ⨁_{i⩽2n} H^{i}(M)

belong to the different isotypical components for different N’s. By Theorem 4.6, a product of two such monomials P_{k_{1},k_{2},k_{3},{n_{i}}} and P_{k′_{1},k′_{2},k′_{3},{n′_{i}}} is equal to P_{k_{1}+k′_{1},k_{2}+k′_{2},k_{3}+k′_{3},{n_{i}+n′_{i}}}, assuming that

P_{k_{1},k_{2},k_{3},{n_{i}}}P_{k′_{1},k′_{2},k′_{3},{n′_{i}}} ∈ ⨁_{i⩽2n} H^{i}(M).

Thus, the isotypical decomposition associated with the 𝔞_{H}-action is compatible with multiplicative structure on H^{∗}(M), for low-dimensional cycles. This implies Lemma 4.9.

We shall use the following corollary of Lemma 4.9.

Corollary 4.10: Let M be a simple hyperkähler manifold, dim_{ℍ} M > 1, and ω_{1}, ω_{2} ∈ H^{2}(M) cohomology classes which are SU(2)-invariant. Then, for all induced complex structures I, we have deg_{I}(ω_{1}ω_{2}) = 0.

Proof: By definition, the classes ω_{1}, ω_{2} satisfy ω_{i} ∈ ker p. By Lemma 4.9, we have ω_{1}ω_{2} ∈ ker p. By Proposition 4.5, deg_{I} ω_{1}ω_{2} = 0.

Let ω be a rational Kähler form. The corresponding 𝔰𝔩(2)-action on H^{∗}(M) is clearly compatible with the rational structure on H^{∗}(M). It is easy to see (using, for instance, Lemma 4.2) that 𝔤 is generated by 𝔰𝔩(2)-triples associated with rational Kähler forms ω. Therefore, the action of 𝔤 on H^{∗}(M) is compatible with the rational structure on H^{∗}(M). Using the isotypic decomposition, we define a natural 𝔤-invariant map r : H^{∗}(M) ⟶ H_{r}^{∗}(M). Further on, we shall use the following properties of this map.

Claim 4.11:

(i) The map r : H^{∗}(M) ⟶ H_{r}^{∗}(M) is compatible with the rational structure on H^{∗}(M).

(ii) For every x ∈ ker r, and every hyperkähler structure H, the corresponding map p : H^{∗}(M) ⟶ H_{o}^{∗}(M) satisfies p(x) = 0.

(iii) For every x ∈ ker r, every hyperkähler structure H, and every induced complex structure I on M, we have deg_{I} x = 0.

Proof: Claim 4.11 (i) is clear, because the action of 𝔤 on H^{∗}(M) is compatible with the rational structure on H^{∗}(M). To prove Claim 4.11 (ii), we notice that the space H_{r}^{∗}(M) is generated from H^{0}(M) by the action of 𝔤, and H_{o}^{∗}(M) is generated from H^{0}(M) by the action of 𝔞_{H}. Since 𝔞_{H} is by definition a subalgebra in 𝔤, we have H_{o}^{∗}(M) ⊂ H_{r}^{∗}(M). The isotypic projection r : H^{∗}(M) ⟶ H_{r}^{∗}(M) is by definition compatible with the 𝔤action. Since 𝔞_{H} ⊂ 𝔤, the map r is also compatible with the 𝔞_{H}-action. Therefore, ker r ⊂ ker p. Claim 4.11 (iii) is implied by Claim 4.11 (ii) and Proposition 4.5.

Let x_{i} be an basis in H^{2}(M, ℚ) which is rational and orthonormal with respect to Bogomolov-Beauville pairing, (x_{i}, x_{i})_{B} = ε_{i} = ±1. Consider the cohomology class θ′ := ε_{i}x_{i}^{2} ∈ H^{4}(M, ℚ). Let θ ∈ H^{4}(M, ℤ) be a non-zero integer cohomology class which is proportional to θ′. From results of [V3] (see also [V3-bis]), the following proposition can be easily deduced.

Proposition 4.12: The cohomology class θ ∈ H^{4}(M, ℤ) is SU(2)- invariant for all hyperkähler structures on M. Moreover, for a generic hyperkähler structure, the group of SU(2)-invariant integer classes α ∈ H_{r}^{4}(M) has rank one, where H_{r}^{∗}(M) is the subalgebra of cohomology generated by H^{2}(M).

Proof: Clearly, if an integer class α is SU(2)-invariant for a generic hyperkähler structure, then α is G-invariant, where G is the group defined in Theorem 4.7. On the other hand, H_{r}^{4}(M) ≅ S^{2}(H^{2}(M)), as follows from Theorem 4.6. Clearly, the vector θ ∈ H_{r}^{4}(M) ≅ S^{2}(H^{2}(M)) is 𝔰𝔬(H^{2}(M))-invariant. Moreover, the space of 𝔰𝔬(H^{2}(M))-invariant vectors in S^{2}(H^{2}(M)) is one-dimensional. Finally, from an explicit computation of G it follows that G acts on H^{4}(M) as SO(H^{2}(M)), and thus, the Lie algebra invariants coincide with invariants of G. We found that the space of G-invariants in H_{r}^{4}(M) is one-dimensional and generated by θ. This proves Proposition 4.12.

Remark 4.13: It is clear how to generalize Proposition 4.12 from dimension 4 to all dimensions. The space H_{r}^{2d}(M)^{G} of G-invariants in H_{r}^{2d}(M) is 1-dimensional for d even and zero-dimensional for d odd.

4.4 Cohomology classes of CA-type

Let M be a compact hyperkähler manifold, and I an induced complex structure. All cohomology classes which appear as fundamental classes of complex subvarieties of (M, I) satisfy certain properties. Classes satisfying these properties are called classes of CA-type, from Complex Analytic. Here is the definition of CA-type.

Definition 4.14: Let η ∈ H_{I}^{2,2}(M)∩H^{4}(M, ℤ) be an integer (2,2)-class. Assume that for all induced complex structures J, satisfying I ∘ J = −J ∘ I, we have deg_{I}(η) ⩾ deg_{J}(η), and the equality is reached only if η is SU(2)- invariant. Assume, moreover, that deg_{I}(η) ⩾ | deg_{J}(η)|. Then η is called a class of CA-type.

Theorem 4.15: Let M be a simple hyperkähler manifold, of dimension dim_{ℍ} M > 1, I an induced complex structure, and η ∈ H_{I}^{2,2}(M) ∩ H^{4}(M, ℤ) an integer (2,2)-class. Assume that one of the following conditions holds.

(i) There exists a complex subvariety X ⊂ (M, I) such that η is the fundamental class of X

(ii) There exists a stable coherent torsion-free sheaf F over (M, I), such that the first Chern class of F is zero, and η = c_{2}(F).

Then η is of CA-type.

Proof: Theorem 4.15 (i) is a direct consequence of Wirtinger’s inequality (Proposition 2.11). It remains to prove Theorem 4.15 (ii).

We assume, temporarily, that F is reflexive. By Corollary 3.24, we have

deg_{I}(2c_{2}(F) − \frac{r − 1}{r}c_{1}(F)^{2}) ⩾ |deg_{J}(2c_{2}(F) − \frac{r − 1}{r}c_{1}(F)^{2})| , (4.2)

and the equality happens only if F is hyperholomorphic. Since c_{1}(F) is SU(2)-invariant, we have deg_{I}(c_{1}(F)^{2}) = deg_{J}(c_{1}(F)^{2}) = 0 (Corollary 4.10). Thus, (4.2) implies that

deg_{I} 2c_{2}(F) ⩾ | deg_{J} 2c_{2}(F)|

and the inequality is strict unless F is hyperholomorphic, in which case, the class c_{2}(F) is SU(2)-invariant by definition. We have proven Theorem 4.15 (ii) for the case of reflexive F.

5. C-RESTRICTED COMPLEX STRUCTURES 59

For F not necessary reflexive sheaf, we have shown in the proof of Claim 3.13 that

c_{2}(F) = c_{2}(F^{∗∗}) + ∑ n_{i}[X_{i}],

where n_{i} are positive integers, and [X_{i}] are the fundamental classes of irreducible components of support of the sheaf F^{∗∗}/F. Therefore, the class c_{2}(F) is a sum of classes of CA-type. Clearly, a sum of cohomology classes of CA-type is again a class of CA-type. This proves Theorem 4.15.

5 C-restricted complex structures on hyperkähler manifolds

5.1 Existence of C-restricted complex structures

We assume from now till the end of this section that the hyperkähler manifold M is simple (Definition 2.7). This assumption can be avoided, but it simplifies notation.

We assume from now till the end of this section that the hyperkähler manifold M is compact of real dimension dim_{ℝ} M ⩾ 8, i. e. dim_{ℍ} M ⩾ 2. This assumption is absolutely necessary. The case of hyperkähler surfaces with dim_{ℍ} M = 1 (torus and K3 surface) is trivial and for our purposes not interesting. It is not difficult to extend our definitions and results to the case of a compact hyperkähler manifold which is a product of simple hyperkähler manifolds with dim_{ℍ} M ⩾ 2.

Definition 5.1: Let M be a compact hyperkähler manifold, and I an induced complex structure. As usually, we denote by deg_{I} : H^{2p}(M) ⟶ ℂ the associated degree map, and by H^{∗}(M) = ⊕H_{I}^{p,q}(M) the Hodge decomposition. Assume that I is algebraic. Let C be a positive real number. We say that the induced complex structure I is C-restricted if the following conditions hold.

(i) For all non-SU(2)-invariant cohomology classes classes η ∈ H_{I}^{1,1}(M) ∩ H^{2}(M, ℤ), we have | deg_{I}(η)| > C.

(ii) Let η ∈ H_{I}^{2,2}(M) be a cohomology class of CA-type which is not SU(2)- invariant. Then | deg_{I}(η)| > C.

The heuristic (completely informal) meaning of this definition is the following. The degree map plays the role of the metric on the cohomology.

Cohomology classes with small degrees are “small”, the rest is “big”. Under reasonably strong assumptions, there are only finitely many “small” integer classes, and the rest is “big”. For each non-SU(2)-invariant cohomology class η there exists at most two induced complex structures for which η is of type (p, p). Thus, for most induced complex structures, all non-SU(2)- invariant integer (p, p) classes are “big”. Intuitively, the C-restriction means that all non-SU(2)-invariant integer (1,1) and (2,2)-cohomology classes are “big”. This definition is needed for the study of first and second Chern classes of sheaves. The following property of C-restricted complex structures is used (see Theorem 4.15): for every subvariety X ⊂ (M, I) of complex codimension 2, either X is trianalytic or deg_{I}(X) > C.

Definition 5.2: Let M be a compact manifold, and H a hyperkähler structure on M. We say that H admits C-restricted complex structures if for all C > 0, the set of all C-restricted algebraic complex structures is dense in the set R_{H} = ℂP^{1} of all induced complex structures.

Proposition 5.3: Let M be a compact simple hyperkähler manifold, dim_{ℍ}(M) > 1, and r : H^{4}(M) ⟶ H_{r}^{4}(M) be the map defined in Claim 4.11. Assume that for all algebraic induced complex structures I, the group H_{I}^{1,1}(M) ∩ H^{2}(M, ℤ) has rank one, and the group

H_{I}^{2,2}(M) ∩ H^{4}(M, ℤ)/(ker r)

has rank 2. Then M admits C-restricted complex structures.

Proof: The proof of Proposition 5.3 takes the rest of this section.

Denote by R the set R ≅ ℂP^{1} of all induced complex structures on M. Consider the set R/{±1} of induced complex structures up to a sign (Lemma 4.2). Let α ∈ H^{2}(M) be a cohomology class which is not SU(2)-invariant. According to Lemma 4.2, there exists a unique element c(α) ∈ R/{±1} such that α ∈ H_{c(α)}^{1,1}(M). This defines a map

c : (H^{2}(M, ℝ)\H_{inv}^{2}(M)) ⟶ R/{±1},

where H_{inv}^{2}(M) ⊂ H^{2}(M) is the set of all SU(2)-invariant cohomology classes. For induced complex structures I and −I, and η ∈ H^{2p}(M), the degree maps satisfy

deg_{I}(η) = (−1)^{p} deg_{−I}(η).  (5.1)

Thus, the number | deg_{I}(η)| is independent from the sign of I.

Let η ∈ H^{∗}(M, ℤ) be a cohomology class. The largest divisor of η is the biggest positive integer number k such that the cohomology class ηk is also integer.

Let α ∈ H^{2}(M, ℤ) be an integer cohomology class, which is not SU(2)- invariant, k its largest divisor and α̃ := αk the corresponding integer class. Denote by d̃eg(α) the number

d̃eg(α) := |deg_{c(α)}(α̃)| .

The induced complex structure c(α) is defined up to a sign, but from (5.1) it is clear that d̃eg(α) is independent from the choice of a sign.

Lemma 5.4: Let M be a compact hyperkähler manifold, and I be an algebraic induced complex structure, such that the group H_{I}^{1,1}(M) ∩ H^{2}(M, ℤ) has rank one, and the group H_{I}^{2,2}(M) ∩ H^{4}(M, ℤ)/(ker r) has rank 2. Denote by α the generator of H_{I}^{1,1}(M) ∩ H^{2}(M, ℤ). Since the class α is proportional to a Kähler form, α is not SU(2)-invariant (Lemma 4.2, (i)). Let d := d̃egα. Then, there exists a positive real constant A depending on volume of M, its topology and its dimension, such that I is d·A-restricted.

Proof: This lemma is a trivial calculation based on results of [V3] (see also [V3-bis] and Subsection 4.3).

Since H_{I}^{1,1}(M)∩H^{2}(M, ℤ) has rank one, for all η ∈ H_{I}^{1,1}(M)∩H^{2}(M, ℤ), η ≠ 0, we have | deg_{I} η| ⩾ d. This proves the first condition of Definition 5.1.

Let θ be the SU(2)-invariant integer cycle θ ∈ H^{4}(M) defined in Proposition 4.12. By Lemma 2.6, θ ∈ H_{I}^{2,2}(M). Consider α^{2} ∈ H_{I}^{2,2}(M), where α is the generator of H_{I}^{1,1}(M) ∩ H^{2}(M, ℤ).

Sublemma 5.5: Let J be an induced complex structure, J ∘I = −J ∘I, and deg_{I}, deg_{J} the degree maps associated with I, J. Then

deg_{I} α^{2} > 0, deg_{J} α^{2} = 0, deg_{I} θ = deg_{J} θ > 0.

Proof: Since α is a Kähler class with respect to I, we have deg_{I} α^{2} > 0. Since the cohomology class θ is SU(2)-invariant, and SU(2) acts transitively on the set of induced complex structures, we have deg_{I} θ = deg_{J} θ. It remains to show that deg_{J} α^{2} = 0 and deg_{J} θ > 0. The manifold M is by our assumptions simple; thus, dim H^{2,0}(M) = 1 ([Bes]). Therefore, in the natural SU(2)-invariant decomposition

H^{2}(M) = H_{inv}^{2}(M) ⊕ H_{+}^{2}(M),  (5.2)

we have dim H_{+}^{2}(M) = 3. In particular, the intersection H_{+}^{2}(M) ∩ H_{I}^{1,1}(M) is 1-dimensional. Consider the decomposition of α, associated with (5.2): α = α_{+} + α_{inv}. Since α is of type (1, 1) with respect to I, the class α_{+} is proportional to the Kähler class ω_{I}, with positive coefficient. A similar argument leads to the following decomposition for θ:

θ = ω_{I}^{2} + ω_{J}^{2} + ω_{K}^{2} + ∑ x_{i}^{2},

where K = I ∘ J is an induced complex structure, and the classes x_{i} belong to H_{inv}^{2}(M). From Corollary 4.10, we obtain that the classes x_{i}^{2} satisfy deg_{I}(x_{i}^{2}) = 0 (here we use dim_{ℍ}(M) > 1). Thus,

deg_{I}(θ) = deg_{I}(ω_{I}^{2} + ω_{J}^{2} + ω_{K}^{2}) = deg_{I}(ω_{I}^{2}) > 0.

Similarly one checks that

deg_{J}(α^{2}) = deg_{J}((α_{+} + α_{inv})^{2}) = deg_{J}(α_{+}^{2}) = deg_{J}(c^{2}ω_{I}) = 0.

This proves Sublemma 5.5.

Return to the proof of Lemma 5.4. Since deg_{I} α^{2} ≠ deg_{J} α^{2}, the class α^{2} is not SU(2)-invariant. Since θ is SU(2)-invariant, θ is not collinear with α^{2}. The degrees deg_{I} of θ and α^{2} are non-zero; we have deg_{I}(θ) = deg_{J}(θ), deg_{I}(α^{2}) ≠ deg_{J}(α^{2}) for J an induced complex structure, J ≠ ±I. By Proposition 4.5, no non-trivial linear combination of θ, α^{2} belongs to ker p. By Claim 4.11 (ii), the classes θ, α^{2} generate a 2-dimensional subspace in H^{4}(M, ℚ)/ ker r.

By assumptions of Lemma 5.4, the group H_{I}^{2,2}(M) ∩ H^{4}(M, ℤ)/(ker r) has rank 2. Therefore ω and α^{2} generate the space

H_{I}^{2,2}(M) ∩ H^{4}(M, ℚ)/(ker r).

To prove Lemma 5.4 it suffices to show that for all integer classes

β = aα^{2} + bθ, a ∈ ℚ\0, deg_{I} β ⩾ deg_{J} β,

we have | deg_{I} β| > A · d, for a constant A depending only on volume, topology and dimension of M. Since deg_{I} β ⩾ | deg_{J} β|, and deg_{J} α^{2} = 0 (Sublemma 5.5), we have

deg_{I}(aα^{2} + bθ) ⩾ | deg_{J} bθ|.

Therefore, either a and b have the same sign, or deg_{I}(aα^{2}) > 2 deg_{I}(bθ). In both cases,

| deg_{I} β| ⩾ \frac{1}{2} deg_{I}(aα^{2}).  (5.3)

Let x ∈ ℚ^{>0} be the smallest positive rational value of a for which there exists an integer class β = aα^{2} + bθ. We have an integer lattice L_{1} in H_{r}^{4}(M) provided by the products of integer classes; the integer lattice L_{2} ⊃ L_{1} provided by integer cycles might be different from that one. Clearly, x is greater than determinant det(L_{1}/L_{2}) of L_{1} over L_{2}, and this determinant is determined by the topology of M.

Form the definition of x and (5.3), we have | deg_{I} β| > x^{2} deg_{I}(α^{2}). On the other hand, deg_{I}(α^{2}) > C deg_{I}(α), where C is a constant depending on volume and dimension of M. Setting A := x^{2} · C, we obtain | deg_{I} β| > x^{2} · C · d. This proves Lemma 5.4.

Consider the maps

d̃eg : H^{2}(M, ℤ)\H_{inv}^{2}(M) ⟶ ℝ,
c : H^{2}(M)\H_{inv}^{2}(M) ⟶ R/{±1}

introduced in the beginning of the proof of Proposition 5.3.

Lemma 5.6: In assumptions of Proposition 5.3, let

O ⊂ H^{2}(M, ℝ)\H_{inv}^{2}(M)

be an open subset of H^{2}(M, ℝ), such that for all x ∈ O, k ∈ ℝ^{>0}, we have k · x ∈ O. Assume that O contains the Kähler class ω_{I} for all induced complex structures I ∈ R. For a positive number C ∈ ℝ^{>0}, consider the set X_{C} ⊂ O

X_{C} := {α ∈ O ∩ H^{2}(M, ℤ) | d̃eg(α) ⩾ C} .

Then c(X_{C}) is dense in R/{±1} for all C ∈ ℝ^{>0}.

Proof: The map d̃eg can be expressed in the following wey. We call an integer cohomology class α ∈ H^{2}(M, ℤ) indivisible if its largest divisor is 1, that is, there are no integer classes α′, and numbers k ∈ ℤ, k > 1, such that α = kα′.

Sublemma 5.7: Let α ∈ H^{2}(M) be an non-SU(2)-invariant cohomology class and α = α_{inv} + α_{+} be a decomposition associated with (5.2). Assume that α is indivisible. Then

d̃eg(α) = C\sqrt{((α_{+}, α_{+})_{B})},  (5.4)

where (·, ·)_{B} is the Bogomolov-Beauville pairing on H^{2}(M) ([V3-bis]; see also Theorem 4.7), and C a constant depending on dim M, Vol M.

Proof: By Proposition 4.5,

deg_{I}(α) = deg_{I}(α_{+})

(clearly, p(α) = α_{+}). By definition of (·, ·)_{B}, we have

deg_{I}(α_{+}) = (α_{+}, ω_{c(α)})_{B}

On the other hand, α_{+} is collinear with ω_{c(α)} by definition of the map c. Now (5.4) follows trivially from routine properties of bilinear forms.

Let I be an induced complex structure such that the cohomology class ω_{I} is irrational: ω_{I} ∉ H^{2}(M, ℚ).

To prove Lemma 5.6, we have to produce a sequence x_{i} ∈ O ∩ H^{2}(M, ℤ) such that

(i) c(x_{i}) converges to I,

(ii) and lim d̃eg(x_{i}) = ∞.

(5.5)

We introduce a metric (·, ·)_{H} on H^{2}(M, ℝ),

(α, β)_{H} := (α_{+}, β_{+})_{B} − (α_{inv}, β_{inv})_{B}.

It is easy to check that (·, ·)_{H} is positive definite ([V3]). For every ε, there exists a rational class ω_{ε} ∈ H^{2}(M, ℚ) which approximates ω_{I} with precision

(ω_{ε} − ω_{I}, ω_{ε} − ω_{I})_{H} < ε.

Since O is open and contains ω_{I}, we may assume that ω_{ε} belongs to O. Take a sequence ε_{i} converging to 0, and let x̃_{i} := ω_{ε_{i}} be the corresponding sequence of rational cohomology cycles. Let x_{i} := λ_{i}x̃_{i} be the minimal positive integer such that x_{i} ∈ H^{2}(M, ℤ). We are going to show that the sequence x_{i} satisfies the conditions of (5.5). First of all, x̃_{i} converges to ω_{I}, and the map

c : H^{2}(M)\H_{inv}^{2}(M) ⟶ R/{±1}

is continuous. Therefore, lim c(x̃_{i}) = c(ω_{I}) = I. By construction of c, c satisfies c(x) = c(λx), and thus, c(x_{i}) = c(x̃_{i}). This proves the condition (i) of (5.5). On the other hand, since ω_{I} is irrational, the sequence λ_{i} goes to infinity. Therefore,

lim(x_{i}, x_{i})_{H} = ∞.

It remains to compare (x_{i}, x_{i})_{H} with d̃egx_{i}. By (5.4),

d̃egx_{i} = \sqrt{((x_{i})_{+}, (x_{i})_{+})_{B}}.

On the other hand, since (x_{i})_{+} ∈ H_{+}^{2}(M), we have

((x_{i})_{+}, (x_{i})_{+})_{B} = ((x_{i})_{+}, (x_{i})_{+})_{H}.

To prove (5.5) (ii), it remains to show that

lim((x_{i})_{+}, (x_{i})_{+})_{H} = lim(x_{i}, x_{i})_{H}.

Since the cohomology class x̃_{i} ∈ H^{2}(M, ℚ) ε-approximates ω_{I}, and ω_{I} belongs to H_{+}^{2}(M), we have

(x̃_{i} − (x̃_{i})_{+}, x̃_{i} − (x̃_{i})_{+})_{H} < ε_{i}.

Therefore,

(x_{i} − (x_{i})_{+}, x_{i} − (x_{i})_{+})_{H} < λ_{i}ε_{i}.  (5.6)

On the other hand, for i sufficiently big, the cohomology class x̃_{i} approaches ω_{I}, and

(x_{i}, x_{i})_{H} > \frac{1}{2}λ_{i}(ω_{I}, ω_{I})_{H}  (5.7)

Comparing (5.6) and (5.7) and using the distance property for the distance given by \sqrt{(·, ·)_{H}}, we find that

\sqrt{(x_{i})_{+}, (x_{i})_{+}} > \sqrt{\frac{1}{2}λ_{i}(ω_{I}, ω_{I})_{H}} − \sqrt{λ_{i}ε_{i}} = \sqrt{λ_{i}} · (\sqrt{\frac{1}{2}(ω_{I}, ω_{I})_{H}} − \sqrt{ε_{i}}) .

(5.8)

Since ε_{i} converges to 0 and λ_{i} converges to infinity, the right hand side of (5.8) converges to infinity. On the other hand, by (5.4) the left hand side of (5.8) is equal constant times d̃egx_{i}, so lim d̃egx_{i} = ∞. This proves the second condition of (5.5). Lemma 5.6 is proven.

We use Lemma 5.4 and Lemma 5.6 in order to finish the proof of Proposition 5.3.

Let M be a compact hyperkähler manifold, and O ⊂ H^{2}(M, ℝ) be the set of all Kähler classes for the Kähler metrics compatible with one of induced complex structures. By Lemma 4.2, O is open in H^{2}(M, ℝ). Applying Lemma 5.6 to O, we obtain the following. In assumptions of Proposition 5.3, let Y_{C} ⊂ R be the set of all algebraic induced complex structures I with d̃egα > C, where α is a rational Kähler class, α ∈ H^{1,1}(M) ∩ H^{2}(M, ℤ). Then Y_{C} is dense in R. Now, Lemma 5.4, implies that for all I ∈ Y_{C}, the induced complex structure I is A · C-restricted, where A is the universal constant of Lemma 5.4. Thus, for all C the set of C-restricted induced complex structures is dense in R. This proves that M admits C-restricted complex structures. We finished the proof of Proposition 5.3.

5.2 Hyperkähler structures admitting C-restricted complex structures

Let M be a compact complex manifold admitting a hyperkähler structure H. Assume that (M, H) is a simple hyperkähler manifold of dimension dim_{ℍ} M > 1. The following definition of (coarse, marked) moduli space for complex and hyperkähler structures on M is standard.

Definition 5.8: Let M_{C^{∞}} be the M considered as a differential manifold, { \atop Comp} be the set of all integrable complex structures, and H̃yp be the set of all hyperkähler structures on M_{C^{∞}}. The set H̃yp is equipped with a natural topology. Let H̃yp^{0} be a connected component of H̃yp containing H and { \atop Comp}^{0} be a set of all complex structures I ∈ { \atop Comp} which are compatible with some hyperkähler structure H_{1} ∈ H̃yp^{0}. Let Diff be the group of diffeomorphisms of M which act trivially on the cohomology. The coarse, marked moduli Hyp of hyperkähler structures on M is the quotient Hyp := H̃yp^{0}/Diff equipped with a natural topology. The coarse, marked moduli

Comp of complex structures on M is defined as Comp := { \atop Comp}^{0}/Diff. For a detailed discussion of various aspects of this definition, see [V3].

Consider the variety

X ⊂ ℙH^{2}(M, ℂ),

consisting of all lines l ∈ ℙH^{2}(M, ℂ) which are isotropic with respect to the Bogomolov-Beauville’s pairing:

X := {l ∈ H^{2}(M, ℂ) | (l, l)_{B} = 0}.

Since M is simple, dim H^{2,0}(M, I) = 1 for all induced complex structures. Let P_{c} : Comp ⟶ ℙH^{2}(M, ℂ) map I to the line H_{I}^{2,0}(M) ⊂ H^{2}(M, ℂ). The map P_{c} is called the period map. It is well known that Comp is equipped with a natural complex structure. From general properties of the period map it follows that P_{c} is compatible with this complex structure. Clearly from the definition of Bogomolov-Beauville’s form, P_{c}(I) ∈ X for all induced complex structures I ∈ Comp (see [Bea] for details).

Theorem 5.9: [Bes] (Bogomolov) The complex analytic map

P_{c} : Comp ⟶ X

is locally an etale covering. ^{1}

It is possible to formulate a similar statement about hyperkähler structures. For a hyperkähler structure H, consider the set R_{H} ⊂ Comp of all induced complex structures associated with this hyperkähler structure. The subset R_{H} ⊂ Comp is a complex analytic subvariety, which is isomorphic to ℂP^{1}. Let S := P_{c}(R_{H}) be the corresponding projective line in X, and \overline{L}(X) be the space of smooth deformations of S in X. The points of L(X) correspond to smooth rational curves of degree 2 in ℙH^{2}(M, ℂ). For every such curve s, there exists a unique 3-dimensional plane L(s) ⊂ H^{2}(M, ℂ), such that s is contained in ℙL. Let Gr be the Grassmanian manifold of all 3-dimensional planes in H^{2}(M, ℂ) and Gr_{0} ⊂ Gr the set of all planes L ∈ Gr such that the restriction of the Bogomolov-Beauville form to L is non-degenerate. Let L(X) ⊂ \overline{L}(X) be the space of all rational curves s ∈ \overline{L}(X) such that the restriction of the Bogomolov-Beauville form to L(s) is non-degenerate: L(s) ∈ Gr_{0}. The correspondence s ⟶ L(s) gives a map κ : L(X) ⟶ Gr_{0}.

Lemma 5.10: The map κ : L(X) ⟶ Gr_{0} is an isomorphism of complex varieties.

Proof: For every plane L ∈ Gr_{0}, consider the set s(L) of all isotropic lines l ∈ L, that is, lines satisfying (l, l)_{B} = 0. Since (·, ·)_{B}|_{L} is nondegenerate, the set s(L) is a rational curve in ℙL. Clearly, this curve has degree 2. Therefore, s(L) belongs to X(L). The map L ⟶ s(L) is inverse to κ.

1The space Comp is smooth, as follows from Theorem 5.9. This space is, however, in most cases not separable ([H]). The space Hyp has no natural complex structures, and can be odd-dimensional.

Consider the standard anticomplex involution

ι : H^{2}(M, ℂ) ⟶ H^{2}(M, ℂ), η ⟶ \overline{η}.

Clearly, ι is compatible with the Bogomolov-Beauville form. Therefore, ι acts on L(X) as an anticomplex involution. Let L(X)_{ι} ⊂ L(X) be the set of all S ∈ L(X) fixed by ι.

Every hyperkähler structure

H ∈ Hyp

gives a rational curve R_{H} ⊂ Comp with points corresponding all induced complex structures. Let P_{h}(H) ⊂ X be the line P_{c}(R_{H}). Clearly from the definition, P_{h}(H) belongs to L(X)_{ι}. We have constructed a map P_{h} : Hyp ⟶ L(X)_{ι}. Let L(Comp) be the space of deformations if R_{H} in Comp. Denote by

γ : Hyp ⟶ L(Comp)

the map H ⟶ R_{H}. The following result gives a hyperkähler analogue of Bogomolov’s theorem (Theorem 5.9).

Theorem 5.11: The map γ : Hyp ⟶ L(Comp) is an embedding. The map P_{h} : Hyp ⟶ L(X)_{ι} is locally a covering.

Proof: The first claim is an immediate consequence of Calabi-Yau Theorem (Theorem 2.4). Now, Theorem 5.11 follows from the Bogomolov’s theorem (Theorem 5.9) and dimension count.

Let I ∈ Comp be a complex structure on M. Consider the groups

H_{h}^{2}(M, I) := H^{1,1}(M, I) ∩ H^{2}(M, ℤ)

and

H_{h}^{2}(M, I) := H_{r}^{2,2}(M, I) ∩ H^{4}(M, ℤ).

For a general I, H_{h}^{2}(M, I) = 0 and H_{h}^{4}(M, I) = ℤ as follows from Proposition 4.12. Therefore, the set of all I with rk H_{h}^{2}(M, I) = 1, rk H_{h}^{4}(M, I) = 2 is a union of countably many subvarieties of codimension 1 in Comp. Similarly, the set V ⊂ Comp of all I with rk H_{h}^{2}(M, I) > 1, rk H_{h}^{4}(M, I) > 2 is a union of countably many subvarieties of codimension more than 1. Together with Theorem 5.11, this implies the following.

Claim 5.12: Let U ⊂ Hyp be the set of all H ∈ Hyp such that R_{H} does not intersect V. Then U is dense in Hyp.

Proof: Consider a natural involution i of Comp which is compatible with the involution ι : X ⟶ X inder the period map P_{c} : Comp ⟶ X. This involution maps the complex structure I to −I.

By Theorem 5.11, Hyp is identified with an open subset in the set L(X)_{ι} of real points of L(Comp).

(5.9)

Let L_{U} ⊂ L(Comp) be the set of all lines which do not intersect V. Since V is a union of subvarieties of codimension at least 2, a general rational line l ∈ L(Comp) does not intersect V. Therefore, L_{U} is dense in L(Comp). Thus, the set of real points of L_{U} is dense L(X)_{ι}. Using the identification (5.9), we obtain the statement of Claim 5.12.

Claim 5.12 together with Proposition 5.3 imply the following theorem.

Theorem 5.13: Let M be a compact simple hyperkähler manifold, dim_{ℍ} M > 1, and Hyp its coarse marked moduli of hyperkähler structures. Let U ⊂ Hyp be the set of all hyperkähler structures which admit C-restricted complex structures (Definition 5.2). Then U is dense in Hyp.

5.3 Deformations of coherent sheaves over manifolds with C-restricted complex structures

The following theorem shows that a semistable deformation of a hyperholomorphic sheaf on (M, I) is again hyperholomorphic, provided that I is a C-restricted complex structure and C is sufficiently big.

Theorem 5.14: Let M be a compact hyperkähler manifold, and F ∈ F(M) a polystable hyperholomorphic sheaf on M (Definition 3.28). Let I be a C-restricted induced complex structure, for C = deg_{I} c_{2}(F),^{2} and F′ be a semistable torsion-free coherent sheaf on (M, I) with the same rank and Chern classes as F. Then the sheaf F′ is hyperholomorphic.

2Clearly, since F is hyperholomorphic, the class c_{2}(F) is SU(2)-invariant, and the number deg_{I} c_{2}(F) independent from I.

Proof: Let F_{1}, ..., F_{n} be the Jordan-Hölder series for the sheaf F′. Since F is hyperholomorphic, we have slope(F) = 0 (Remark 3.12). Therefore, slope(F_{i}) = 0, and deg_{I}(c_{1}(F_{i})) = 0. By Definition 5.1 (i), then, the class c_{1}(F_{i}) is SU(2) invariant for all i. To prove that F′ is hyperholomorphic it remains to show that the classes c_{2}(F_{i}), c_{2}(F_{i}^{∗∗}) are SU(2)-invariant for all i.

Consider an exact sequence

0 ⟶ F_{i} ⟶ F_{i}^{∗∗} ⟶ F_{i}/F_{i}^{∗∗} ⟶ 0.

Let [F_{i}/F_{i}^{∗∗}] ∈ H^{4}(M) be the fundamental class of the union of all components of Sup(F_{i}/F_{i}^{∗∗}) of complex codimension 2, taken with appropriate multiplicities. Clearly, c_{2}(F_{i}) = c_{2}(F_{i}^{∗∗}) + [F_{i}/F_{i}^{∗∗}]. Since [F_{i}/F_{i}^{∗∗}] is an effective cycle, deg_{I}([F_{i}/F_{i}^{∗∗}]) ⩾ 0. By the Bogomolov-Miyaoka-Yau inequality (see Corollary 3.24), we have deg_{I}(c_{2}(F_{i}^{∗∗}) ⩾ 0. Therefore,

deg_{I} c_{2}(F_{i}) ⩾ deg_{I} c_{2}(F_{i}^{∗∗}) ⩾ 0.  (5.10)

Using the product formula for Chern classes, we obtain

c_{2}(F) = ∑_{i} c_{2}(F_{i}) + ∑_{i,j} c_{2}(F_{i}) ∧ c_{2}(F_{j}).  (5.11)

By Corollary 4.10, deg_{I}(∑_{i,j} c_{2}(F_{i}) ∧ c_{2}(F_{j})) = 0. Since the numbers deg_{I} c_{2}(F_{i}) are non-negative, we have deg_{I} c_{2}(F_{i}) ⩽ deg_{I} c_{2}(F) = C. By Theorem 4.15, the classes c_{2}(F_{i}), c_{2}(F_{i}^{∗∗}) are of CA-type. By Definition 5.1 (ii), then, the inequality deg_{I} c_{2}(F_{i}) ⩽ C implies that the class c_{2}(F_{i}) is SU(2)-invariant. By (5.10), deg_{I} c_{2}(F_{i}^{∗∗}) ⩽ deg_{I} c_{2}(F_{i}), so the class c_{2}(F_{i}^{∗∗}) is also SU(2)-invariant. Theorem 5.14 is proven.

6 Desingularization of hyperholomorphic sheaves

The aim of this section is the following theorem.

6. DESINGULARIZATION OF HYPERHOLOMORPHIC SHEAVES 71

Theorem 6.1: Let M be a hyperkähler manifold, not necessarily compact, I an induced complex structure, and F a reflexive coherent sheaf over (M, I) equipped with a hyperholomorphic connection (Definition 3.15). Assume that F has isolated singularities. Let M̃ ⟶^{σ} M be a blow-up of (M, I) in the singular set of F, and σ^{∗}F the pullback of F. Then σ^{∗}F is a locally trivial sheaf, that is, a holomorphic vector bundle.

We prove Theorem 6.1 in Subsection 6.4.

The idea of the proof is the following. We apply to F the methods used in the proof of Desingularization Theorem (Theorem 2.16). The main ingredient in the proof of Desingularization Theorem is the existence of a natural ℂ^{∗}-action on the completion Ô_{x}(M, I) of the local ring O_{x}(M, I), for all x ∈ M. This ℂ^{∗}-action identifies Ô_{x}(M, I) with a completion of a graded ring. Here we show that a sheaf F is ℂ^{∗}-equivariant. Therefore, a germ of F at x has a grading, which is compatible with the natural ℂ^{∗}-action on Ô_{x}(M, I). Singularities of such reflexive sheaves can be resolved by a single blow-up.

6.1 Twistor lines and complexification

Further on, we need the following definition.

Definition 6.2: Let X be a real analytic variety, which is embedded to a complex variety X_{ℂ}. Assume that the sheaf of complex-valued real analytic functions on X coincides with the restriction of O_{X_{ℂ}} to X ⊂ X_{ℂ}. Then X_{ℂ} is called a complexification of X.

For more details on complexification, the reader is referred to [GMT]. There are the most important properties.

Claim 6.3: In a neighbourhood of X, the manifold X_{ℂ} has an anticomplex involution. The variety X is identified with the set of fixed points of this involution, considered as a real analytic variety.

Let Y be a complex variety, and X the underlying real analytic variety. Then the product of Y and its complex conjugate is a complexification of X, with embedding X ↪ Y × \overline{Y} given by the diagonal.

The complexification is unique in the following weak sense. For X_{ℂ}, X′_{ℂ} complexifications of C, the complex manifolds X_{ℂ}, X′_{ℂ} are naturally identified in a neighbourhood of X.

Let M be a hyperkähler manifold, Tw(M) its twistor space, and π : Tw(M) ⟶ ℂP^{1} the twistor projection. Let l ⊂ Tw(M) be a rational curve, such that the restriction of π to l is an identity. Such a curve gives a section of π, and vice versa, every section of π corresponds to such a curve. The set of sections of the projection π is called the space of twistor lines, denoted by Lin, or Lin(M). This space is equipped with complex structure, by Douady ([Do]).

Let m ∈ M be a point. Consider a twistor line s_{m} : I ⟶ (I × m) ∈ ℂP^{1} × M = Tw. Then s_{m} is called a horizontal twistor line. The space of horizontal twistor lines is a real analytic subvariety in Lin, denoted by Hor, or Hor(M). Clearly, the set Hor is naturally identified with M.

Proposition 6.4: ( Hitchin, Karlhede, Lindström, Roček) Let M be a hyperkähler manifold, Tw(M) its twistor space, I, J ∈ ℂP^{1} induced complex structures, and Lin the space of twistor lines. The complex manifolds (M, I) and (M, J) are naturally embedded to Tw(M):

(M, I) = π^{−1}(I), (M, J) = π^{−1}(J).

Consider a point s ∈ Lin, s : ℂP^{1} ⟶ Tw(M). Let

ev_{I,J} : Lin(M) ⟶ (M, I) × (M, J)

be the map defined by ev_{I,J}(s) = (s(I), s(J)). Assume that I ≠ J. Then there exists a neighbourhood U of Hor ⊂ Lin, such that the restriction of ev_{I,J} to U is an open embedding.

Proof: [HKLR], [V-d3].

Consider the anticomplex involution i of ℂP^{1} ≅ S^{2} which corresponds to the central symmetry of S^{2}. Let ι : Tw ⟶ Tw be the corresponding involution of the twistor space Tw(M) = ℂP^{1} × M, (x, m) ⟶ (i(x), m). It is clear that ι maps holomorphic subvarieties of Tw(M) to holomorphic subvarieties. Therefore, ι acts on Lin as an anticomplex involution. For J = −I, we obtain a local identification of Lin in a neighbourhood of Hor with (M, I) × (M, −I), that is, with (M, I) times its complex conjugate. Therefore, the space of twistor lines is a complexification of (M, I). The natural anticomplex involution of Claim 6.3 coincides with ι. This gives an identification of Hor and the real analytic manifold underlying (M, I).

We shall explain how to construct the natural ℂ^{∗}-action on a local ring of a hyperkähler manifold, using the machinery of twistor lines.

Fix a point x_{0} ∈ M and induced complex structures I, J, such that I ≠ ±J. Let V_{1}, V_{2} be neighbourhoods of s_{x_{0}} ∈ Lin, and U_{1}, U_{2} be neighbourhoods of (x_{0}, x_{0}) in (M, I) × (M, −I), (M, J) × (M, −J), such that the evaluation maps ev_{I,−I}, ev_{J,−J} induce isomorphisms

ev_{I,−I} : V_{1} ˜ ⟶ U_{1}, ev_{J,−J} : V_{2} ˜ ⟶ U_{2}.

Let B be an open neighbourhood of x_{0} ∈ M, such that (B, I)×(B, −I) ⊂ U_{1} and (B, I) × (B, −I) ⊂ U_{2}. Denote by V_{I} ⊂ V_{1} be the preimage of (B, I) × (B, −I) under ev_{I,−I}, and by V_{J} ⊂ V_{2} be the preimage of (B, J) × (B, −J) under ev_{J,−J}. Let p_{I} : V_{I} ⟶ (B, I) be the evaluation, s ⟶ s(I), and e_{I} : (B, I) ⟶ V_{1} the map associating to x ∈ B the unique twistor line passing through (x, x_{0}) ⊂ (B, I) × (B, −I). In the same fashion, we define e_{J} and p_{J}. We are interested in the composition

Ψ_{I,J} := e_{I} ∘ p_{J} ∘ e_{J} ∘ p_{I} : (B_{0}, I) ⟶ (B, I)

which is defined in a smaller neighbourhood B_{0} ⊂ B of x_{0} ∈ M.

The following proposition is the focal point of this Subsection: we explain the map Ψ_{I,J} of [V-d2], [V-d3] is geometric terms (in [V-d2], [V-d3] this map was defined algebraically).

Proposition 6.5: Consider the map Ψ_{I,J} : (B_{0}, I) ⟶ (B, I) defined above. By definition, Ψ_{I,J} preserves the point x_{0} ∈ B_{0} ⊂ B. Let dΨ_{I,J} be the differential of Ψ_{I,J} acting on the tangent space T_{x_{0}}B_{0}. Assume that I ≠ ±J. Then dΨ_{I,J} is a multiplication by a scalar λ ∈ ℂ, 0 < |λ| < 1.

Proof: The map Ψ_{I,J} was defined in [V-d2], [V-d3] using the identifications between the real analytic varieties underlying (M, I) and (M, J). We proved that Ψ_{I,J} defined this way acts on T_{x_{0}}B_{0} as a multiplication by the scalar λ ∈ ℂ, 0 < |λ| < 1. It remains to show that the map Ψ_{I,J} defined in [V-d2], [V-d3] coincides with Ψ_{I,J} defined above.

Consider the natural identification

(B, I) × (B, −I) ∼ (B, J) × (B, −J),

which is defined in a neighbourhood B_{ℂ} of (x_{0}, x_{0}). There is a natural projection a_{I} : B_{ℂ} ⟶ (M, I). Consider the embedding b_{I} : (B, I) ⟶ B_{ℂ}, x ⟶ (x, x_{0}), defined in a neighbourhood of x_{0} ∈ (B, I). In a similar way we define a_{J}, b_{J}. In [V-d2], [V-d3] we defined Ψ_{I,J} as a composition b_{I} ∘ a_{J} ∘ b_{J} ∘ a_{I}. Earlier in this Subsection, we described a local identification of (B, I) × (B, −I) and Lin(B). Clearly, under this identification, the maps a_{I}, b_{I} correspond to p_{I}, e_{I}. Therefore, the definition of Ψ_{I,J} given in this paper is equivalent to the definition given in [V-d2], [V-d3].

6.2 The automorphism Ψ_{I,J} acting on hyperholomorphic sheaves

In this section, we prove that hyperholomorphic sheaves are equivariant with respect to the map Ψ_{I,J}, considered as an automorphism of the local ring O_{x_{0}}(M, I).

Theorem 6.6: Let M be a hyperkähler manifold, not necessarily compact, x_{0} ∈ M a point, I an induced complex structure and F a reflexive sheaf over (M, I) equipped with a hyperholomorphic connection. Let J ≠ ±I be another induced complex structure, and B_{0}, B the neighbourhoods of x_{0} ∈ M for which the map Ψ_{I,J} : B_{0} ⟶ B was defined in Proposition 6.5. Assume that Ψ_{I,J} : B_{0} ⟶ B is an isomorphism. Then there exists a canonical functorial isomorphism of coherent sheaves

Ψ_{I,J}^{F} : F |_{B_{0}} ⟶ Ψ_{I,J}^{∗}(F |_{B}).

Proof: Return to the notation introduced in Subsection 6.1. Let W := V_{I} ∩ V_{J}. By definition of V_{I}, V_{J}, the evaluation maps produce open embeddings

ev_{I,−I} : Lin(W) ↪ (W, I) × (W, −I),

and

ev_{J,−J} : Lin(W) ↪ (W, J) × (W, −J),

Let S ⊂ W be the singular set of F |_{W} , Tw(S) ⊂ Tw(W) the corresponding embedding, and L_{0} ⊂ Lin(W) be the set of all lines l ∈ Lin(W) which do not intersect Tw(S). Consider the maps

p_{I} : L_{0} ↪ (W, I)\S

and

p_{J} : L_{0} ↪ (W, J)\S

obtained by restricting the evaluation map p_{I} : Lin(M) ⟶ (M, I) to L_{0} ⊂ Lin(M). Since F is equipped with a hyperholomorphic connection, the vector bundle F |_{(M,J)\S} has a natural holomorphic structure. Let \underline{F}_{1} := p_{I}^{∗} (F |_{(M,I)\S} ) and \underline{F}_{2} := p_{J}^{∗} (F |_{(M,J)\S} ) be the corresponding pullback sheaves over L_{0}, and F_{1}, F_{2} the sheaves on Lin(W) obtained as direct images of \underline{F}_{1}, \underline{F}_{2} under the open embedding L_{0} ↪ Lin(W).

Lemma 6.7: Under these assumptions, the sheaves F_{1}, F_{2} are coherent reflexive sheaves. Moreover, there exists a natural isomorphism of coherent sheaves Ψ_{1,2} : F_{1} ⟶ F_{2}.

Proof: The complex codimension of the singular set S in (M, I) is at least 3, because F is reflexive ([OSS], Ch. II, 1.1.10). Since S is trianalytic (Claim 3.16), this codimension is even. Thus, codim_{ℂ}(S, (M, I)) ⩾ 4. Therefore,

codim_{ℂ}(Tw(S), Tw(M)) ⩾ 4.

Consider the set L_{S} of all twistor lines l ∈ Lin(W) passing through Tw(S). For generic points x, y ∈ Tw(W), there exists a unique line l ∈ Lin(W) passing through x, y. Therefore,

codim_{ℂ}(L_{S}, Lin(W)) = codim_{ℂ}(Tw(S), Tw(M)) − 1 ⩾ 3.

By definition, L_{0} := Lin(W)\L_{S}. Since F_{1}, F_{2} are direct images of bundles \underline{F}_{1}, \underline{F}_{2} over a subvariety L_{S} of codimension 3, these sheaves are coherent and reflexive ([OSS], Ch. II, 1.1.12; see also Lemma 9.2). To show that they are naturally isomorphic it remains to construct an isomorphism between \underline{F}_{1} and \underline{F}_{2}.

Let F be a coherent sheaf on Tw(W) obtained from F |_{W} as in the proof of Proposition 3.17. The singular set of F is Tw(S) ⊂ Tw(W). Therefore, the restriction F|_{Tw(W)\ Tw(S)} is a holomorphic vector bundle. For all hor-

izontal twistor lines l_{x} ⊂ Tw(W)\ Tw(S), the restriction F|_{lx} is clearly a trivial vector bundle over l_{x} ≅ ℂP^{1}. A small deformation of a trivial vector bundle is again trivial. Shrinking W if necessary, we may assume that for all lines l ∈ L_{0}, the restriction of F to l ≅ ℂP^{1} is a trivial vector bundle.

The isomorphism \underline{Ψ}_{1,2} : \underline{F}_{1} ⟶ \underline{F}_{2} is constructed as follows. Let l ∈

L_{0} be a twistor line. The restriction F|_{l} is trivial. Consider l as a map l : ℂP^{1} ⟶ Tw(M). We identify ℂP^{1} with the set of induced complex structures on M. By definition, the fiber of F_{1} in l is naturally identified with the space F|_{l(I)} , and the fiber of F_{2} in l is identifies with F|_{l(J)} . Since F|_{l}

is trivial, the fibers of the bundle F|_{l} are naturally identified. This provides a vector bundle isomorphism \underline{Ψ}_{1,2} : \underline{F}_{1} ⟶ \underline{F}_{2} mapping \underline{F}_{1}|_{l} = F|_{l(I)} to

\underline{F}_{2}|_{l} = F|_{l(J)} . It remains to show that this isomorphism is compatible with the holomorphic structure.

Since the bundle F|_{l} is trivial, we have an identification

F|_{l(I)} ≅ F|_{l(J)} = Γ(F|_{l}),

where Γ(F|_{l}) is the space of global sections of F|_{l}. Thus, F_{i}|_{l} = Γ(F|_{l}), and this identification is clearly holomorphic. This proves Lemma 6.7.

We return to the proof of Theorem 6.6. Denote by F_{J} the restriction of F to (M, J) = π^{−1}(J) ⊂ Tw(M). The map Ψ_{I,J} was defined as a composition e_{I} ∘ p_{J} ∘ e_{J} ∘ p_{I}. The sheaf p_{I}^{∗}F is by definition isomorphic to F_{1}, and p_{J}^{∗}F_{J} to F_{2}. On the other hand, clearly, e_{J}^{∗}F_{2} = F_{J}. Therefore, (p_{J} ∘ e_{J})^{∗}F_{2} ≅ F_{2}. Using the isomorphism F_{1} ≅ F_{2}, we obtain (p_{J} ∘ e_{J})^{∗}F_{1} ≅ F_{1}. To sum it up, we have the following isomorphisms:

\begin{aligned}p_{I}^{∗}F & ≅ F_{1}, \\ (p_{J} ∘ e_{J})^{∗}F_{1} & ≅ F_{1}, \\ e_{I}^{∗}F_{1} & ≅ F. \end{aligned}

A composition of these isomorphisms gives an isomorphism

Ψ_{I,J}^{F} : F |_{B_{0}} ⟶ Ψ_{I,J}^{∗}(F |_{B}).

This proves Theorem 6.6.

6.3 A ℂ^{∗}-action on a local ring of a hyperkähler manifold

Let M be a hyperkähler manifold, non necessarily compact, x ∈ M a point and I, J induced complex structures, I ≠ J. Consider the complete local ring O_{x,I} := Ô_{x}(M, I). Throughout this section we consider the map Ψ_{I,J} (Proposition 6.5) as an automorphism of the ring O_{x,I}. Let 𝔪 be the maximal ideal of O_{x,I}, and 𝔪/𝔪^{2} the Zariski cotangent space of (M, I) in x.

By Proposition 6.5, Ψ_{I,J} acts on 𝔪/𝔪^{2} as a multiplication by a number λ ∈ ℂ, 0 < |λ| < 1.

(6.1)

Let V_{λ^{n}} be the eigenspace corresponding to the eigenvalue λ^{n},

V_{λ^{n}} := {v ∈ O_{x,I} | Ψ_{I,J}(v) = λ^{n}v}.

Clearly, ⊕V_{λ^{i}} is a graded subring in O_{x,I}. In [V-d2], (see also [V-d3]) we proved that the ring ⊕V_{λ^{i}} is dense in O_{x,I} with respect to the adic topology. Therefore, the ring O_{x,I} is identified with the adic completion of ⊕V_{λ^{i}}.

Consider at action of ℂ^{∗} on ⊕V_{λ^{i}}, with z ∈ ℂ^{∗} acting on V_{λ^{i}} as a multiplication by z^{i}. This ℂ^{∗}-action is clearly continuous, with respect to the adic topology. Therefore, it can be extended to

O_{x,I} = { \atop ⊕V_{λ^{i}}}.

Definition 6.8: Let M, I, J, x, O_{x,I} be as in the beginning of this Subsection. Consider the ℂ^{∗}-action

Ψ_{I,J}(z) : O_{x,I} ⟶ O_{x,I}

constructed as above. Then Ψ_{I,J}(z) is called the canonical ℂ^{∗}-action associated with M, I, J, x.

In the above notation, consider a reflexive sheaf F on (M, I) equipped with a hyperholomorphic connection. Denote the germ of F at x by F_{x}, F_{x} := F ⊗_{O_{(M,I)}} O_{x,I}. From Theorem 6.6, we obtain an isomorphism F_{x} ≅ Ψ_{I,J}^{∗}F_{x}. This isomorphism can be interpreted as an automorphism

Ψ_{I,J}^{F} : F_{x} ⟶ F_{x}

satisfying

Ψ_{I,J}^{F}(αv) = Ψ_{I,J}(α)v,  (6.2)

for all α ∈ O_{x,I}, v ∈ F_{x}.

By (6.2), the automorphism Ψ_{I,J}^{F} respects the filtration

F_{x} ⊃ 𝔪F_{x} ⊃ 𝔪^{2}F_{x} ⊃ ...

Thus, it makes sense to speak of Ψ_{I,J}^{F}-action on 𝔪^{i}F_{x}/𝔪^{i+1}F_{x}.

Lemma 6.9: The automorphism Ψ_{I,J}^{F} acts on 𝔪^{i}F_{x}/𝔪^{i+1}F_{x} as a multiplication by λ^{i}, where λ ∈ ℂ is the number considered in (6.1).

Proof: By (6.2), it suffices to prove Lemma 6.9 for i = 0. In other words, we have to show that Ψ_{I,J}^{F} acts as identity on F_{x}/𝔪F_{x}. We reduced Lemma 6.9 to the following claim.

Claim 6.10: In the above assumptions, the automorphism Ψ_{I,J}^{F} acts as identity on F_{x}/𝔪F_{x}.

Proof: In the course of defining the map Ψ_{I,J}^{F}, we identified the space Lin(M) with a complexification of (M, I), and defined the maps

p_{I} : Lin(M) ⟶ (M, I), p_{J} : Lin(M) ⟶ (M, I)

(these maps are smooth, in a neighbourhood of Hor ⊂ Lin(M), by Proposition 6.4), and

e_{I} : (B, I) ⟶ Lin(M), e_{J} : (B, J) ⟶ Lin(M)

(these maps are locally closed embeddings). Consider (M, J) as a subvariety of Tw(M), (M, J) = π^{−1}(J). Let F be the lift of F to Tw(M) (see the proof of Proposition 3.17 for details). Denote the completion of O_{x}(M, J) by O_{x,J}. Let F_{J} denote the O_{x,J}-module (F|_{(M,J)} ) ⊗_{O_{(M,J)}} Ô_{x,J}. Consider the horizontal twistor line l_{x} ∈ Lin(M). Let Lin_{x}(M) be the spectre of the completion O_{x,Lin} of the local ring of holomorphic functions on Lin(M) in l_{x}. The maps p_{I}, p_{J}, e_{I}, e_{J} can be considered as maps of corresponding formal manifolds:

p_{I} : Lin_{x}(M) ⟶ Spec(O_{x,I}),
p_{J} : Lin_{x}(M) ⟶ Spec(O_{x,J}),
e_{I} : Spec(O_{x,I}) ⟶ Lin_{x}(M),
e_{J} : Spec(O_{x,J}) ⟶ Lin_{x}(M),

As in Subsection 6.2, we consider the O_{x,Lin}-modules F_{1} := p_{I}^{∗}F_{x} and F_{2} := p_{J}^{∗}F_{J}. By Lemma 6.7, there exists a natural isomorphism Ψ_{1,2} : F_{1} ⟶ F_{2}.

Let 𝔪_{l_{x}} be the maximal ideal of O_{x,Lin}. Since the morphism p_{I} is smooth, the space F_{1}/𝔪_{l_{x}}F_{1} is naturally isomorphic to F_{x}/𝔪F_{x}. Similarly, the space F_{2}/𝔪_{l_{x}}F_{2} is isomorphic to F_{J}/𝔪_{J}F_{J}, where 𝔪_{J} is the maximal ideal of O_{x,J}. We have a chain of isomorphisms

\begin{aligned}F_{x}/𝔪F_{x} & ⟶^{p_{I}^{∗}} F_{1}/𝔪_{l_{x}}F_{1} ⟶^{Ψ_{1,2}} F_{2}/𝔪_{l_{x}}F_{2} \\ & ⟶^{e_{J}^{∗}} F_{J}/𝔪_{J}F_{J} ⟶^{p_{J}^{∗}} F_{2}/𝔪_{l_{x}}F_{2} \\ & ⟶^{Ψ_{1,2}^{−1}} F_{1}/𝔪_{l_{x}}F_{1} ⟶^{e_{I}^{∗}} F_{x}/𝔪F_{x}. \end{aligned}  (6.3)

By definition, for any f ∈ F_{x}/𝔪F_{x}, the value of Ψ_{I,J}^{F}(f) is given by the composide map of (6.3) applied to f. The composition

F_{2}/𝔪_{l_{x}}F_{2} ⟶^{e_{J}^{∗}} F_{J}/𝔪_{J}F_{J} ⟶^{p_{J}^{∗}} F_{2}/𝔪_{l_{x}}F_{2}  (6.4)

is identity, because the spaces F_{2}/𝔪_{l_{x}}F_{2} and F_{J}/𝔪_{J}F_{J} are canonically identified, and this identification can be performed via e_{J}^{∗} or p_{J}^{∗}. Thus, the map (6.3) is a composition

F_{x}/𝔪F_{x} ⟶^{p_{I}^{∗}} F_{1}/𝔪_{l_{x}}F_{1} ⟶^{Ψ_{1,2}} F_{2}/𝔪_{l_{x}}F_{2} ⟶^{Ψ_{1,2}^{−1}} F_{1}/𝔪_{l_{x}}F_{1} ⟶^{e_{I}^{∗}} F_{x}/𝔪F_{x}.

This map is clearly equivalent to a composition

F_{x}/𝔪F_{x} ⟶^{p_{I}^{∗}} F_{1}/𝔪_{l_{x}}F_{1} ⟶^{e_{I}^{∗}} F_{x}/𝔪F_{x},

which is identity according to the same reasoning which proved that (6.4) is identity. We proved Claim 6.10 and Lemma 6.9.

Consider the λ^{n}-eigenspaces F_{λ^{n}} of F_{x}. Consider the ⊕V_{λ^{n}}-submodule ⊕F_{λ^{n}} ⊂ F_{x}, where ⊕V_{λ^{n}} ⊂ O_{x,I} is the ring defined in Subsection 6.3. From Claim 6.10 and (6.1) it follows that ⊕F_{λ^{n}} is dense in F_{x}, with respect to the adic topology on F_{x}. For z ∈ ℂ^{∗}, let Ψ_{I,J}^{F}(z) : ⊕F_{λ^{n}} ⟶ ⊕ F_{λ^{n}} act on F_{λ^{n}} as a multiplication by z^{n}. As in Definition 6.8, we extend Ψ_{I,J}^{F}(z) to F_{x} = { \atop ⊕F_{λ^{n}}}. This automorphism makes F_{x} into a ℂ^{∗}-equivariant module over O_{x,I}

Definition 6.11: The constructed above ℂ^{∗}-equivariant structure on F_{x} is called the canonical ℂ^{∗}-equivariant structure on F_{x} associated with J.

6.4 Desingularization of ℂ^{∗}-equivariant sheaves

Let M be a hyperkähler manifold, I an induced complex structure and F a reflexive sheaf with isolated singularities over (M, I), equipped with a hyperholomorphic connection. We have shown that the sheaf F admits a ℂ^{∗}-equivariant structure compatible with the canonical ℂ^{∗}-action on the local ring of (M, I). Therefore, Theorem 6.1 is implied by the following proposition.

Proposition 6.12: Let B be a complex manifold, x ∈ B a point. Assume that there is an action Ψ(z) of ℂ^{∗} on B which fixes x and acts on T_{x}B be dilatations. Let F be a reflexive coherent sheaf on B, which is locally trivial outside of x. Assume that the germ F_{x} of F in x is equipped with a ℂ^{∗}-equivariant structure, compatible with Ψ(z). Let B̃ be a blow-up of B in x, and π : B̃ ⟶ B the standard projection. Then the pullback sheaf F̃ := π^{∗}F is locally trivial on B̃.

Proof: Let C := π^{−1}(x) be the singular locus of π. The sheaf F is locally trivial outside of x. Let d be the rank of F |_{B\x}. To prove that F̃ is

locally trivial, we need to show that for all points y ∈ B̃, the fiber π^{∗}F |_{y} is d-dimensional. Therefore, to prove Proposition 6.12 it suffices to show that π^{∗}F |_{C} is a vector bundle of dimension d.

The variety C is naturally identified with the projectivization ℙT_{x}B of the tangent space T_{x}B. The total space of T_{x}B is equipped with a natural action of ℂ^{∗}, acting by dilatations. Clearly, coherent sheaves on ℙT_{x}B are in one-to-one correspondence with ℂ^{∗}-equivariant coherent sheaves on T_{x}B. Consider a local isomorphism φ : T_{x}B ⟶ B which is compatible with ℂ^{∗}-action, maps 0 ∈ T_{x}B to x and acts as identity on the tangent space T_{0}(T_{x}B) = T_{x}B. The sheaf φ^{∗}F is ℂ^{∗}-equivariant. Clearly, the corresponding sheaf on ℙT_{x}B is canonically isomorphic with π^{∗}F |_{C}. Let l ∈ T_{x}B be a line passing through 0, and l\0 its complement to 0. Denote the corresponding point of ℙT_{x}B by y. The restriction φ^{∗}F |_{l\0} is a ℂ^{∗}-equivariant vector bundle. The ℂ^{∗}-equivariant structure identifies all the fibers of the bundle φ^{∗}F |_{l\0} . Let F_{l} be one of these fibers. Clearly, the fiber of π^{∗}F |_{C}

in y is canonically isomorphic to F_{l}. Therefore, the fiber of π^{∗}F |_{C} in y is d-dimensional. We proved that π^{∗}F is a bundle.

7 Twistor transform and quaternionic-Kähler geometry

This Section is a compilation of results known from the literature. Subsection 7.1 is based on [KV] and the results of Subsection 7.2 are implicit in [KV]. Subsection 7.3 is based on [Sal], [N1] and [N2], and Subsection 7.4 is a recapitulation of the results of A. Swann ([Sw]).

7.1 Direct and inverse twistor transform

In this Subsection, we recall the definition and the main properties of the direct and inverse twistor transform for bundles over hyperkähler manifolds ([KV]).

The following definition is a non-Hermitian analogue of the notion of a hyperholomorphic connection.

Definition 7.1: Let M be a hyperkähler manifold, not necessarily compact, and (B, ∇) be a vector bundle with a connection over M, not necessarily Hermitian. Assume that the curvature of ∇ is contained in the space Λ_{inv}^{2}(M, End(B)) of SU(2)-invariant 2-forms with coefficients in End(B). Then (B, ∇) is called an autodual bundle, and ∇ an autodual connection.

Let Tw(M) be the twistor space of M, equipped with the standard maps π : Tw(M) ⟶ ℂP^{1}, σ : Tw(M) ⟶ M.

We introduce the direct and inverse twistor transforms which relate autodual bundles on the hyperkähler manifold M and holomorphic bundles on its twistor space Tw(M).

Let B be a complex vector bundle on M equipped with a connection ∇. The pullback σ^{∗}B of B to Tw(M) is equipped with a pullback connection σ^{∗}∇.

Lemma 7.2: ([KV], Lemma 5.1) The connection ∇ is autodual if and only if the connection σ^{∗}∇ has curvature of Hodge type (1, 1).

Proof: Follows from Lemma 2.6.

In assumptions of Lemma 7.2, consider the (0, 1)-part (σ^{∗}∇)^{0,1} of the connection σ^{∗}∇. Since σ^{∗}∇ has curvature of Hodge type (1, 1), we have

((σ^{∗}∇)^{0,1})^{2} = 0,

and by Proposition 2.19, this connection is integrable. Consider (σ^{∗}∇)^{0,1} as a holomorphic structure operator on σ^{∗}B.

Let A be the category of autodual bundles on M, and C the category of holomorphic vector bundles on Tw(M). We have constructed a functor

(σ^{∗}•)^{0,1} : A ⟶ C,

∇ ⟶ (σ^{∗}∇)^{0,1}. Let s ∈ Hor ⊂ Tw(M) be a horizontal twistor line (Subsection 6.1). For any (B, ∇) ∈ A, consider corresponding holomorphic vector bundle (σ^{∗}B, (σ^{∗}∇)^{0,1}). The restriction of (σ^{∗}B, (σ^{∗}∇)^{0,1}) to s ≅ ℂP^{1} is a trivial vector bundle. A converse statement is also true. Denote by C_{0} the category of holomorphic vector bundles C on Tw(M), such that the restriction of C to any horizontal twistor line is trivial.

Theorem 7.3: Consider the functor

(σ^{∗}•)^{0,1} : A ⟶ C_{0}

constructed above. Then it is an equivalence of categories.

Proof: [KV], Theorem 5.12.

Definition 7.4: Let M be a hyperkähler manifold, Tw(M) its twistor space and F a holomorphic vector bundle. We say that F is compatible with twistor transform if the restriction of C to any horizontal twistor line s ∈ Tw(M) is a trivial bundle on s ≅ ℂP^{1}.

Recall that a connection ∇ in a vector bundle over a complex manifold is called (1, 1)-connection if its curvature is of Hodge type (1, 1).

Remark 7.5: Let F be a holomorphic bundle over Tw(M) which is compatible with twistor transform. Then F is equipped with a natural (1, 1)-connection ∇_{F} = σ^{∗}∇, where (B, ∇) is the corresponding autodual bundle over M. The connection ∇_{F} is not, generally speaking, Hermitian, or compatible with a Hermitian structure.

7.2 Twistor transform and Hermitian structures on vector bundles

Results of this Subsection were implicit in [KV], but in this presentation, they are new.

Let M be a hyperkähler manifold, not necessarily compact, and Tw(M) its twistor space. In Subsection 7.1, we have shown that certain holomorphic vector bundles over Tw(M) admit a canonical (1,1)-connection ∇_{F} (Remark 7.5). This connection can be non-Hermitian. Here we study the Hermitian structures on (F, ∇_{F}) in terms of holomorphic properties of F.

Definition 7.6: Let F be a real analytic complex vector bundle over a real analytic manifold X_{ℝ}, and h : F × F ⟶ ℂ a O_{X_{ℝ}}-linear pairing on F. Then h is called semilinear if for all α ∈ O_{X_{ℝ}} ⊗_{ℝ} ℂ, we have

h(αx, y) = α · h(x, y), and h(x, αy) = \overline{α} · h(x, y).

For X a complex manifold and F a holomorphic vector bundle, by a semilinear pairing on F we understand a semilinear pairing on the underlying real analytic bundle. Clearly, a real analytic Hermitian metric is always semilinear.

Let X be a complex manifold, I : T X ⟶ T X the complex structure operator, and i : X ⟶ X a real analytic map. We say that X is anticomplex if the induced morphism of tangent spaces satisfies i ∘ I = −I ∘ I. For a complex vector bundle F on X, consider the complex adjoint vector bundle \overline{F}, which coincides with F as a real vector bundle, with ℂ-action which is conjugate to that defined on F. Clearly, for every holomorphic vector bundle F, and any anticomplex map i : X ⟶ X, the bundle i^{∗}\overline{F} is equipped with a natural holomorphic structure.

Let M be a hyperkähler manifold, and Tw(M) its twistor space. Recall that Tw(M) = ℂP^{1}×M is equipped with a canonical anticomplex involution ι, which acts as identity on M and as central symmetry I ⟶ −I on ℂP^{1} = S^{2}. For any holomorphic bundle F on Tw(M), consider the corresponding holomorphic bundle ι^{∗}\overline{F}.

Let M be a hyperkähler manifold, I an induced complex structure, F a vector bundle over M, equipped with an autodual connection ∇, and F the corresponding holomorphic vector bundle over Tw(M), equipped with a canonical connection ∇_{F}. As usually, we identify (M, I) and the fiber π^{−1}(I) of the twistor projection π : Tw(M) ⟶ ℂP^{1}. Let ∇_{F} = ∇_{I}^{1,0} + ∇_{I}^{0,1} be the Hodge decomposition of ∇ with respect to I.

Clearly, the operator ∇_{I}^{1,0} can be considered as a holomorphic structure operator on F, considered as a complex vector bundle over (M, −I).

(7.1)

Then the holomorphic structure operator on F|_{(M,I)} is equal to ∇_{I}^{0,1}, and

the holomorphic structure operator on F|_{(M,−I)} is equal to ∇_{I}^{1,0}.

Assume that the bundle (F, ∇_{F}) is equipped with a non-degenerate semilinear pairing h which is compatible with the connection. Consider the natural connection ∇_{F^{∗}} on the dual bundle to F, and its Hodge decomposition (with respect to I)

∇_{F^{∗}} = ∇_{F^{∗}}^{1,0} + ∇_{F^{∗}}^{0,1}.

Clearly, the pairing h gives a C^{∞}-isomorphism of F and the complex conjugate of its dual bundle, denoted as \overline{F}^{∗}. Since h is semilinear and compatible with the connection, it maps the holomorphic structure operator ∇_{I}^{0,1} to the complex conjugate of ∇_{F^{∗}}^{1,0}. On the other hand, the operator ∇_{F^{∗}}^{1,0} is a holomorphic structure operator in F^{∗}|_{(M,−I)} , as (7.1) claims. We obtain that the map h can be considered as an isomorphism of holomorphic vector bundles

h : F ⟶ (ι^{∗}\overline{F})^{∗}.

This correspondence should be thought of as a (direct) twistor transform for bundles with a semilinear pairing.

Proposition 7.7: (direct and inverse twistor transform for bundles with semilinear pairing) Let M be a hyperkähler manifold, and C_{sl} the category of autodual bundles over M equipped with a non-degenerate semilinear pairing. Consider the category C_{hol,sl} of holomorphic vector bundles F on Tw(M), compatible with twistor transform and equipped with an isomorphism

𝔥 : F ⟶ (ι^{∗}\overline{F})^{∗}.

Let T : C_{sl} ⟶ C_{hol,sl} be the functor constructed above. Then T is an isomorphism of categories.

Proof: Given a pair F, 𝔥 : F ⟶ (ι^{∗}\overline{F})^{∗}, we need to construct a nondegenerate semilinear pairing h on F|_{(M,I)} , compatible with a connection. Since F is compatible with twistor transform, it is a pullback of a bundle (F, ∇) on M. This identifies the real analytic bundles F|_{(M,I′)} , for all induced complex structures I′. Taking I′ = ±I, we obtain an identification of the C^{∞}-bundles F|_{(M,I)} , F|_{(M,−I)} . Thus, 𝔥 can be considered as an iso-

morphism of F = F|_{(M,I)} and (\overline{F})^{∗}|_{(M,I)} . This allows one to consider 𝔥 as a semilinear form h on F. We need only to show that h is compatible with the connection ∇. Since ∇_{F} is an invariant of holomorphic structure, the map 𝔥 : F ⟶ (ι^{∗}\overline{F})^{∗} is compatible with the connection ∇_{F}. Thus, the obtained above form h is compatible with the connection ∇_{F} |_{(M,I)} = ∇. This proves Proposition 7.7.

7.3 B_{2}-bundles on quaternionic-Kähler manifolds

Definition 7.8: ([Sal], [Bes]) Let M be a Riemannian manifold. Consider a bundle of algebras End(T M), where T M is the tangent bundle to M. Assume that End(T M) contains a 4-dimensional bundle of subalgebras W ⊂ End(T M), with fibers isomorphic to a quaternion algebra ℍ. Assume, moreover, that W is closed under the transposition map ⊥ : End(T M) ⟶ End(T M) and is preserved by the Levi-Civita connection. Then M is called quaternionic-Kähler.

Example 7.9: Consider the quaternionic projective space

ℍP^{n} = (ℍ^{n}\0)/ℍ^{∗}.

It is easy to see that ℍP^{n} is a quaternionic-Kähler manifold. For more examples of quaternionic-Kähler manifolds, see [Bes].

A quaternionic-Kähler manifold is Einstein ([Bes]), i. e. its Ricci tensor is proportional to the metric: Ric(M) = c · g, with c ∈ ℝ. When c = 0, the manifold M is hyperkähler, and its restricted holonomy group is Sp(n); otherwise, the restricted holonomy is Sp(n) · Sp(1). The number c is called the scalar curvature of M. Further on, we shall use the term quaternionic-Kähler manifold for manifolds with non-zero scalar curvature.

The quaternionic projective space ℍP^{n} has positive scalar curvature.

The quaternionic projective space is the only example of quaternionic-Kähler manifold which we need, in the course of this paper. However, the formalism of quaternionic-Kähler manifolds is very beautiful and significantly simplifies the arguments, so we state the definitions and results for a general quaternionic-Kähler manifold whenever possible.

Let M be a quaternionic-Kähler manifold, and W ⊂ End(T M) the corresponding 4-dimensional bundle. For x ∈ M, consider the set R_{x} ⊂ W |_{x}, consisting of all I ∈ W |_{x} satisfying I^{2} = −1. Consider R_{x} as a Rieman-

nian submanifold of the total space of W |_{x}. Clearly, R_{x} is isomorphic to a 2-dimensional sphere. Let R = ∪_{x}R_{x} be the corresponding spherical fibration over M, and Tw(M) its total space. The manifold Tw(M) is equipped with an almost complex structure, which is defined in the same way as the almost complex structure for the twistor space of a hyperkähler manifold. This almost complex structure is known to be integrable (see [Sal]).

Definition 7.10: ([Sal], [Bes]) Let M be a quaternionic-Kähler manifold. Consider the complex manifold Tw(M) constructed above. Then Tw(M) is called the twistor space of M.

Note that (unlike in the hyperkähler case) the space Tw(M) is Kähler. For quaternionic-Kähler manifolds with positive scalar curvature, the anticanonical bundle of Tw(M) is ample, so Tw(M) is a Fano manifold.

Quaternionic-Kähler analogue of a twistor transform was studied by T. Nitta in a serie of papers ([N1], [N2] etc.) It turns out that the picture given in [KV] for Kähler manifolds is very similar to that observed by T. Nitta.

A role of SU(2)-invariant 2-forms is played by the so-called B_{2}-forms.

Definition 7.11: Let SO(T M) ⊂ End(T M) be a group bundle of all orthogonal automorphisms of T M, and G_{M} := W ∩ SO(T M). Clearly, the fibers of G_{M} are isomorphic to SU(2). Consider the action of G_{M} on the bundle of 2-forms Λ^{2}(M). Let Λ_{inv}^{2}(M) ⊂ Λ^{2}(M) be the bundle of G_{M}- invariants. The bundle Λ_{inv}^{2}(M) is called the bundle of B_{2}-forms. In a similar fashion we define B_{2}-forms with coefficients in a bundle.

Definition 7.12: In the above assumptions, let (B, ∇) be a bundle with connection over M. The bundle B is called a B_{2}-bundle, and ∇ is called a B_{2}-connection, if its curvature is a B_{2}-form.

Consider the natural projection σ : Tw(M) ⟶ M. The proof of the following claim is completely analogous to the proof of Lemma 2.6 and Lemma 7.2.

Claim 7.13:

(i) Let ω be a 2-form on M. The pullback σ^{∗}ω is of type (1, 1) on Tw(M) if and only if ω is a B_{2}-form on M.

(ii) Let B be a complex vector bundle on M equipped with a connection ∇, not necessarily Hermitian. The pullback σ^{∗}B of B to Tw(M) is equipped with a pullback connection σ^{∗}∇. Then, ∇ is a B_{2}-connection if and only if σ^{∗}∇ has curvature of Hodge type (1, 1).

There exists an analogue of direct and inverse twistor transform as well.

Theorem 7.14: For any B_{2}-connection (B, ∇), consider the corresponding holomorphic vector bundle

(σ^{∗}B, (σ^{∗}∇)^{0,1}).

The restriction of (σ^{∗}B, (σ^{∗}∇)^{0,1}) to a line σ^{−1}(m) ≅ ℂP^{1} is a trivial vector bundle, for any point m ∈ M. Denote by C_{0} the category of holomorphic vector bundles C on Tw(M), such that the restriction of C to σ^{−1}(m) is trivial, for all m ∈ M, and by A the category of B_{2}-bundles (not necessarily Hermitian). Consider the functor

(σ^{∗}•)^{0,1} : A ⟶ C_{0}

constructed above. Then it is an equivalence of categories.

Proof: It is easy to modify the proof of the direct and inverse twistor transform theorem from [KV] to work in quaternionic-Kähler situation.

We will not use Theorem 7.14, except for its consequence, which was proven in [N1].

Corollary 7.15: Consider the functor

(σ^{∗}•)^{0,1} : A ⟶ C_{0}

constructed in Theorem 7.14. Then (σ^{∗}•)^{0,1} gives an injection κ from the set of equivalence classes of Hermitian B_{2}-connections over M to the set of equivalence classes of holomorphic connections over Tw(M).

Let M be a quaternionic-Kähler manifold. The space Tw(M) has a natural Kähler metric g, such that the standard map σ : Tw(M) ⟶ M is a Riemannian submersion, and the restriction of g to the fibers σ^{−1}(m) of σ is a metric of constant curvature on σ^{−1}(m) = ℂP^{1} ([Sal], [Bes]).

Example 7.16: In the case M = ℍP^{n}, we have Tw(M) = ℂP^{2n+1}, and the Kähler metric g is proportional to the Fubini-Study metric on ℂP^{2n+1}.

Theorem 7.17: (T. Nitta) Let M be a quaternionic-Kähler manifold of positive scalar curvature, Tw(M) its twistor space, equipped with a natural Kähler structure, and B a Hermitian B_{2}-bundle on M. Consider the pullback σ^{∗}B, equipped with a Hermitian connection. Then σ^{∗}B is a Yang-Mills bundle on Tw(M), and deg c_{1}(σ^{∗}B) = 0.

Proof: [N2].

Let κ be the map considered in Corollary 7.15. Assume that M is a compact manifold. In [N2], T. Nitta defined the moduli space of Hermitian B_{2}-bundles. By Uhlenbeck-Yau theorem, Yang-Mills bundles are polystable. Then the map κ provides an embedding from the moduli of non-decomposable Hermitian B_{2}-bundles to the moduli M of stable bundles on Tw(M). The image of κ is a totally real subvariety in M ([N2]).

7.4 Hyperkähler manifolds with special ℍ^{∗}-action and quaternionic-Kähler manifolds of positive scalar curvature

Further on, we shall need the following definition.

Definition 7.18: An almost hypercomplex manifold is a smooth manifold M with an action of quaternion algebra in its tangent bundle For each L ∈ ℍ, L^{2} = −1, L gives an almost complex structure on M. The manifold M is caled hypercomplex if the almost complex structure L is integrable, for all possible choices L ∈ ℍ.

The twistor space for a hypercomplex manifold is defined in the same way as for hyperkähler manifolds. It is also a complex manifold ([K]). The formalism of direct and inverse twistor transform can be repeated for hypercomplex manifolds verbatim.

Let ℍ^{∗} be the group of non-zero quaternions. Consider an embedding SU(2) ↪ ℍ^{∗}. Clearly, every quaternion h ∈ ℍ^{∗} can be uniquely represented as h = |h| · g_{h}, where g_{h} ∈ SU(2) ⊂ ℍ^{∗}. This gives a natural decomposition ℍ^{∗} = SU(2) × ℝ^{>0}. Recall that SU(2) acts naturally on the set of induced complex structures on a hyperkähler manifold.

Definition 7.19: Let M be a hyperkähler manifold equipped with a free smooth action ρ of the group ℍ^{∗} = SU(2) × ℝ^{>0}. The action ρ is called special if the following conditions hold.

(i) The subgroup SU(2) ⊂ ℍ^{∗} acts on M by isometries.

(ii) For λ ∈ ℝ^{>0}, the corresponding action ρ(λ) : M ⟶ M is compatible with the hyperholomorphic structure (which is a fancy way of saying that ρ(λ) is holomorphic with respect to any of induced complex structures).

(iii) Consider the smooth ℍ^{∗}-action ρ_{e} : ℍ^{∗} × End(T M) ⟶ End(T M) induced on End(T M) by ρ. For any x ∈ M and any induced complex structure I, the restriction I|_{x} can be considered as a point in the total space of End(T M). Then, for all induced complex structures I, all g ∈ SU(2) ⊂ ℍ^{∗}, and all x ∈ M, the map ρ_{e}(g) maps I|_{x} to

g(I)|_{ρe(g)(x)} .

Speaking informally, this can be stated as “ℍ^{∗}-action interchanges the induced complex structures”.

(iv) Consider the automorphism of S^{2}T^{∗}M induced by ρ(λ), where λ ∈ ℝ^{>0}. Then ρ(λ) maps the Riemannian metric tensor s ∈ S^{2}T^{∗}M to λ^{2}s.

Example 7.20: Consider the flat hyperkähler manifold M_{fl} = ℍ^{n}\0. There is a natural action of ℍ^{∗} on ℍ^{n}\0. This gives a special action of ℍ^{∗} on M_{fl}.

The case of a flat manifold M_{fl} = ℍ^{n}\0 is the only case where we apply the results of this section. However, the general statements are just as difficult to prove, and much easier to comprehend.

Definition 7.21: Let M be a hyperkähler manifold with a special action ρ of ℍ^{∗}. Assume that ρ(−1) acts non-trivially on M. Then M/ρ(±1) is also a hyperkähler manifold with a special action of ℍ^{∗}. We say that the manifolds (M, ρ) and (M/ρ(±1), ρ) are hyperkähler manifolds with special action of ℍ^{∗} which are special equivalent. Denote by H_{sp} the category of hyperkähler manifolds with a special action of ℍ^{∗} defined up to special equivalence.

A. Swann ([Sw]) developed an equivalence between the category of quaternionic-Kähler manifolds of positive scalar curvature and the category H_{sp}. The purpose of this Subsection is to give an exposition of Swann’s formalism.

Let Q be a quaternionic-Kähler manifold. The restricted holonomy group of Q is Sp(n) · Sp(1), that is, (Sp(n) × Sp(1))/{±1}. Consider the principal bundle G with the fiber Sp(1)/{±1}, corresponding to the subgroup

Sp(1)/{±1} ⊂ (Sp(n) × Sp(1))/{±1}.

of the holonomy. There is a natural Sp(1)/{±1}-action on the space ℍ^{∗}/{±1}. Let

U(Q) := G ×_{Sp(1)/{±1}} ℍ^{∗}/{±1}.

Clearly, U(Q) is fibered over Q, with fibers which are isomorphic to ℍ^{∗}/{±1}. We are going to show that the manifold U(Q) is equipped with a natural hypercomplex structure.

There is a natural smooth decomposition U(Q) ≅ G × ℝ^{>0} which comes from the isomorphism ℍ^{∗} ≅ Sp(1) × ℝ^{>0}.

Consider the standard 4-dimensional bundle W on Q. Let x ∈ Q be a point. The fiber W |_{q} is isomorphic to ℍ, in a non-canonical way. The

choices of isomorphism W |_{q} ≅ ℍ are called quaternion frames in q. The set of quaternion frames gives a fibration over Q, with a fiber Aut(ℍ) ≅ Sp(1)/{±1}. Clearly, this fibration coincides with the principal bundle G constructed above. Since U(Q) ≅ G×ℝ^{>0}, a choice of u ∈ U(Q)|_{q} determines

an isomorphism W |_{q} ≅ ℍ.

Let (q, u) be the point of U(Q), with q ∈ Q, u ∈ U(Q)|_{q}. The natural connection in U(Q) gives a decomposition

T_{(q,u)}U(Q) = T_{u}(U(Q)|_{q} ) ⊕ T_{q}Q.

The space U(Q)|_{q} ≅ ℍ^{∗}/{±1} is equipped with a natural hypercomplex

structure. This gives a quaternion action on T_{u}(U(Q)|_{q} ) The choice of

u ∈ U(Q)|_{q} determines a quaternion action on T_{q}Q, as we have seen above. We obtain that the total space of U(Q) is an almost hypercomplex manifold.

Proposition 7.22: (A. Swann) Let Q be a quaternionic-Kähler manifold. Consider the manifold U(Q) constructed as above, and equipped with a quaternion algebra action in its tangent space. Then U(Q) is a hypercomplex manifold.

Proof: Clearly, the manifold U(Q) is equipped with a ℍ^{∗}-action, which is related with the almost hypercomplex structure as prescribed by Definition 7.19 (ii)-(iii). Pick an induced complex structure I ∈ ℍ. This gives an algebra embedding ℂ ⟶ ℍ. Consider the corresponding ℂ^{∗}-action ρ_{I} on an almost complex manifold (U(Q), I). This ℂ^{∗}-action is compatible with the almost complex structure. The quotient U(Q)/ρ(I) is an almost complex manifold, which is naturally isomorphic to the twistor space Tw(Q). Let L^{∗} be a complex vector bundle of all (1, 0)-vectors v ∈ T (Tw(Q)) tangent to the fibers of the standard projection σ : Tw(Q) ⟶ Q, and L be the dual vector bundle. Denote by T ot_{≠0}(L) the complement Tot(L)\N, where N = Tw(Q) ⊂ Tot(L) is the zero section of L. Using the natural connection in L, we obtain an almost complex structure on Tot(L).

Consider the natural projection φ : T ot_{≠0}(L) ⟶ Q. The fibers φ^{−1}(q) of φ are identified with the space of non-zero vectors in the total space of the cotangent bundle T^{∗}σ^{−1}(q) ≅ T^{∗}(ℂP^{1}). This space is naturally isomorphic to

G|_{q} × ℝ^{>0} = U(Q)|_{q} ≅ ℍ^{∗}/{±1}.

This gives a canonical isomorphism of almost complex manifolds

(U(Q), I) ⟶ T ot_{≠0}(L).

Therefore, to prove that (U(Q), I) is a complex manifold, it suffices to show that the natural almost complex structure on T ot_{≠0}(L) ⊂ Tot(L) is integrable. Consider the natural connection ∇_{L} on L. To prove that Tot(L) is a complex manifold, it suffices to show that ∇_{L} is a holomorphic connection. The bunlde L is known under the name of holomorphic contact bundle, and it is known to be holomorphic ([Sal], [Bes]).

Remark 7.23: The result of Proposition 7.22 is well known. We have given its proof because we shall need the natural identification T ot_{≠0}(L) ≅ U(Q) further on in this paper.

Theorem 7.24: Let Q be a quaternionic-Kähler manifold of positive scalar curvature, and U(Q) the hypercomplex manifold constructed above. Then U(Q) admits a unique (up to a scaling) hyperkähler metric compatible with the hypercomplex structure.

Proof: [Sw].

Consider the action of ℍ^{∗} on U(M) defined in the proof of Proposition 7.22. This action satisfies the conditions (ii) and (iii) of Definition 7.19. The conditions (i) and (iv) of Definition 7.19 are easy to check (see [Sw] for details). This gives a functor from the category C of quaternionic-Kähler manifolds of positive scalar curvature to the category H_{sp} of Definition 7.21.

Theorem 7.25: The functor Q ⟶ U(Q) from C to H_{sp} is an equivalence of categories.

Proof: [Sw].

The inverse functor from H_{sp} to C is constructed by taking a quotient of M by the action of ℍ^{∗}. Using the technique of quaternionic-Kähler reduction anf hyperkähler potentials ([Sw]), one can equip the quotient M/ℍ^{∗} with a natural quaternionic-Kähler structure.

8 ℂ^{∗}-equivariant twistor spaces

In Section 7, we gave an exposition of the twistor transform, B_{2}-bundles and Swann’s formalism. In the present Section, we give a synthesis of these theories, obtaining a construction with should be thought of as Swann’s formalism for vector bundles.

Consider the equivalence of categories Q ⟶ U(Q) constructed in Theorem 7.25 (we call this equivalence “Swann’s formalism”). We show that B_{2}- bundles on Q are in functorial bijective correspondence with ℂ^{∗}-equivariant holomorphic bundles on Tw(U(Q)) (Theorem 8.5).

8. ℂ^{∗}-EQUIVARIANT TWISTOR SPACES 93

In Subsection 8.4, this equivalence is applied to the vector bundle π^{∗}(F) of Theorem 6.1. We use it to construct a canonical Yang-Mills connection on π^{∗}(F)|_{C}, where C is a special fiber of π : M̃ ⟶ (M, I) (see Theorem 6.1 for details and notation). This implies that the holomorphic bundle π^{∗}(F)|_{C} is polystable (Theorem 8.15).

8.1 B_{2}-bundles on quaternionic-Kähler manifolds and ℂ^{∗}-equivariant holomorphic bundles over twistor spaces

For the duration of this Subsection, we fix a hyperkähler manifold M, equipped with a special ℍ^{∗}-action ρ, and the corresponding quaternionic-Kähler manifold Q = M/ℍ^{∗}. Denote the natural quotient map by φ : M ⟶ Q.

Lemma 8.1: Let ω be a 2-form over Q, and φ^{∗}ω its pullback to M. Then the following conditions are equivalent

(i) ω is a B_{2}-form

(ii) φ^{∗}ω is of Hodge type (1, 1) with respect to some induced complex structure I on M

(iii) φ^{∗}ω is SU(2)-invariant.

Proof: Let I be an induced complex structure on M. As we have shown in the proof of Proposition 7.22, the complex manifold (M, I) is identified with an open subset of the total space Tot(L) of a holomorphic line bundle L over Tw(Q). The map φ is represented as a composition of the projections h : Tot(L) ⟶ Tw(Q) and σ_{Q} : Tw(Q) ⟶ Q. Since the map h is smooth and holomorphic, the form φ^{∗}ω is of Hodge type (1, 1) if and only if σ_{Q}^{∗}ω is of type (1, 1). By Claim 7.13 (i), this happens if and only if ω is a B_{2}-form. This proves an equivalence (i) ⇔ (ii). Since the choice of I is arbitrary, the pullback φ^{∗}ω of a B_{2}-form is of Hodge type (1, 1) with respect to all induced complex structures. By Lemma 2.6, this proves the implication (i) ⇒ (iii). The implication (iii) ⇒ (ii) is clear.

Proposition 8.2: Let (B, ∇) be a complex vector bundle with connection over Q, and (φ^{∗}B, φ^{∗}∇) its pullback to M. Then the following conditions are equivalent

(i) (B, ∇) is a B_{2}-form

(ii) The curvature of (φ^{∗}B, φ^{∗}∇) is of Hodge type (1,1) with respect to some induced complex structure I on M

(iii) The bundle (φ^{∗}B, φ^{∗}∇) is autodual Proof: Follows from Lemma 8.1 applied to ω = ∇^{2}.

For any point I ∈ ℂP^{1}, consider the corresponding algebra embedding ℂ ↪^{c_{I}} ℍ. Let ρ_{I} be the action of ℂ^{∗} on (M, I) obtained as a restriction of ρ to c_{I}(ℂ^{∗}) ⊂ ℍ^{∗}. Clearly from Definition 7.19 (ii), ρ_{I} acts on (M, I) by holomorphic automorphisms.

Consider Tw(M) as a union

Tw(M) = ⋃_{I∈ℂP^{1}} π^{−1}(I), π^{−1}(I) = (M, I)

Gluing ρ(I) together, we obtain a smooth ℂ^{∗}-action ρ_{ℂ} on Tw(M).

Claim 8.3: Consider the action ρ_{ℂ} : ℂ^{∗} × Tw(M) ⟶ Tw(M) constructed above. Then ρ_{ℂ} is holomorphic.

Proof: It is obvious from construction that ρ_{ℂ} is compatible with the complex structure on Tw(M).

Example 8.4: Let M = ℍ^{n}\0. Since Tw(ℍ^{n}) is canonically isomorphic to a total space of the bundle O(1)^{n} over ℂP^{1}, the twistor space Tw(M) is Tot(O(1)^{n}) without zero section. The group ℂ^{∗} acts on Tot(O(1)^{n}) by dilatation, and the restriction of this action to Tw(M) coincides with ρ_{ℂ}.

Consider the map σ : Tw(M) ⟶ M. Let (B, ∇) be a B_{2}-bundle over Q. Since the bundle (φ^{∗}B, φ^{∗}∇) is autodual, the curvature of σ^{∗}φ^{∗}∇ has type (1, 1). Let (σ^{∗}φ^{∗}B, (σ^{∗}φ^{∗}∇)^{0,1}) be the holomorphic bundle obtained from (φ^{∗}B, φ^{∗}∇) by twistor transform. Clearly, this bundle is ℂ^{∗}-equivariant, with respect to the natural ℂ^{∗}-action on Tw(M). It turns out that any ℂ^{∗}-equivariant bundle F on Tw(M) can be obtained this way, assuming that F is compatible with twistor transform.

Theorem 8.5: In the above assumptions, let C_{B_{2}} be the category of of B_{2}-bundles on Q, and C_{Tw,ℂ^{∗}} the category of ℂ^{∗}-equivariant holomorphic bundles on Tw(M) which are compatible with the twistor transform. Consider the functor

(σ^{∗}φ^{∗})^{0,1} : C_{B_{2}} ⟶ C_{Tw,ℂ^{∗}},

(B, ∇) ⟶ (σ^{∗}φ^{∗}B, (σ^{∗}φ^{∗}∇)^{0,1}), constructed above. Then (σ^{∗}φ^{∗})^{0,1} establishes an equivalence of categories.

We prove Theorem 8.5 in Subsection 8.3.

Remark 8.6: Let Q be an arbitrary quaternionic-Kähler manifold, and M = U(Q) the corresponding fibration. Then M is hypercomplex, and its twistor space is equipped with a natural holomorphic action of ℂ^{∗}. This gives necessary ingredients needed to state Theorem 8.5 for Q with negative scalar curvature. The proof which we give for Q with positive scalar curvature will in fact work for all quaternionic-Kähler manifolds.

Question 8.7: What happens with this construction when Q is a hyperkähler manifold?

In this paper, we need Theorem 8.5 only in the case Q = ℍP^{n}, M = ℍ^{n}\0, but the general proof is just as difficult.

8.2 ℂ^{∗}-equivariant bundles and twistor transform

Let M be a hyperkähler manifold, and Tw(M) its twistor space. Recall that Tw(M) = ℂP^{1} × M is equipped with a canonical anticomplex involution ι, which acts as identity on M and as central symmetry I ⟶ −I on ℂP^{1} = S^{2}.

Proposition 8.8: Let M be a hyperkähler manifold, and Tw(M) its twistor space. Assume that Tw(M) is equipped with a free holomorphic action ρ(z) : Tw(M) ⟶ Tw(M) of ℂ^{∗}, acting along the fibers of π : Tw(M) ⟶ ℂP^{1}. Assume, moreover, that ι ∘ ρ(z) = ρ(\overline{z}) ∘ ι, where ι is the natural anticomplex involution of Tw(M).^{1} Let F be a ℂ^{∗}-equivariant holomorphic vector bundle on Tw(M). Assume that F is compatible with the twistor transform. Let ∇_{F} be the natural connection on F (Remark 7.5). Then ∇_{F} is flat along the leaves of ρ.

Proof: First of all, let us recall the construction of the natural connection ∇_{F}. Let F be an arbitrary bundle compatible with the twistor transform. We construct ∇_{F} in terms of the isomorphism Ψ_{1,2} defined in Lemma 6.7.

1These assumptions are automatically satisfied when M is equipped with a special ℍ^{∗}-action, and ρ(z) is the corresponding ℂ^{∗}-action on Tw(M).

Consider an induced complex structure I. Let F_{I} be the restriction of F to (M, I) = π^{−1}(I) ⊂ Tw(M). Consider the evaluation map

p_{I} : Lin(M) ⟶ (M, I)

(Subsection 6.1). In a similar way we define the holomorphic vector bundle F_{−I} on (M, −I) and the map p_{−I} : Lin(M) ⟶ (M, −I). Denote by F_{1}, F_{−1} the sheaves p_{I}^{∗}(F_{I}), p_{−I}^{∗}(F_{−I}). In Lemma 6.7, we constructed an isomorphism Ψ_{1,−1} : F_{1} ⟶ F_{−1}.

Let us identify Lin(M) with (M, I) × (M, I) (this identification is naturally defined in a neighbourhood of Hor ⊂ Lin(M) – see Proposition 6.4). Then the maps p_{I}, p_{−I} became projections to the relevant components. Let

\begin{matrix}\overline{∂} : F_{1} & ⟶ F_{1} ⊗ p_{I}^{∗}Ω^{1}(M, −I), \\ ∂ : F_{−1} & ⟶ F_{−1} ⊗ p_{−I}^{∗}Ω^{1}(M, I), \end{matrix}

be the sheaf maps obtained as pullbacks of de Rham differentials (the tensor product is taken in the category of coherent sheaves over Lin(M)). Twisting ∂ by an isomorphism Ψ_{1,−1} : F_{1} ⟶ F_{−1}, we obtain a map

∂^{Ψ} : F_{1} ⟶ F_{1} ⊗ p_{I}^{∗}Ω^{1}(M, I).

Adding \overline{∂} and ∂^{Ψ}, we obtain

∇ : F_{1} ⟶ F_{1} ⊗ (p_{I}^{∗}Ω^{1}(M, I) ⊕ p_{I}^{∗}Ω^{1}(M, −I)).

Clearly, ∇ satisfies the Leibniz rule. Moreover, the sheaf p_{I}^{∗}Ω^{1}(M, I) ⊕ p_{I}^{∗}Ω^{1}(M, −I) is naturally isomorphic to the sheaf of differentials over

Lin(M) = (M, I) × (M, −I).

Therefore, ∇ can be considered as a connection in F_{1}, or as a real analytic connection in a real analytic complex vector bundle underlying F_{I}. From the definition of ∇_{F} ([KV]), it is clear that ∇_{F} |_{(M,I)} equals ∇.

Return to the proof of Proposition 8.8. Consider a ℂ^{∗}-action ρ_{I}(z) on (M, I), (M, −I) induced from the natural embeddings (M, I) ↪ Tw(M), (M, −I) ↪ Tw(M). Then F_{I} is a ℂ^{∗}-equivariant bundle. Since ι ∘ ρ(z) = ρ(\overline{z}) ∘ ι, the identification Lin(M) = (M, I) × (M, I) is compatible with ℂ^{∗}-action. Let r = \frac{d}{dr} be the holomorphic vector field on (M, I) corresponding to the ℂ^{∗}-action. To prove Proposition 8.8, we have to show that the operator

[∇_{r}, ∇\overline{r}] : F_{I} ⟶ F_{I} ⊗ Λ^{1,1}(M, I)

vanishes.

Consider the equivariant structure operator

ρ(z)^{F} : ρ_{I}(z)^{∗}F_{I} ⟶ F_{I}.

Let U be a ℂ^{∗}-invariant Stein subset of (M, I). Consider ρ(z)^{F} an an endomorphism of the space of global holomorphic sections Γ_{U}(F_{I}). Let

D_{r}(f) := lim_{ε→0} \frac{ρ_{I}(1 + ε)}{ε},

for f ∈ Γ_{U}(F_{I}). Clearly, D_{r} is a well defined sheaf endomorphism of F_{I}, satisfying

D_{r}(α · f) = \frac{d}{dr} α · f + α · D_{r}(f),

for all α ∈ O_{(M,I)}. We say that a holomorphic section f of F_{I} is ℂ^{∗}- invariant if D_{r}(f) = 0. Clearly, the O_{(M,I)}-sheaf F_{I} is generated by ℂ^{∗}- invariant sections. Therefore, it suffices to check the equality

[∇_{r}, ∇\overline{r}](f) = 0

for holomorphic ℂ^{∗}-invariant f ∈ F_{I}.

Since f is holomorphic, we have ∇\overline{r}f = 0. Thus,

[∇_{r}, ∇\overline{r}](f) = ∇\overline{r}∇_{r}(f).

We obtain that Proposition 8.8 is implied by the following lemma.

Lemma 8.9: In the above assumptions, let f be a ℂ^{∗}-invariant section of F_{I}. Then ∇_{r}(f) = 0.

Proof: Return to the notation we used in the beginning of the proof of Proposition 8.8. Then, ∇(f) = \overline{∂}(f) + ∂^{Ψ}(f). Since f is holomorphic, \overline{∂}(f) = 0, so we need to show that ∂^{Ψ}(f)(r) = 0. By definition of ∂^{Ψ}, this is equivalent to proving that

∂Ψ_{1,−1}(f)(r) = 0.

Consider the ℂ^{∗}-action on Lin(M) which is induced by the ℂ^{∗}-action on Tw(M). Since the maps p_{I}, p_{−I} are compatible with the ℂ^{∗}-action, the sheaves F_{1}, F_{−1} are ℂ^{∗}-equivariant. We can repeat the construction of the operator D_{r} for the sheaf F_{−I}. This allows one to speak of holomorphic ℂ^{∗}-invariant sections of F_{−I}. Pick a ℂ^{∗}-invariant Stein subset U ⊂ (M, −I).

Since the statement of Lemma 8.9 is local, we may assume that M = U. Let g_{1}, ..., g_{n} be a set of ℂ^{∗}-invariant sections of F_{I} which generated F_{I}. Then, the sections p_{−I}^{∗}(g_{1}), ..., p_{−I}^{∗}(g_{n}) generate F_{−1}. Consider the section Ψ_{1,−1}(f) of F_{−1}. Clearly, Ψ_{1,−1} commutes with the natural ℂ^{∗}-action. Therefore, the section Ψ_{1,−1}(f) is ℂ^{∗}-invariant, and can be written as

Ψ_{1,−1}(f) = ∑ α_{i}p_{−I}^{∗}(g_{i}),

where the functions α_{i} are ℂ^{∗}-invariant. By definition of ∂ we have

∂ (∑ α_{i}p_{−I}^{∗}(g_{i})) = ∑ ∂(α_{i}p_{−I}^{∗}(g_{i})) + ∑ α_{i}∂(p_{−I}^{∗}(g_{i})).

On the other hand, g_{i} is a holomorphic section of F_{−I}, so ∂p_{−I}^{∗}(g_{i}) = 0. We obtain

∂ (∑ α_{i}p_{−I}^{∗} · (g_{i})) = ∑ ∂α_{i}p_{−I}^{∗}(g_{i}).

Thus,

∂Ψ_{1,−1}(f)(r) = ∑ \frac{∂α_{i}}{∂r} p_{−I}^{∗}(g_{i}),

but since the functions α_{i} are ℂ^{∗}-invariant, their derivatives along r vanish. We obtain ∂Ψ_{1,−1}(f)(r) = 0. This proves Lemma 8.9. Proposition 8.8 is proven.

8.3 Twistor transform and the ℍ^{∗}-action

For the duration of this Subsection, we fix a hyperkähler manifold M, equipped with a special ℍ^{∗}-action ρ, and the corresponding quaternionic-Kähler manifold Q = M/ℍ^{∗}. Denote the natural quotient map by φ : M ⟶ Q. Clearly, Theorem 8.5 is an immediate consequence of the following theorem.

Theorem 8.10: Let F be a ℂ^{∗}-equivariant holomorphic bundle over Tw(M), which is compatible with the twistor transform. Consider the natural connection ∇_{F} on F. Then ∇_{F} is flat along the leaves of ℍ^{∗}-action.

Proof: The leaves of ℍ^{∗}-action are parametrized by the points of q ∈ Q. Consider such a leaf M_{q} := φ^{−1}(q) ⊂ M. Clearly, M_{q} is a hyperkähler submanifold in M, equipped with a special action of ℍ^{∗}. Moreover, the restriction of F to Tw(M_{q}) ⊂ Tw(M) satisfies assumptions of Theorem 8.10. To prove that ∇_{F} is flat along the leaves of ℍ^{∗}-action, we have to show that F|_{Tw(Mq)} is flat, for all q. Therefore, it suffices to prove Theorem 8.10 for dim_{ℍ} M = 1.

Lemma 8.11: We work in notation and assumptions of Theorem 8.10. Assume that dim_{ℍ} M = 1. Then the connection ∇_{F} is flat.

Proof: Let I be an induced complex structure, and F_{I} := F |_{(M,I)} the corresponding holomorphic bundle on (M, I). Denote by z_{I} the vector field corresponding to the ℂ^{∗}-action ρ_{I} on (M, I). By definition, the connection ∇|_{F_{I}} has SU(2)-invariant curvature Θ_{I}. On the other hand, Θ_{I}(z_{I}, \overline{z}_{I}) = 0 by Proposition 8.8. Since ∇_{F} = σ^{∗}∇ is a pullback of an autodual connection ∇ on M, its curvature is a pullback of Θ_{I}. In particular, Θ = Θ_{I} is independent from the choice of induced complex structure I. We obtain that Θ(z_{I}, \overline{z}_{I}) = 0 for all induced complex structures I on M.

Now Lemma 8.11 is implied by the following linear-algebraic claim.

Claim 8.12: Let M be a hyperkähler manifold equipped with a special ℍ^{∗}-action, dim_{ℍ} M = 1. Consider the vectors z_{I}, \overline{z}_{I} defined above. Let Θ be a smooth SU(2)-invariant 2-form, such that for all induced complex structures, I, we have Θ(z_{I}, \overline{z}_{I}) = 0. Then Θ = 0.

Proof: The proof of Claim 8.12 is an elementary calculation. Fix a point m_{0} ∈ M. Consider the flat hyperkähler manifold ℍ\0, equipped with a natural special action of ℍ^{∗}. From the definition of a special action, it is clear that the map ρ defines a covering ℍ\0 ⟶ M, h ⟶ ρ(h)m_{0} of hyperkähler manifolds, and this covering is compatible with the special action. Therefore, the hyperkähler manifold M is flat, and the ℍ^{∗}-action is linear in the flat coordinates.

Let

Λ^{2}(M) = Λ^{+}(M) ⊕ Λ^{−}(M)

be the standard decomposition of Λ^{2}(M) according to the eigenvalues of the Hodge ∗ operator. Consider the natural Hermitian metric on Λ^{2}(M). Then Λ^{−}(M) is the bundle of SU(2)-invariant 2-forms (see, e. g., [V1]), and Λ^{+}(M) is its orthogonal complement. Consider the corresponding orthogonal projection Π : Λ^{2}(M) ⟶ Λ^{−}(M). Denote by dz_{I} ∧ d\overline{z}_{I} the differential form which is dual to the bivector z_{I} ∧ \overline{z}_{I}. Let R ⊂ Λ^{−}(M) be the C^{∞}(M)- subsheaf of Λ^{−}(M) generated by Π(dz_{I} ∧d\overline{z}_{I}), for all induced complex structures I on M. Clearly, Θ ∈ Λ^{−}(M) and Θ is orthogonal to R ⊂ Λ^{−}(M).

Therefore, to prove that Θ = 0 it suffices to show that R = Λ^{−}(M). Since M is covered by ℍ\0, we may prove R = Λ^{−}(M) in assumption M = ℍ\0.

Let γ be the real vector field corresponding to dilatations of M = ℍ\0, and dγ the dual 1-form. Clearly,

dz_{I} ∧ d\overline{z}_{I} = 2\sqrt{−1} dγ ∧ I(dγ).

Averaging dγ ∧ I(dγ) by SU(2), we obtain

Π(dz_{I} ∧ d\overline{z}_{I}) = \sqrt{−1} (dγ ∧ I(dγ) − J(dγ) ∧ K(dγ))

where I, J, K is the standard triple of generators for quaternion algebra. Similarly,

Π(dz_{J} ∧ d\overline{z}_{J}) = \sqrt{−1} (dγ ∧ J(dγ) + K(dγ) ∧ I(dγ))

and

Π(dz_{K} ∧ d\overline{z}_{K}) = \sqrt{−1} (dγ ∧ K(dγ) + I(dγ) ∧ J(dγ))

Thus, Π(R) is a 3-dimensional sub-bundle of Λ^{−}(M). Since dim Λ^{−}(M) = 3, we have Π(R) = Λ^{−}(M). This proves Claim 8.12. Lemma 8.11 and Theorem 8.10 is proven.

8.4 Hyperholomorphic sheaves and ℂ^{∗}-equivariant bundles over M_{fl}

Let M be a hyperkähler manifold, I an induced complex structure and F a reflexive sheaf over (M, I), equipped with a hyperholomorphic connection. Assume that F has an isolated singularity in x ∈ M. Consider the sheaf F on Tw(M) corresponding to F as in the proof of Proposition 3.17. Let s_{x} ⊂ Tw(M) be the horizontal twistor line corresponding to x, and 𝔪 its ideal. Consider the associated graded sheaf of 𝔪. Denote by Tw^{gr} the spectre of this associated graded sheaf. Clearly, Tw^{gr} is naturally isomorphic to Tw(T_{x}M), where T_{x}M is the flat hyperkähler manifold corresponding to the space T_{x}M with induced quaternion action. Consider the natural ℍ^{∗}- action on T_{x}M. This provides the hyperkähler manifold T_{x}M\0 with a special ℍ^{∗}-action.

Let s_{0} ⊂ Tw^{gr} be the horizontal twistor line corresponding to s_{x}. The space Tw^{gr} \s_{0} is equipped with a holomorphic ℂ^{∗}-action (Claim 8.3). Denote by F^{gr} the sheaf on Tw^{gr} associated with F. Clearly, F^{gr} is ℂ^{∗}- equivariant. In order to be able to apply Theorem 8.5 and Theorem 8.10 to F^{gr}|_{Tw^{gr} \s_{0}} , we need only to show that F^{gr} is compatible with twistor transform.

Proposition 8.13: Let M be a hyperkähler manifold, I an induced complex structure and F a reflexive sheaf over (M, I), equipped with a hyperholomorphic connection. Assume that F has an isolated singularity in x ∈ M. Let F^{gr} be the ℂ^{∗}-equivariant bundle on Tw^{gr} \s_{0} constructed above. Then

(i) the bundle F^{gr} is compatible with twistor transform.

(ii) Moreover, the natural connection ∇_{F^{gr}} (Remark 7.5) is Hermitian.

Proof: The argument is clear, but cumbersome, and essentially hinges on taking associate graded quotients everywhere and checking that all equations remain true. We give a simplified version of the proof, which omits some details and notation.

Consider the bundle F|_{M\sx}. This bundle is compatible with twistor transform, and therefore, is equipped with a natural connection ∇_{F}. This connection is constructed using the isomorphism Ψ_{1,−1} : F_{1} ⟶ F_{−1} (see the proof of Proposition 8.8). We apply the same consideration to F^{gr}|_{(TxM,I)} , and show that the resulting connection ∇_{F^{gr}} is hyperholomorphic. This implies that F^{gr} admits a (1, 1)-connection which is a pullback of some connection on F^{gr}|_{(TxM,I)} . This argument is used to prove that F^{gr} is compatible with the twistor transform.

We use the notation introduced in the proof of Proposition 8.8. Let Lin^{gr} be the space of twistor maps in Tw^{gr}. Consider the maps p_{±I}^{gr} : Lin^{gr} ⟶ (T_{x}M, ±I) and the sheaves F_{±1}^{gr} := (p_{±I}^{gr})^{∗}F_{±I}^{gr} obtained in the same way as the maps p_{±I} and the sheaves F_{±1} from the corresponding associated graded objects. Taking the associated graded of Ψ_{1,−1} gives an isomorphism Ψ_{1,−1}^{gr} : F_{1}^{gr} ⟶ F_{−1}^{gr}. Using the same construction as in the proof of Proposition 8.8, we obtain a connection operator

\overline{∂}^{gr}+∂^{Ψ^{gr}} = ∇_{I}^{gr} : F_{1}^{gr} ⟶ F_{1}^{gr}⊗((p_{−I}^{gr})^{∗}Ω^{1}(T_{x}M, I)⊕(p_{I}^{gr})^{∗}Ω^{1}(T_{x}M, −I)).

Since (\overline{∂}^{gr})^{2} = (∂^{Ψ^{gr}})^{2} = 0, the curvature of ∇_{I}^{gr} has Hodge type (1, 1) with respect to I. To prove that ∇_{I}^{gr} is hyperholomorphic, we need to show that the curvature of ∇_{I}^{gr} has type (1, 1) with respect to every induced complex structure. Starting from another induced complex structure J, we obtain a connection ∇_{J}^{gr}, with the curvature of type (1, 1) with respect to J. To prove that ∇_{J}^{gr} is hyperholomorphic it remains to show that ∇_{J}^{gr} = ∇_{I}^{gr}.

Let ∇_{I}, ∇_{J} be the corresponding operators on F_{1}. From the construction, it is clear that ∇_{I}^{gr}, ∇_{J}^{gr} are obtained from ∇_{I}, ∇_{J} by taking the associated graded quotients. On the other hand, ∇_{I} = ∇_{J}. Therefore, the connections ∇_{I}^{gr} and ∇_{J}^{gr} are equal. We proved that the bundle F^{gr}|_{Tw^{gr} \s_{0}} is compatible with the twistor transform. To prove Proposition 8.13, it remains to show that the natural connection on F^{gr} is Hermitian.

The bundle F|_{Tw(M\x_{0})} is by definition Hermitian. Consider the corre-

sponding isomorphism F ⟶ (ι^{∗}\overline{F})^{∗} (Proposition 7.7). Taking an associate graded map, we obtain an isomorphism

F^{gr} ˜→(ι^{∗}\overline{F}^{gr})^{∗}.

This gives a non-degenerate semilinear form h^{gr} on F^{gr}. It remains only to show that h^{gr} is pseudo-Hermitian (i. e. satisfies h(x, y) = \overline{h(y, x)}) and positive definite.

Let M_{ℂ}^{gr} be a complexification of M^{gr} = T_{x}M, M_{ℂ}^{gr} = Lin(M^{gr}). Consider the corresponding complex vector bundle F_{ℂ}^{gr} over M_{ℂ}^{gr} underlying F^{gr}. The metric h^{gr} can be considered as a semilinear form F_{ℂ}^{gr} × F_{ℂ}^{gr} ⟶ O_{M_{ℂ}^{gr}}. This semilinear form is obtained from the corresponding form h on F by taking the associate graded quotients. Since h is Hermitian, the form h^{gr} is pseudo-Hermitian. To prove that h^{gr} is positive semidefinite, we need to show that for all f ∈ F_{ℂ}^{gr}, the function h^{gr}(f, \overline{f}) belongs to O_{M_{ℂ}^{gr}} · \overline{O}_{M_{ℂ}^{gr}}, where O_{M_{ℂ}^{gr}}·\overline{O}_{M_{ℂ}^{gr}} denotes the ℝ^{>0}-semigroup of O_{M_{ℂ}^{gr}} generated by x·\overline{x}, for all x ∈ O_{M_{ℂ}^{gr}}. A similar property for h holds, because h is positive definite. Clearly, taking associated graded quotient of the semigroup O_{M_{ℂ}} · \overline{O}_{M_{ℂ}}, we obtain O_{M_{ℂ}^{gr}} · \overline{O}_{M_{ℂ}^{gr}}. Thus,

h^{gr}(f, \overline{f}) ∈ (O_{M_{ℂ}} · \overline{O}_{M_{ℂ}})^{gr} = O_{M_{ℂ}^{gr}} · \overline{O}_{M_{ℂ}^{gr}}

This proves that h^{gr} is positive semidefinite. Since h^{gr} is non-degenerate, this form in positive definite. Proposition 8.13 is proven.

Remark 8.14: Return to the notations of Theorem 6.1. Consider the bundle π^{∗}F |_{C}, where C = ℙT_{x}M is the blow-up divisor. Clearly, this bundle

corresponds to the graded sheaf F_{I}^{gr} = F^{gr}|_{(M,I)} on (T_{x}M, I). By Proposi-

tion 8.13 (see also Theorem 8.10), the bundle π^{∗}F |_{C} is equipped with a nat-

ural ℍ^{∗}-invariant connection and Hermitian structure.^{2} The sheaf π^{∗}F |_{M̃\C}

is a hyperholomorphic bundle over M̃\C ≅ M\x_{0}. Therefore, π^{∗}F |_{M̃\C} is equipped with a natural metric and a hyperholomorphic connection. It is expected that the natural connection and metric on π^{∗}F |_{M̃\C} can be extended to π^{∗}F, and the rectriction of the resulting connection and metric to π^{∗}F |_{C} coincides with that given by Proposition 8.13 and Theorem 8.10. This will give an alternative proof of Proposition 8.13 (ii), because a continuous extension of a positive definite Hermitian metric is a positive semidefinite Hermitian metric.

8.5 Hyperholomorphic sheaves and stable bundles on ℂP^{2n+1}

The purpose of the current Section was to prove the following result, which is a consequence of Proposition 8.13 and Theorem 8.10.

Theorem 8.15: Let M be a hyperkähler manifold, I an induced complex structure and F a reflexive sheaf on (M, I) admitting a hyperholomorphic connection. Assume that F has an isolated singularity in x ∈ M, and is locally trivial outside of x. Let π : M̃ ⟶ (M, I) be the blow-up of (M, I) in x. Consider the holomorphic vector bundle π^{∗}F on M̃ (Theorem 6.1). Let C ⊂ (M, I) be the blow-up divisor, C = ℙT_{x}M. Then the holomorphic bundle π^{∗}F |_{C} admits a natural Hermitian connection ∇ which is flat along the leaves of the natural ℍ^{∗}-action on ℙT_{x}M. Moreover, the connection ∇ is Yang-Mills, with respect to the Fubini-Study metric on C = ℙT_{x}M, the degree deg c_{1} (π^{∗}F |_{C} ) vanishes, and the holomorphic vector bundle π^{∗}F |_{C} is polystable.

Proof: By definition, coherent sheaves on C = ℙT_{x}M correspond bijectively to ℂ^{∗}-equivariant sheaves on T_{x}M\0. Let F^{gr} be the associated graded sheaf of F (Subsection 8.4). Consider F^{gr} as a bundle on T_{x}M\0.

2As usually, coherent sheaves over projective variety X correspond to finitely generated graded modules over the graded ring ⊕Γ(O_{X}(i)).

In the notation of Proposition 8.13, F^{gr} = F^{gr}|_{(M,I)} . By Proposition 8.13, the sheaf F^{gr} is ℂ^{∗}-equivariant and compatible with the twistor transform. According to the Swann’s formalism for bundles (Theorem 8.10), the bundle F^{gr}|_{TxM\0} is equipped with a natural Hermitian connection ∇_{F^{gr}} which is flat along the leaves of ℍ^{∗}-action. Let (B, ∇_{ℍ}) be the corresponding B_{2}-bundle on

ℙ_{ℍ}T_{x}M := (T_{x}M\0)/ℍ^{∗} ≅ ℍP^{n}.

Then π^{∗}F |_{C} is a holomorphic bundle over the corresponding twistor space C = Tw(ℙ_{ℍ}T_{x}M), obtained as a pullback of (B, ∇_{ℍ}) as in Claim 7.13. The natural Kähler metric on the twistor space C = Tw(ℙ_{ℍ}T_{x}M) is the Fubini-Study metric (Example 7.16). By Theorem 7.17, the bundle π^{∗}F |_{C} is Yang-

Mills and has deg c_{1} (π^{∗}F |_{C} ) = 0. Finally, by Uhlenbeck-Yau theorem

(Theorem 2.24), the bundle π^{∗}F |_{C} is polystable.

9 Moduli spaces of hyperholomorphic sheaves and bundles

9.1 Deformation of hyperholomorphic sheaves with isolated singularities

The following theorem is an elementary consequence of Theorem 8.15. The proof uses well known results on stability and reflexization (see, for instance, [OSS]). The main idea of the proof is the following. Given a family of hyperholomorphic sheaves with an isolated singularity, we blow-up this singularity and restrict the obtained family to a blow-up divisor. We obtain a family of coherent sheaves 𝔙_{s}, s ∈ S over ℂP^{2n+1}, with fibers semistable of slope zero. Assume that for all s ∈ S, s ≠ s_{0}, the sheaf 𝔙_{s} is trivial. Then the family 𝔙 is also trivial, up to a reflexization.

We use the following property of reflexive sheaves.

Definition 9.1: Let X be a complex manifold, and F a torsion-free coherent sheaf. We say that F is normal if for all open subvarieties U ⊂ X, and all closed subvarieties Y ⊂ U of codimension 2, the restriction

Γ_{U}(F) ⟶ Γ_{U\Y} (F)

is an isomorphism.

Lemma 9.2: Let X be a complex manifold, and F a torsion-free coherent sheaf. Then F is reflexive if and only if F is normal.

Proof: [OSS], Lemma 1.1.12.

Theorem 9.3: Let M be a hyperkähler manifol, I an induced complex structure, S a complex variety and 𝔉 a family of coherent sheaves over (M, I) × S. Consider the sheaf F_{s_{0}} := 𝔉|_{(M,I)×{s_{0}}}. Assume that the sheaf F_{s_{0}} is equipped with a filtration ξ. Let F_{i}, i = 1, ..., m denote the associated graded components of ξ, and F_{i}^{∗∗} denote their reflexizations. Assume that 𝔉 is locally trivial outside of (x_{0}, s_{0}) ∈ (M, I) × S. Assume, moreover, that all sheaves F_{i}^{∗∗}, i = 1, ..., m admit a hyperholomorphic connection. Then the reflexization 𝔉^{∗∗} is locally trivial.

Proof: Clearly, it suffices to prove Theorem 9.3 for 𝔉 reflexive. Let X̃ be the blow-up of (M, I)×S in {x_{0}}×S, and 𝔉̃ the pullback of 𝔉 to X̃. Clearly, X̃ = M̃ × S, where M̃ is a blow-up of (M, I) in x_{0}. Denote by C ⊂ M̃ the blow-up divisor of M̃. Taking S sufficiently small, we may assume that the bundle 𝔉|_{{x_{0}}×(S\{s_{0}})} is trivial. Thus, the bundle 𝔉̃|_{(C×S)\(C×{s_{0}})} , which

is a pullback of 𝔉|_{{x_{0}}×(S\{s_{0}})} under the natural projection (C × S)\(C × {s_{0}}) ⟶ {x_{0}} × (S\{s_{0}} is trivial. To prove that 𝔉 is locally trivial, we have to show that 𝔉̃ is locally trivial, and that the restriction of 𝔉̃ to C × S is trivial along the fibers of the natural projection C × S ⟶ S. Clearly, to show that 𝔉̃ is locally trivial we need only to prove that the fiber 𝔉̃|_{z}

has constant dimension for all z ∈ C × S. Thus, 𝔉̃ is locally trivial if and only if 𝔉̃|_{C×S} is locally trivial. This sheaf is reflexive, since it corresponds to an associate graded sheaf of a reflexive sheaf, in the sense of Footnote to Remark 8.14. It is non-singular in codimension 2, because all reflexive sheaves are non-singular in codimension 2 ([OSS], Ch. II, Lemma 1.1.10). By Theorem 8.15, the sheaf 𝔉̃|_{C×{s}} is semistable of slope zero. Theorem

9.3 is implied by the following lemma, applied to the sheaf 𝔉̃|_{C×S}.

Lemma 9.4: Let C be a complex projective space, S a complex variety and 𝔉 a torsion-free sheaf over C × S. Consider an open set U ↪^{j} C × S, which is a complement of C × {s_{0}} ⊂ C × S. Assume that the sheaf 𝔉|_{U} is trivial: 𝔉|_{U} ≅ O_{U}^{n}. Assume, moreover, that 𝔉 is non-singular in codimension

2, the sheaf (𝔉|_{C×{s_{0}}} )^{∗∗} is semistable of slope zero and

rk 𝔉 = rk 𝔉|_{C×{s_{0}}} .

Then the reflexization 𝔉^{∗∗} of 𝔉 is a trivial bundle.

Proof: Using induction, it suffices to prove Lemma 9.4 assuming that it is proven for all 𝔉′ with rk 𝔉′ < rk 𝔉. We may also assume that S is Stein, smooth and 1-dimensional.

Step 1: We construct an exact sequence

0 ⟶ 𝔉_{2} ⟶ 𝔉 ⟶ im p_{O_{1}} ⟶ 0

of sheaves of positive rank, which, as we prove in Step 3, satisfy assumptions of Lemma 9.4.

Consider the pushforward sheaf j_{∗}O_{U}^{n}. From the definition of j_{∗}, we obtain a canonical map

𝔉 ⟶ j_{∗}O_{U}^{n},  (9.1)

and the kernel of this map is a torsion subsheaf in 𝔉.

Let f be a coordinate function on S, which vanishes in s_{0} ∈ S. Clearly,

j_{∗}O_{U}^{n} ≅ O_{C×S}^{n} [\frac{1}{f} ] .

On the other hand, the sheaf O_{C×S} [\frac{1}{f} ] is a direct limit of the following diagram:

O_{C×S}^{n} ⟶^{·f} O_{C×S}^{n} ⟶^{·f} O_{C×S}^{n} ⟶^{·f} ...,

where ·f is the injection given by the multiplication by f. Thus, the map (9.1) gives an embedding

𝔉 ↪^{p} O_{C×S}^{n},

which is identity outside of (x_{0}, s_{0}). Multiplying p by \frac{1}{f} if necessary, we may

assume that the restriction p|_{C×{s_{0}}} is non-trivial. Thus, p gives a map

𝔉|_{C×{s_{0}}} ⟶ O_{C×{s_{0}}}^{n}.  (9.2)

with image of positive rank. Since both sides of (9.2) are semistable of slope zero, and O_{C×{s_{0}}}^{n} is polystable, the map (9.2) satisfies the following conditions. (see [OSS], Ch. II, Lemma 1.2.8 for details).

Let F_{1} := im p|_{C×{s_{0}}}, and F_{2} := ker p|_{C×{s_{0}}}. Then the

reflexization of F_{1} is a trivial bundle O_{C×{s_{0}}}^{k}, and p maps

F_{1} to the direct summand O′_{1} = O_{C×{s_{0}}}^{k} ⊂ O_{C×{s_{0}}}^{n}.

Let O_{1} = O_{C×S}^{k} ⊂ O_{C×S}^{n} be the corresponding free subsheaf of O_{C×S}^{n}. Consider the natural projection π_{O_{1}} of O_{C×S}^{n} to O_{1}. Let p_{O_{1}} be the composition of p and π_{O_{1}}, 𝔉_{1} the image of p_{O_{1}}, and 𝔉_{2} the kernel of p_{O_{1}}.

Step 2: We show that the sheaves 𝔉_{2} and 𝔉_{1} and non-singular in codimension 2.

Consider the exact sequence

T or^{1}(O_{C×{s_{0}}}, 𝔉_{1}) ⟶ 𝔉_{2}|_{C×{s_{0}}} ⟶ 𝔉|_{C×{s_{0}}} ⟶ 𝔉_{1}|_{C×{s_{0}}} ⟶ 0

obtained by tensoring the sequence

0 ⟶ 𝔉_{2} ⟶ 𝔉 ⟶ 𝔉_{1} ⟶ 0

with O_{C×{s_{0}}}. From this sequence, we obtain an isomorphism 𝔉_{1}|_{C×{s_{0}}} ≅ F_{1}.

A torsion-free coherent sheaf over a smooth manifold is non-singular in codimension 1 ([OSS], Ch. II, Corollary 1.1.8).

Since 𝔉 is non-singular in codimension 2, the restriction 𝔉|_{C×{s_{0}}} is non-

singular in codimension 1. Therefore, the torsion of 𝔉|_{C×{s_{0}}} has support of codimension at least 2 in C × {s_{0}}. Since the sheaf F_{2} is a subsheaf of 𝔉|_{C×{s_{0}}}, its torsion has support of codimension at least 2. Therefore, the singular set of F_{2} has codimension at least 2 in C × {s_{0}}. The rank of F_{2} is by definition equal to n − k.

Since F_{1} has rank k, the singular set of 𝔉_{1} coincides with the singular set of F_{1}. Since the restriction 𝔉_{1}|_{C×{s_{0}}} = F_{1}, is a subsheaf of a trivial bundle of dimension k on C × {s_{0}}, it is torsion-free. Therefore, the singularities of 𝔉_{1} have codimension at least 2 in C × {s_{0}}.

We obtain that the support of T or^{1}(O_{C×{s_{0}}}, 𝔉_{1}) has codimension at least 2 in C × {s_{0}}. Since the quotient sheaf

𝔉_{2}|_{C×{s_{0}}} /T or^{1}(O_{C×{s_{0}}}, 𝔉_{1}) ≅ F_{2}  (9.3)

is isomorphic to the sheaf F_{2}, this quotient is non-singular in codimension 1. Since we proved that F_{2} is non-singular in codimension 1, the sheaf 𝔉_{2}|_{C×{s_{0}}} is also non-singular in codimension 1, and its rank is equal to the rank of F_{2}.

Let R be the union of singular sets of the sheaves 𝔉_{2}, 𝔉, 𝔉_{1}. Clearly, R is contained in C × {s_{0}}, and R coincides with the set of all x ∈ C × {s_{0}} where the dimension of the fiber of the sheaves 𝔉_{2}, 𝔉, 𝔉_{1} is not equal to n − k, n, k. We have seen that the restrictions of 𝔉_{2}, 𝔉_{1} to C × {s_{0}} have ranks n − k, k. Therefore, the singular sets of 𝔉_{2}, 𝔉_{1} coincide with the singular sets of 𝔉_{2}|_{C×{s_{0}}}, 𝔉_{1}|_{C×{s_{0}}}. We have shown that these singular sets have codimension at least 2 in C × {s_{0}}. On the other hand, 𝔉 is nonsingular in codimension 2, by the conditions of Lemma 9.4. Therefore, R has codimension at least 3 in C × S.

Step 3: We check the assumptions of Lemma 9.4 applied to the sheaves 𝔉_{2}, 𝔉_{1}.

Since the singular set of 𝔉_{1} has codimension 2 in C × {s_{0}}, the O_{C×{s_{0}}}- sheaf T or^{1}(O_{C×{s_{0}}}, 𝔉_{1}) is a torsion sheaf with support of codimension 2

in C × {s_{0}}. By (9.3), the reflexization of 𝔉_{2}|_{C×{s_{0}}} coincides with the

reflexization of F_{2}. Thus, the sheaf (𝔉_{2}|_{C×{s_{0}}} )^{∗∗} is semistable. On the other hand, outside of C × {s_{0}}, the sheaf 𝔉_{2} is a trivial bundle. Thus, 𝔉_{2} satisfies assumptions of Lemma 9.4. Similarly, the sheaf 𝔉_{1} is non-singular in codimension 2, its restriction to C × {s_{0}} has trivial reflexization, and it is free outside of C × {s_{0}}.

Step 4: We apply induction and prove Lemma 9.4.

By induction assumption, the reflexization of 𝔉_{2} is isomorphic to a trivial bundle O_{C×S}^{n−k}. and reflexization of 𝔉_{1} is O_{C×S}^{k}. We obtain an exact sequence

0 ⟶ 𝔉_{2} ⟶ 𝔉 ⟶ 𝔉_{1} ⟶ 0,  (9.4)

where the sheaves 𝔉_{2} and 𝔉_{1} have trivial reflexizations.

Let V := C × S\R. Restricting the exact sequence (9.4) to V, we obtain an exact sequence

0 ⟶ O_{V}^{n−k} ⟶^{a} 𝔉|_{V} ⟶^{b} O_{V}^{k} ⟶ 0.  (9.5)

Since V is a complement of a codimension-3 complex subvariety in a smooth Stein domain, the first cohomology of a trivial sheaf on V vanish. Therefore, the sequence (9.5) splits, and the sheaf 𝔉|_{V} is a trivial bundle. Consider the

pushforward ζ_{∗}𝔉|_{V} , where ζ : V ⟶ C×S is the standard map. Then ζ_{∗}𝔉|_{V} is a reflexization of 𝔉 (a pushforward of a reflexive sheaf over a subvariety of codimension 2 or more is reflexive – see Lemma 9.2). On the other hand, since the sheaf 𝔉|_{V} is a trivial bundle, its push-forward over a subvariety of codimension at least 2 is also a trivial bundle over C × S. We proved that the sheaf 𝔉^{∗∗} = ζ_{∗}𝔉|_{V} is a trivial bundle over C × S. The push-forward

ζ_{∗}𝔉|_{V} coincides with reflexization of 𝔉, by Lemma 9.2. This proves Lemma 9.4 and Theorem 9.3.

9.2 The Maruyama moduli space of coherent sheaves

This Subsection is a compilation of results of Gieseker and Maruyama on the moduli of coherent sheaves over projective manifolds. We follow [OSS], [Ma2].

To study the moduli spaces of holomorphic bundles and coherent sheaves, we consider the following definition of stability.

Definition 9.5: (Gieseker–Maruyama stability) ([Gi], [OSS]) Let X be a projective variety, O(1) the standard line bundle and F a torsion-free coherent sheaf. The sheaf F is called Gieseker–Maruyama stable (resp. Gieseker–Maruyama semistable) if for all coherent subsheaves E ⊂ F with 0 < rk E < rk F, we have

p_{F} (k) < p_{E}(k) (resp., p_{F} (k) ⩽ p_{E}(k))

for all sufficiently large numbers k ∈ ℤ. Here

p_{F} (k) = \frac{dim Γ_{X}(F ⊗ O(k))}{rk F}.

Clearly, Gieseker–Maruyama stability is weaker than the Mumford-Ta-kemoto stability. Every Gieseker–Maruyama semistable sheaf F has a socalled Jordan-Hölder filtration F_{0} ⊂ F_{1} ⊂ ... ⊂ F with Gieseker–Maruyama stable successive quotients F_{i}/F_{i−1}. The corresponding associated graded sheaf

⊕F_{i}/F_{i−1}

is independent from a choice of a filtration. It is called the associate graded quotient of the Jordan-Hölder filtration on F.

Definition 9.6: Let F, G be Gieseker–Maruyama semistable sheaves on X. Then F, G are called S-equivalent if the corresponding associate graded quotients ⊕F_{i}/F_{i−1}, ⊕G_{i}/G_{i−1} are isomorphic.

Definition 9.7: Let X be a complex manifold, F a torsion-free sheaf on X, and Y a complex variety. Consider a sheaf F on X ×Y which is flat over Y. Assume that for some point s_{0} ∈ Y, the sheaf F|_{X×{s_{0}}} is isomorphic to F. Then F is called a deformation of F parametrized by Y. We say that a sheaf F′ on X is deformationally equivalent to F if for some s ∈ Y, the restriction F|_{X×{s}} is isomorphic to F′. Slightly less formally, such sheaves are called deformations of F. If F′ is a (semi-)stable bundle, it is called a (semi-) stable bundle deformation of F.

Remark 9.8: Clearly, the Chern classes of deformationally equivalent sheaves are equal.

Definition 9.9: Let X be a complex manifold, and F a torsion-free sheaf on X, and M_{mar} a complex variety. We say that M_{mar} is a coarse moduli space of deformations of F if the following conditions hold.

(i) The points of s ∈ M_{mar} are in bijective correspondence with S-equiva-lence classes of coherent sheaves F_{s} which are deformationally equivalent to F.

(ii) For any flat deformation F of F parametrized by Y, there exists a unique morphism φ : Y ⟶ M_{mar} such that for all s ∈ Y, the restriction F|_{X×{s}} is S-equivalent to the sheaf F_{φ(s)} corresponding to φ(s) ∈ M_{mar}.

Clearly, the coarse moduli space is unique. By Remark 9.8, the Chern classes of F_{s} are equal for all s ∈ M_{mar}.

It is clear how to define other kinds of moduli spaces. For instance, replacing the word sheaf by the word bundle throughout Definition 9.9, we obtain a definition of the coarse moduli space of semistable bundle deformations of F. Further on, we shall usually omit the word “coarse” and say “moduli space” instead.

Theorem 9.10: (Maruyama) Let X be a projective manifold and F a coherent sheaf over X. Then the Maruyama moduli space M_{mar} of deformations of F exists and is compact.

Proof: See, e. g., [Ma2].

9.3 Moduli of hyperholomorphic sheaves and C-restricted comples structures

Usually, the moduli space of semistable bundle deformations of a bundle F is not compact. To compactify this moduli space, Maruyama adds points corresponding to the deformations of F which are singular (these deformations can be non-reflexive and can have singular reflexizations). Using the desingularization theorems for hyperholomorphic sheaves, we were able to obtain Theorem 9.3, which states (roughly speaking) that a deformation of a semistable hyperholomorphic bundle is again a semistable bunlde, assuming that all its singularities are isolated. In Section 5, we showed that under certain conditions, a deformation of a hyperholomorphic sheaf is again hyperholomorphic (Theorem 5.14). This makes it possible to prove that a deformation of a semistable hyperholomorphic bundle is locally trivial.

In [V5], we have shown that a Hilbert scheme of a K3 surface has no non-trivial trianalytic subvarieties, for a general hyperkähler structure.

Theorem 9.11: Let M be a compact hyperkähler manifold without non-trivial trianalytic subvarieties, dim_{ℍ} ⩾ 2, and I an induced complex structure. Consider a hyperholomorphic bundle F on (M, I) (Definition 3.11). Assume that I is a C-restricted complex structure, C = deg_{I} c_{2}(F). Let M be the moduli space of semistable bundle deformations of F over (M, I). Then M is compact.

Proof: The complex structure I is by definition algebraic, with unique polarization. This makes it possible to speak of Gieseker–Maruyama stability on (M, I). Denote by M_{mar} the Maruyama moduli of deformations of F. Then M is naturally an open subset of M_{mar}. Let s ∈ M_{mar} be an arbitrary point and F_{s} the corresponding coherent sheaf on (M, I), defined up to S-equivalence. According to Remark 9.8, the Chern classes of F and F_{s} are equal. Thus, by Theorem 5.14, the sheaf F_{s} is hyperholomorphic. Therefore, F_{s} admits a filtration with hyperholomorphic stable quotient sheaves F_{i}, i = 1, ..., m. By Claim 3.16, the singular set S of F_{s} is trianalytic. Since M has no proper trianalytic subvarieties, S is a collection of points. We obtain that F_{s} has isolated singularities. Let 𝔉 be a family of deformations of F, parametrized by Y. The points y ∈ Y correspond to deformations F_{y} of F_{s}. Assume that for all y ∈ Y, y ≠ s, the sheaf F_{y} is a bundle. Since M is open in M_{mar}, such a deformation always exists.

The sheaf F_{s} has isolated singularities and admits a filtration with hyperholomorphic stable quotient sheaves. This implies that the family 𝔉 satisfies the conditions of Theorem 9.3. By Theorem 9.3, the reflexization 𝔉^{∗∗} is locally trivial. To prove that M = M_{mar}, we have to show that for all s ∈ M_{mar}, the corresponding coherent sheaf F_{s} is locally trivial. Therefore, to finish the proof of Theorem 9.11, it remains to prove the following algebro-geometric claim.

Claim 9.12: Let X be a compact complex manifold, dim_{ℂ} X > 2, and 𝔉 a torsion-free coherent sheaf over X ×Y which is flat over Y. Assume that the reflexization of 𝔉 is locally trivial, 𝔉 has isolated singularities, and for some point s ∈ Y, the restriction of 𝔉 to the complement (X ×Y)\(X ×{s}) is locally trivial. Then the reflexization (𝔉|_{X} × {s})^{∗∗} is locally trivial.

Remark 9.13: We say that a kernel of a map from a bundle to an Artinian sheaf is a bundle with holes. In slightly more intuitive terms, Claim 9.12 states that a flat deformation of a bundle with holes is again a bundle with holes, and cannot be smooth, assuming that dim_{ℂ} X > 2.

Proof of Claim 9.12: Claim 9.12 is well known. Here we give a sketch of a proof. Consider a coherent sheaf F_{s} = 𝔉|_{X×{s}}, and an exact sequence

0 ⟶ F_{s} ⟶ F_{s}^{∗∗} ⟶ k ⟶ 0,

where k is an Artinian sheaf. By definition, the sheaf F_{s}^{∗∗} is locally trivial. The flat deformations of F_{s} are infinitesimally classified by Ext^{1}(F_{s}, F_{s}). Replacing F_{s} by a quasi-isomorphic complex of sheaves F_{s}^{∗∗} ⟶ k, we obtain a spectral sequence converging to Ext^{•}(F_{s}, F_{s}). In the E_{2}-term of this sequence, we observe the group

Ext^{1}(F_{s}^{∗∗}, F_{s}^{∗∗}) ⊕ Ext^{1}(k, k) ⊕ Ext^{2}(k, F_{s}^{∗∗}) ⊕ Ext^{0}(F_{s}^{∗∗}, k).

which is responsible for Ext^{1}(F_{s}, F_{s}). The term Ext^{1}(F_{s}^{∗∗}, F_{s}^{∗∗}) is responsible for deformations of the bundle F_{s}^{∗∗}, the term Ext^{0}(F_{s}^{∗∗}, k) for the deformations of the map F_{s}^{∗∗} ⟶ k, and the term Ext^{1}(k, k) for the deformations of the Artinian sheaf k. Thus, the term Ext^{2}(k, F_{s}^{∗∗}) is responsible for the deformations of F_{s} which change the dimension of the cokernel of the embedding F_{s} ⟶ F_{s}^{∗∗}. We obtain that whenever Ext^{2}(k, F_{s}^{∗∗}) = 0, all deformations of F_{s} are singular. On the other hand, Ext^{2}(k, F_{s}^{∗∗}) = 0, because the i-th Ext from the skyscraper to a free sheaf on a manifold of dimension more than i vanishes (this is a basic result of Grothendieck’s duality, [H-Gro]).

10 New examples of hyperkähler manifolds

10.1 Twistor paths

This Subsection contains an exposition and further elaboration of the results of [V3-bis] concerning the twistor curves in the moduli space of complex structures on a complex manifold of hyperkähler type.

Let M be a compact manifold admitting a hyperkähler structure. In Definition 5.8, we defined the coarse, marked moduli space of complex structures on M, denoted by Comp. For the duration of this section, we fix a compact simple hyperkähler manifold M, and its moduli Comp.

Further on, we shall need the following fact.

Claim 10.1: Let M be a hyperkähler manifold, I an induced complex structure of general type, and B a holomorphic vector bundle over (M, I). Then B is stable if an only if B is simple.^{1}

Proof: By Lemma 2.26, for all ω ∈ P ic(M, I), we have deg_{I}(ω) = 0. Therefore, every subsheaf of B is destabilising.

Remark 10.2: In assumptions of Claim 10.1, all stable bundles are hyperholomorphic (Theorem 2.27). Therefore, Claim 10.1 implies that B is hyperholomorphic if it is simple.

1Simple sheaves are coherent sheaves which have no proper subsheaves

In Subsection 5.2, we have shown that every hyperkähler structure H corresponds to a holomorphic embedding

κ(H) : ℂP^{1} ⟶ Comp, L ⟶ (M, L).

Definition 10.3: A projective line C ⊂ Comp is called a twistor curve if C = κ(H) for some hyperkähler structure H on M.

The following theorem was proven in [V3-bis].

Theorem 10.4: ([V3-bis], Theorem 3.1) Let I_{1}, I_{2} ∈ Comp. Then there exist a sequence of intersecting twistor curves which connect I_{1} with I_{2}.

Definition 10.5: Let P_{0}, ..., P_{n} ⊂ Comp be a sequence of twistor curves, supplied with an intersection point x_{i+1} ∈ P_{i} ∩ P_{i+1} for each i. We say that γ = P_{0}, ..., P_{n}, x_{1}, ..., x_{n} is a twistor path. Let I, I′ ∈ Comp. We say that γ is a twistor path connecting I to I′ if I ∈ P_{0} and I′ ∈ P_{n}. The lines P_{i} are called the edges, and the points x_{i} the vertices of a twistor path.

Recall that in Definition 2.13, we defined induced complex structures which are generic with respect to a hyperkähler structure.

Given a twistor curve P, the corresponding hyperkähler structure H is unique (Theorem 5.11). We say that a point x ∈ P is of general type, or generic with respect to P if the corresponding complex structure is generic with respect to H.

Definition 10.6: Let I, J ∈ Comp and γ = P_{0}, ..., P_{n} be a twistor path from I to J, which corresponds to the hyperkähler structures H_{0}, ..., H_{n}. We say that γ is admissible if all vertices of γ are of general type with respect to the corresponding edges.

Remark 10.7: In [V3-bis], admissible twistor paths were defined slightly differently. In addition to the conditions above, we required that I, J are of general type with respect to H_{0}, H_{n}.

Theorem 10.4 proves that every two points I, I′ in Comp are connected with a twistor path. Clearly, each twistor path induces a diffeomorphism μ_{γ} : (M, I) ⟶ (M, I′). In [V3-bis], Subsection 5.2, we studied algebrogeometrical properties of this diffeomorphism.

Theorem 10.8: Let I, J ∈ Comp, and γ be an admissible twistor path from I to J. Then

(i) There exists a natural isomorphism of tensor cetegories

Φ_{γ} : Bun_{I}(H_{0}) ⟶ Bun_{J}(H_{n}),

where Bun_{I}(H_{0}), Bun_{J}(H_{n}) are the categories of polystable hyperholomorphic vector bundles on (M, I), (M, J), taken with respect to H_{0}, H_{n} respectively.

(ii) Let B ∈ Bun_{I}(H_{0}) be a stable hyperholomorphic bundle, and

M_{I,H_{0}}(B)

the moduli of stable deformations of B, where stability is taken with respect to the Kähler metric induced by H_{0}. Then Φ_{γ} maps stable bundles which are deformationally equivalent to B to the stable bundles which are deformationally equivalent to Φ_{γ}(B). Moreover, obtained this way bijection

Φ_{γ} : M_{I,H_{0}}(B) ⟶ M_{J,H_{n}}(Φ_{γ}(B))

induces a real analytic isomorphism of deformation spaces.

Proof: Theorem 10.8 (i) is a consequence of [V3-bis], Corollary 5.1. Here we give a sketch of its proof.

Let I be an induced complex structure of general type. By Claim 10.1, a bundle B over (M, I) is stable if and only if it is simple. Thus, the category Bun_{I}(H) is independent from the choice of H (Claim 10.1).

In Theorem 3.27, we constructed the equivalence of categories Φ_{I,J}, which gives the functor Φ_{γ} for twistor path which consists of a single twistor curve. This proves Theorem 10.8 (i) for n = 1. A composition of isomorphisms Φ_{I,J} ∘ Φ_{J,J′} is well defined, because the category Bun_{I}(H) is independent from the choice of H. Taking successive compositions of the maps Φ_{I,J}, we obtain an isomorphism Φ_{γ}. This proves Theorem 10.8 (i).

The variety M_{I,H}(B) is singular hyperkähler ([V1]), and the variety M_{J,H}(B) is the same singular hyperkähler variety, taken with another induced complex structure. By definition of singular hyperkähler varieties, this implies that M_{I,H}(B), M_{J,H}(B) are real analytic equivalent, with equivalence provided by Φ_{I,J}. This proves Theorem 10.8 (ii).

For I ∈ Comp, denote by P ic(M, I) the group H^{1,1}(M, I) ∩ H^{2}(M, ℤ), and by P ic(I, ℚ) the space H^{1,1}(M, I) ∩ H^{2}(M, ℚ) ⊂ H^{2}(M). Let Q ⊂ H^{2}(M, ℚ) be a subspace of H^{2}(M, ℚ), and

Comp_{Q} := {I ∈ Comp | P ic(I, ℚ) = Q}.

Theorem 10.9: Let H, H′ be hyperkähler structures, and I, I′ be complex structures of general type to and induced by H, H′. Assume that P ic(I, ℚ) = P ic(I′, ℚ) = Q, and I, I′ lie in the same connected component of Comp_{Q}. Then I, I′ can be connected by an admissible path.

Proof: This is [V3-bis], Theorem 5.2.

For a general Q, we have no control over the number of connected components of Comp_{Q} (unless global Torelli theorem is proven), and therefore we cannot directly apply Theorem 10.9 to obtain results from algebraic geometry.^{2} However, when Q = 0, Comp_{Q} is clearly connected and dense in Comp. This is used to prove the following corollary.

Corollary 10.10: Let I, I′ ∈ Comp_{0}. Then I can be connected to I′ by an admissible twistor path.

Proof This is [V3-bis], Corollary 5.2.

Definition 10.11: Let I ∈ Comp be a complex structure, ω be a Kähler form on (M, I), and H the corresponding hyperkähler metric, which exists by Calabi-Yau theorem. Then ω is called a generic polarization if any of the following conditions hold

(i) For all a ∈ P ic(M, I), the degree deg_{ω}(a) ≠ 0, unless a = 0.

(ii) For all SU(2)-invariant integer classes a ∈ H^{2}(M, ℤ), we have a = 0.

The conditions (i) and (ii) are equivalent by Lemma 2.26.

Claim 10.12: Let I ∈ Comp be a complex structure, ω be a Kähler form on (M, I), and H the corresponding hyperkähler structure, which exists by Calabi-Yau theorem. Then ω is generic if and only if for all integer classes a ∈ H^{1,1}(M, I), the class a is not orthogonal to ω with respect to the Bogomolov-Beauville pairing.

2Exception is a K3 surface, where Torelli holds. For K3, Comp_{Q} is connected for all Q ⊂ H^{2}(M, ℚ).

Proof: Clearly, the map deg_{ω} : H^{2}(M) ⟶ ℝ is equal (up to a scalar multiplier) to the orthogonal projection onto the line ℝ · ω. Then, Claim 10.12 is equivalent to Definition 10.11, (i).

From Claim 10.12 it is clear that the set of generic polarizations is a complement to a countable union of hyperplanes. Thus, generic polarizations are dense in the Kähler cone of (M, I), for all I.

Claim 10.13: Let I, J ∈ Comp, and a, b be generic polarizations on (M, I). Consider the corresponding hyperkähler structures H_{0} and H_{n} inducing I and J. Then there exists an admissible twistor path starting from I, H_{0} and ending with H_{n}, J.

Proof: Consider the twistor curves P_{0}, P_{n} corresponding to H_{0}, H_{n}. Since a, b are generic, the curves P_{0}, P_{n} intersect with Comp_{0}. Applying Corollary 10.10, we connect the curves P_{0} and P_{n} by an admissible path.

Putting together Claim 10.13 and Theorem 10.8, we obtain the following result.

Theorem 10.14: Let I, J ∈ Comp be complex structures, and a, b be generic polarizations on (M, I), (M, J). Then

(i) There exist an isomorphism of tensor cetegories

Φ_{γ} : Bun_{I}(a) ⟶ Bun_{J}(a),

where Bun_{I}(a), Bun_{J}(b) are the categories of polystable hyperholomorphic vector bundles on (M, I), (M, J), taken with respect to the hyperkähler structures defined by the Kähler classes a, b as in Theorem 2.4.

(ii) Let B ∈ Bun_{I}(a) be a stable hyperholomorphic bundle, and

M_{I,a}(B)

the moduli of stable deformations of B, where stability is taken with respect to the polarization a. Then Φ_{γ} maps stable bundles which are deformationally equivalent to B to the stable bundles which are deformationally equivalent to Φ_{γ}(B). Moreover, obtained this way bijection

Φ_{γ} : M_{I,a}(B) ⟶ M_{J,b}(Φ_{γ}(B))

induces a real analytic isomorphism of deformation spaces.

Lemma 10.15: In assumptions of Theorem 10.8, let B be a holomorphic tangent bundle of (M, I). Then Φ_{γ}(B) is a holomorphic tangent bundle of (M, J).

Proof: Clear.

Corollary 10.16: Let I, J ∈ Comp be complex structures, and a, b generic polarizations on (M, I), (M, J). Assume that the moduli of stable deformations M_{I,a}(T (M, I)) of the holomorphic tangent bundle T^{1,0}(M, I) is compact. Then the space M_{J,b}(T (M, J)) is also compact.

Proof: Let γ be the twistor path of Claim 10.13. By Lemma 10.15, Φ_{γ}(T (M, I)) = T (M, J). Applying Theorem 10.8, we obtain a real analytic equivalence from M_{I,a}(T (M, I)) to M_{J,b}(T (M, J)).

10.2 New examples of hyperkähler manifolds

Theorem 10.17: Let M be a compact hyperkähler manifold without nontrivial trianalytic subvarieties, dim_{ℍ} M ⩾ 2, and I an induced complex structure. Consider a hyperholomorphic bundle F on M (Definition 3.28). Let F_{I} be the corresponding holomorphic bundle over (M, I). Assume that I is a C-restricted complex structure, C = deg_{I} c_{2}(F). Assume, moreover, that all semistable bundle deformations of F_{I} are stable.^{3} Denote by M_{F}^{I} the moduli of stable bundle deformations of F_{I} over (M, I). Then

(i) the normalization M̃_{F}^{I} is a compact and smooth complex manifold equipped with a natural hyperkähler structure.

(ii) Moreover, for all induced complex structures J on M, the the variety M_{F}^{J} is compact, and has a smooth normalization M̃_{F}^{J}, which is also equipped with a natural hyperkähler structure.

(iii) Finally, the hyperkähler manifolds M̃_{F}^{J}, M̃_{F}^{I} are naturally isomorphic.

Proof: The variety M_{F}^{I} is compact by Theorem 9.11. In [V1], it was proven that the space M_{F}^{I} of stable deformations of F is a singular hyperkähler variety (see also [KV] for an explicit construction of the twistor space of M_{F}^{I}). Then Theorem 10.17 is a consequence of the Desingularization Theorem for singular hyperkähler varietiess (Theorem 2.16).

3This may happen, for instance, when rk F = dim_{ℂ} M = n, and the number c_{n}(F) is prime.

The assumptions of Theorem 10.17 are quite restrictive. Using the technique of twistor paths, developed in Subsection 10.1, it is possible to prove a more accessible form of Theorem 10.17.

Let M be a hyperkähler manifold, and I, J induced complex structures. Given an admissible twistor path from I to J, we obtain an equivalence Φ_{γ} between the category of hyperholomorphic bundles on (M, I) and (M, J).

Theorem 10.18: Let M be a compact simple hyperkähler manifold, dim_{ℍ} M > 1, and I a complex structure on M. Consider a generic polarization a on (M, I). Let H be the corresponding hyperkähler structure, and F a hyperholomorphic bundle on (M, I). Fix a hyperkähler structure H′ on M admitting C-restricted complex structures, such that M has no trianalytic subvarieties with respect to H′. Assume that for some C-restricted complex structure J induced by H′, C = deg_{I} c_{2}(F), all admissible twistor paths γ from I to J, and all semistable bundles F′ which are deformationally equivalent to Φ_{γ}(F), the bundle F′ is stable. Then the space of stable deformations of F is compact.

Remark 10.19: The space of stable deformations of F is singular hyperkähler ([V1]) and its normalization is smooth and hyperkähler (Theorem 2.16).

Proof of Theorem 10.18: Clearly, F′ satisfies assumptions of Theorem 10.17, and the moduli space of its stable deformations is compact. Since Φ_{γ} induces a homeomorphism of moduli spaces (Theorem 10.8), the space of stable deformations of F is also compact.

Applying Theorem 10.18 to the holomorphic tangent bundle T (M, I), we obtain the following corollary.

Theorem 10.20: Let M be a compact simple hyperkähler manifold, dim_{ℍ}(M) > 1. Assume that for a generic hyperkähler structure H on M, this manifold admits no trianalytic subvarieties.^{4} Assume, moreover, that for some C-restricted induced complex structure I, all semistable bundle deformations of T (M, I) are stable, for C > deg_{I} c_{2}(M). Then, for all complex structures J on M and all generic polarizations ω on (M, J), the deformation space M_{J,ω}(T (M, J)) is compact.

4This assumption holds for a Hilbert scheme of points on a K3 surface.

Proof: Follows from Theorem 10.18 and Corollary 10.16.

10.3 How to check that we obtained new examples of hyperkähler manifolds?

A. Beauville [Bea] described two families of compact hyperkähler manifolds, one obtained as the Hilbert scheme of points on a K3-surface, another obtained as the Hilbert scheme of a 2-dimensional torus factorized by the free torus action.

Conjecture 10.21: There exist compact simple hyperkähler manifolds which are not isomorphic to deformations of these two fundamental examples.

Here we explain our strategy of a proof of Conjecture 10.21 using results on compactness of the moduli space of hyperholomorphic bundles.

The results of this subsection are still in writing, so all statements below this line should be considered as conjectures. We give an idea of a proof for each result and label it as “proof”, but these “proofs” are merely sketches.

First of all, it is possible to prove the following theorem.

Theorem 10.22: Let M be a complex K3 surface without automorphisms. Assume that M is Mumford-Tate generic with respect to some hyperkähler structure. Consider the Hilbert scheme M^{[n]} of points on M, n > 1. Pick a hyperkähler structure H on M^{[n]} which is compatible with the complex structure. Let B be a hyperholomorphic bundle on (M^{[n]}, H), rk B = 2. Then B is a trivial bundle.

Proof: The proof of Theorem 10.22 is based on the same ideas as the proof of Theorem 2.17.

For a compact complex manifold X of hyperkähler type, denote its coarse, marked moduli space (Definition 5.8) by Comp(X).

Corollary 10.23: Let M be a K3 surface, I ∈ Comp(X) an arbitrary complex structure on X = M^{[n]}, n > 1, and a a generic polarization on (X, J). Consider the hyperkähler structure H which corresponds to (I, a) as in Theorem 2.4. Let B, rk B = 2 be a hyperholomorphic bundle over (X, H). Then B is trivial.

Proof: Follows from Theorem 10.22 and Theorem 10.14.

Corollary 10.24: Let M be a K3 surface, I ∈ Comp(X) an arbitrary complex structure on X = M^{[n]}, n > 1, and a a generic polarization on (X, I). Consider the hyperkähler structure H which corresponds to (J, a) (Theorem 2.4). Let B, rk B ⩽ 6 be a stable hyperholomorphic bundle on (X, H). Assume that the Chern class c_{rk B}(B) is non-zero. Assume, moreover, that I is C-restricted, C = deg_{I}(c_{2}(B)). Let B′ be a semistable deformation of B over (X, I). Then B′ is stable.

Proof: Consider the Jordan–Hölder serie for B′. Let Q_{1} ⊕ Q_{2} ⊕ ... be the associated graded sheaf. By Theorem 5.14, the stable bundles Q_{i} are hyperholomorphic. Since c_{rk B}(B) ≠ 0, we have c_{rk Q_{i}}(Q_{i}) ≠ 0. Therefore, the bundles Q_{i} are non-trivial. By Corollary 10.23, rk Q_{i} > 2. Since all the Chern classes of the bundles Q_{i} are SU(2)-invariant, the odd Chern classes of Q_{i} vanish (Lemma 2.6). Therefore, rk Q_{i} ⩾ 4 for all i. Since rk B ⩽ 6, we have i = 1 and the bundle B′ is stable.

Let M be a K3 surface, X = M^{[i]}, i = 2, 3 be its second or third Hilbert scheme of points, I ∈ Comp(X) arbitrary complex structure on X, and a a generic polarization on (X, I). Consider the hyperkähler structure H which corresponds to J and a by Calabi-Yau theorem (Theorem 2.4). Denote by T X the tangent bundle of X, considered as a hyperholomorphic bundle. Let Def(T X) denote the hyperkähler desingularization of the moduli of stable deformations of T X. By Theorem 10.14, the real analytic subvariety underlying Def(T X) is independent from the choice of I. Therefore, its dimension is also independent from the choice of I. The dimension of the deformation space Def(T X) can be estimated by a direct computation, for X a Hilbert scheme. We obtain that dim Def(T X) > 40.

Claim 10.25: In these assumptions, the space Def(T X) is a compact hyperkähler manifold.

Proof: By Corollary 10.24, all semistable bundle deformations of T X are stable. Then Claim 10.25 is implied by Theorem 10.20.

Clearly, deforming the complex structure on X, we obtain a deformation of complex structures on Def(T X). This gives a map

Comp(X) ⟶ Comp(Def(T X)).  (10.1)

It is easy to check that the map (10.1) is complex analytic, and maps twistor curves to twistor curves.

Claim 10.26: Let X, Y be hyperkähler manifolds, and

φ : Comp(X) ⟶ Comp(Y)

be a holomorphic map of corresponding moduli spaces which maps twistor curves to twistor curves. Then φ is locally an embedding.

Proof: An elementary argument using the period maps, in the spirit of Subsection 5.2.

The following result, along with Theorem 10.22, is the major stumbling block on the way to proving Conjecture 10.21. The other results of this Subsection are elementary or routinely proven, but the complete proof of Theorem 10.22 and Theorem 10.27 seems to be difficult.

Theorem 10.27: Let X be a simple hyperkähler manifold without proper trianalytic subvarieties, B a hyperholomorphic bundle over X, and I an induced complex structure. Denote the corresponding holomorphic bundle over (X, I) by B_{I}. Assume that the space M of stable bundle deformations of B is compact. Let Def(B) be the hyperkähler desingularization of M. Then Def(M) is a simple hyperkähler manifold.

Proof: Given a decomposition Def(M) = M_{1} ×M_{2}, we obtain a parallel 2-form on Ω_{1} on Def(B), which is a pullback of the holomorphic symplectic form on M_{1}. Consider the space A of connections on B, which is an infinitely-dimensional complex analytic Banach manifold. Then Ω_{1} corresponds to a holomorphic 2-form Ω̃_{1} on A. Since Ω_{1} is parallel with respect to the natural connection on Def(B), the form Ω̃_{1} is also a parallel 2-form on the tangent space to A, which is identified with Ω^{1}(X, End(B)). It is possible to prove that this 2-form is obtained as

A, B ⟶ ∫_{Y} Θ (A|_{Y} , B|_{Y} ) Vol(Y)

where

Θ : Ω^{1}(Y, End(B)) × Ω^{1}(Y, End(B)) ⟶ O_{Y}

is a certain holomorphic pairing on the bundle Ω^{1}(Y, End(B)), and Y is a trianalytic subvariety of X. Since X has no trianalytic subvarieties, Ω̃_{1} is obtained from a O_{X}-linear pairing

Ω^{1}(X, End(B)) × Ω^{1}(X, End(B)) ⟶ O_{X}.

Using stability of B, it is possible to show that such a pairing is unique, and thus, Ω_{1} coincides with the holomorphic symplectic form on Def(B). Therefore, Def(B) = M_{1}, and this manifold is simple.

Return to the deformations of tangent bundles on X = M^{[i]}, i = 2, 3. Recall that the second Betti number of a Hilbert scheme of points on a K3 surface is equal to 23, and that of the generalized Kummer variety is 7 ([Bea]). Consider the map (10.1). By Theorem 10.27, the manifold Def(T X) is simple. By Bogomolov’s theorem (Theorem 5.9), we have

dim Comp(Def(T X)) = dim H^{2}(Def(T X)) − 2.

Therefore, either dim H^{2}(Def(T X)) > dim H^{2}(X) = 23, or the map (10.1) is etale. In the first case, the second Betti number of Def(T X) is bigger than that of known simple hyperkähler manifolds, and thus, Def(T X) is a new example of a simple hyperkähler manifold; this proves Conjecture 10.21. Therefore, to prove Conjecture 10.21, we may assume that dim H^{2}(Def(T X)) = 23, the map (10.1) is etale, and Def(T X) is a deformation of a Hilbert scheme of points on a K3 surface.

Consider the universal bundle B̃ over X × Def(T X). Restricting B̃ to {x} × Def(T X), we obtain a bundle B on Def(T X). Let Def(B) be the hyperkähler desingularization of the moduli space of stable deformations of B. Clearly, the manifold Def(B) is independent from the choice of x ∈ X. Taking the generic hyperkähler structure on X, we may assume that the hyperkähler structure H on Def(T X) is also generic. Thus, (Def(T X), H) admits C-restricted complex structures and has no trianalytic subvarieties. In this situation, Corollary 10.24 implies that the hyperkähler manifold Def(B) is compact. Applying Claim 10.26 again, we obtain a sequence of maps

Comp(X) ⟶ Comp(Def(T X)) ⟶ Comp(Def(B))

which are locally closed embeddings. By the same argument as above, we may assume that the composition Comp(X) ⟶ Comp(Def(B)) is etale, and the manifold Def(B) is a deformation of a Hilbert scheme of points on K3. Using Mukai’s version of Fourier transform ([O], [BBR]), we obtain an embedding of the corresponding derived categories of coherent sheaves,

D(X) ⟶ D(Def(T X)) ⟶ D(Def(B)).

Using this approach, it is easy to prove that

dim X ⩽ dim Def(T X) ⩽ dim Def(B).

Let x ∈ X be an arbitrary point. Consider the complex C_{x} ∈ D(Def(B)) of coherent sheaves on Def(B), obtained as a composition of the Fourier-Mukai transform maps. It is easy to check that the lowest non-trivial cohomology sheaf of C_{x} is a skyscraper sheaf in a point F (x) ∈ Def(B). This gives an embedding

F : X ⟶ Def(B).

The map F is complex analytic for all induced complex structure. We obtained the following result.

Lemma 10.28: In the above assumptions, the embedding

F : X ⟶ Def(B)

is compatible with the hyperkähler structure.

By Lemma 10.28, the manifold Def(B) has a trianalytic subvariety F (X), of dimension 0 < dim F (X) < 40 < dim Def(B). On the other hand, for a hyperkähler structure on X generic, the corresponding hyperkähler structure on Def(B) is also generic, so this manifold has no trianalytic subvarieties. We obtained a contradiction. Therefore, either Def(T X) or Def(B) is a new example of a simple hyperkähler manifold. This proves Conjecture 10.21.

Acknowledegments: I am grateful to V. Batyrev, A. Beilinson, P. Deligne, D. Gaitsgory, D. Kaledin, D. Kazhdan, M. Koncevich and T. Pantev for valuable discussions. My gratitude to D. Kaledin, who explained to me the results of [Sw]. This paper uses many ideas of our joint work on direct and inverse twistor transform ([KV]).

[BS] Bando, S., Siu, Y.-T, Stable sheaves and Einstein-Hermitian metrics, In: Geometry and Analysis on Complex Manifolds, Festschrift for Professor S. Kobayashi’s 60th Birthday, ed. T. Mabuchi, J. Noguchi, T. Ochiai, World Scientific, 1994, pp. 39-50.

[BBR] Bartocci, C., Bruzzo, U., Hernandez Ruiperez, D., A Fourier-Mukai transform for stable bundles on K3 surfaces, J. Reine Angew. Math. 486, pp. 1-16 (1997).

[Bea] Beauville, A. Varietes Kähleriennes dont la première classe de Chern est nulle. J. Diff. Geom. 18, pp. 755-782 (1983).

[Bes] Besse, A., Einstein Manifolds, Springer-Verlag, New York (1987)

[Bo] Bogomolov, F.A., Hamiltonian Kähler manifolds, (English. Russian original) Sov. Math., Dokl. 19, 1462-1465 (1978); translation from Dokl. Akad. Nauk SSSR 243, 1101-1104 (1978).

[C] Calabi, E., Metriques kähleriennes et fibrès holomorphes, Ann. Ecol. Norm. Sup. 12 (1979), 269-294.

[D] Deligne, P., Categories tannakiennes, The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II, Prog. Math. 87, 111-195 (1990).

[Do] Douady, A., Le probleme des modules pour les sous-espaces analitiques compactes d’un espace analitique donné, Ann. Inst. Fourier, 16 (1966), vol. 1, pp. 1-95.

[O’G] O’Grady, K. G., Desingularized moduli spaces of sheaves on a K3, alggeom/9708009, 41 pages, preliminary version.

[Gi] Gieseker, D. On the moduli of vector bundles on an algebraic surface, Ann. Math., II. Ser. 106, 45-60 (1977).

[GH] Griffiths, Ph., Harris, J., Principles of Algebraic Geometry, Wiley-Intersience, New York, 1978.

[GMT] Guaraldo, F., Macri, P., Tancredi, A., Topics on real analytic spaces, Advanced lectures in mathematics, Braunschweig: F. Vieweg, 1986.

[F] Fujiki, A., On primitively symplectic compact Kähler V-manifolds of dimension four, in Classification of algebraic and analytic manifolds, (Katata, 1982), 71–250, Progr. Math., 39, Birkhäuser Boston, 1983.

[H-Gro] Hartshorne, R., Local cohomology; a seminar given by A. Grothendieck, Harvard University, Fall, 1961. // Lecture notes in mathematics, vol. 41, Springer-Verlag, 1967.

[HKLR] N.J. Hitchin, A. Karlhede, U. Lindström, M. Roček, Hyperkähler metrics and supersymmetry, Comm. Math. Phys (1987).

[H] Huybrechts, D., Birational symplectic manifolds and their deformations, alggeom/9601015.

[K] Kaledin, D., Integrability of the twistor space for a hypercomplex manifold, alg-geom/9612016, 9 pages, Latex2e.

[KV] Kaledin, D., Verbitsky, M., Non-Hermitian Yang-Mills connections, // alggeom 9606019 (1996), 48 pages, LaTeX 2e.

[Ko] Kobayashi S., Differential geometry of complex vector bundles, // Princeton University Press, 1987.

[Ma1] Maruyama, Masaki On boundedness of families of torsion free sheaves, J. Math. Kyoto Univ. 21, 673-701 (1981)

[Ma2] Maruyama, Masaki, Construction of moduli spaces of stable sheaves via Simpson’s idea, Lect. Notes Pure Appl. Math. 179, 147-187 (1996).

[LY] Li, Jun, and Yau, S.-T., Hermitian Yang-Mills connections on non-Kahler manifolds, in “Mathematical aspects of string theory” (S.T. Yau ed.), World Scientific Publ., London, 1987, pp. 560-573.

[N1] Nitta, Takashi, Vector bundles over quaternionic Kähler manifolds, Tohoku Math. J., II. Ser. 40, No.3, 425-440 (1988).

[N2] Nitta, Takashi, Moduli spaces of Yang-Mills connections over quaternionic Kähler manifolds, Osaka J. Math. 26, No.2, 287-298 (1989).

[OSS] Christian Okonek, Michael Schneider, Heinz Spindler, Vector bundles on complex projective spaces. Progress in mathematics, vol. 3, Birkhauser, 1980.

[O] Orlov, D. Equivalences of derived categories and K3 surfaces, alg-geom electronic preprint 9606006 (1996).

[Sal] Salamon, S., Quaternionic Kähler manifolds, Inv. Math. 67 (1982), 143-171.

[Sw] Swann, A., HyperKähler and quaternionic Kähler geometry, Math. Ann. 289, No. 3, 421-450 (1991).

[UY] Uhlenbeck K., Yau S. T., On the existence of Hermitian Yang-Mills connections in stable vector bundles, // Comm. on Pure and Appl. Math., 39, p. S257-S293 (1986).

[V0] M. Verbitsky, On the action of the Lie algebra 𝔰𝔬(5) on the cohomology of a hyperkähler manifold, Func. Anal. and Appl. 24 (1990), 70-71.

[V1] Verbitsky M., Hyperholomorphic bundles over a hyperkähler manifold, alggeom electronic preprint 9307008 (1993), 43 pages, LaTeX, also published in: Journ. of Alg. Geom., 5 no. 4 (1996) pp. 633-669.

[V2] Verbitsky M., Hyperkähler and holomorphic symplectic geometry I, Journ. of Alg. Geom., 5 no. 3 (1996) pp. 401-415. A different version appeared as alg-geom/9307009

[V2(II)] Verbitsky M., Hyperkähler embeddings and holomorphic symplectic geometry II, alg-geom electronic preprint 9403006 (1994), 14 pages, LaTeX, also published in: GAFA 5 no. 1 (1995), 92-104.

[V3] Verbitsky, M., Cohomology of compact hyperkähler manifolds. alg-geom electronic preprint 9501001, 89 pages, LaTeX.

[V3-bis] Verbitsky, M., Cohomology of compact hyperkähler manifolds and its applications, alg-geom electronic preprint 9511009, 12 pages, LaTeX, also published in: GAFA vol. 6 (4) pp. 601-612 (1996).

[V4] Verbitsky M., Deformations of trianalytic subvarieties of hyperkähler manifolds, alg-geom electronic preprint 9610010 (1996), 51 pages, LaTeX2e.

[V5] Verbitsky M., Trianalytic subvarieties of the Hilbert scheme of points on a K3 surface, alg-geom/9705004.

[V-a] Verbitsky M., Algebraic structures on hyperkähler manifolds, alg-geom electronic preprint 9609011, also published in: Math. Res. Lett., 3, 763-767 (1996).

[V-d] Verbitsky M., Desingularization of singular hyperkähler varieties I, electronic preprint 9611015 (1996), 13 pages, LaTeX 2e, also published in: Math. Res. Lett. 4 (1997), no. 2-3, pp. 259–271.

[V-d2] Verbitsky M., Desingularization of singular hyperkähler varieties II, alggeom/9612013 (1996), 15 pages, LaTeX 2e.

[V-d3] Verbitsky M., Hypercomplex Varieties, alg-geom/9703016 (1997), 40 pages, LaTeX (to appear in Comm. in An. and Geom.)

[Y] Yau, S. T., On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I. // Comm. on Pure and Appl. Math. 31, 339-411 (1978).

E-mail: verbit@@thelema.dnttm.rssi.ru, verbit@@ihes.fr

128 BIBLIOGRAPHY

Part II. Hyperkähler structures on total spaces of holomorphic cotangent bundles

D. Kaledin

Introduction

A hyperkähler manifold is by definition a Riemannian manifold M equipped with two anti-commuting almost complex structures I, J parallel with respect to the Levi-Civita connection. Hyperkähler manifolds were introduced by E. Calabi in [C]. Since then they have become the topic of much research. We refer the reader to [B] and to [HKLR] for excellent overviews of the subject.

Let M be a hyperkähler manifold. The almost complex structures I and J generate an action of the quaternion algebra ℍ in the tangent bundle Θ(M) to the manifold M. This action is parallel with respect to the Levi-Civita connection. Every quaternion h ∈ ℍ with h^{2} = −1, in particular, the product K = IJ ∈ ℍ, defines by means of the ℍ-action an almost complex structure M_{h} on M. This almost complex structure is also parallel, hence integrable and Kähler. Thus every hyperkähler manifold M is canonically Kähler, and in many different ways. For the sake of convenience, we will consider M as a Kähler manifold by means of the complex structure M_{I}, unless indicated otherwise.

One of the basic facts about hyperkähler manifolds is that the Kähler manifold M_{I} underlying a hyperkähler manifold M is canonically holomorphically symplectic. To see this, let ω_{J}, ω_{K} be the Kähler forms for the complex structures M_{J}, M_{K} on the manifold M, and consider the 2-form Ω = ω_{J} + \sqrt{−1}ω_{K} on M. It is easy to check that the form Ω is of Hodge type (2, 0) for the complex structure M_{I} on M. Since it is obviously nondegenerate and closed, it is holomorphic, and the Kähler manifold M_{I} equipped with the form Ω is a holomorphically symplectic manifold.

It is natural to ask whether every holomorphically symplectic manifold ⟨M, Ω⟩ underlies a hyperkähler structure on M, and if so, then how many such hyperkähler structures are there. Note that if such a hyperkähler structure exists, it is completely defined by the Kähler metric h on M. Indeed, the Kähler forms ω_{J} and ω_{K} are by definition the real and imaginary parts of the form Ω, and the forms ω_{J} and ω_{K} together with the metric define the complex structures J and K on M and, consequently, the whole ℍ-action in the tangent bundle Θ(M). For the sake of simplicity, we will call a metric h on a holomorphically symplectic manifold ⟨M, Ω⟩ hyperkähler if the Riemannian manifold ⟨M, h⟩ with the quaternionic action associated to the pair ⟨Ω, h⟩ is a hyperkähler manifold.

It is known (see, e.g., [Beau]) that if the holomorphically symplectic manifold M is compact, for example, if M is a K3-surface, then every Kähler class in H^{1,1}(M) contains a unique hyperkähler metric. This is, in fact, a consequence of the famous Calabi-Yau Theorem, which provides the canonical Ricci-flat metric on M with the given cohomology class. This Ricci-flat metric turns out to be hyperkähler. Thus in the compact case holomorphically symplectic and hyperkähler manifolds are essentially the same.

The situation is completely different in the general case. For example, all holomorphically symplectic structures on the formal neighborhood of the origin 0 ∈ ℂ^{2n} in the 2n-dimensional complex vector space ℂ^{2n} are isomorphic by the appropriate version of the Darboux Theorem. On the other hand, hyperkähler structures on this formal neighborhood form an infinitedimensional family (see, e.g., [HKLR], where there is a construction of a smaller, but still infinite-dimensional family of hyperkähler metrics defined on the whole ℂ^{2n}). Thus, to obtain meaningful results, it seems necessary to restrict our attention to holomorphically symplectic manifolds belonging to some special class.

The simplest class of non-compact holomorphically symplectic manifolds is formed by total spaces T^{∗}M to the cotangent bundle to complex manifolds M. In fact, the first examples of hyperkähler manifolds given by Calabi in [C] were of this type, with M being a Kähler manifold of constant holomorphic sectional curvature (for example, a complex projective space). It has been conjectured for some time that every total space T^{∗}M of the cotangent bundle to a Kähler manifold admits a hyperkähler structure. The goal of this paper is to prove that this is indeed the case, if one agrees to consider only an open neighborhood U ⊂ T^{∗}M of the zero section M ⊂ T^{∗}M. Our main result is the following.

Theorem 1 Let M be a complex manifold equipped with a Kähler metric. The metric on M extends to a hyperkähler metric h defined in the formal neighborhood of the zero section M ⊂ T^{∗}M in the total space T^{∗}M to the holomorphic cotangent bundle to M. The extended metric h is invariant under the action of the group U(1) on T^{∗}M given by dilatations along the fibers of the canonical projection ρ : T^{∗}M → M. Moreover, every other U(1)-invariant hyperkähler metric on the holomorphically symplectic manifold T^{∗}M becomes equal to h after a holomorphic symplectic U(1)- equivariant automorphism of T^{∗}M over M. Finally, if the Kähler metric on M is real-analytic, then the formal hyperkähler metric h converges to a real-analytic metric in an open neighborhood U ⊂ T^{∗}M of the zero section M ⊂ T^{∗}M.

Many of the examples of hyperkähler metrics obtained by Theorem 1 are already known. (See, e.g., [Kr1], [Kr2], [N], [H2], [BG], [DS].) In these examples M is usually a generalized flag manifold or a homogeneous space of some kind. On the other hand, very little is known for manifolds of general type. In particular, it seems that even for curves of genus g ≥ 2 Theorem 1 is new.

We would like to stress the importance of the U(1)-invariance condition on the metric in the formulation of Theorem 1. This condition for a total space T^{∗}M of a cotangent bundle is equivalent to a more general compatibility condition between a U(1)-action and a hyperkähler structure on a smooth manifold introduced by N.J. Hitchin (see, e.g., [H2]). Thus Theorem 1 can be also regarded as answering a question of Hitchin’s in [H2], namely, whether every Kähler manifold can be embedded as the submanifold of U(1)-fixed points in a U(1)-equivariant hyperkähler manifold. On the other hand, it is this U(1)-invariance that guarantees the uniqueness of the metric h claimed in Theorem 1.

We also prove a version of Theorem 1 “without the metrics”. The Kähler metric on M in this theorem is replaced with a holomorphic connection ∇ on the cotangent bundle to M without torsion and (2, 0)-curvature. We call such connections Kählerian. The total space of the cotangent bundle T^{∗}M is replaced with the total space \overline{T} M of the complex-conjugate to the tangent bundle to M. (Note that a priori there is no complex structure on \overline{T} M, but the U(1)-action by dilatations on this space is well-defined.) The analog of the notion of a hyperkähler manifold “without the metric” is the notion if a hypercomplex manifold (see, e.g., [B]). We define a version of Hitchin’s condition on the U(1)-action for hypercomplex manifolds and prove the following.

Theorem 2 Let M be a complex manifold, and let \overline{T} M be the total space of the complex-conjugate to the tangent bundle to M equipped with an action of the group U(1) by dilatation along the fibers of the projection \overline{T} M → M. There exists a natural bijection between the set of all Kählerian connections on the cotangent bundle to M and the set of all isomorphism classes of U(1)- equivariant hypercomplex structures on the formal neighborhood of the zero section M ⊂ \overline{T} M in \overline{T} M such that the projection ρ : \overline{T} M → M is holomorphic. If the Kählerian connection on M is real-analytic, the corresponding hypercomplex structure is defined in an open neighborhood U ⊂ \overline{T} M of the zero section.

Our main technical tool in this paper is the relation between U(1)-equiva-riant hyperkähler manifolds and the theory of ℝ-Hodge structures discovered by P. Deligne and C. Simpson (see [D2], [S1]). To emphasize this relation, we use the name Hodge manifolds for the hypercomplex manifolds equipped with a compatible U(1)-action.

It must be noted that many examples of hyperkähler manifolds equipped with a compatible U(1)-action are already known. Such are, for example, many of the manifolds constructred by the so-called hyperkähler reduction from flat hyperkähler spaces (see [H2] and [HKLR]). An important class of such manifolds is formed by the so-called quiver varieties, studied by H. Nakajima ([N]). On the other hand, the moduli spaces M of flat connections on a complex manifold M, studied by Hitchin ([H1]) when M is a curve and by Simpson ([S2]) in the general case, also belong to the class of Hodge manifolds, as Simpson has shown in [S1]. Some parts of our theory, especially the uniqueness statement of Theorem 1, can be applied to these known examples.

We now give a brief outline of the paper. Sections 1-3 are preliminary and included mostly to fix notation and terminology. Most of the facts in these sections are well-known.

• In Section 1 we have collected the necessary facts from linear algebra about quaternionic vector spaces and ℝ-Hodge structures. Everything is standard, with an exception, perhaps, of the notion of weakly Hodge map, which we introduce in Subsection 1.4.

• We begin Section 2 with introducing a technical notion of a Hodge bundle on a smooth manifold equipped with a U(1)-action. This notion will be heavily used throughout the paper. Then we switch to our main subject, namely, various differential-geometric objects related to a quaternion action in the tangent bundle. The rest of Section 2 deals with almost quaternionic manifolds and the compatibility conditions between an almost quaternionic structure and a U(1)-action on a smooth manifold M.

• In Section 3 we describe various integrability conditions on an almost quaternionic structure. In particular, we recall the definition of a hypercomplex manifold and introduce U(1)-equivariant hypercomplex manifolds under the name of Hodge manifolds. We then rewrite the definition of a Hodge manifold in the more convenient language of Hodge bundles, to be used throughout the rest of the paper. Finally, in Subsection 3.3 we discuss metrics on hypercomplex and Hodge manifolds. We recall the definition of a hyperkähler manifold and define the notion of a polarization of a Hodge manifold. A polarized Hodge manifold is the same as a hyperkähler manifold equipped with a U(1)-action compatible with the hyperkähler structure of the sense of Hitchin, [H2].

• The main part of the paper begins in Section 4. We start with arbitrary Hodge manifolds and prove that in a neighborhood of the subset M^{U(1)} of “regular” U(1)-fixed points

• termsregular fixed point every such manifold M is canonically isomorphic to an open neighborhood of the zero section in a total space \overline{T} M^{U(1)} of the tangent bundle to the fixed point set. A fixed point m ∈ M is “regular” if the group U(1) acts on the tangent space T_{m}M with weights 0 and 1. We call this canonical isomorphism the linearization of the regular Hodge manifold.

The linearization construction can be considered as a hyperkähler analog of the Darboux-Weinstein Theorem in the symplectic geometry. Apart from the cotangent bundles, it can be applied to the Hitchin-Simpson moduli space M of flat connections on a Kähler manifold X. The regular points in this case correspond to stable flat connections such that the underlying holomorphic bundle is also stable. The linearization construction provides a canonical embedding of the subspace M^{reg} ⊂ M of regular points into the total space T^{∗}M_{0} of the cotangent bundle to the space M_{0} of stable holomorphic bundles on X. The unicity statement of Theorem 1 guarantees that the hyperkähler metric on M^{reg} provided by the Simpson theory is the same as the canonical metric constructed in Theorem 1.

• Starting with Section 5, we deal only with total spaces \overline{T} M of the complex-conjugate to tangent bundles to complex manifolds M. In Section 5 we describe Hodge manifolds structures on \overline{T} M in terms of certain connections on the locally trivial fibration \overline{T} M → M. “Connection” here is understood as a choice of the subbundle of horizontal vectors, regardless of the vector bundle structure on the fibration \overline{T} M → M. We establish a correspondence between Hodge manifold structures on \overline{T} M and connections on \overline{T} M over M of certain type, which we call Hodge connection.

• In Section 6 we restrict our attention to the formal neighborhood of the zero section M ⊂ \overline{T} M. We introduce the appropriate “formal” versions of all the definitions and then establish a correspondence between formal Hodge connections on \overline{T} M over M and certain differential operators on the manifold M itself, which we call extended connections

• termsextended connection. We also introduce a certain canonical algebra bundle B (M, ℂ) on the complex manifold M, which we call the Weil algebra of the manifold M. Extended connections give rise to natural derivations of the Weil algebra.

• Before we can proceed with classification of extended connections

• termsextended connection on the manifold M and therefore of regular Hodge manifolds, we need to derive some linear-algebraic facts on the Weil algebra B (M, ℂ). This is the subject of Section 7. We begin with introducing a certain version of the de Rham complex of a smooth complex manifold, which we call the total de Rham complex. Then we combine it the material of Section 6 to define the so-called total Weil algebra of the manifold M and establish some of its properties. Section 7 is the most technical part of the whole paper. The reader is advised to skip reading it until needed.

• Section 8 is the main section of the paper. In this section we prove, using the technical results of Section 7, that extended • termsextended connection connections on M are in a natural one-

to-one correspondence with Kählerian connections on the cotangent bundle to M (Theorem 8.1). This proves the formal part of Theorem 2.

• In Section 9 we deal with polarizations. After some preliminaries, we use Theorem 2 to deduce the formal part of Theorem 1 (see Theorem 9.1).

• Finally, in Section 10 we study the convergence of our formal series and prove Theorem 1 and Theorem 2 in the real-analytic case.

• We have also attached a brief Appendix, where we sketch a more conceptual approach to some of the linear algebra used in the paper, in particular, to our notion of a weakly Hodge map. This approach also allows us to describe a simple and conceptual proof of Proposition 7.1, the most technical of the facts proved in Section 7. The Appendix is mostly based on results of Deligne and Simpson ([D2], [S1]).

Acknowledgment. I would like to thank A. Beilinson, D. Kazhdan, A. Levin, L. Posicelsky, A. Shen and A. Tyurin for stimulating discussions. I am especially grateful to my friends Misha Verbitsky and Tony Pantev for innumerable discussions, constant interest and help, without which this paper most probably would not have been written. I would also like to mention here how much I have benefited from a course on moduli spaces and Hodge theory given by Carlos Simpson at MIT in the Fall of 1993. On a different note, I would like to express my dearest gratitude to Julie Lynch, formerly at International Press in Cambridge, and also to the George Soros’s Foundation and to CRDF for providing me with a source of income during the preparation of this paper.

1. Preliminary facts from linear algebra

1.1. Quaternionic vector spaces

1.1.1. Throughout the paper denote by ℍ the ℝ-algebra of quaternions. Definition. A quaternionic vector space is a finite-dimensional left module over the algebra ℍ.

Let V be a quaternionic vector space. Every algebra embedding I : ℂ → ℍ defines by restriction an action of ℂ on V. Denote the corresponding complex vector space by V_{I}.

Fix once and for all an algebra embedding I : ℂ → ℍ and call the complex structure V_{I} the preferred complex structure on V.

1.1.2. Let the group ℂ^{∗} act on the algebra ℍ by conjugation by I(ℂ^{∗}). Since I(ℝ^{∗}) ⊂ I(ℂ^{∗}) lies in the center of the algebra ℍ, this action factors through the map N : ℂ^{∗} → ℂ^{∗}/ℝ^{∗} ≅ U(1) from ℂ^{∗} to the one-dimensional unitary group defined by N(a) = a^{−1}\overline{a}. Call this action the standard action of U(1) on ℍ.

The standard action commutes with the multiplication and leaves invariant the subalgebra I(ℂ). Therefore it extends to an action of the complex algebraic group ℂ^{∗} ⊃ U(1) on the algebra ℍ considered as a right complex vector space over I(ℂ). Call this action the standard action of ℂ^{∗} on ℍ. 1.1.3. Definition. An equivariant quaternionic vector space is a quaternionic vector space V equipped with an action of the group U(1) such that the action map

ℍ ⊗_{ℝ} V → V

is U(1)-equivariant.

The U(1)-action on V extends to an action of ℂ^{∗} on the complex vector space V_{I}. The action map ℍ ⊗_{ℝ} V → V factors through a map

Mult : ℍ ⊗_{ℂ} V_{I} → V_{I}

of complex vector spaces. This map is ℂ^{∗}-equivariant if and only if V is an equivariant quaternionic vector space. 1.1.4. The category of complex algebraic representations V of the group ℂ^{∗} is equivalent to the category of graded vector spaces V = ⊕V . We will say that a representation W is of weight i if W = W^{i}, that is, if an element z ∈ ℂ^{∗} acts on W by multiplication by z^{k}. For every representation W we will denote by W (k) the representation corresponding to the grading

W (k)^{i} = W^{k+i}.

1.1.5. The algebra ℍ considered as a complex vector space by means of right multiplication by I(ℂ) decomposes ℍ = I(ℂ) ⊕ \overline{ℂ} with respect to the standard ℂ^{∗}-action. The first summand is of weight 0, and the second is of weight 1. This decomposition is compatible with the left I(ℂ)-actions as well and induces for every complex vector space W a decomposition

ℍ ⊗_{ℂ} W = W ⊕ \overline{ℂ} ⊗_{ℂ} W.

If W is equipped with an ℂ^{∗}-action, the second summand is canonically isomorphic to \overline{W} (1), where \overline{W} is the vector space complex-conjugate to W. 1.1.6. Let V be an equivariant quaternionic vector space. The action map

Mult : ℍ ⊗_{ℂ} V_{I} ≅ V_{I} ⊕ \overline{ℂ} ⊗ V_{I} → V_{I}

decomposes Mult = id ⊕ j for a certain map j : \overline{V_{I}}(1) → V_{I}. The map j satisfies j ∘ \overline{j} = −id, and we obviously have the following.

Lemma. The correspondence V ⟼ ⟨V_{I}, j⟩ is an equivalence of categories between the category of equivariant quaternionic vector spaces and the category of pairs ⟨W, j⟩ of a graded complex vector space W and a map j : \overline{W}→ W^{1−}satisfying j ∘ \overline{j} = −id.

1.1.7. We will also need a particular class of equivariant quaternionic vector spaces which we will call regular. Definition. An equivariant quaternionic vector space V is called regular if every irreducible ℂ^{∗}-subrepresentation of V_{I} is either trivial or of weight 1.

Lemma. Let V be a regular ℂ^{∗}-equivariant quaternionic vector space and let V_{I}^{0} ⊂ V_{I} be the subspace of ℂ^{∗}-invariant vectors. Then the action map

V_{I}^{0} ⊕ \overline{V_{I}^{0}} ≅ ℍ ⊗_{ℂ} V_{I}^{0} → V_{I}

is invertible.

Proof. Let V_{I}^{1} ⊂ V_{I} be the weight 1 subspace with respect to the gmaction. Since V is regular, V_{I} = V_{I}^{0} ⊕ V_{I}^{1}. On the other hand, j : \overline{V_{I}} → V_{I} interchanges V_{I}^{0} and V_{I}^{1}. Therefore V_{I}^{1} ≅ \overline{V_{I}^{0}} and we are done. □

Thus every regular equivariant quaternionic vector space is a sum of several copies of the algebra ℍ itself. The corresponding Hodge structure has Hodge numbers h^{1,0} = h^{0,1}, h^{p,q} = 0 otherwise.

1.2. The complementary complex structure

1.2.1. Let J : ℂ → ℍ be another algebra embedding. Say that embeddings I and J are complementary if

J(\sqrt{−1})I(\sqrt{−1}) = −I(\sqrt{−1})J(\sqrt{−1}).

Let V be an equivariant quaternionic vector space. The standard U(1)- action on ℍ induces an action of U(1) on the set of all algebra embeddings ℂ → ℍ. On the subset of embeddings complementary to I this action is transitive. Therefore for every two embeddings J_{1}, J_{2} : ℂ → ℍ complementary to I the complex vector spaces V_{J_{1}} and V_{J_{2}} are canonically isomorphic. We will from now on choose for convenience an algebra embedding J : ℂ → ℍ complementary to I and speak of the complementary complex structure V_{J} on V; however, nothing depends on this choice.

1.2.2. For every equivariant quaternionic vector space V the complementary embedding J : ℂ → ℍ induces an isomorphism

J ⊗ id : ℂ ⊗_{ℝ} V → ℍ ⊗_{I(ℂ)} V_{I}

of complex vector spaces. Let Mult : ℍ ⊗_{I(ℂ)} V_{I} → V_{I}, Mult : ℂ ⊗_{ℝ} V → V_{J} be the action maps. Then there exists a unique isomorphism H : V_{J} → V_{I} of complex vector spaces such that the diagram

ℂ ⊗_{ℝ} V ⟶^{J⊗id} ℍ ⊗_{I(ℂ)} V_{I}
Mult↓  _{Mult}↓
V_{J}  ⟶^{H}  V_{I}

is commutative. Call the map H : V_{J} → V_{I} the standard isomorphism between the complementary and the preferred complex structures on the equivariant quaternionic vector space V. 1.2.3. Note that both V_{I} and V_{J} are canonically isomorphic to V as real vector spaces; therefore the map H : V_{J} → V_{I} is in fact an automorphism of the real vector space V. Up to a constant this automorphism is given by the action of the element I(\sqrt{−1}) + J(\sqrt{−1}) ∈ ℍ on the ℍ-module V.

1.3. ℝ-Hodge structures

1.3.1. Recall that a pure ℝ-Hodge structure W of weight i is a pair of a graded complex vector space W = ⊕_{p+q=i}W^{p,q} and a real structure map

: \overline{W^{p,q}} → W^{q,p} satisfying ∘ = id. The bigrading W^{p,q} is called the Hodge type bigrading. The dimensions h^{p,q} = dim_{ℂ} W^{p,q} are called the Hodge numbers of the pure ℝ-Hodge structure W. Maps between pure Hodge structures are by definition maps of their underlying complex vector spaces compatible with the bigrading and the real structure maps. 1.3.2. Recall also that for every k the Hodge-Tate pure ℝ-Hodge structure ℝ(k) of weight −2k is by definition the 1-dimensional complex vector space with complex conjugation equal to (−1)^{k} times the usual one, and with Hodge bigrading

ℝ(k)^{p,q} = \begin{cases}ℝ(k), p = q = −k, \\ 0, otherwise. \end{cases}

For a pure ℝ-Hodge structure V denote, as usual, by V (k) the tensor product V (k) = V ⊗ ℝ(k).

1.3.3. We will also need special notation for another common ℝ-Hodge structure, which we now introduce. Note that for every complex V be a complex vector space the complex vector space V ⊗_{ℝ} ℂ carries a canonical ℝ-Hodge structure of weight 1 with Hodge bigrading given by

V^{1,0} = V ⊂ V ⊗_{ℝ} ℂ V^{0,1} = \overline{V} ⊗_{ℝ} ℂ.

In particular, ℂ ⊗_{ℝ} ℂ carries a natural ℝ-Hodge structure of weight 1 with Hodge numbers h^{1,0} = h^{0,1} = 1. Denote this Hodge structure by W_{1}. 1.3.4. Let ⟨W, ⟩ be a pure Hodge structure, and denote by W_{ℝ} ⊂ W the ℝ-vector subspace of elements preserved by . Define the Weil operator C : W → W by

C = (\sqrt{−1})^{p−q} : W^{p,q} → W^{q,p}.

The operator C : W → W preserves the ℝ-Hodge structure, in particular, the subspace W_{ℝ} ⊂ W. On pure ℝ-Hodge structures of weight 0 the Weil operator C corresponds to the action of −1 ∈ U(1) ⊂ ℂ^{∗} in the corresponding representation. 1.3.5. For a pure Hodge structure W of weight i let

j = C ∘ : \overline{W ^{,1−}}→ W^{1− ,}.

If i is odd, in particular, if i = 1, then j∘\overline{j} = −id. Together with Lemma 1.1.6 this gives the following.

Lemma. The category of equivariant quaternionic vector spaces is equivalent to the category of pure ℝ-Hodge structures of weight 1.

1.3.6. Let V be an equivariant quaternionic vector space, and let ⟨W, ⟩ be ℝ-Hodge structure of weight 1 corresponding to V under the equivalence of Lemma 1.3.5. By definition the complex vector space W is canonically isomorphic to the complex vector space V_{I} with the preferred complex structure on V. It will be more convenient for us to identify W with the complementary complex vector space V_{J} by means of the standard isomorphism H : V_{J} → V_{I}. The multiplication map

Mult : V_{I} ⊗_{ℝ} ℂ ≅ V ⊗_{ℂ} ℍ → V_{J}

is then a map of ℝ-Hodge structures. The complex conjugation : W → \overline{W} is given by

= C ∘ H ∘ j ∘ H^{−1} = C ∘ i : V_{J} → \overline{V_{J}},  (1.1)

where i : V_{J} → \overline{V_{J}} is the action of the element I(\sqrt{−1}) ⊂ ℍ.

1.4. Weakly Hodge maps

1.4.1. Recall that the category of pure ℝ-Hodge structures of weight i is equivalent to the category of pairs ⟨V, F ⟩ of a real vector space V and a decreasing filtration F on the complex vector space V_{ℂ} = V ⊗_{ℝ}ℂ satisfying

V_{ℂ} = ⨁_{p+q=i} F^{p}V_{ℂ} ∩ \overline{F^{q}V_{ℂ}}.

The filtration F is called the Hodge filtration. The Hodge type bigrading and the Hodge filtration are related by V^{p,q} = F^{p}V_{ℂ} ∩ \overline{F^{q}V_{ℂ}} and F^{p}V_{ℂ} = ⊕_{k≥p}V^{k,i−k}. 1.4.2. Let ⟨V, F ⟩ and ⟨W, F ⟩ be pure ℝ-Hodge structures of weights n and m respectively. Usually maps of pure Hodge structures are required to preserve the weights, so that Hom(V, W) = 0 unless n = m. In this paper we will need the following weaker notion of maps between pure ℝ-Hodge structures. Definition. An ℝ-linear map f : V → W is said to be weakly Hodge if it preserves the Hodge filtrations.

Equivalently, the complexified map f : V_{ℂ} → W_{ℂ} must decompose

f = ∑_{0≤p≤m−n} f^{p,m−n−p},  (1.2)

where the map f^{p,m−n−p} : V_{ℂ} → W_{ℂ} is of Hodge type (p, m − n − p). Note that this condition is indeed weaker than the usual definition of a map of Hodge structures: a weakly Hodge map f : V → W can be non-trivial if m is strictly greater than n. If m < n, then f must be trivial, and if m = n, then weakly Hodge maps from V to W are the same as usual maps of ℝ-Hodge structures. 1.4.3. We will denote by WHodge the category of pure ℝ-Hodge structures of arbitrary weight with weakly Hodge maps as morphisms. For every integer n let WHodge_{n} be the full subcategory in WHodge consisting of pure ℝ-Hodge structures of weight n, and let WHodge_{≥n} be the full subcategory of ℝ-Hodge structures of weight not less than n. Since weakly Hodge maps between ℝ-Hodge structures of the same weight are the same as usual maps of ℝ-Hodge structures, the category WHodge_{n} is the usual category of pure ℝ-Hodge structures of weight n. 1.4.4. Let W_{1} = ℂ ⊗_{ℝ} ℂ be the pure Hodge structure of weight 1 with Hodge numbers h^{1,0} = h^{0,1} = 1, as in 1.3.3. The diagonal embedding ℂ → ℂ⊗_{ℝ}ℂ considered as a map w_{1} : ℝ → W_{1} from the trivial pure ℝ-Hodge structure ℝ of weight 0 to W_{1} is obviously weakly Hodge. It decomposes w_{1} = w_{1}^{1,0} + w_{1}^{0,1} as in (1.2), and the components w_{1}^{1,0} : ℂ → W_{1}^{1,0} and w_{1}^{0,1} : ℂ → W_{1}^{0,1} are isomorphisms of complex vectors spaces. Moreover, for every pure ℝ-Hodge structure V of weight n the map w_{1} induces a weakly Hodge map w_{1} : V → W_{1} ⊗ V, and the components w_{1}^{1,0} : V_{ℂ} → V_{ℂ} ⊗ W_{1}^{1,0} and w_{1}^{0,1} : V_{ℂ} → V_{ℂ}⊗W_{1}^{0,1} are again isomorphisms of complex vector spaces.

More generally, for every k ≥ 0 let W_{k} = S^{k}W_{1} be the k-th symmetric power of the Hodge structure W_{1}. The space W_{k} is a pure ℝ-Hodge structure of weight k, with Hodge numbers h^{k,0} = h^{k−1,1} = … = h^{0,k} = 1. Let w_{k} : ℝ → W_{k} be the k-th symmetric power of the map w_{1} : ℝ → W_{1}. For every pure ℝ-Hodge structure V of weight n the map w_{k} induces a weakly Hodge map w_{k} : V → W_{k} ⊗ V, and the components

w_{k}^{p,q} : V_{ℂ} → V_{ℂ} ⊗ W_{k}^{p,k−p}, 0 ≤ p ≤ k

are isomorphisms of complex vector spaces. 1.4.5. The map w_{k} is a universal weakly Hodge map from a pure ℝ-Hodge structures V of weight n to a pure ℝ-Hodge structure of weight n + k. More precisely, every weakly Hodge map f : V → V′ from V to a pure ℝ-Hodge structure V′ of weight n + k factors uniquely through w_{k} : V → W_{k} ⊗ V by means of a map f′ : W_{k} ⊗ V → V′ preserving the pure ℝ-Hodge structures. Indeed, V_{ℂ} ⊗ W_{k} = ⨁_{0≤p≤k} V_{ℂ} ⊗ W_{k}^{p,k−p}, and the maps w_{k}^{p,k−p} : V_{ℂ} → V_{ℂ} ⊗ W_{k}^{p,k−p} are invertible. Hence to obtain the desired factorization it is necessary and sufficient to set

f′ = f^{p,k−p} ∘ (w_{k}^{p,k−p})^{−1} : V_{ℂ} ⊗ W_{k}^{p,k−p} → V_{ℂ} → V′_{ℂ},

where f = ∑_{0≤p≤k} f^{p,k−p} is the Hodge type decomposition (1.2). 1.4.6. It will be convenient to reformulate the universal properties of the maps w_{k} as follows. By definition W_{k} = S^{k}W_{1}, therefore the dual ℝ-Hodge structures equal W_{k}^{∗} = S^{k}W_{1}^{∗}, and for every n, k ≥ 0 we have a canonical projection can : W_{n}^{∗} ⊗ W_{k}^{∗} → W_{n+k}^{∗}. For every pure ℝ-Hodge structure V of weight k ≥ 0 let Γ(V) = V ⊗ W_{k}^{∗}.

Lemma. The correspondence V ↦ Γ(V) extends to a functor

Γ : WHodge_{≥0} → WHodge_{0}

adjoint on the right to the canonical embedding WHodge_{0} ↪ WHodge_{≥0}.

Proof. Consider a weakly Hodge map f : V_{n} → V_{n+k} from ℝ-Hodge structure V_{n} of weight n to a pure ℝ-Hodge structure V_{n+k} of weight n + k. By the universal property the map f factors through the canonical map w_{k} : V_{n} → V_{n} ⊗W_{k} by means of a map f_{k} : V_{n} ⊗W_{k} → V_{n+k}. Let f′_{k} : V_{k} → V_{n+k} ⊗W_{k}^{∗} be the adjoint map, and let

Γ(f) = can ∘ f′_{k} :Γ(V_{n}) = V_{n} ⊗ W_{k}^{∗} → V_{n+k} ⊗ W_{n}^{∗} ⊗ W_{k}^{∗} →
→Γ(V_{n+k}) = V ⊗ W_{n+k}^{∗}.

This defines the desired functor Γ : WHodge_{≥0} → WHodge_{0}. The adjointness is obvious. □ Remark. See Appendix for a more geometric description of the functor Γ : WHodge_{≥0} → WHodge_{0}. 1.4.7. The complex vector space W_{1} = ℂ ⊗_{ℝ}ℂ is equipped with a canonical skew-symmetric trace pairing W_{1} ⊗_{ℂ} W_{1} → ℂ. Let γ : ℂ → W_{1}^{∗} be the map dual to w_{1} : ℝ → W_{1} under this pairing. The map γ is not weakly Hodge, but it decomposes γ = γ^{−1,0} + γ^{0,−1} with respect to the Hodge type bigrading. Denote γ_{l} = γ^{−1,0}, γ_{r} = γ^{0,−1}. For every 0 ≤ p ≤ k the symmetric powers of the maps γ_{l}, γ_{r} give canonical complex-linear embeddings

γ_{l}, γ_{r} : W_{p}^{∗} → W_{k}^{∗}.

1.4.8. The map γ_{l} if of Hodge type (p − k, 0), while γ_{r} is of Hodge type (0, p−k), and the maps γ_{l}, γ_{r} are complex conjugate to each other. Moreover, they are each compatible with the natural maps can : W_{p}^{∗} ⊗ W_{q}^{∗} → W_{p+q}^{∗} in the sense that can ∘ (γ_{l} ⊗ γ_{l}) = γ_{l} ∘ can and can ∘ (γ_{r} ⊗ γ_{r}) = γ_{r} ∘ can. For every p, q, k such that p + q ≥ k we have a short exact sequence

0 ⟶ W_{p+q−k} ⟶^{γ_{r}−γ_{l}} W_{p}^{∗} ⊕ W_{q}^{∗} ⟶^{γ_{l}+γ_{l}} W_{k}^{∗} ⟶ 0  (1.3)

of complex vector spaces. We will need this exact sequence in 7.1.7. 1.4.9. The functor Γ is, in general, not a tensor functor. However, the canonical maps can : W_{n}^{∗} ⊗ W_{k}^{∗} → W_{n+k}^{∗} define for every two pure ℝ-Hodge structures V_{1}, V_{2} of non-negative weight a surjective map

Γ(V_{1}) ⊗ Γ(V_{2}) → Γ(V_{1} ⊗ V_{2}).

These maps are functorial in V_{1} and V_{2} and commute with the associativity and the commutativity morphisms. Moreover, for every algebra A in the tensor category WHodge_{≥0} they turn Γ(A) into an algebra in WHodge_{0}.

1.5. Polarizations

1.5.1. Consider a quaternionic vector space V, and let h be a Euclidean metric on V. Definition. The metric h is called Quaternionic-Hermitian if for any algebra embedding I : ℂ → ℍ the metric h is the real part of an Hermitian metric on the complex vector space V_{I}.

Equivalently, a metric is Quaternionic-Hermitian if it is invariant under the action of the group SU(2) ⊂ ℍ of unitary quaternions. 1.5.2. Assume that the quaternionic vector space V is equivariant. Say that a metric h on V is Hermitian-Hodge if it is Quaternionic-Hermitian and, in addition, invariant under the U(1)-action on V.

Let V_{I} be the vector space V with the preferred complex structure I, and let

V_{I} = ⨁ V

and j : \overline{V}→ V^{1−}be as in Lemma 1.1.6. The metric h is Hermitian-Hodge if and only if

(i) it is the real part of an Hermitian metric on V_{I},

(ii) h(V^{p}, V^{q}) = 0 whenever p ≠ q, and

(iii) h(j(a), b) = −h(a, j(b)) for every a, b ∈ V.

1.5.3. Recall that a polarization S on a pure ℝ-Hodge structure W of weight i is a bilinear form S : W ⊗ W → ℝ(−i) which is a morphism of pure Hodge structures and satisfies

S(a, b) = (−1)^{i}S(b, a)
S(a, Ca) > 0

for every a, b ∈ W. (Here C : W → W is the Weil operator.) 1.5.4. Let V be an equivariant quaternionic vector space equipped with an Euclidean metric h, and let ⟨W, ⟩ be the pure ℝ-Hodge structure of weight 1 associated to V by Lemma 1.3.5. Recall that W = V_{J} as a complex vector space. Assume that h extends to an Hermitian metric h_{J} on V_{J}, and let S : W ⊗ W → ℝ(−1) be the bilinear form defined by

S(a, b) = h(a, C\overline{b}), a, b ∈ W_{ℝ} ⊂ W.

The form S is a polarization if and only if the metric h is Hermitian-Hodge. This gives a one-to-one correspondence between the set of Hermitian-Hodge metrics on V and the set of polarizations on the Hodge structure W.

1.5.5. Let W^{∗} be the Hodge structure of weight −1 dual to W. The sets of polarizations on W and on W^{∗} are, of course, in a natural one-to-one correspondence. It will be more convenient for us to identify the set of Hermitian-Hodge metrics on V with the set of polarizations on W^{∗} rather then on W.

Assume that the metric h on the equivariant quaternionic vector space V is Hermitian-Hodge, and let S ∈ Λ^{2}(W) ⊂ Λ^{2}(V ⊗ ℂ) be the corresponding polarization. Extend h to an Hermitian metric h_{I} on the complex vector space V with the preferred complex structure V_{I}, and let

ω_{I} ∈ V_{I} ⊗ \overline{V_{I}} ⊂ Λ^{2}(V ⊗_{ℝ} ℂ)

be the imaginary part of the corresponding Hermitian metric on the dual space V_{I}^{∗}. Let i : V_{J} → \overline{V_{J}} the action of the element I(\sqrt{−1}) ∈ ℍ. By (1.1) we have

ω_{I}(a, b) = h(a, i(b)) = h(a, C\overline{b}) = S(a, b)

for every a, b ∈ V_{J} ⊂ V ⊗ ℂ. Since ω_{I} is real, and V ⊗ ℂ = V_{J} ⊕ \overline{V_{J}}, the 2-forms ω_{I} and S are related by

ω_{I} = \frac{1}{2}(S + ν(S)),  (1.4)

where ν : V ⊗ ℂ → \overline{V ⊗ ℂ} is the usual complex conjugation.

2. Hodge bundles and quaternionic manifolds

2.1. Hodge bundles

2.1.1. Throughout the rest of the paper, our main tool in studying hyperkähler structures on smooth manifolds will be the equivalence between equivariant quaternionic vector spaces and pure ℝ-Hodge structures of weight 1, established in Lemma 1.1.6. In order to use it, we will need to generalize this equivalence to the case of vector bundles over a smooth manifold M, rather than just vector spaces. We will also need to consider manifolds equipped with a smooth action of the group U(1), and we would like our generalization to take this U(1)-action into account.

Such a generalization requires, among other things, an appropriate notion of a vector bundle equipped with a pure ℝ-Hodge structure. We introduce and study one version of such a notion in this section, under the name of a Hodge bundle (see Definition 2.1.2).

2.1.2. Let M be a smooth manifold equipped with a smooth U(1)-action (or a U(1)-manifold, to simplify the terminology), and let ι : M → M be the action of the element −1 ∈ U(1) ⊂ ℂ^{∗}. Definition. An Hodge bundle of weight k on M is a pair ⟨E, ⟩ of a U(1)- equivariant complex vector bundle E on M and a U(1)-equivariant bundle map : \overline{ι^{∗}E}(k) → E satisfying ∘ ι^{∗} = id.

Hodge bundles of weight k over M form a tensor ℝ-linear additive category, denoted by WHodge_{k}(M). 2.1.3. Let E, F be two Hodge bundles on M of weights m and n. Say that a bundle map. or, more generally, a differential operator f : E → F is weakly Hodge if

(i) f = \overline{ι^{∗}f}, and

(ii) there exists a decomposition f = ∑_{0≤n−m} f_{i} with f_{i} being of degree i with respect to the U(1)-equivariant structure. (In particular, f = 0 unless n ≥ m, and we always have f_{k} = \overline{ι^{∗}f_{n−m−k}}.)

Denote by WHodge(M) the category of Hodge bundles of arbitrary weight on M, with weakly Hodge bundle maps as morphisms. For every i the category WHodge_{i}(M) is a full subcategory in WHodge(M). Introduce also the category WHodge^{D}(M) with the same objects as WHodge(M) but with weakly Hodge differential operators as morphisms. Both the categories WHodge(M) and WHodge^{D}(M) are additive ℝ-linear tensor categories. 2.1.4. For a weakly Hodge map f : E → F call the canonical decomposition

f = ∑_{0≤i≤m−n} f_{i}

the H-type decomposition. For a Hodge bundle E on M of non-negative weight k let Γ(E) = E ⊗ W_{k}^{∗}, where W_{k} is the canonical pure ℝ-Hodges structure introduced in 1.4.4. The universal properties of the Hodge structures W_{k} and Lemma 1.4.6 generalize immediately to Hodge bundles. In particular, Γ extends to a functor Γ : WHodge_{≥0}(M) → WHodge_{0}(M) adjoint on the right to the canonical embedding. 2.1.5. If the U(1)-action on M is trivial, then Hodge bundles of weight i are the same as real vector bundles E equipped with a Hodge type bigrading E = ⊕_{p+q=i}E^{p,q} on the complexification E_{ℂ} = E ⊗_{ℝ} ℂ. In particular, if M = pt is a single point, then WHodge(M) ≅ WHodge^{D}(M) is the category of pure ℝ-Hodge structures. Weakly Hodge bundle map are then the same as weakly Hodge maps of ℝ-Hodge structures considered in 1.4.2. (Thus the notion of a Hodge bundle is indeed a generalization of the notion of a pure ℝ-Hodge structure.) 2.1.6. The categories of Hodge bundles are functorial in M, namely, for every smooth map f : M_{1} → M_{2} of smooth U(1)-manifolds M_{1}, M_{2} there exist pull-back functors

f^{∗} : WHodge(M_{1}) → WHodge(M_{1})
f^{∗} : WHodge^{D}(M_{1}) → WHodge^{D}(M_{1}).

In particular, let M be a smooth U(1)-manifold and let π : M → pt be the canonical projection. Then every ℝ-Hodge structure V of weight i defines a constant Hodge bundle π^{∗}V on M, which we denote for simplicity by the same letter V. Thus the trivial bundle ℝ = Λ^{0}(M) = π^{∗}ℝ(0) has a natural structure of a Hodge bundle of weight 0. 2.1.7. To give a first example of Hodge bundles and weakly Hodge maps, consider a U(1)-manifold M equipped with a U(1)-invariant almost complex structure M_{I}. Let Λ^{i}(M, ℂ) = ⊕Λ^{,i−}(M_{I}) be the usual Hodge type decomposition of the bundles Λ^{i}(M, ℂ) of complex valued differential forms on M. The complex vector bundles Λ^{p,q}(M_{I}) are naturally U(1)-equivariant. Let

: Λ^{p,q}(M_{I}) → ι^{∗}\overline{Λ^{q,p}(M_{I})}

be the usual complex conjugation, and introduce a U(1)-equivariant structure on Λ (M, ℂ) by setting

Λ^{i}(M, ℂ) = ⨁_{0≤j≤i} Λ^{j,i−j}(M)(j).

The bundle Λ^{i}(M, ℂ) with these U(1)-equivariant structure and complex conjugation is a Hodge bundle of weight i on M. The de Rham differential d_{M} is weakly Hodge, and the H-type decomposition for d_{M} is in this case the usual Hodge type decomposition d = ∂ + ¯∂. 2.1.8. Remark. Definition 2.1.2 is somewhat technical. It can be heuristically rephrased as follows. For a complex vector bundle E on M the space of smooth global section C^{∞}(M, E) is a module over the algebra C^{∞}(M, ℂ) of smooth ℂ-valued functions on M, and the bundle E is completely defined by the module C^{∞}(M, E). The U(1)-action on M induces a representation of U(1) on the algebra C^{∞}(M, ℂ). Let ν : C^{∞}(M, ℂ) → C^{∞}(M, ℂ) be composition of the complex conjugation and the map ι^{∗} : C^{∞}(M, ℂ) → C^{∞}(M, ℂ). The map ν is an anti-complex involution; together with the U(1)-action it defines a pure ℝ-Hodge structure of weight 0 on the algebra C^{∞}(M, ℂ).

Giving a weight i Hodge bundle structure on E is then equivalent to giving a weight i pure ℝ-Hodge structure on the module C^{∞}(M, E) such that the multiplication map

C^{∞}(M, ℂ) ⊗ C^{∞}(M, E) → C^{∞}(M, E)

is a map of ℝ-Hodge structures.

2.2. Equivariant quaternionic manifolds

2.2.1. We now turn to our main subject, namely, various differential-geo-metric structures on smooth manifolds associated to actions of the algebra ℍ of quaternions. 2.2.2. Definition. A smooth manifold M is called quaternionic if it is equipped with a smooth action of the algebra ℍ on its cotangent bundle Λ^{1}(M).

Let M be a quaternionic manifold. Every algebra embedding J : ℂ → ℍ defines by restriction an almost complex structure on the manifold M. Denote this almost complex structure by M_{J}. 2.2.3. Assume that the manifold M is equipped with a smooth action of the group U(1), and consider the standard action of U(1) on the vector space ℍ. Call the quaternionic structure and the U(1)-action on M compatible if the action map

ℍ ⊗_{ℝ} Λ^{1}(M) → Λ^{1}(M)

is U(1)-equivariant.

Equivalently, the quaternionic structure and the U(1)-action are compatible if the action preserves the almost complex structure M_{I}, and the action map

ℍ ⊗_{ℂ} Λ^{1,0}(M_{I}) → Λ^{1,0}(M_{I})

is U(1)-equivariant. 2.2.4. Definition. A quaternionic manifold M equipped with a compatible smooth U(1)-action is called an equivariant quaternionic manifold.

For a U(1)-equivariant complex vector bundle E on M denote by E(k) the bundle E with U(1)-equivariant structure twisted by the 1-dimensional representation of weight k. Lemma 1.1.6 immediately gives the following.

Lemma. The category of quaternionic manifolds is equivalent to the category of pairs ⟨M_{I}, j⟩ of an almost complex manifold M_{I} and a ℂ-linear U(1)- equivariant smooth map j : Λ^{0,1}(M_{I})(1) → Λ^{0,1}(M_{I}) satisfying j ∘ \overline{j} = −id.

2.3. Quaternionic manifolds and Hodge bundles

2.3.1. Let M be a smooth U(1)-manifold. Recall that we have introduced in Subsection 2.1 a notion of a Hodge bundle on M. Hodge bundles arise naturally in the study of quaternionic structures on M in the following way. Define a quaternionic bundle on M as a real vector bundle E equipped with a left action of the algebra ℍ, and let Bun(M, ℍ) be the category of smooth quaternionic vector bundles on the manifold M. Let also Bun^{U(1)}(M, ℍ) be the category of smooth quaternionic bundles E on M equipped with a U(1)-equivariant structure such that the ℍ-action map ℍ → End (E) is U(1)- equivariant. Lemma 1.3.5 immediately generalizes to give the following.

Lemma. The category Bun^{U(1)}(M, ℍ) is equivalent to the category of Hodge bundles of weight 1 on M.

2.3.2. Note that if the U(1)-manifold M is equipped with an almost complex structure, then the decomposition ℍ = \overline{ℂ} ⊕ I(ℂ) (see 1.1.5) induces an isomorphism

can : ℍ ⊗_{I(ℂ)} Λ^{0,1} ≅ Λ^{1,0}(M) ⊕ Λ^{0,1}(M) ≅ Λ^{1}(M, ℂ).

The weight 1 Hodge bundle structure on Λ^{1}(M, ℂ) corresponding to the natural quaternionic structure on ℍ ⊗_{I(ℂ)} Λ^{0,1}(M) is the same as the one considered in 2.1.7. 2.3.3. Assume now that the smooth U(1)-manifold M is equipped with a compatible quaternionic structure, and let M_{I} be the preferred almost complex structure on M. Since M_{I} is preserved by the U(1)-action on M, the complex vector bundle Λ^{0,1}(M_{I}) of (0, 1)-forms on M_{I} is naturally U(1)- equivariant.

The quaternionic structure on Λ^{1}(M) induces by Lemma 2.3.1 a weight-1 Hodge bundle structure on Λ^{0,1}(M_{I}). The corresponding U(1)-action on Λ^{0,1}(M_{I}) is induced by the action on M_{I}, and the real structure map

: Λ^{1,0}(M_{I})(1) → Λ^{0,1}(M_{I})

is given by = \sqrt{−1} (ι^{∗} ∘ j). (Here j is induced by quaternionic structure, as in Lemma 2.2.4). 2.3.4. Let M_{J} be the complementary almost complex structure on the equivariant quaternionic manifold M. Recall that in 1.2 we have defined for every equivariant quaternionic vector space V the standard isomorphism H : V_{J} → V_{I}. This construction can be immediately generalized to give a complex bundle isomorphism

H : Λ^{0,1}(M_{J}) → Λ^{0,1}(M_{I}).

Let P : Λ^{1}(M, ℂ) → Λ^{0,1}(M_{J}) be the natural projection, and let Mult : ℍ ⊗_{I(ℂ)}Λ^{0,1}(M_{I}) → Λ^{0,1}(M_{I}) be the action map. By definition the diagram

Λ^{1}(M, ℂ) ⟶^{can} ℍ ⊗_{I(ℂ)} Λ^{0,1}(M_{I})
P ↓  _{Mult}↓
Λ^{0,1}(M_{J}) ⟶^{H}  Λ^{0,1}(M_{I})

is commutative. Since the map Mult is compatible with the Hodge bundle structures, so is the projection P. Remark. This may seems paradoxical, since the complex conjugation on Λ^{1}(M, ℂ) does not preserve Ker P = Λ^{0,1}(M_{J}). However, under our definition of a Hodge bundle the conjugation on Λ^{1}(M, ℂ) is ι^{∗} ∘ rather than . Both and ι^{∗} interchange Λ^{1,0}(M_{J}) and Λ^{0,1}(M_{J}). 2.3.5. The standard isomorphism H : Λ^{0,1}(M_{J}) → Λ^{0,1}(M_{I}) does not commute with the Dolbeult differentials. They are, however, related by means of the Hodge bundle structure on Λ^{0,1}(M_{I}). Namely, we have the following.

Lemma. The Dolbeult differential D : Λ^{0}(M, ℂ) → Λ^{0,1}(M_{I}) for the almost complex structure M_{J} is weakly Hodgeweakly Hodge map. The U(1)- invariant component D_{0} in the H-type decomposition D = D_{0} + \overline{D_{0}} of the map D coincides with the Dolbeult differential for the almost complex structure M_{I}.

Proof. The differential D is the composition D = P ∘ d_{M} of the de Rham differential d_{M} : Λ^{0}(M, ℂ) → Λ^{1}(M, ℂ) with the canonical projection P. Since both P and d_{M} are weakly Hodge, so is D. The rest follows from the construction of the standard isomorphism H. □

2.4. Holonomic derivations

2.4.1. Let M be a smooth U(1)-manifold. In order to give a Hodge-theoretic description of the set of all equivariant quaternionic structures on M, it is convenient to work not with various complex structures on M, but with associated Dolbeult differentials. To do this, recall the following universal property of the cotangent bundle Λ^{1}(M).

Lemma. Let M be a smooth manifold, and let E be a complex vector bundle on M. Every differential operator ∂ : Λ^{0}(M, ℂ) → E which is a derivation with respect to the algebra structure on Λ^{0}(M, ℂ) factors uniquely through the de Rham differential d_{M} : Λ^{0}(M, ℂ) → Λ^{1}(M, ℂ) by means of a bundle map P : Λ^{1}(M, ℂ) → E.

2.4.2. We first use this universal property to describe almost complex structures. Let M be a smooth manifold equipped with a complex vector bundle E. Definition. A derivation D : Λ^{0}(M, ℂ) → E is called holonomic if the associated bundle map P : Λ^{1}(M, ℂ) → E induces an isomorphism of the subbundle Λ^{1}(M, ℝ) ⊂ Λ^{1}(M, ℂ) of real 1-forms with the real vector bundle underlying E.

By Lemma 2.4.1 the correspondence

M_{I} ↦ ⟨Λ^{0,1}(M_{I}), ¯∂⟩

identifies the set of all almost complex structures M_{I} on M with the set of all pairs ⟨E, D⟩ of a complex vector bundle E and a holonomic derivation D : Λ^{0}(M, ℂ) → E. 2.4.3. Assume now that the smooth manifold M is equipped with smooth action of the group U(1). Then we have the following.

Lemma. Let E be a weight 1 Hodge bundle on the smooth U(1)-manifold M, and let

D : Λ^{0}(M, ℂ) → E

be a weakly Hodge holonomic derivation. There exists a unique U(1)-equi-variant quaternionic structure on M such that E ≅ Λ^{0,1}(M_{J}) and D is the Dolbeult differential for the complementary almost complex structures M_{J} on M.

Proof. Since the derivation M is holonomic, it induces an almost complex structure M_{J} on M. To construct an almost complex structure M_{I} complementary to M_{J}, consider the H-type decomposition D = D_{0} + \overline{D_{0}} of the derivation D : Λ^{0}(M, ℂ) → E (defined in 2.1.4).

The map D_{0} is also a derivation. Moreover, it is holonomic. Indeed, by dimension count it is enough to prove that the associated bundle map P : Λ^{1}(M, ℝ) → E is injective. Since the bundle Λ^{1}(M, ℝ) is generated by exact 1-forms, it is enough to prove that any real valued function f on M with D_{0}f = 0 is constant. However, since D is weakly Hodge,

Df = D_{0}f + \overline{D_{0}}f = D_{0}f + \overline{D_{0}\overline{f}} = D_{0}f + \overline{D_{0}f},

hence D_{0}f = 0 if and only if Df = 0. Since D is holonomic, f is indeed constant.

The derivation D_{0}, being holonomic, is the Dolbeult differential for an almost complex structure M_{I} on M. Since D_{0} is by definition U(1)- equivariant, the almost complex structure M_{I} is U(1)-invariant. Moreover, E ≅ Λ^{0,1}(M_{I}) as U(1)-equivariant complex vector bundles. By Lemma 2.3.1 the weight 1 Hodge bundle structure on E induces an equivariant quaternionic bundle structure on E and, in turn, a structure of an equivariant quaternionic manifold on M. The almost complex structure M_{I} coincides by definition with the preferred almost complex structure.

It remains to notice that by Lemma 2.3.5 the Dolbeult differential ¯∂_{J} for the complementary almost complex structure on M indeed equals D = D_{0} + \overline{D_{0}}. □

Together with Lemma 2.3.5, this shows that the set of equivariant quaternionic structures on the U(1)-manifold M is naturally bijective to the set of pairs ⟨E, D⟩ of a weight 1 Hodge bundle E on M and a weakly Hodge holonomic derivation D : Λ^{0}(M, ℂ) → E.

3. Hodge manifolds

3.1. Integrability

3.1.1. There exists a notion of integrability for quaternionic manifolds analogous to that for the almost complex ones. Definition. A quaternionic manifold M is called hypercomplex if for two complementary algebra embeddings I, J : ℂ → ℍ the almost complex structures M_{I}, M_{J} are integrable.

In fact, if M is hypercomplex, then M_{I} is integrable for any algebra embedding I : ℂ → ℍ. For a proof see, e.g., [K]. 3.1.2. When a quaternionic manifold M is equipped with a compatible U(1)- action, there exist a natural choice for a pair of almost complex structures on M, namely, the preferred and the complementary one. Definition. An equivariant quaternionic manifold M is called a Hodge manifold if both the preferred and the equivariant almost complex structures M_{I}, M_{J} are integrable.

Hodge manifolds are the main object of study in this paper. 3.1.3. There exists a simple Hodge-theoretic description of Hodge manifolds based on Lemma 2.4.3. To give it (see Proposition 3.1), consider an equivariant quaternionic manifold M, and let M_{J} and M_{I} be the complementary and the preferred complex structures on M. The weight 1 Hodge bundle structure on Λ^{0,1}(M_{J}) induces a weight i Hodge bundle structure on the bundle Λ^{0,i}(M_{J}) of (0, i)-forms on M_{J}. The standard identification H : Λ^{0,1}(M_{J}) → Λ^{0,1}(M_{I}) given in 2.3.4 extends uniquely to an algebra isomorphism H : Λ^{0,i}(M_{J}) → Λ^{0,i}(M_{I}).

Let D : Λ^{0}(M_{J}) → Λ^{0, +1}(M_{J}) be the Dolbeault differential for the almost complex manifold M_{J}.

Lemma. The equivariant quaternionic manifold M is Hodge if and only if the following holds.

(i) M_{J} is integrable, that is, D ∘ D = 0, and

(ii) the differential D : Λ^{0,i}(M_{J}) → Λ^{0,i+1}(M_{J}) is weakly Hodge for every i ≥ 0.

Proof. Assume first that the conditions (i), (ii) hold. Condition (i) means that the complementary almost complex structure M_{J} is integrable. By (ii) the map D is weakly Hodge.

Let D = D_{0} + \overline{D_{0}} be the H-type decomposition. The map D_{0} is an algebra derivation of Λ^{0,}(M_{I}). Moreover, by Lemma 2.3.5 the map D_{0} : Λ^{0}(M, ℂ) → Λ^{0,1}(M_{J}) is the Dolbeault differential ¯∂_{I} for the preferred almost complex structure M_{I} on M. (Or, more precisely, is identified with ¯∂_{I} under the standard isomorphism H.) But the Dolbeult differential admits at most one extension to a derivation of the algebra Λ^{0,}(M_{J}). Therefore D_{0} = ¯∂_{I} everywhere.

The composition D_{0} ∘ D_{0} is the (2, 0)-component in the H-type decomposition of the map D ∘ D. Since D ∘ D = 0,

D_{0} ∘ D_{0} = ¯∂_{I} ∘ ¯∂_{I} = 0.

Therefore the preferred complex structure M_{I} is also integrable, and the manifold M is indeed Hodge.

Assume now that M is Hodge. The canonical projection P : Λ^{1}(M, ℂ) → Λ^{0,1}(M_{J}) extends then to a multiplicative projection

P : Λ (M, ℂ) → Λ^{0,}(M_{J})

from the de Rham complex of the complex manifold M_{I} to the Dolbeault complex of the complex manifold M_{J}. The map P is surjective and weakly Hodge, moreover, it commutes with the differentials. Since the de Rham differential preserves the pre-Hodge structures, so does the Dolbeault differential D. □ 3.1.4. Lemma 3.1.3 and Lemma 2.4.3 together immediately give the following.

Proposition 3.1 The category of Hodge manifolds is equivalent to the category of triples ⟨M, E, D⟩ of a smooth U(1)-manifold M, a weight 1 Hodge bundle E on M, and a weakly Hodge algebra derivation

D = D : Λ (E) → Λ^{+1}(E)

such that D ∘ D = 0, and the first component

D^{0} : Λ^{0}(M, ℂ) = Λ^{0}(E) → E = Λ^{1}(E)

is holonomic in the sense of 2.4.2.

3.2. The de Rham complex of a Hodge manifold

3.2.1. Let M be a Hodge manifold. In this subsection we study in some detail the de Rham complex Λ (M, ℂ) of the manifold M, in order to obtain information necessary for the discussion of metrics on M given in the Subsection 3.3. The reader is advised to skip this subsection until needed. 3.2.2. Let Λ^{0,}(M_{J}) be the Dolbeault complex for the complementary complex structure M_{J} on M. By Proposition 3.1 the complex vector bundle Λ^{0,i}(M_{J}) is a Hodge bundle of weight i on M, and the Dolbeult differential D : Λ^{0,}(M_{J}) → Λ^{0, +1}(M_{J}) is weakly Hodge. Therefore D admits an H-type decomposition D = D_{0} + \overline{D_{0}}. 3.2.3. Consider the de Rham complex Λ^{i}(M, ℂ) of the smooth manifold M. Let Λ^{i}(M, ℂ) = ⊕_{p+q}Λ^{p,q}(M_{J}) be the Hodge type decomposition for the complementary complex structure M_{J} on M, and let ν : Λ^{p,q}(M_{J}) → \overline{Λ^{q,p}(M_{J})} be the usual complex conjugation. Denote also

f^{ν} = ν ∘ f ∘ ν^{−1}

for any map f : Λ (M, ℂ) → Λ (M, ℂ).

Let d_{M} : Λ (M, ℂ) → Λ^{+1}(M, ℂ) be the de Rham differential, and let d_{M} = D + D^{ν} be the Hodge type decomposition for the complementary complex structure M_{J} on M. Since the Dolbeult differential, in turn, equals D = D_{0} + \overline{D_{0}}, we have

d_{M} = D_{0} + \overline{D_{0}} + D_{0}^{ν} + \overline{D_{0}}^{ν}.

3.2.4. Let ¯∂_{I} : Λ (M, ℂ) → Λ^{+1}(M, ℂ) be the Dolbeult differential for the preferred complex structure M_{I} on M. As shown in the proof of Lemma 3.1.3, the (0, 1)-component of the differential ¯∂_{I} with respect to the complex structure M_{J} equals D_{0}. Therefore the (1, 0)-component of the complex-conjugate map ∂_{I} = ¯∂_{I}^{ν} equals D_{0}^{ν}. Since d_{M} = ¯∂_{I} + ∂_{I}, we have

¯∂_{I} = D_{0} + \overline{D_{0}}^{ν}
∂_{I} = \overline{D_{0}} + D_{0}^{ν}

3.2.5. The standard isomorphism H : Λ^{0,1}(M_{J}) → Λ^{0,1}(M_{I}) introduced in 3.1.3 extends uniquely to a bigraded algebra isomorphism H : Λ^{,}(M_{J}) → Λ^{,}(M_{I}). By definition of the map H, on Λ^{0}(M, ℂ) we have

\begin{aligned}¯∂_{I} & = H ∘ D_{0} ∘ H^{−1} \\ ∂_{I} & = H ∘ D_{0}^{ν} ∘ H^{−1} \\ d_{M} & = ∂_{I} + ¯∂_{I} = H ∘ (D_{0} + D_{0}^{ν})H^{−1}. \end{aligned}  (3.1)

The right hand side of the last identity is the algebra derivation of the de Rham complex Λ (M, ℂ). Therefore, by Lemma 2.4.1 it holds not only on Λ^{0}(M, ℂ), but on the whole Λ (M, ℂ). The Hodge type decomposition for the preferred complex structure M_{I} then shows that the first two identities also hold on the whole de Rham complex Λ (M, ℂ). 3.2.6. Let now ξ = I(\sqrt{−1}) : Λ^{0,1}(M_{J}) → Λ^{1,0}(M_{J}) be the operator corresponding to the preferred almost complex structure M_{I} on M. Let also ξ = 0 on Λ^{0}(M, ℂ) and Λ^{1,0}(M_{J}), and extend ξ to a derivation ξ : Λ^{,}(M_{J}) → Λ^{−1, +1}(M_{J}) by the Leibnitz rule. We finish this subsection with the following simple fact.

Lemma. On Λ^{,0}(M_{J}) ⊂ Λ (M, ℂ) we have

\begin{aligned}ξ ∘ D_{0} + D_{0} ∘ ξ & = \overline{D}_{0}^{ν} \\ ξ ∘ \overline{D}_{0} + \overline{D}_{0} ∘ ξ & = −D_{0}^{ν}. \end{aligned}  (3.2)

Proof. It is easy to check that both identities hold on Λ^{0}(M, ℂ). But both sides of these identities are algebra derivations of Λ^{,0}(M_{J}), and the right hand sides are holonomic in the sense of 2.4.2. Therefore by Lemma 2.4.1 both identities hold on the whole Λ^{,0}(M_{J}). □

3.3. Polarized Hodge manifolds

3.3.1. Let M be a quaternionic manifold. A Riemannian metric h on M is called Quaternionic-Hermitian if for every point m ∈ M the induced metric h_{m} on the tangent bundle T_{m}M is Quaternionic-Hermitian in the sense of 1.5.1.

Definition. A hyperkähler manifold is a hypercomplex manifold M equipped with a Quaternionic-Hermitian metric h which is Kähler for both integrable almost complex structures M_{I}, M_{J} on M. Remark. In the usual definition (see, e.g., [B]) the integrability of the almost complex structures M_{I}, M_{J} is omitted, since it is automatically implied by the Kähler condition. 3.3.2. Let M be a Hodge manifold equipped with a Riemannian metric h. The metric h is called Hermitian-Hodge if it is Quaternionic-Hermitian and, in addition, invariant under the U(1)-action on M. Definition. Say that the manifold M is polarized by the Hermitian-Hodge metric h if h is not only Quaternionic-Hermitian, but also hyperkähler. 3.3.3. Let M be a Hodge manifold. By Proposition 3.1 the holomorphic cotangent bundle Λ^{1,0}(M_{J}) for the complementary complex structure M_{J} on M is a Hodge bundle of weight 1. The holomorphic tangent bundle Θ(M_{J}) is therefore a Hodge bundle of weight −1. By 1.5.4 the set of all Hermitian-Hodge metrics h on M is in natural one-to-one correspondence with the set of all polarizations on the Hodge bundle Θ(M_{J}).

Since θ(M) is of odd weight, its polarizations are skew-symmetric as bilinear forms and correspond therefore to smooth (2, 0)-forms on the complex manifold M_{J}. A (2, 0)-form Ω defines a polarization on Θ(M_{J}) if and only if Ω ∈ C^{∞}(M, Λ^{2,0}(M_{J})) considered as a map

Ω : ℝ(−1) → Λ^{2,0}(M_{J})

is a map of weight 2 Hodge bundles, and for an arbitrary smooth section χ ∈ C^{∞}(M, Θ(M_{J})) we have

Ω(χ, \overline{ι^{∗}(χ)}) > 0.  (3.3)

3.3.4. Assume that the Hodge manifold M is equipped with an Hermitian-Hodge metric h. Let Ω ∈ C^{∞}(M, Λ^{2,0}(M_{J})) be the corresponding polarization , and let ω_{I} ∈ C^{∞}(M, Λ^{1,1}(M_{I})) be the (1, 1)-form on the complex manifold M_{I} associated to the Hermitian metric h. Either one of the forms Ω, ω_{I} completely defines the metric h, and by (1.4) we have

Ω + ν(Ω) = ω_{I},

where ν : Λ (M, ℂ) → Λ (M, ℂ) is the complex conjugation.

Lemma. The Hermitian-Hodge metric h polarizes M if and only if the corresponding (2, 0)-form Ω on M_{J} is holomorphic, that is,

DΩ = 0,

where D is the Dolbeult differential for complementary complex structure M_{J}.

Proof. Let ω_{I}, ω_{J} ∈ Λ^{2}(M, ℂ) be the Kähler forms for the metric h and complex structures M_{I}, M_{J} on M. The metric h is hyperkähler, hence polarizes M, if and only if d_{M}ω_{I} = d_{M}ω_{J} = 0.

Let D = D_{0} + \overline{D_{0}} be the H-type decomposition and let H : Λ^{,}(M_{J}) → Λ^{,}(M_{J}) be the standard algebra identification introduced in 3.2.5. By definition H(ω_{I}) = ω_{J}. Moreover, by (3.1) H^{−1} ∘ d_{M} ∘ H = D_{0} + \overline{D_{0}}^{ν}, hence

H(d_{M}ω_{J}) = D_{0}ω_{I} + \overline{D_{0}}^{ν}ω_{I},

and the metric h is hyperkähler if and only if

d_{M}ω_{I} = (D_{0} + \overline{D_{0}}^{ν})ω_{I} = 0  (3.4)

But 2ω_{I} = Ω + ν(Ω). Since Ω is of Hodge type (2, 0) with respect to the complementary complex structure M_{J}, (3.4) is equivalent to

\overline{D_{0}}Ω = D_{0}Ω = \overline{D_{0}}^{ν}Ω = D_{0}^{ν}Ω = 0.

Moreover, let ξ be as in Lemma 3.2.6. Then ξ(Ω) = 0, and by (3.2) D_{0}Ω = \overline{D}_{0}Ω = 0 implies that D_{0}^{ν}Ω = \overline{D}_{0}^{ν}Ω = 0 as well. It remains to notice that since the metric h is Hermitian-Hodge, Ω is of H-type (1, 1) as a section of the weight 2 Hodge bundle Λ^{2,0}(M_{J}). Therefore DΩ = 0 implies \overline{D_{0}}Ω = D_{0}Ω = 0, and this proves the lemma. □ Remark. This statement is wrong for general hyperkähler manifolds (everything in the given proof carries over, except for the implication DΩ = 0 ⇒ D_{0}Ω = \overline{D_{0}}Ω = \overline{D_{0}}^{ν}Ω = D_{0}^{ν}Ω = 0, which depends substantially on the U(1)-action). To describe general hyperkähler metrics in holomorphic terms, one has to introduce the so-called twistor spaces (see, e.g., [HKLR]).

4. Regular Hodge manifolds

4.1. Regular stable points

4.1.1. Let M be a smooth manifold equipped with a smooth U(1)-action with differential φ_{M} (thus φ_{M} is a smooth vector field on M). Since the group U(1) is compact, the subset M^{U(1)} ⊂ M of points fixed under U(1) is a smooth submanifold.

Let m ∈ M^{U(1)} ⊂ M be a point fixed under U(1). Consider the representation of U(1) on the tangent space T_{m} to M at m. Call the fixed point m regular if every irreducible subrepresentation of T_{m} is either trivial or isomorphic to the representation on ℂ given by embedding U(1) ⊂ ℂ^{∗}. (Here ℂ is considered as a 2-dimensional real vector space.) Regular fixed points form a union of connected component of the smooth submanifold M^{U(1)} ⊂ M. 4.1.2. Assume that M is equipped with a complex structure preserved by the U(1)-action. Call a point m ∈ M stable if for any t ∈ ℝ, t ≥ 0 there exists exp(\sqrt{−1}tφ_{M})m, and the limit

m_{0} ∈ M, m_{0} = lim_{t→+∞} exp(\sqrt{−1}tφ_{M})m

also exists. 4.1.3. For every stable point m ∈ M the limit m_{0} is obviously fixed under U(1). Call a point m ∈ M regular stable if it is stable and the limit m_{0} ∈ M^{U(1)} is a regular fixed point.

Denote by M^{reg} ⊂ M the subset of all regular stable points. The subset M^{reg} is open in M. Example. Let Y be a complex manifold with a holomorphic bundle E and let E be the total space of E. Let ℂ^{∗} act on M by dilatation along the fibers. Then every point e ∈ E is regular stable. 4.1.4. Let M be a Hodge manifold. Recall that the U(1)-action on M preserves the preferred complex structure M_{I}. Definition. A Hodge manifold M is called regular if M_{I}^{reg} = M_{I}.

4.2. Linearization of regular Hodge manifolds

4.2.1. Consider a regular Hodge manifold M. Let Δ ⊂ ℂ be the unit disk equipped with the multiplicative semigroup structure. The group U(1) ⊂ Δ is embedded into Δ as the boundary circle.

Lemma. The action a : U(1) × M → M extends uniquely to a holomorphic action ã : Δ × M_{I} → M_{I}. Moreover, for every x ∈ Δ \ {0} the action map ã(x) : M_{I} → M_{I} is an open embedding.

Proof. Since M is regular, the exponential flow exp(itφ_{M}) of the differential φ_{M} of the action is defined for all positive t ∈ ℝ. Therefore a : U(1) × M → M extends uniquely to a holomorphic action

ã : Δ^{∗} × M_{I} → M_{I},

where Δ^{∗} = Δ\{0} is the punctured disk. Moreover, the exponential flow converges as t → +∞, therefore ã extends to Δ × M_{I} continuously. Since this extension is holomorphic on a dense open subset, it is holomorphic everywhere. This proves the first claim.

To prove the second claim, consider the subset Δ̃ ⊂ Δ^{∗} of points x ∈ Δ such that ã(x) is injective and étale. The subset Δ̃ is closed under multiplication and contains the unit circle U(1) ⊂ Δ^{∗}. Therefore to prove that Δ̃ = Δ^{∗}, it suffices to prove that Δ̃ contains the interval ]0, 1] ⊂ Δ^{∗}.

By definition we have ã(h) = exp(−\sqrt{−1} log hφ_{M}) for every h ∈]0, 1] ⊂ Δ^{∗}. Thus we have to prove that if for some t ∈ ℝ, t ≥ 0 and for two points m_{1}, m_{2} ∈ M we have

exp(\sqrt{−1}tφ_{M})(m_{1}) = exp(\sqrt{−1}tφ_{M})(m_{2}),

then m_{1} = m_{2}. Let m_{1}, m_{2} be such two points and let

t = inf\{t ∈ ℝ, t ≥ 0, exp(\sqrt{−1}tφ_{M})(m_{1}) = exp(\sqrt{−1}tφ_{M})(m_{2})\}.

If the point m_{0} = exp(\sqrt{−1}tφ_{M})(m_{1}) = exp(\sqrt{−1}tφ_{M})(m_{2}) ∈ M is not U(1)-invariant, then it is a regular point for the vector field \sqrt{−1}φ_{M}, and by the theory of ordinary differential equations we have t = 0 and m_{1} = m_{2} = m_{0}.

Assume therefore that m_{0} ∈ M^{U(1)} is U(1)-invariant. Since the group U(1) is compact, the vector field \sqrt{−1}φ_{M} has only a simple zero at m_{0} ⊂ M^{U(1)} ⊂ M. Therefore m_{0} = exp(\sqrt{−1}tφ_{M})m_{1} implies that the point m_{1} ∈ M also is U(1)-invariant, and the same is true for the point m_{2} ∈ M. But ã(exp t) acts by identity on M^{U(1)} ⊂ M. Therefore in this case we also have m_{1} = m_{2} = m_{0}. □ 4.2.2. Let V = M_{I}^{U(1)} ⊂ M_{I} be the submanifold of fixed points of the U(1) action. Since the action preserves the complex structure on M_{I}, the submanifold V is complex.

Lemma. There exists a unique U(1)-invariant holomorphic map

ρ_{M} : M_{I} → V

such that ρ_{M}|_{V} = id.

Proof. For every point m ∈ M we must have ρ_{M}(m) = lim_{t→+∞} exp(itφ_{M}), which proves uniqueness.

To prove that ρ_{M} thus defined is indeed holomorphic, notice that the diagram

M_{I} ⟶^{0×id} Δ × M
ρ_{M} ↓  ↓ã
V ⟶ M_{I}

is commutative. Since the action ã : Δ × M_{I} → M_{I} is holomorphic, so is the map ρ_{M}. □ 4.2.3. Call the canonical map ρ_{M} : M_{I} → M_{I}^{U(1)} the canonical projection of the regular Hodge manifold M onto the submanifold V ⊂ M of fixed points.

Lemma. The canonical projection ρ_{M} : M → M^{U(1)} is submersive, that is, for every point m ∈ M the differential dρ_{M} : T_{m}M → T_{ρ(m)}M^{U(1)} of the map ρ_{M} at m is surjective.

Proof. Since ρ_{M}|_{M^{U(1)}} = id, the differential dρ_{M} is surjective at points m ∈ V ⊂ M. Therefore it is surjective on an open neighborhood U ⊃ V of V in M. For any point m ∈ M there exists a point x ∈ Δ such that x m ∈ U. Since ρ_{M} is Δ-invariant, this implies that dρ_{M} is surjective everywhere on M. □ 4.2.4. Let Θ(M/V) be the relative tangent bundle of the holomorphic map ρ : M → V. Let Θ(M) and Θ(V) be the tangent bundles of M and V and consider the canonical exact sequence of complex bundles

0 ⟶ Θ(M/V) ⟶ Θ(M) ⟶^{dρ_{M}} ρ^{∗}Θ(V) ⟶ 0,

where dρ_{M} is the differential of the projection ρ_{M} : M → V.

The quaternionic structure on M defines a ℂ-linear map j : Θ(M) → \overline{Θ}(M). Restricting to Θ(M/V) and composing with dρ_{M}, we obtain a ℂlinear map j : Θ(M/V) → ρ^{∗}\overline{Θ}(V). 4.2.5. Let \overline{T} V be the total space of the bundle \overline{Θ}(V) complex-conjugate to the tangent bundle Θ(V), and let ρ : \overline{T} V → V be the projection. Let the group U(1) act on \overline{T} V by dilatation along the fibers of the projection ρ.

Since the canonical projection ρ_{M} : M → V is U(1)-invariant, the differential φ_{M} of the U(1)-action defines a section

φ_{M} ∈ C^{∞}(M, Θ(V/M)).

The section j(φ_{M}) ∈ C^{∞}(M, ρ_{M}^{∗} \overline{Θ}(V)) defines a map Lin_{M} : M → \overline{T} V such that Lin_{M} ∘ρ = ρ_{M} : M → V. Call the map Lin_{M} the linearization of the regular Hodge manifold M.

Proposition 4.1 The linearization map Lin_{M} is a U(1)-equivariant open embedding.

Proof. The map j : Θ(M/V) → ρ^{∗}\overline{Θ}(V) is of degree 1 with respect to the U(1)-action, while the section φ_{M} ∈ C^{∞}(M, Θ(M/V)) is U(1)-invariant. Therefore the map Lin_{M} is U(1)-equivariant.

Consider the differential d Lin_{M} : T_{m}(M) → T_{m}(\overline{T} V) at a point m ∈ V ⊂ M. We have

d Lin_{M} = dρ_{M} ⊕ dρ_{M} ∘ j : T_{m}(M) → T_{m}(V) ⊕ \overline{T}_{m}(V)

with respect to the decomposition T_{m}(\overline{T} V) = T_{m}(V) ⊕ \overline{T} (V). The tangent space T_{m} is a regular equivariant quaternionic vector space. Therefore the map d Lin_{M} is bijective at m by Lemma 1.1.7. Since Lin_{M} is bijective on V, this implies that Lin_{M} is an open embedding on an open neighborhood U ⊂ M of the submanifold V ⊂ M.

To finish the proof of proposition, it suffices prove that the linearization map Lin_{M} : M_{I} → \overline{T} V is injective and étale on the whole M_{I}. To prove injectivity, consider arbitrary two points m_{1}, m_{2} ∈ M_{I} such that Lin_{M}(m_{1}) = Lin_{M}(m_{2}). There exists a point x ∈ Δ \ {0} such that x m_{1}, x m_{2} ∈ U. The map Lin_{M} is U(1)-equivariant and holomorphic, therefore it is Δ-equivariant, and we have

Lin_{M}(x m_{1}) = x Lin_{M}(m_{1}) = x Lin_{M}(m_{2}) = Lin_{M}(x m_{2}).

Since the map Lin_{M} : U → \overline{T} V is injective, this implies that x m_{1} = x m_{2}. By Lemma 4.2.1 the action map x : M_{I} → M_{I} is injective. Therefore this is possible only if m_{1} = m_{2}, which proves injectivity.

To prove that the linearization map is étale, note that by Lemma 4.2.1 the action map x : M_{I} → M_{I} is not only injective, but also étale. Since Lin_{M} is ëtale on U, so is the composition Lin_{M} ∘x : M_{I} → U → \overline{T} V is étale. Since Lin_{M} ∘x = x ∘ Lin_{M}, the map Lin_{M} : M_{I} → \overline{T} V is étale at the point m_{1} ∈ M_{I}.

Thus the linearization map is also injective and étale on the whole of M_{I}. Hence it is indeed an open embedding, which proves the propostion. □

4.3. Linear Hodge manifold structures

4.3.1. By Proposition 4.1 every regular Hodge manifold M admits a canonical open embedding Lin_{M} : M → \overline{T} V into the total space \overline{T} V of the (complex-conjugate) tangent bundle to its fixed points submanifold V ⊂ M.

This embedding induces a Hodge manifold structure on a neighborhood of the zero section V ⊂ \overline{T} V.

In order to use the linearization construction, we will need a characterization of all Hodge manifold structures on neighborhoods of V ⊂ \overline{T} V obtained in this way (see 4.3.5). It is convenient to begin with an invariant characterization of the linearization map Lin_{M} : M → \overline{T} V. 4.3.2. Let V be an arbitrary complex manifold, let \overline{T} V be the total space of the complex-conjugate to the tangent bundle Θ(V) to V, and let ρ : \overline{T} V → V be the canonical projection. Contraction with the tautological section of the bundle ρ^{∗}\overline{Θ(V)} defines for every p a bundle map

τ : ρ^{∗}Λ^{p+1}(V, ℂ) → ρ^{∗}Λ^{p}(V, ℂ),

which we call the tautological map. In particular, the induced map

τ : C^{∞}(V, Λ^{0,1}(V)) → C^{∞}(\overline{T} V , ℂ)

identifies the space C^{∞}(V, Λ^{0,1}(V)) of smooth (0, 1)-forms on V with the subspace in C^{∞}(\overline{T} V , ℂ) of function linear along the fibers of the projection \overline{T} V → V. 4.3.3. Let now M be a Hodge manifold. Let V ⊂ M_{I} be the complex submanifold of U(1)-fixed points, and let ρ_{M} : M → V be the canonical projection. Assume that M is equipped with a smooth U(1)-equivariant map f : M → \overline{T} V such that ρ_{M} = ρ ∘ f. Let ¯∂_{I} be the Dolbeult differential for the preferred complex structure M_{I} on M, and let φ ∈ Θ(M/V) be the differential of the U(1)-action on M. Let also j : Λ^{0,1}(M_{I}) → Λ^{1,0}(M_{I}) be the map induced by the quaternionic structure on M.

Lemma. The map f : M → \overline{T} V coincides with the linearization map if and only if for every (0, 1)-form α ∈ C^{∞}(V, Λ^{0,1}(V)) we have

f^{∗}τ(α) = ⟨φ, j(ρ_{M}^{∗}α)⟩.  (4.1)

Moreover, if f = Lin_{M}, then we have

f^{∗}τ(β) = ⟨φ, j(f^{∗}β)⟩  (4.2)

for every smooth section β ∈ C^{∞}(\overline{T} V , ρ^{∗}Λ^{1}(V, ℂ)).

Proof. Since functions on \overline{T} V linear along the fibers separate points, the correspondence

f^{∗} ∘ τ : C^{∞}(V, Λ^{0,1}(V)) → C^{∞}(M, ℂ)

characterizes the map f uniquely, which proves the “only if” part of the first claim. Since by assumption ρ_{M} = ρ∘f, the equality (4.1) is a particular case of (4.2) with β = ρ^{∗}α. Therefore the “if” part of the first claim follows from the second claim, which is a rewriting of the definition of the linearization map Lin_{M} : M → \overline{T} V (see 4.2.5). □ 4.3.4. Let now Lin_{M} : M → \overline{T} V be the linearization map for the regular Hodge manifold M. Denote by U ⊂ \overline{T} V the image of Lin_{M}. The subset U ⊂ \overline{T} V is open and U(1)-invariant. In addition, the isomorphism Lin_{M} : M → U induces a regular Hodge manifold structure on U.

Denote by Lin_{U} the linearization map for the regular Hodge manifold U. Lemma 4.3.3 implies the following.

Corollary. We have Lin_{M} ∘ Lin_{U} = Lin_{M}, thus the linearization map

Lin_{U} : U → \overline{T} V

coincides with the given embedding U ↪ \overline{T} V.

Proof. Let α ∈ C^{∞}(V, Λ^{0,1}(V) be a (0, 1)-form on V. By Lemma 4.3.3 we have Lin_{U}^{∗} τ(α) = ⟨φ_{U}, j_{U}(ρ_{U}^{∗}α)⟩, and it suffices to prove that

Lin_{M}^{∗}(Lin_{U}^{∗}(τ(α))) = ⟨φ_{M}, j_{M}(ρ_{M}^{∗}α)⟩.

By definition we have ρ_{M} = ρ_{U} ∘ Lin_{M}. Moreover, the map Lin_{M} is U(1)- equivariant, therefore it sends φ_{M} to φ_{U}. Finally, by definition it commutes with the quaternionic structure map j. Therefore

Lin_{M}^{∗}(Lin_{U}^{∗}(τ(α))) = Lin_{M}^{∗}(⟨φ_{U}, j_{U}(ρ_{U}^{∗}α)⟩) = ⟨φ_{M}, j_{M}(ρ_{M}^{∗}α)⟩,

which proves the corollary. □ 4.3.5. Definition. Let U ⊂ \overline{T} V be an open U(1)-invariant neighborhood of the zero section V ⊂ \overline{T} V. A Hodge manifold structure on \overline{T} V is called linear if the associated linearization map Lin_{U} : U → \overline{T} V coincides with the given embedding U ↪ \overline{T} V.

By Corollary 4.3.4 every Hodge manifold structure on a subset U ⊂ \overline{T} V obtained by the linearization construction is linear. 4.3.6. We finish this section with the following simple observation, which we will need in the next section.

Lemma. Keep the notations of Lemma 4.3.3. Moreover, assume given a subspace A ⊂ C^{∞}(\overline{T} V , ρ^{∗}Λ^{1}(V, ℂ)) such that the image of A under the restriction map

Res : C^{∞}(\overline{T} V , ρ^{∗}Λ^{1}(V, ℂ)) → C^{∞}(V, Λ^{1}(V, ℂ))

onto the zero section V ⊂ \overline{T} V is the whole space C^{∞}(V, Λ^{1}(V, ℂ)). If (4.2) holds for every section β ∈ A, then it holds for every smooh section

β ∈ C^{∞}(\overline{T} V , ρ^{∗}Λ^{1}(V, ℂ)).

Proof. By assumptions sections β ∈ A generate the restriction of the bundle ρ^{∗}Λ^{1}(V, ℂ) onto the zero section V ⊂ \overline{T} V. Therefore there exists an open neighborhood U ⊂ \overline{T} V of the zero section V ⊂ \overline{T} V such that the C^{∞}(U, ℂ)- submodule

C^{∞}(U, ℂ) A ⊂ C^{∞}(U, ρ^{∗}Λ^{1}(V, ℂ))

is dense in the space C^{∞}(U, ρ^{∗}Λ^{1}(V, ℂ)) of smooth sections of the pullback bundle ρ^{∗}Λ^{1}(V, ℂ). Since the equality (4.2) is continuous and linear with respect to multiplication by smooth functions, it holds for all sections β ∈ C^{∞}(U, ρ^{∗}Λ^{1}(V, ℂ)). Since it is also compatible with the natural unit disc action on M and \overline{T} V, it holds for all sections β ∈ C^{∞}(\overline{T} V , ρ^{∗}Λ^{1}(V, ℂ)) as well. □

5. Tangent bundles as Hodge manifolds

5.1. Hodge connections

5.1.1. The linearization construction reduces the study of arbitrary regular Hodge manifolds to the study of linear Hodge manifold structures on a neighborhood U ⊂ \overline{T} V of the zero section V ⊂ \overline{T} V of the total space of the complex conjugate to the tangent bundle of a complex manifold V. In this section we use the theory of Hodge bundles developed in Subsection 2.1 in order to describe Hodge manifold structures on U in terms of connections on the locally trivial fibration U → V of a certain type, which we call Hodge connections (see 5.1.7). It is this description, given in Proposition 5.1, which we will use in the latter part of the paper to classify all such Hodge manifold structures. 5.1.2. We begin with some preliminary facts about connections on locally trivial fibrations. Let f : X → Y be an arbitrary smooth map of smooth manifolds X and Y. Assume that the map f is submersive, that is, the codifferential δ_{f} : f^{∗}Λ^{1}(Y) → Λ^{1}(X) is an injective bundle map. Recall that a connection on f is by definition a splitting Θ : Λ^{1}(X) → f^{∗}Λ^{1}(Y) of the canonical embedding δ_{f}.

Let d_{X} be the de Rham differential on the smooth manifold X. Every connection Θ on f : X → Y defines an algebra derivation

D = Θ ∘ d_{X} : Λ^{0}(X) → f^{∗}Λ^{1}(Y),

satisfying

Dρ^{∗}h = ρ^{∗}d_{Y} h  (5.1)

for every smooth function h ∈ C^{∞}(Y, ℝ). Vice versa, by the universal property of the cotangent bundle (Lemma 2.4.1) every algebra derivation D : Λ^{0}(X) → Λ^{1}(Y) satisfying (5.1) comes from a unique connection Θ on f. 5.1.3. Recall also that a connection Θ is called flat if the associated derivation D extends to an algebra derivation

D : f^{∗}Λ (Y) → f^{∗}Λ^{+1}(Y)

so that D ∘ D = 0. The splitting Θ : Λ^{1}(X) → f^{∗}Λ^{1}(Y) extends in this case to an algebra map

Θ : Λ (X) → f^{∗}Λ (Y)

compatible with the de Rham differential d_{X} : Λ (X) → Λ^{+1}(X). 5.1.4. We will need a slight generalization of the notion of connection. Definition. Let f : X → Y be a smooth submersive morphism of complex manifolds. A ℂ-valued connection Θ on f is a splitting Θ : Λ^{1}(Y, ℂ) → f^{∗}Λ^{1}(X, ℂ) of the codifferential map δf : f^{∗}Λ^{1}(Y, ℂ) → Λ^{1}(X, ℂ) of complex vector bundles. A ℂ-valued connection Θ is called flat if the associated algebra derivation

D = Θ ∘ d_{X} : Λ^{0}(X, ℂ) → f^{∗}Λ^{1}(Y, ℂ)

extends to an algebra derivation

D : f^{∗}Λ (Y, ℂ) → f^{∗}Λ^{+1}(Y, ℂ)

satisfying D ∘ D = 0.

As in 5.1.2, every derivation D : Λ^{0}(X, ℂ) → f^{∗}Λ^{1}(Y, ℂ) satisfying (5.1) comes from a unique ℂ-valued connection Θ on f : X → Y. Remark. By definition for every flat connection on f : X → Y the subbundle of horizontal vectors in the the tangent bundle Θ(X) is an involutive distribution. By Frobenius Theorem this implies that the connection defines locally a trivialization of the fibration f.

This is no longer true for flat ℂ-valued connections: the subbundle of horizontal vectors in Θ(X) ⊗ ℂ is only defined over ℂ, and the Frobenius Theorem does not apply. One can try to correct this by replacing the splitting Θ : Λ^{1}(X, ℂ) → f^{∗}Λ^{1}(Y, ℂ) with its real part Re Θ : Λ^{1}(X) → Λ^{1}(Y), but this real part is, in general, no longer flat.

5.1.5. For every ℂ-valued connection Θ : Λ^{1}(X, ℂ) → f^{∗}Λ^{1}(Y, ℂ) on a fibration f : X → Y the kernel Ker Θ ⊂ Λ^{1}(X, ℂ) is canonically isomorphic to the quotient Λ^{1}(X, ℂ)/δ_{f}(f^{∗}Λ^{1}(Y, ℂ)), and the composition

R = Θ ∘ d_{X} : Λ^{1}(X, ℂ)/δ_{f}(f^{∗}Λ^{1}(Y, ℂ) ≅ Ker Θ → f^{∗}Λ^{2}(Y, ℂ)

is in fact a bundle map. This map is called the curvature of the ℂ-valued connection Θ. The connection Θ is flat if and only if its curvature R vanishes. 5.1.6. Let now M be a complex manifold, and let U ⊂ \overline{T} M be an open neighborhood of the zero section M ⊂ \overline{T} M in the total space \overline{T} M of the complex-conjugate to the tangent bundle to M. Let ρ : U → M be the natural projection. Assume that U is invariant with respect to the natural action of the unit disc Δ ⊂ ℂ on \overline{T} M. 5.1.7. Since M is complex, by 2.1.7 the bundle Λ^{1}(M, ℂ) is equipped with a Hodge bundle structure of weight 1. The pullback bundle ρ^{∗}Λ^{1}(M, ℂ) is then also equipped with a weight 1 Hodge bundle structure.

Our description of the Hodge manifold structures on the subset U ∈ \overline{T} M is based on the following notion. Definition. A Hodge connection on the pair ⟨M, U⟩ is a ℂ-valued connection on ρ : U → M such that the associated derivation

D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ)

is weakly Hodge in the sense of 2.1.3. A Hodge connection is called flat if it extends to a weakly Hodge derivation

D : ρ^{∗}Λ (M, ℂ) → ρ^{∗}Λ^{+1}(M, ℂ)

satisfying D ∘ D = 0. 5.1.8. Assume given a flat Hodge connection D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) on the pair ⟨U, M⟩, and assume in addition that the derivation D is holonomic in the sense of 2.4.2. Then the pair ⟨D, ρ^{∗}Λ^{1}(M, ℂ)⟩ defines by Proposition 3.1 a Hodge manifold structure on U.

It turns out that every Hodge manifold structure on U can be obtained in this way. Namely, we have the following.

Proposition 5.1 There correspondence D ↦ ⟨ρ^{∗}Λ^{1}(M, ℂ), D⟩ is a bijection between the set of all flat Hodge connections D on the pair ⟨U, M⟩ such that D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) is holonomic in the sense of 2.4.2, and the set of all Hodge manifold structures on the U(1)-manifold U such that the projection ρ : U_{I} → M is holomorphic for the preferred complex structure U_{I} on U.

5.1.9. The crucial part of the proof of Proposition 5.1 is the following observation.

Lemma. Assume given a Hodge manifold structure on the U(1)-manifold U ⊂ \overline{T} M. Let δ_{ρ} : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{1}(U, ℂ) be the codifferential of the projection ρ : U → M, and let P : Λ^{1}(U, ℂ) → Λ^{0,1}(U_{J}) be the canonical projection. The bundle map given by the composition

P ∘ δ_{ρ} : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{0,1}(U_{J})

is an isomorphism of complex vector bundles.

Proof. Since the bundles ρ^{∗}Λ^{1}(M, ℂ) and Λ^{0,1}(U_{J}) are of the same rank, and the maps δ_{ρ} : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{1}(U, ℂ) and P ∘ δ_{ρ} are equivariant with respect to the action of the unit disc on U, it suffices to prove the claim on M ⊂ U. Let m ∈ M be an arbitrary point, and let V = T_{m}^{∗}\overline{T} M be the cotangent bundle at m to the Hodge manifold U ⊂ \overline{T} M. Let also V^{0} ⊂ V be the subspace of U(1)-invariant vectors in V.

The space V is an equivariant quaternionic vector space. Moreover, the fibers of the bundles ρ^{∗}Λ^{1}(M, ℂ) and Λ^{0,1}(U_{J}) at the point m are complex vector spaces, and we have canonical identifications

ρ^{∗}Λ^{1}(M, ℂ)|_{m} ≅ V_{I}^{0} ⊕ \overline{V_{I}^{0}},
Λ^{0,1}(U_{J})|_{m} ≅ V_{J}.

Under these identifications the map P ∘ δ_{ρ} at the point m coincides with the action map V_{I}^{0} ⊕ \overline{V_{I}^{0}} → V_{J}, which is invertible by Lemma 1.1.7. □ 5.1.10. By Proposition 3.1 every Hodge manifold structure on U is given by a pair ⟨E, D⟩ of a Hodge bundle E on U of weight 1 and a holonomic derivation D : Λ^{0}(U, ℂ) → E. Lemma 5.1.9 gives an isomorphism E ≅ ρ^{∗}Λ^{1}(M, ℂ), so that D becomes a flat ℂ-valued connection on U over M. To prove Proposition 5.1 it suffices now to prove the following.

Lemma. The complex vector bundle isomorphism

P ∘ δ_{ρ} : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{0,1}(U_{J})

associated to a Hodge manifold structure on U is compatible with the Hodge bundle structures if and only if the projection ρ : U_{I} → M is holomorphic for the preferred complex structure U_{I} on U.

Proof. The preferred complex structure U_{I} induces a Hodge bundle structure of weight 1 on Λ^{1}(U, ℂ) by 2.1.7, and the canonical projection P : Λ^{1}(U, ℂ) → Λ^{0,1}(U_{J}) is compatible with the Hodge bundle structures by 2.3.4. If the projection ρ : U_{I} → M is holomorphic, then the codifferential δ_{ρ} : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{1}(U, ℂ) sends the subbundles ρ^{∗}Λ^{1,0}(M), ρ^{∗}Λ^{0,1}(M) ⊂ ρ^{∗}Λ^{1}(M, ℂ) into, respectively, the subbundles Λ^{1,0}(U_{I}), Λ^{0,1}(U_{I}) ⊂ Λ^{1}(U, ℂ). Therefore the map δ_{ρ} : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{1}(U, ℂ) is compatible with the Hodge bundle structures, which implies the “if” part of the lemma.

To prove the “only if” part, assume that P ∘δ_{ρ} is a Hodge bundle isomorphism. Since the complex conjugation ν : \overline{Λ^{0,1}(U_{I})} → Λ^{1,0}(U_{J}) is compatible with the Hodge bundle structures, the projection \overline{P} : Λ^{1}(U, ℂ) → Λ^{1,0}(U_{J}) and the composition \overline{P} ∘ δ_{ρ} : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{1,0}(U_{J}) are also compatible with the Hodge bundle structures. Therefore the map

P ⊕ \overline{P} : Λ^{1}(U, ℂ) → Λ^{1,0}(U_{J}) ⊕ Λ^{0,1}(U_{J})

is a Hodge bundle isomorphism, and the composition

δ_{ρ} ∘ (P ⊕ \overline{P}) : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{1}(U, ℂ)

is a Hodge bundle map. Therefore the codifferential δ_{ρ} : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{1}(U, ℂ) is compatible with the Hodge bundle structures. This means precisely that the projection ρ : U_{I} → M is holomorphic, which finishes the proof of the lemma and of Proposition 5.1. □

5.2. The relative de Rham complex of U over M

5.2.1. Keep the notation of the last subsection. To use Proposition 5.1 in the study of Hodge manifold structures on the open subset U ⊂ \overline{T} M, we will need a way to check whether a given Hodge connection on the pair ⟨U, M⟩ is holonomic in the sense of 2.4.2. We will also need to rewrite the linearity condition 4.3.5 for a Hodge manifold structure on U in terms of the associated Hodge connection D. To do this, we will use the so-called relative de Rham complex of U over M. For the convenience of the reader, and to fix notation, we recall here its definition and main properties. 5.2.2. Since the projection ρ : U → M is submersive, the codifferential

δ_{ρ} : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{1}(U, ℂ)

is injective. The relative cotangent bundle Λ^{1}(U/M, C) is by definition the quotient bundle

Λ^{1}(U/M, ℂ) = Λ^{1}(U, ℂ)/δ_{ρ}(ρ^{∗}Λ^{1}(M, ℂ)).

Let π : Λ^{1}(U, ℂ) → Λ^{1}(U/M, ℂ) be the natural projection. We have by definition the short exact sequence

0 ⟶ ρ^{∗}Λ^{1}(M, ℂ) ⟶^{δ_{ρ}} Λ^{1}(U, ℂ) ⟶^{π} Λ^{1}(U/M, ℂ) ⟶ 0

(5.2)

of complex vector bundles on U. 5.2.3. The composition d^{r} = π ∘ d_{U} of the de Rham differential d_{U} with the projection π is an algebra derivation

d^{r} : Λ^{0}(U, ℂ) → Λ^{1}(U/M, ℂ),

called the relative de Rham differential. It is a first order differential operator, and d^{r}f = 0 if and only if the smooth function f : U → ℂ factors through the projection ρ : U → M.

Let Λ (U/M, ℂ) be the exterior algebra of the bundle Λ^{1}(U/M, ℂ). The projection π extends to an algebra map

π : Λ (U, ℂ) → Λ (U/M, ℂ).

The differential d^{r} extends to an algebra derivation

d^{r} : Λ (U/M, ℂ) → Λ^{+1}(U/M, ℂ)

satisfying d^{r} ∘ d^{r} = 0, and we have π ∘ d_{U} = d^{r} ∘ π. The differential graded algebra ⟨Λ (U/M, ℂ), d^{r}⟩ is called the relative de Rham complex of U over M. 5.2.4. Since the relative de Rham differential d^{r} is linear with respect to multiplication by functions of the form ρ^{∗}f with f ∈ C^{∞}(M, ℂ), it extends canonically to an operator

d^{r} : ρ^{∗}Λ^{i}(M, ℂ) ⊗ Λ (U/M, ℂ) → ρ^{∗}Λ^{i}(M, ℂ) ⊗ Λ^{+1}(U/M, ℂ).

The two-step filtration ρ^{∗}Λ^{1}(M, ℂ) ⊂ Λ^{1}(U, ℂ) induces a filtration on the de Rham complex Λ (U, ℂ), and the i-th associated graded quotient of this filtration is isomorphic to the complex ⟨ρ^{∗}Λ^{i}(M, ℂ) ⊗ Λ (U/M, ℂ), d^{r}⟩. 5.2.5. Since U ⊂ \overline{T} M lies in the total space of the complex-conjugate to the tangent bundle to M, we have a canonical algebra isomorphism

can : \overline{ρ^{∗}Λ (M, ℂ)} → Λ (U/M, ℂ).

Let τ : C^{∞}(M, Λ^{1}(M, ℂ)) → C^{∞}(U, ℂ) be the tautological map sending a 1-form to the corresponding linear function on \overline{T} M, as in 4.3.2. Then for every smooth 1-form α ∈ C^{∞}(M, Λ^{1}(M, ℂ)) we have

can(ρ^{∗}α) = d^{r}τ(α).  (5.3)

5.2.6. The complex vector bundle Λ^{1}(U/M, ℂ) has a natural real structure, and it is naturally U(1)-equivariant. Moreover, the decomposition Λ^{1}(M, ℂ) = Λ^{1,0}(M) ⊕ Λ^{0,1}(M) induces a decomposition

Λ^{1}(U/M, ℂ) = can(Λ^{1,0}(M)) ⊕ can(Λ^{0,1}(M)).

This allows to define, as in 2.1.7, a canonical Hodge bundle structure of weight 1 on Λ^{1}(U/M, ℂ). It gives rise to a Hodge bundle structure on Λ^{i}(U/M, ℂ) of weight i, and the relative de Rham differential

d^{r} : Λ (U/M, ℂ) → Λ^{+1}(U/M, ℂ)

is weakly Hodge. 5.2.7. The canonical isomorphism can : \overline{ρ^{∗}Λ^{1}(M, ℂ)} → Λ^{1}(U/M, ℂ) is not compatible with the Hodge bundle structures. The reason for this is that the real structure on the Hodge bundles Λ (U/M, ℂ) is, by definition 2.1.7, twisted by ι^{∗}, where ι : \overline{T} M → \overline{T} M is the action of −1 ∈ U(1) ⊂ ℂ. Therefore, while can is U(1)-equivariant, it is not real. To correct this, introduce an involution ζ : Λ^{1}(M, ℂ) → Λ^{1}(M, ℂ) by

ζ = \begin{cases}id & on Λ^{1,0}(M) ⊂ Λ^{1}(M, ℂ) \\ −id & on Λ^{0,1}(M) ⊂ Λ^{1}(M, ℂ) \end{cases}  (5.4)

and set

η = can ∘ ρ^{∗}\overline{ζ} : ρ^{∗}\overline{Λ^{1}(M, ℂ)} → ρ^{∗}\overline{Λ^{1}(M, ℂ)} → Λ^{1}(U/M, ℂ)  (5.5)

Unlike can, the map η preserves the Hodge bundle structures.

It will also be convenient to twist the tautological map τ : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{0}(U, ℂ) by the involution ζ. Namely, define a map σ : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{0}(U, ℂ) by

σ = τ ∘ ρ^{∗}\overline{ζ} : ρ^{∗}\overline{Λ^{1}(M, ℂ)} → ρ^{∗}\overline{Λ^{1}(M, ℂ)} → Λ^{0}(U/M, ℂ)  (5.6)

By (5.3) the twisted tautological map σ and the canonical map η satisfy

η(ρ^{∗}α) = d^{r}σ(α)  (5.7)

for every smooth 1-form α ∈ C^{∞}(M, Λ^{1}(M, ℂ)). 5.2.8. Let φ ∈ Θ(U) be the differential of the canonical U(1)-action on U ⊂ \overline{T} M. The vector field φ is real and tangent to the fibers of the projection ρ : U → M. Therefore the contraction with φ defines an algebra derivation

Λ^{+1}(U/M, ℂ) → Λ (U/M, ℂ)
α ↦ ⟨φ, α⟩

The following lemma gives a relation between this derivation, the canonical weakly Hodge map η : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{1}(U/M, ℂ) given by (5.5), and the tautological map τ : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{0}(U, ℂ).

Lemma. For every smooth section α ∈ C^{∞}(U, ρ^{∗}Λ^{1}(M, ℂ)) we have

\sqrt{−1}τ(α) = ⟨φ, η(α)⟩ ∈ C^{∞}(U, ℂ).

Proof. Since the equality that we are to prove is linear with respect to multiplication by smooth functions on U, we may assume that the section α is the pull-back of a smooth 1-form α ∈ C^{∞}(M, Λ^{1}(M, ℂ)). The Lie derivative L_{φ} : Λ (U, ℂ) → Λ (U, ℂ) with respect to the vector field φ is compatible with the projection π : Λ (U, ℂ) → Λ (U/M, ℂ) and defines therefore an algebra derivation L_{φ} : Λ (U/M, ℂ) → Λ (U/M, ℂ). The Cartan homotopy formula gives

L_{φ}τ(α) = ⟨φ, d^{r}τ(α)⟩.  (5.8)

The function τ(α) on \overline{T} M is by definition ℝ-linear along the fibers of the projection ρ : \overline{T} M → M. The subspace τ(C^{∞}(M, Λ^{1}(M, ℂ))) ⊂ C^{∞}(U, ℂ) of such functions decomposes as

τ(C^{∞}(M, Λ^{1}(M, ℂ))) = τ(C^{∞}(M, Λ^{1,0}(M))) ⊕ τ(C^{∞}(M, Λ^{0,1}(M, ℂ))),

and the group U(1) acts on the components with weight 1 and −1. Therefore the derivative L_{φ} of the U(1)-action acts on the components by multiplication with \sqrt{−1} and −\sqrt{−1}. By definition of the involution ζ (see (5.4)) this can be written as

L_{φ}τ(α) = \sqrt{−1}τ(ζ(α)).  (5.9)

On the other hand, by (5.3) and the definition of the map η we have

d^{r}τ(α) = can(α) = η(ζ(α)).  (5.10)

Substituting (5.9) and (5.10) into (5.8) gives

\sqrt{−1}τ(ζ(α)) = ⟨φ, η(ζ(α)),

which is equivalent to the claim of the lemma. □

5.3. Holonomic Hodge connections

5.3.1. We will now describe a convenient way to check whether a given Hodge connection D on the pair ⟨U, M⟩ is holonomic in the sense of 2.4.2. To do this, we proceed as follows.

Consider the restriction Λ^{1}(U, ℂ)|_{M} of the bundle Λ^{1}(U, ℂ) to the zero section M ⊂ U ⊂ \overline{T} M, and let

Res : Λ^{1}(U, ℂ)|_{M} → Λ^{1}(M, ℂ)

be the restriction map. The kernel of the map Res coincides with the conormal bundle to the zero section M ⊂ U, which we denote by S^{1}(M, ℂ). The map Res splits the restriction of exact sequence (5.2) onto the zero section M ⊂ U, and we have the direct sum decomposition

Λ^{1}(U, ℂ)|_{M} = S^{1}(M, ℂ) ⊕ Λ^{1}(M, ℂ).  (5.11)

5.3.2. The U(1)-action on U ⊂ \overline{T} M leaves the zero section M ⊂ U invariant and defines therefore a U(1)-action on the conormal bundle S^{1}(M, ℂ). Together with the usual real structure twisted by the action of the map ι : \overline{T} M → \overline{T} M, this defines a Hodge bundle structure of weight 0 on the bundle S^{1}(M, ℂ).

Note that the automorphism ι : \overline{T} M → \overline{T} M acts as −id on the Hodge bundle S^{1}(M, ℂ), so that the real structure on S^{1}(M, ℂ) is minus the usual one. Moreover, as a complex vector bundle the conormal bundle S^{1}(M, ℂ) to M ⊂ \overline{T} M is canonically isomorphic to the cotangent bundle Λ^{1}(M, ℂ). The Hodge type bigrading on S^{1}(M, ℂ) is given in terms of this isomorphism by

S^{1}(M, ℂ) = S^{1,−1}(M) ⊕ S^{−1,1}(M) ≅ Λ^{1,0}(M) ⊕ Λ^{0,1}(M) = Λ^{1}(M, ℂ).

5.3.3. Let

C_{lin}^{∞}(U, ℂ) = τ(C^{∞}(M, Λ^{1}(M, ℂ))) ⊂ C^{∞}(U, ℂ)

be the subspace of smooth functions linear along the fibers of the canonical projection ρ : U ⊂ \overline{T} M → M. The relative de Rham differential defines an isomorphism

d^{r} : C_{lin}^{∞}(U, ℂ) → C^{∞}(M, S^{1}(M, ℂ)).  (5.12)

This isomorphism is compatible with the canonical Hodge structures of weight 0 on both spaces, and it is linear with respect to multiplication by smooth functions f ∈ C^{∞}(M, ℂ). 5.3.4. Let now D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) be a Hodge connection on the pair ⟨U, M⟩, and let Θ : Λ^{1}(U, ℂ) → ρ^{∗}Λ^{(}M, ℂ) be the corresponding bundles map. Since D is a ℂ-valued connection, the restriction Θ|_{M} onto the zero section M ⊂ M decomposes as

Θ = D_{0} ⊕ id : S^{1}(M, ℂ) ⊕ Λ^{1}(M, ℂ) → Λ^{1}(M, ℂ)  (5.13)

with respect to the direct sum decomposition (5.11) for a certain bundle map D_{0} : S^{1}(M, ℂ) → Λ^{1}(M, ℂ). Definition. The bundle map D_{0} : S^{1}(M, ℂ) → Λ^{1}(M, ℂ) is called the principal part of the Hodge connection D. 5.3.5. Consider the map D_{0} : C^{∞}(M, S^{1}(M, ℂ)) → C^{∞}(M, Λ^{1}(M, ℂ)) on the spaces of smooth sections induced by the principal part D_{0} of a Hodge connection D. Under the isomorphism (5.12) this map coincides with the restriction of the composition

Res ∘D : C^{∞}(U, ℂ) → C^{∞}(U, ρ^{∗}Λ^{1}(M, ℂ)) → C^{∞}(M, Λ^{1}(M, ℂ))

onto the subspace C_{lin}^{∞}(U, ℂ) ⊂ C^{∞}(U, ℂ). Each of the maps Res, D is weakly Hodge, so that this composition also is weakly Hodge. Since the isomorphism (5.12) is compatible with the Hodge bundle structures, this implies that the principal part D_{0} of the Hodge connection D is a weakly Hodge bundle map. In particular, it is purely imaginary with respect to the usual real structure on the conormal bundle S^{1}(M, ℂ). 5.3.6. We can now formulate the main result of this subsection.

Lemma. A Hodge connection D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) on the pair ⟨U, M⟩ is holonomic in the sense of 2.4.2 on an open neighborhood U_{0} ⊂ U of the zero section M ⊂ U if and only if its principal part D_{0} : S^{1}(M, ℂ) → Λ^{1}(M, ℂ) is a complex vector bundle isomorphism.

Proof. By definition the derivation D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) is holonomic in the sense of 2.4.2 if and only if the corresponding map

Θ : Λ^{1}(U, ℝ) → ρ^{∗}Λ^{1}(M, ℂ)

is an isomorphism of real vector bundles. This is an open condition. Therefore the derivation D is holonomic on an open neighborhood U_{0} ⊃ M of the zero section M ⊂ U if and only if the map Θ is an isomorphism on the zero section M ⊂ U itself.

According to (5.13), the restriction Θ|_{M} decomposes as Θ_{M} = D_{0} + id, and the principal part D_{0} : S^{1}(M, ℂ) → Λ^{1}(M, ℂ) of the Hodge connection D is purely imaginary with respect to the usual real structure on Λ^{1}(U, ℂ)|_{M}, while the identity map id : ρ^{∗}Λ^{1}(M, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) is, of course, real. Therefore Θ_{M} is an isomorphism if and only is D_{0} is an isomorphism, which proves the lemma. □

5.4. Hodge connections and linearity

5.4.1. Assume now given a Hodge manifold structure on the subset U ⊂ \overline{T} M, and let D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) be the associated Hodge connection on the pair ⟨U, M⟩ given by Proposition 5.1. We now proceed to rewrite the linearity condition 4.3.5 in terms of the Hodge connection D.

Let j : Λ^{1}(U, ℂ) → \overline{Λ^{1}(U, ℂ)} be the canonical map defined by the quaternionic structure on U, and let ι^{∗} : Λ^{1}(U, ℂ) → ι^{∗}Λ^{1}(U, ℂ) be the action of the canonical involution ι : U → U. Let also D^{ι} : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) be the operator ι^{∗}-conjugate to the Hodge connection D.

We begin with the following identity.

Lemma. For every smooth function f ∈ C^{∞}(U, ℂ) we have

d^{r}f = \frac{\sqrt{−1}}{2}π(j(δ_{ρ}(D − D^{ι})(f))),

where π : Λ^{1}(U, ℂ) → Λ^{1}(U/M, ℂ) is the canonical projection, and

δ_{ρ} : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{1}(U, ℂ)

is the codifferential of the projection ρ : U → M.

Proof. By definition of the Hodge connection D the Dolbeault derivative ¯∂_{J}f coincides with the (0, 1)-component of the 1-form δ_{ρ}(Df) ∈ Λ^{1}(U, ℂ) with respect to the complementary complex structure U_{J} on U. Therefore

¯∂_{J}f = \frac{1}{2}δ_{ρ}(Df) + \frac{\sqrt{−1}}{2}j(δ_{ρ}(Df)).

Applying the complex conjugation ν : Λ (U, ℂ) → \overline{Λ (U, ℂ)} to this equation, we get

\begin{aligned}∂_{J}f & = ν (\frac{1}{2}δ_{ρ}(Dν(f)) + \frac{\sqrt{−1}}{2}j(δ_{ρ}(Dν(f))))) = \\ & = \frac{1}{2}ν(δ_{ρ}(Dν(f))) − \frac{\sqrt{−1}}{2}j(ν(δ_{ρ}(Dν(f)))). \end{aligned}

Since the map δ_{ρ} ∘ D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) is weakly Hodge, we have

δ_{ρ}(D(ι^{∗}ν(f))) = ι^{∗}ν(δ_{ρ}(Df)).

Therefore ν(δ_{ρ}(D(f))) = δ_{ρ}(D^{ι}(ν(f))), and we have

∂_{J}f = \frac{1}{2}δ_{ρ}(D^{ι}f) − \frac{\sqrt{−1}}{2}j(δ_{ρ}(D^{ι}f)).

Thus the de Rham derivative d_{U}f equals

d_{U}f = ∂_{J}f + ¯∂_{J}f = \frac{1}{2}δ_{ρ}((D + D^{ι})f) + \frac{\sqrt{−1}}{2}j(δ_{ρ}((D − D^{ι})f)).

Now, by definition δ_{ρ} ∘ π = 0. Therefore

d^{r}f = π(d_{U}f) = \frac{\sqrt{−1}}{2}π(j(δ_{ρ}((D − D^{ι})f))),

which is the claim of the lemma. □ 5.4.2. We will also need the following fact. It can be derived directly from Lemma 5.4.1, but it is more convenient to use Lemma 5.3.6 and the fact that the Hodge connection D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) is holonomic.

Lemma. In the notation of Lemma 5.4.1, let

A = δ_{ρ} ((D − D^{ι}) (C_{lin}^{∞}(U, ℂ))) ⊂ C^{∞}(U, ρ^{∗}Λ^{1}(M, ℂ))

be the subspace of sections α ∈ C^{∞}(U, ρ^{∗}Λ^{1}(M, ℂ)) of the form α = δ_{ρ}((D− D^{ι})f), where f ∈ C^{∞}(U, ℂ) lies in the subspace C_{lin}^{∞}(U, ℂ) ⊂ C^{∞}(U, ℂ) of smooth functions on U linear along the fibers of the projection ρ : U → M. The restriction Res(A) ⊂ C^{∞}(M, Λ^{1}(M, ℂ)) of the subspace A onto the zero section M ⊂ U is the whole space C^{∞}(M, Λ^{1}(M, ℂ)).

Proof. Let D_{0} = Res ∘D : C_{lin}^{∞}(U, ℂ) → C^{∞}(M, Λ^{1}(M, ℂ)) be the principal part of the Hodge connection D in the sense of Definition 5.3.4. Since the canonical automorphism ι : \overline{T} M → \overline{T} M acts as −id on C_{lin}^{∞}(U, ℂ), we have D_{0}^{ι} = −D_{0}. Therefore

Res(A) = Res ∘(D − D^{ι}) (C_{lin}^{∞}(U, ℂ)) = = (D_{0} − D_{0}^{ι}) (C_{lin}^{∞}(U, ℂ)) = D_{0} (C_{lin}^{∞}(U, ℂ)) .

Since the Hodge connection D is holonomic, this space coincides with the whole C^{∞}(M, Λ^{1}(M, ℂ)) by Lemma 5.3.6. □ 5.4.3. We now apply Lemma 5.4.1 to prove the following criterion for the linearity of the Hodge manifold structure on U defined by the Hodge connection D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ).

Lemma. The Hodge manifold structure on U ⊂ \overline{T} M corresponding to a Hodge connection D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) is linear in the sense of 4.3.5 if and only if for every smooth function f ∈ C^{∞}(U, ℂ) linear along the fibers of the projection ρ : U ⊂ \overline{T} M → M we have

f = \frac{1}{2}σ ((D − D^{ι})f) ,  (5.14)

where σ : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{0}(U, ℂ) is the twisted tautological map introduced in (5.6), and D^{ι} : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) is the operator ι^{∗}-conjugate to D, as in 5.4.1.

Proof. By Lemma 4.3.3 the Hodge manifold structure on U is linear if and only if for every α ∈ C^{∞}(U, ρ^{∗}Λ^{1}(M, ℂ)) we have

⟨φ, j(α)⟩ = τ(α),  (5.15)

where φ is the differential of the U(1)-action on U, j : Λ^{1}(U, ℂ) → \overline{Λ^{1}(U, ℂ)} is the operator given by the quaternionic structure on U, and τ : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{0}(U, ℂ) is the tautological map sending a 1-form on M to the corresponding linear function on \overline{T} M, as in 4.3.2. Moreover, by Lemma 5.4.2 and Lemma 4.3.6 the equality (5.15) holds for all smooth sections α ∈ C^{∞}(U, ρ^{∗}Λ^{1}(M, ℂ)) if and only if it holds for sections of the form

α = \frac{\sqrt{−1}}{2}δ_{ρ}((D − D^{ι})f),  (5.16)

where f ∈ C_{lin}^{∞}(U, ℂ) ⊂ C^{∞}(U, ℂ) is linear along the fibers of ρ : U → M.

Let now f ∈ C^{∞}(U, ℂ) be a smooth function on U linear along the fibers of ρ : U → M, and let α be as in (5.16). Since φ is a vertical vector field on U over M, we have ⟨φ, j(α)⟩ = ⟨φ, π(j(α))⟩, where π : Λ^{1}(U, ℂ) → Λ^{1}(U/M, ℂ) is the canonical projection. By Lemma 5.4.1

⟨φ, j(α)⟩ = ⟨φ, π(j(α))⟩ = ⟨φ, d^{r}f⟩.  (5.17)

Since the function f is linear along the fibers of ρ : U → M, we can assume that f = σ(β) for a smooth 1-form β ∈ C^{∞}(M, Λ^{1}(M, ℂ). Then by (5.7) and by Lemma 5.2.8 the right hand side of (5.17) is equal to

⟨φ, d^{r}f⟩ = ⟨φ, d^{r}(σ(β))⟩ = ⟨φ, η(β)⟩ = \sqrt{−1}τ(β).

Therefore, (5.15) is equivalent to

\sqrt{−1}τ(β) = τ (\frac{\sqrt{−1}}{2}δ_{ρ}((D − D^{ι})σ(β))) .  (5.18)

But we have τ = σ∘ζ, where ζ : ρ^{∗}Λ^{1}(M, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) is the invulution introduced in (5.4). In particular, the map ζ is invertible, so that (5.18) is in turn equivalent to

σ(β) = \frac{1}{2}σ(δ_{ρ}((D − D^{ι})σ(β))),

or, substituting back f = σ(β), to

f = \frac{1}{2}σ(δ_{ρ}((D − D^{ι})f)),

which is exactly the condition (5.14). □ 5.4.4. Definition. A Hodge connection D on the pair ⟨U, M⟩ is called linear if it satisfies the condition 5.14.

We can now formulate and prove the following more useful version of Proposition 5.1.

Proposition 5.2 Every linear Hodge connection

D:Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ)

on the pair ⟨U, M⟩ defines a linear Hodge manifold structure on an open neighborhood V ⊂ U of the zero section M ⊂ U, and the canonical projection ρ : V_{I} → M is holomorphic for the preferred complex structure V_{I} on V. Vice versa, every such linear Hodge manifold structure on U comes from a unique linear Hodge connection D on the pair ⟨U, M⟩.

Proof. By Proposition 5.1 and Lemma 5.4.3, to prove this proposition suffices to prove that if a Hodge connection D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) is linear, then it is holonomic in the sense of 2.4.2 on a open neighborhood V ⊂ U of the zero section M ⊂ U. Lemma 5.3.6 reduces this to proving that the principal part D_{0} : S^{1}(M, ℂ) → Λ^{1}(M, ℂ) of a linear Hodge connection D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) is a bundle isomorphism.

Let D : Λ^{0}(U, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) be such connection. By (5.14) we have

\frac{1}{2}σ ∘ (D_{0} − D_{0}^{ι}) = id : S^{1}(M, ℂ) → Λ^{1}(M, ℂ) → S^{1}(M, ℂ).

Since σ : Λ^{1}(M, ℂ) → S^{1}(M, ℂ) is a bundle isomorphism, so is the bundle map D_{0} − D_{0}^{ι} : S^{1}(M, ℂ) → Λ^{1}(M, ℂ). As in the proof of Lemma 5.4.2, we have D_{0} = −D_{0}^{ι}. Thus D_{0} = \frac{1}{2}(D_{0} − D_{0}^{ι}) : S^{1}(M, ℂ) → Λ^{1}(M, ℂ) also is a bundle isomorphism, which proves the proposition. □

6. Formal completions

6.1. Formal Hodge manifolds

6.1.1. Proposition 5.2 reduces the study of arbitrary regular Hodge manifolds to the study of connections of a certain type on a neighborhood U ⊂ \overline{T} M of the zero section M ⊂ \overline{T} M in the total space \overline{T} M of the tangent bundle to a complex manifold M. To obtain further information we will now restrict our attention to the formal neighborhood of this zero section. This section contains the appropriate definitions. We study the convergence of our formal series in Section 10. 6.1.2. Let X be a smooth manifold and let Bun(X) be the category of smooth real vector bundles over X. Let also Diff(X) be the category with the same objects as Bun(X) but with differential operators as morphisms.

Consider a closed submanifold Z ⊂ X. For every two real vector bundles E and F on X the vector space Hom(E, F) of bundle maps from E to F is naturally a module over the ring C^{∞}(X) of smooth functions on X. Let 𝔍_{Z} ⊂ C^{∞}(X) be the ideal of functions that vanish on Z and let Hom_{Z}(E, F) be the 𝔍_{Z}-adic completion of the C^{∞}(X)-module Hom(E, F).

For any three bundles E, F, G the composition map

Mult : Hom(E, F) ⊗ Hom(F, G) → Hom(E, G)

is C^{∞}(X)-linear, hence extends to a map

Mult : Hom_{Z}(E, F) ⊗ Hom_{Z}(F, G) → Hom_{Z}(E, G).

Let Bun_{Z}(X) be the category with the same objects as Bun(X) and for every two objects E, F ∈ Ob Bun(X) with Hom_{Z}(E, F) as the space of maps between F anf F. The category Bun_{Z}(X), as well as Bun(X), is an additive tensor category. 6.1.3. The space of differential operators Diff(E, F) is also a C^{∞}(X) module, say, by left multiplication. Let Diff_{Z}(E, F) be its 𝔍_{Z}-completion. The composition maps in Diff(X) are no longer C^{∞}(X)-linear. However, they still are compatible with the 𝔍_{Z}-adic topology, hence extend to completions. Let Diff_{Z}(X) be the category with the same objects as Bun(X) and with Diff_{Z}(E, F) as the space of maps between two objects E, F ∈ Ob Bun(X).

By construction we have canonical Z-adic completion functors

Bun(X) → Bun_{Z}(X) and Diff(X) → Diff_{Z}(X).

Call the categories Bun_{Z}(X) and Diff_{Z}(X) the Z-adic completions of the categories Bun(X) and Diff(X). 6.1.4. When the manifold X is equipped with a smooth action of compact Lie group G fixing the submanifold Z, the completion construction extends to the categories of G-equivariant bundles on M. When G = U(1), the categories WHodge(X) and WHodge^{D}(X) defined in 2.1.3 also admit canonical completions, denoted by WHodge_{Z}(X) and WHodge_{Z}^{D}(X). 6.1.5. Assume now that the manifold X is equipped with a smooth U(1)- action fixing the smooth submanifold Z ⊂ X. Definition. A formal quaternionic structure on X along the submanifold Z ⊂ X is given by an algebra map

Mult : ℍ → End_{Bun_{Z}(X)} (Λ^{1}(X))

from the algebra ℍ to the algebra End_{Bun_{Z}(X)} (Λ^{1}(X)) of endomorphisms of the cotangent bundle Λ^{1}(X) in the category Bun_{Z}(X). A formal quaternionic structure is called equivariant if the map Mult is equivariant with respect to the natural U(1)-action on both sides. 6.1.6. Note that Lemma 2.4.1 still holds in the situation of formal completions. Consequently, everything in Section 2 carries over word-by-word to the case of formal quaternionic structures. In particular, by Lemma 2.4.3 giving a formal equivariant quaternionic structure on X along Z is equivalent to giving a pair ⟨E, D⟩ of a Hodge bundle E on X and a holonomic algebra derivation D : Λ^{0}(X) → E in WHodge_{Z}^{D}(X). 6.1.7. The most convenient way to define Hodge manifold structures on X in a formal neighborhood of Z is by means of the Dolbeault complex, as in Proposition 3.1.

Definition. A formal Hodge manifold structure on X along Z is a pair of a Hodge bundle E ∈ Ob WHodge_{Z}(X) of weight 1 and an algebra derivation D : Λ E → Λ^{+1}E in WHodge_{Z}^{D}(X) such that D^{0} : Λ^{0}(E) → E is holonomic and D^{0} ∘ D^{1} = 0. 6.1.8. Let U ⊂ X be an open subset containing Z ⊂ X. For every Hodge manifold structure on U the Z-adic completion functor defines a formal Hodge manifold structure on X along Z. Call it the Z-adic completion of the given structure on U. Remark. Note that a Hodge manifold structure on U is completely defined by the preferred and the complementary complex structures U_{I}, U_{J}, hence always real-analytic by the Newlander-Nirenberg Theorem. Therefore, if two Hodge manifold structures on U have the same completion, they coincide on every connected component of U intersecting Z.

6.2. Formal Hodge manifold structures on tangent bundles

6.2.1. Let now M be a complex manifold, and let \overline{T} M be the total space of the complex-conjugate to the tangent bundle to M equipped with an action of U(1)by dilatation along the fibers of the projection ρ : \overline{T} M → M. All the discussion above applies to the case X = \overline{T} M, Z = M ⊂ \overline{T} M. Moreover, the linearity condition in the form given in Lemma 4.3.3 generalizes immediately to the formal case. Definition. A formal Hodge manifold structure on \overline{T} M along M is called linear if for every smooth (0, 1)-form α ∈ C^{∞}(M, Λ^{0,1}(M)) we have

τ(α) = ⟨φ, j(ρ^{∗})⟩ ∈ C_{M}^{∞}(\overline{T} M, ℂ),

where j is the map induced by the formal quaternionic structure on \overline{T} M and φ and τ are as in Lemma 4.3.3. 6.2.2. As in the non-formal case, linear Hodge manifold structures on \overline{T} M along M ⊂ \overline{T} M can be described in terms of differential operators of certain type. Definition. A formal Hodge connection on \overline{T} M along M ⊂ \overline{T} M is an algebra derivation

D : Λ^{0}(\overline{T} M, ℂ) → ρ^{∗}Λ^{1}(M, ℂ)

in WHodge_{M}^{D}(\overline{T} M) such that for every smooth function f ∈ C^{∞}(M, ℂ) we have Dρ^{∗} = ρ^{∗}d_{M}f, as in (5.1). A formal Hodge connection is called flat if it extends to an algebra derivation

D : ρ^{∗}Λ (M, ℂ) → Λ^{+1}(M, ℂ)

in WHodge_{M}^{D}(\overline{T} M) such that D ∘ D = 0. A formal Hodge connection is called linear if it satisfies the condition (5.14) of Lemma 5.4.3, that is, for every function f ∈ C_{lin}^{∞}(\overline{T} M, ℂ) linear along the fibers of the projection ρ : \overline{T} M → M we have

f = \frac{1}{2}σ ((D − D^{ι})f) ,

where σ : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{0}(\overline{T} M, ℂ) is the twisted tautological map introduced in (5.6), the automorphism ι : \overline{T} M → \overline{T} M is the multiplication by −1 ∈ ℂ on every fiber of the projection ρ : \overline{T} M → M, and D^{ι} : Λ^{0}(\overline{T} M, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) is the operator ι^{∗}-conjugate to D, as in 5.4.1.

The discussion in Section 5 generalizes immediately to the formal case and gives the following.

Lemma. Linear formal Hodge manifold structures on \overline{T} M along the zero section M ⊂ \overline{T} M are in a natural one-to-one correspondence with linear flat formal Hodge connections on \overline{T} M along M.

6.3. The Weil algebra

6.3.1. Let, as before, M be a complex manifold and let \overline{T} M be the total space of the complex conjugate to its tangent bundle, as in 4.2.5. In the remaining part of this section we give a description of the set of all formal Hodge connections on \overline{T} M along M in terms of certain differential operators on M rather than on \overline{T} M. We call such operators extended connections on M (see 6.4.1 for the definition). Together with a complete classification of extended connections given in the next Section, this description provides a full classification of regular Hodge manifolds “in the formal neighborhood of the subset of U(1)-fixed points”. 6.3.2. Before we define extended connections in Subsection 6.4), we need to introduce a certain algebra bundle in WHodge(M) which we call the Weil algebra. We begin with some preliminary facts.

Recall (see, e.g., [D1]) that every additive category A admits a canonical completion Lim_{⟵} A with respect to filtered projective limits. The category Lim_{⟵} A is also additive, and it is tensor if A was tensor. Objects of the canonical completion Lim_{⟵} A are called pro-objects in A.

6.3.3. Let ρ : \overline{T} M → M be the canonical projection. Extend the pullback functor ρ^{∗} : Bun(M) → Bun(\overline{T} M) to a functor

ρ^{∗} : Bun(M) → Bun_{M}(\overline{T} M)

to the M-adic completion Bun_{M}(\overline{T} M). The functor ρ^{∗} admits a right adjoint direct image functor

ρ_{∗} : Bun_{M}(\overline{T} M) → Lim_{⟵} Bun(M).

Moreover, the functor ρ_{∗} extends to a functor

ρ_{∗} : Diff_{M}(\overline{T} M) → Lim_{⟵} Diff(M).

Denote by B^{0}(M, ℂ) = ρ_{∗}Λ^{0}(\overline{T} M) the direct image under the projection ρ : \overline{T} M → M of the trivial bundle Λ^{0}(\overline{T} M) on \overline{T} M.

The compact Lie group U(1) acts on \overline{T} M by dilatation along the fibers, and the functor ρ_{∗} : Diff_{M}(\overline{T} M) → Lim_{⟵} Diff(M) obviously extends to a

functor ρ_{∗} : WHodge_{M}^{D}(\overline{T} M) → Lim_{⟵} WHodge(M). The restriction of ρ_{∗} to

the subcategory WHodge_{M}(\overline{T} M) ⊂ WHodge_{M}^{D}(\overline{T} M) is adjoint on the right to the pullback functor ρ^{∗} : WHodge(M) → WHodge_{M}(\overline{T} M). 6.3.4. The constant bundle Λ^{0}(\overline{T} M) is canonically a Hodge bundle of weight 0. Therefore B^{0}(M, ℂ) = ρ_{∗}Λ^{0}(M, ℂ) is also a Hodge bundle of weight 0. Moreover, it is a commutative algebra bundle in Lim_{⟵} WHodge_{0}(M). Let

S^{1}(M, ℂ) be the conormal bundle to the zero section M ⊂ \overline{T} M equipped with a Hodge bundle structure of weight 0 as in 5.3.2, and denote by S^{i}(M, ℂ) the i-th symmetric power of the Hodge bundle S^{1}(M, ℂ). Then the algebra bundle B^{0}(M, ℂ) in Lim_{⟵} WHodge_{0}(M) is canonically isomorphic

B^{0}(M, ℂ) ≅ Ŝ(M, ℂ)

to the completion Ŝ(M, ℂ) of the symmetric algebra S (M, ℂ) of the Hodge bundle S^{1}(M, ℂ) with respect to the augmentation ideal S^{>0}(M, ℂ).

Since the U(1)-action on M is trivial, the category WHodge(M) of Hodge bundles on M is equivalent to the category of pairs ⟨E, ⟩ of a complex bundle E equipped with a Hodge type bigrading

E = ⨁_{p,q} E^{p,q}

and a real structure : E^{p,q} → \overline{E^{q,p}}. The Hodge type bigrading on B^{0}(M, ℂ) is induced by the Hodge type bograding S^{1}(M, ℂ) = S^{1,−1}(M) ⊕ S^{−1,1}(M) on the generators subbundle S^{1}(M, ℂ) ⊂ B^{0}(M, ℂ), which was described in 5.3.2.

Remark. The complex vector bundle S^{1}(M, ℂ) is canonically isomorphic to the cotangent bundle Λ^{1}(M, ℂ). However, the Hodge bundle structures on these two bundles are different (in fact, they have different weights). 6.3.5. Consider the pro-bundles

B (M, ℂ) = ρ_{∗}ρ^{∗}Λ (M, ℂ)

on M. The direct sum ⊕B (M, ℂ) is a graded algebra in Lim_{⟵} Bun(M, ℂ).

Moreover, since for every i ≥ 0 the bundle Λ^{i}(M, ℂ) is a Hodge bundle of weight i (see 2.1.7), B^{i}(M, ℂ) is also a Hodge bundle of weight i. Denote by

B^{i}(M, ℂ) = ⨁_{p+q=i} B^{p,q}(M, ℂ)

the Hodge type bigrading on B^{i}(M, ℂ).

The Hodge bundle structures on B (M, ℂ) are compatible with the multiplication. By the projection formula

B (M, ℂ) ≅ B^{0}(M, ℂ) ⊗ Λ (M, ℂ),

and this isomorphism is compatible with the Hodge bundle structures on both sides. Definition. Call the algebra B (M, ℂ) in Lim_{⟵} WHodge(M) the Weil algebra of the complex manifold M. 6.3.6. The canonical involution ι : \overline{T} M → \overline{T} M induces an algebra involution ι^{∗} : B (M, ℂ) → B (M, ℂ). It acts on generators as follows

ι^{∗} = −id : S^{1}(M, ℂ) → S^{1}(M, ℂ) ι^{∗} = id : Λ^{1}(M, ℂ) → Λ^{1}(M, ℂ).

For every operator N : B^{p}(M, ℂ) → B^{q}(M, ℂ), p and q arbitrary, we will denote by

N^{ι} = ι^{∗} ∘ N ∘ ι^{∗} : B^{p}(M, ℂ) → B^{q}(M, ℂ)

the operator ι^{∗}-conjugate to N. 6.3.7. The twisted tautological map σ : ρ^{∗}Λ^{1}(M, ℂ) → Λ^{0}(\overline{T} M, ℂ) introduced in 5.6 induces via the functor ρ_{∗} a map σ : B^{1}(M, ℂ) → B^{0}(M, ℂ). Extend this map to a derivation

σ : B^{+1}(M, ℂ) → B (M, ℂ)

by setting σ = 0 on S^{1}(M, ℂ) ⊂ B^{0}(M, ℂ). The derivation B^{+1}(M, ℂ) → B (M, ℂ) is not weakly Hodge. However, it is real with respect to the real structure on the Weil algebra B (M, ℂ).

6.3.8. By definition of the twisted tautological map (5.6, 4.3.2), the derivation σ : B^{+1}(M, ℂ) → B (M, ℂ) maps the subbundle Λ^{1}(M, ℂ) ⊂ B^{1}(M, ℂ) to the subbundle S^{1}(M, ℂ) ⊂ B^{0}(M, ℂ) and defines a complex vector bundle isomorphism σ : Λ^{1}(M, ℂ) → S^{1}(M, ℂ). To describe this isomorphism explicitly, recall that sections of the bundle B^{0}(M, ℂ) are the same as formal germs along M ⊂ \overline{T} M of smooth functions on the manifold \overline{T} M. The sections of the subbundle S^{1}(M, ℂ) ⊂ B^{0}(M, ℂ) form the subspace of functions linear along the fibers of the canonical projection ρ : \overline{T} M → M. The isomorphism σ : Λ^{1}(M, ℂ) → S^{1}(M, ℂ) induces an isomorphism between the space of smooth 1-forms on the manifold M and the space of smooth functions on \overline{T} M linear long the fibers of ρ : \overline{T} M → M. This isomorphism coincides with the tautological one on the subbundle Λ^{1,0} ⊂ Λ^{1}(M, ℂ), and it is minus the tautological isomorphism on the subbundle Λ^{0,1} ⊂ Λ^{1}(M, ℂ).

Denote by

C = σ^{−1} : S^{1}(M, ℂ) → Λ^{1}(M, ℂ)

the bundle isomorphism inverse to σ. Note that the complex vector bundle isomorphism σ : Λ^{1}(M, ℂ) → S^{1}(M, ℂ) is real. Moreover, it sends the subbundle Λ^{1,0}(M) ⊂ Λ^{1}(M, ℂ) to S^{1,−1}(M) ⊂ S^{1}(M, ℂ), and it sends Λ^{0,1}(M) to S^{−1,1}(M). Therefore the inverse isomorphism C : S^{1}(M, ℂ) → Λ^{1}(M, ℂ) is weakly Hodge. It coincides with the tautological isomorphism on the subbundle S^{1,−1} ⊂ S^{1}(M, ℂ), and it equals minus the tautological isomorphism on the subbundle S^{−1,1} ⊂ S^{1}(M, ℂ).

6.4. Extended connections

6.4.1. We are now ready to introduce the extended connections. Keep the notation of the last subsection. Definition. An extended connection on a complex manifold M is a differential operator D : S^{1}(M, ℂ) → B^{1}(M, ℂ) which is weakly Hodge in the sense of 2.1.3 and satisfies

D(fa) = fDa + a ⊗ df  (6.1)

for any smooth function f and a local section a of the pro-bundle B^{0}(M, ℂ). 6.4.2. Let D be an extended connection on the manifold M. By 6.3.5 we have canonical bundle isomorphisms

B^{1}(M, ℂ) ≅ B^{0}(M, ℂ) ⊗ Λ^{1}(M, ℂ) ≅ ⨁_{i≥0} S^{i}(M, ℂ) ⊗ Λ^{1}(M, ℂ).

Therefore the operator D : S^{1} → B^{1} decomposes

D = ∑_{p≥0} D_{p}, D_{p} : S^{1}(M, ℂ) → S^{i}(M, ℂ) ⊗ Λ^{1}(M, ℂ).  (6.2)

By (6.1) all the components D_{p} except for the D_{1} are weakly Hodge bundle maps on M, while

D_{1} : S^{1}(M, ℂ) → S^{1}(M, ℂ) ⊗ Λ^{1}(M, ℂ)

is a connection in the usual sense on the Hodge bundle S^{1}(M, ℂ). Definition. The weakly Hodge bundle map D_{0} : S^{1}(M, ℂ) → Λ^{1}(M, ℂ) is called the principal part of the extended connection D on M. The connection D_{1} is called the reduction of the extended connection D. 6.4.3. Extended connection on M are related to formal Hodge connections on the total space \overline{T} M of the complex-conjugate to the tangent bundle to M by means of the direct image functor

ρ_{∗} : WHodge_{M}^{D}(\overline{T} M) → Lim_{⟵} WHodge^{D}(M).

Namely, let D : Λ^{0}(M, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) be a formal Hodge connection on \overline{T} M along M in the sense of 6.2.2. The restriction of the operator

ρ_{∗}D : B^{0}(M, ℂ) → B^{1}(M, ℂ)

to the generators subbundle S^{1}(M, ℂ) ⊂ B^{0}(M, ℂ) is then an extended connection on M in the sense of 6.4.1. The principal part D_{0} : S^{1}(M, ℂ) → Λ^{1}(M, ℂ) of the Hodge connection D in the sense of 5.3.4 coincides with the principal part of the extended connection ρ_{∗}D. 6.4.4. We now write down the counterparts of the flatness and linearity conditions on a Hodge connection on \overline{T} M for the associated extended connection on M. We begin with the linearity condition 5.4.4. Let D : S^{1}(M, ℂ) → B^{1}(M, ℂ) be an extended connection on M, let σ : B^{+1}(M, ℂ) → B (M, ℂ) be the algebra derivation introduced in 6.3.7, and let

D^{ι} : B (M, ℂ) → B^{+1}(M, ℂ)

be the operator ι^{∗}-conjugate to D as in 6.3.6. Definition. An extended connection D is called linear if for every local section f of the bundle S^{1}(M, ℂ) we have

f = \frac{1}{2}σ((D − D^{ι})f).

This is, of course, the literal rewriting of Definition 5.4.4. In particular, a formal Hodge connection D on \overline{T} M is linear if and only if so is the extended connection ρ_{∗}D on M. 6.4.5. Next we rewrite the flatness condition 5.1.7. Again, let

D : S^{1}(M, ℂ) → B^{1}(M, ℂ)

be an extended connection on M. Since the algebra pro-bundle B^{0}(M, ℂ) is freely generated by the subbundle S^{1}(M, ℂ) ⊂ B^{1}(M, ℂ), by (6.1) the operator D : S^{1}(M, ℂ) → B^{1}(M, ℂ) extends uniquely to an algebra derivation

D : B^{0}(M, ℂ) → B^{1}(M, ℂ).

Moreover, we can extend this derivation even further to a derivation of the Weil algebra

D : B (M, ℂ) → B^{+1}(M, ℂ)

by setting

D(f ⊗ α) = Df ⊗ α + f ⊗ dα  (6.3)

for any smooth section f ∈ C^{∞}(M, B^{0}(M, ℂ)) and any smooth form α ∈ C^{∞}(M, Λ (M, ℂ)). We will call this extension the derivation, associated to the extended connection D.

Vice versa, the Weil algebra B (M, ℂ) is generated by the subbundles

S^{1}(M, ℂ), Λ^{1}(M, ℂ) ⊂ B (M, ℂ).

Moreover, for every algebra derivation D : B (M, ℂ) → B^{+1}(M, ℂ) the condition (6.3) completely defines the restriction of D to the generator subbundle Λ^{1}(M, ℂ) ⊂ B^{1}(M, ℂ). Therefore an algebra derivation D : B (M, ℂ) → B (M, ℂ) satisfying (6.3) is completely determined by its restriction to the generators subbundle S^{1}(M, ℂ) → B^{1}(M, ℂ). If the derivation D is weakly Hodge, then this restriction is an extended connection on M. 6.4.6. Definition. The extended connection D is called flat if the associated derivation satisfies D ∘ D = 0.

If a formal Hodge connection D on \overline{T} M is flat in the sense of 5.1.7, then we have a derivation D : ρ^{∗}Λ (M, ℂ) → ρ^{∗}Λ^{+1}(M, ℂ) such that D ∘ D = 0. The associated derivation ρ_{∗}D : B (M, ℂ) → B^{+1}(M, ℂ) satisfies (6.3). Therefore the extended connection ρ_{∗}D : S^{1}(M, ℂ) → B^{1}(M, ℂ) is also flat. 6.4.7. It turns out that one can completely recover a Hodge connection D on \overline{T} M from the corresponding extended connection ρ_{∗}D on M. More precisely, we have the following.

Lemma. The correspondence D ↦ ρ_{∗}D is a bijection between the set of formal Hodge connections on \overline{T} M along M ⊂ \overline{T} M and the set of extended connections on M. A connection D is flat, resp. linear if and only if ρ_{∗}D is flat, resp. linear.

Proof. To prove the first claim of the lemma, it suffices to prove that every extended connection on M comes from a unique formal Hodge connection on the pair ⟨\overline{T} M, M⟩. In general, the functor ρ_{∗} is not fully faithful on the category Diff(M), in other words, it does not induce an isomorphism on the spaces of differential operators between vector bundles on \overline{T} M. However, for every complex vector bundle E on \overline{T} M the functor ρ_{∗} does induce an isomorphism

ρ_{∗} : Der_{M}(Λ^{0}(M, ℂ), E) ≅ Der_{B^{0}(M,ℂ)}(B^{0}(M, ℂ), ρ_{∗}E)

between the space of derivations from Λ^{0}(M, ℂ) to F completed along M ⊂ \overline{T} M and the space of derivations from the algebra B^{0}(M, ℂ) = ρ_{∗}Λ^{0}(M, ℂ) to the B^{0}(M, ℂ)-module ρ_{∗}E. Therefore every derivation

D′ : B^{0}(M, ℂ) → B^{1}(M, ℂ) = ρ_{∗}ρ^{∗}Λ^{1}(M, ℂ)

must be of the form D′ = ρ_{∗}D for some derivation

D : Λ^{0}(\overline{T} M, ℂ) → ρ^{∗}Λ^{1}(M, ℂ)

It is easy to check that D is a Hodge connection if and only if D′ = ρ_{∗}D is weakly Hodge and satisfies (6.1). By 6.4.5 the space of all such derivations D′ : B^{0}(M, ℂ) → B^{1}(M, ℂ) coincides with the space of all extended connections on M, which proves the first claim of the lemma.

Analogously, for every extended connection D′ = ρ_{∗}D on M, the canonical extension of the operator D′ to an algebra derivation D′ : B (M, ℂ) → B^{+1}(M, ℂ) constructed in 6.4.5 must be of the form ρ_{∗}D for a certain weakly Hodge differential operator D : ρ^{∗}Λ (M, ℂ) → Λ^{+1}(M, ℂ). If the extended connection D′ is flat, then D′ ∘ D′ = 0. Therefore D ∘ D = 0, which means that the Hodge connection D is flat. Vice versa, if the Hodge connection D is flat, then it extends to a weakly Hodge derivation D : ρ^{∗}Λ (M, ℂ) → Λ^{+1}(M, ℂ) so that D∘D = 0. The equality D∘D = 0 implies, in particular, that the operator ρ_{∗}D vanishes on the sections of the form

Df = df ∈ C^{∞}(M, Λ^{1}(M, ℂ)) ⊂ C^{∞}(M, B^{1}(M, ℂ)),

where f ∈ C^{∞}(M, ℂ) is a smooth function on M. Therefore ρ_{∗}D coincides with the de Rham differential on the subbundle Λ^{1}(M, ℂ) ⊂ B^{1}(M, ℂ).

Hence by 6.4.5 it is equal to the canonical derivation D′ : B (M, ℂ) → B^{+1}(M, ℂ). Since D ∘ D = 0, we have D′ ∘ D′ = 0, which means that the extended connection D′ is flat.

Finally, the equivalence of the linearity conditions on the Hodge connection D and on the extended connection D′ = ρ_{∗}D is trivial and has already been noted in 6.4.4. □

This lemma together with Lemma 6.2.2 reduces the classification of linear formal Hodge manifold structures on \overline{T} M along the zero section M ⊂ \overline{T} M to the classification of extended connections on the manifold M itself.

7. Preliminaries on the Weil algebra

7.1. The total de Rham complex

7.1.1. Before we proceed further in the study of extended connections on a complex manifold M, we need to establish some linear-algebraic facts on the structure of the Weil algebra B (M, ℂ) defined in 6.3.5. We also need to introduce an auxiliary Hodge bundle algebra on M which we call the total Weil algebra. This is the subject of this section. Most of the facts here are of a technical nature, and the reader is advised to skip this section until needed. 7.1.2. We begin with introducing and studying a version of the de Rham complex of a complex manifold M which we call the total de Rham complex. Let M be a smooth complex U(1)-manifold. Recall that by 2.1.7 the de Rham complex Λ (M, ℂ) of the complex manifold M is canonically a Hodge bundle algebra on M. Let Λ_{tot}(M) = Γ(Λ (M, ℂ)) be the weight 0 Hodge bundle obtained by applying the functor Γ defined in 2.1.4 to the de Rham algebra Λ (M, ℂ). By 1.4.9 the bundle Λ_{tot}(M) carries a canonical algebra structure. By 2.1.7 the de Rham differential d_{M} is weakly Hodge. Therefore it induces an algebra derivation d_{M} : Λ_{tot}(M) → Λ_{tot}^{+1}(M) which is compatible with the Hodge bundle structure and satisfies d_{M} ∘ d_{M} = 0. Definition. The weight 0 Hodge bundle algebra Λ_{tot}(M) is called the total de Rham complex of the complex manifold M. 7.1.3. By definition

Λ_{tot}^{i}(M) = Γ(Λ^{i}(M, ℂ)) = Λ^{i}(M, ℂ) ⊗ W_{i}^{∗},

where W_{i}^{∗} = S^{i}W_{1}^{∗} is the symmetric power of the ℝ-Hodge structure W_{1}^{∗}, as in 1.4.4. To describe the structure of the algebra Λ_{tot}(M), we will use the following well-known general fact. (For the sake of completeness, we have included a sketch of its proof, see 7.1.12.)

Lemma. Let A, B be two objects in an arbitrary ℚ-linear symmetric tensor category A, and let C = S (A ⊗ B) be the sum of symmetric powers of the object A ⊗ B. Note that the object C is naturally a commutative algebra in A in the obvious sense. Let also C̃= ⨁_{k} S^{k}A ⊗ S^{k}B with the obvious commutative algebra structure. The isomorphism C^{1} ≅ C̃^{1} ≅ A ⊗ B extends to a surjective algebra map C → C̃, and its kernel 𝔍 ⊂ C is the ideal generated by the subobject 𝔍^{2} = Λ^{2}(A) ⊗ Λ^{2}(B) ⊂ S^{2}(A ⊗ B).

7.1.4. The category of complexes of Hodge bundles on M is obviously ℚlinear and tensor. Applying Lemma 7.1.3 to A = W_{1}^{∗}, B = Λ^{1}(M, ℂ)[1] immediately gives the following.

Lemma. The total de Rham complex Λ_{tot}(M) of the complex manifold M is generated by its first component Λ_{tot}^{1}(M), and the kernel of the canonical surjective algebra map

Λ (Λ_{tot}^{1}(M)) → Λ_{tot}(M)

from the exterior algebra of the bundle Λ_{tot}^{1}(M) to Λ_{tot}(M) is the ideal generated by the subbundle

Λ^{2}W_{1} ⊗ S^{2}(Λ^{1}(M, ℂ)) ⊂ S^{2}(Λ_{tot}^{1}(M)).

7.1.5. We can describe the Hodge bundle Λ_{tot}^{1}(M) more explicitly in the following way. By definition, as a U(1)-equivariant complex vector bundle it equals

Λ_{tot}^{1}(M) = Λ^{1}(M, ℂ)⊗W_{1}^{∗} = (Λ^{1,0}(M)(1) ⊕ Λ^{0,1}(M)(0))⊗(ℂ(0) ⊕ ℂ(−1)) ,

where Λ^{p,q}(M)(i) is the U(1)-equivariant bundle Λ^{p,q}(M) tensored with the 1-dimensional representation of weight i, and ℂ(i) is the constant U(1)- bundle corresponding to the representation of weight i. If we denote

S^{1}(M, ℂ) = Λ^{1,0}(M)(1) ⊕ Λ^{0,1}(M)(−1) ⊂ Λ_{tot}^{1}(M),
Λ_{ll}^{1}(M) = Λ^{1,0}(M) ⊂ Λ_{tot}^{1}(M),
Λ_{rr}^{1}(M) = Λ^{0,1}(M) ⊂ Λ_{tot}^{1}(M),

then we have

Λ_{tot}^{1}(M) = S^{1}(M, ℂ) ⊕ Λ_{ll}^{1}(M) ⊕ Λ_{rr}^{1}(M).

The complex conjugation : Λ_{tot}^{1}(M) → ι^{∗}\overline{Λ_{tot}^{1}(M)} preserves the subbundle

S^{1}(M, ℂ) ⊂ Λ_{tot}^{1}(M, ℂ)

and interchanges Λ_{ll}^{1}(M) and Λ_{rr}^{1}(M). 7.1.6. If the U(1)-action on the manifold M is trivial, then Hodge bundles are the same as bigraded complex vector bundles with a real structure. In this case the Hodge bigrading on the Hodge bundle Λ_{tot}^{1}(M, ℂ) is given by

(Λ_{tot}^{1}(M))^{1,−1} = S^{1,−1}(M, ℂ) = Λ^{1,0}(M)(1),
(Λ_{tot}^{1}(M))^{−1,1} = S^{−1,1}(M, ℂ) = Λ^{0,1}(M)(−1),
(Λ_{tot}^{1}(M))^{0,0} = Λ_{ll}^{1}(M) ⊕ Λ_{rr}^{1}(M) = Λ^{1}(M, ℂ).

Under these identifications, the real structure on Λ_{tot}^{1}(M, ℂ) is minus the one induced by the usual real structure on the complex vector bundle Λ^{1}(M, ℂ). Remark. The Hodge bundle S^{1}(M, ℂ) is canonically isomorphic to the conormal bundle to the zero section M ⊂ \overline{T} M, which we have described in 5.3.2. 7.1.7. Recall now that we have defined in 1.4.7 canonical embeddings γ_{l}, γ_{r} : W_{p}^{∗} → W_{k}^{∗} for every 0 ≤ p ≤ k. Since W_{0}^{∗} = ℂ, for every p, q ≥ 0 we have by (1.3) a short exact sequence

0 ⟶ ℂ ⟶ W_{p}^{∗} ⊕ W_{q}^{∗} ⟶^{γ_{l}⊕γ_{r}} W_{p+q}^{∗} ⟶ 0  (7.1)

of complex vector spaces. Recall also that the embeddings γ_{l}, γ_{r} are compatible with the natural maps can : W_{p}^{∗} ⊗ W_{q}^{∗} → W_{p+q}^{∗}. Therefore the subbundles defined by

Λ_{l}^{k}(M) = ⨁_{0≤p≤k} γ_{l}(W_{p}^{∗}) ⊗ Λ^{p,k−p}(M) ⊂ Λ_{tot}^{k}(M) = ⨁_{0≤p≤k} W_{k}^{∗} ⊗ Λ^{p,k−p}(M)
Λ_{r}^{k}(M) = ⨁_{0≤p≤k} γ_{r}(W_{p}^{∗}) ⊗ Λ^{k−p,p}(M) ⊂ Λ_{tot}^{k}(M) = ⨁_{0≤p≤k} W_{k}^{∗} ⊗ Λ^{k−p,p}(M)

are actually subalgebras in the total de Rham complex Λ_{tot}(M). 7.1.8. To describe the algebras Λ_{l}(M) and Λ_{r}(M) explicitly, note that we obviously have Λ_{tot}(M) = Λ_{l}(M) + Λ_{r}(M). Moreover, in the notation of 7.1.5 we have

Λ_{l}^{1}(M) = S^{1}(M, ℂ) ⊕ Λ_{ll}^{1}(M) ⊂ Λ_{tot}^{1}(M),
Λ_{r}^{1}(M) = S^{1}(M, ℂ) ⊕ Λ_{rr}^{1}(M) ⊂ Λ_{tot}^{1}(M).

By Lemma 7.1.3, the algebra

Λ_{l}(M) = (⨁_{p} W_{p}^{∗} ⊗ Λ^{p,0}(M)) ⊗ (⨁_{q} Λ^{0,q}(M))

is the subalgebra in the total de Rham complex Λ_{tot}(M) generated by Λ_{l}^{1}(M), and the ideal of relations is generated by the subbundle

S^{2}(Λ^{1,0}(M)) ⊗ Λ^{2}(W_{1}^{∗}) ⊂ Λ^{2}(Λ_{l}^{1}(M)).

Analogously, the subalgebra Λ_{r}(M) ⊂ Λ_{tot}^{1}(M) is generated by Λ_{r}^{1}(M), and the relations are generated by

S^{2}(Λ^{0,1}(M)) ⊗ Λ^{2}(W_{1}^{∗}) ⊂ Λ^{2}(Λ_{r}^{1}(M)).

7.1.9. We will also need to consider the ideals in these algebras defined by

Λ_{ll}^{k}(M) = ⨁_{1≤p≤k} γ_{l}(W_{p}^{∗}) ⊗ Λ^{p,k−p}(M) ⊂ Λ_{l}^{k}(M)
Λ_{rr}^{k}(M) = ⨁_{1≤p≤k} γ_{r}(W_{p}^{∗}) ⊗ Λ^{k−p,p}(M) ⊂ Λ_{r}^{k}(M)

The ideal Λ_{ll}(M) ⊂ Λ_{l}^{1}(M) is generated by the subbundle Λ_{ll}^{1}(M) ⊂ Λ_{l}^{1}(M), and the ideal Λ_{rr}(M) ⊂ Λ_{r}^{1}(M) is generated by the subbundle Λ_{rr}^{1}(M) ⊂ Λ_{r}^{1}(M). 7.1.10. Denote by Λ_{o}(M) = Λ_{l}(M) ∩ Λ_{r}(M) ⊂ Λ_{tot}(M) the intersection of the subalgebras Λ_{l}(M) and Λ_{r}(M). Unlike either of these subalgebras, the subalgebra Λ_{o}(M) ⊂ Λ_{tot}(M) is compatible with the weight 0 Hodge bundle structure on the total de Rham complex. By (7.1) we have a short exact sequence

0 ⟶ Λ (M, ℂ) ⟶ Λ_{l}(M) ⊕ Λ_{r}(M) ⟶ Λ_{tot}(M) ⟶ 0

(7.2)

of complex vector bundles on M. Therefore the algebra Λ_{o}(M) is isomorphic, as a complex bundle algebra, to the usual de Rham complex Λ (M, ℂ). As a Hodge bundle algebra it is canonically isomorphic to the exterior algebra of the Hodge bundle S^{1}(M, ℂ) of weight 0 on the manifold M.

Finally, note that the short exact sequence (7.2) induces a direct sum decomposition

Λ_{tot}(M) ≅ Λ_{ll}(M) ⊕ Λ_{o}(M) ⊕ Λ_{rr}(M).

7.1.11. Remark. The total de Rham complex Λ_{tot}(M) is related to Simpson’s theory of Higgs bundles (see [S3]) in the following way. Recall that Simpson has proved that every (sufficiently stable) complex bundle E on a compact complex manifold M equipped with a flat connection ∇ admits a unique Hermitian metric h such that ∇ and the 1-form θ = ∇ − ∇^{h} ∈ C^{∞}(M, Λ^{1}(End E)) satisfy the so-called harmonicity condition. He also has shown that this condition is equivalent to the vanishing of a certain curvature-like tensor R ∈ Λ^{2}(M, End E) which he associated canonically to every pair ⟨∇, θ⟩.

Recall that flat bundles ⟨E, ∇⟩ on the manifold M are in one-to-one correspondence with free differential graded modules E ⊗ Λ (M, ℂ) over the de Rham complex Λ (M, ℂ). It turns out that complex bundles E equipped with a flat connection ∇ and a 1-form θ ∈ C^{∞}(M, Λ^{1}(End E)) such that Simpson’s tensor R vanishes are in natural one-to-one correspondence with free differential graded modules E ⊗Λ_{tot}(M) over the total de Rham complex Λ_{tot}(M). Moreover, a pair ⟨θ, ∇⟩ comes from a variation of pure ℝ-Hodge structure on E if and only if there exists a Hodge bundle structure on E such that the product Hodge bundle structure on the free module E ⊗ Λ_{tot}(M) is compatible with the differential. 7.1.12. Proof of Lemma 7.1.3. For every k ≥ 0 let G = Σ_{k} × Σ_{k} be the product of two copies of the symmetric group Σ_{k} on k letters. Let V_{k} be the ℚ-representation of G_{k} induced from the trivial representation of the diagonal subgroup Σ_{k} ⊂ G_{k}. The representation V_{k} decomposes as

V_{k} = ⨁_{V} V ⊠ V,

where the sum is over the set of irreducible representations V of Σ_{k}. We obviously have

C^{k} = Hom_{G_{k}} (V_{k}, A^{⊗k} ⊗ B^{⊗k}) = ⨁_{V} Hom_{Σ_{k}} (V, A^{⊗k}) ⊗ Hom_{Σ_{k}} (V, A^{⊗k}) .

Let 𝔍 ⊂ C be the ideal generated by Λ^{2}A ⊗ Λ^{2}B ⊂ S^{2}(A ⊗ B). It is easy to see that

𝔍^{k} = ∑_{1≤l≤k−1} ⨁_{V} Hom_{Σ_{k}} (V, A^{⊗k}) ⊗ Hom_{Σ_{k}} (V, A^{⊗k}) ⊂ C^{k},

where the first sum is taken over the set of k − 1 subgroups Σ_{2} ⊂ Σ_{k}, the l-th one transposing the l-th and the l + 1-th letter, while the second sum is taken over all irreducible constituents V of the representation of Σ_{k} induced from the sign representation of the corresponding Σ_{2} ⊂ Σ_{k}. Now, there is obviously only one irreducible representation of Σ_{k} which is not encountered as an index in this double sum, namely, the trivial one. Hence C^{k}/𝔍^{k} = S^{k}A ⊗ S^{k}B, which proves the lemma. □

7.2. The total Weil algebra

7.2.1. Assume from now on that the U(1)-action on the complex manifold M is trivial. We now turn to studying the Weil algebra of the manifold M. Let S^{1}(M, ℂ) = S^{1,−1}(M, ℂ) ⊕ S^{−1,1}(M, ℂ) be the weight 0 Hodge bundle on M introduced in 5.3.2. To simplify notation, denote

S = Ŝ(S^{1}(M, ℂ))
Λ = Λ (M, ℂ),

where Ŝis the completed symmetric power, and let

B = B (M, ℂ) = S ⊗ Λ

be the Weil algebra of the complex manifold M introduced in 6.3.5. Recall that the algebra B carries a natural Hodge bundle structure. In particular, it is equipped with a Hodge type bigrading B^{i} = ∑_{p+q=i} B^{p,q}. 7.2.2. We now introduce a different bigrading on the Weil algebra B . The commutative algebra B is freely generated by the subbundles

S^{1} = S^{1,−1} ⊕ S^{−1,1} ⊂ B^{0} and Λ^{1} = Λ^{1,0} ⊕ Λ^{0,1} ⊂ B^{1},

therefore to define a multiplicative bigrading on the algebra B it suffices to assign degrees to these generator subbundles S^{1,−1}, S^{−1,1}, Λ^{1,0}, Λ^{0,1} ⊂ B . Definition. The augmentation bigrading on B is the multiplicative bigrading defined by setting

deg S^{1,−1} = deg Λ^{1,0} = (1, 0)
deg S^{−1,1} = deg Λ^{0,1} = (0, 1)

on generators S^{1,−1}, S^{−1,1}, Λ^{1,0}, Λ^{0,1} ⊂ B^{,}.

We will denote by B_{p,q}^{,} the component of the Weil algebra of augmentation bidegree (p, q). For any linear map a : B → B we will denote by a = ∑_{p,q} a_{p,q} its decomposition with respect to the augmentation bidegree.

It will also be useful to consider a coarser augmentation grading on B , defined by deg B_{p,q} = p + q. We will denote by B_{k} = ⨁_{p+q=k} B_{p,q} the component of B of augmentation degree k.

7.2.3. Note that the Hodge bidegree and the augmentation bidegree are, in general, independent. Moreover, the complex conjugation : B → \overline{B} sends B_{p,q} to \overline{B_{q,p}}. Therefore the augmentation bidegree components B_{p,q} ⊂ B are not Hodge subbundles. However, the coarser augmentation grading is compatible with the Hodge structures, and the augmentation degree kcomponent B_{k}^{i} ⊂ B^{i} carries a natural Hodge bundle structure of weight i. Moreover, the sum B_{p,q} + B_{q,p} ⊂ B is also a Hodge subbundle. 7.2.4. We now introduce an auxiliary weight 0 Hodge algebra bundle on M, called the total Weil algebra. Recall that we have defined in 2.1.4 a functor Γ : WHodge_{≥0}(M) → WHodge_{0}(M) adjoint on the right to the canonical embedding. Consider the Hodge bundle B_{tot} = Γ(B ) of weight 0 on M. By 1.4.9 the multiplication on B induces an algebra structure on Γ(B ). Definition. The Hodge algebra bundle B_{tot} of weight 0 is called the total Weil algebra of the complex manifold M. Remark. For a more conceptual description of the functor Γ and the total Weil algebra, see Appendix. 7.2.5. By definition of the functor Γ we have B_{tot}^{k} = B^{k} ⊗ W_{k}^{∗} = S ⊗ Λ^{k} ⊗ W_{k}^{∗} = S ⊗ Λ_{tot}^{k}, where Λ_{tot}^{k} = Λ^{k} ⊗ W^{∗} = Γ(Λ^{k}) is the total de Rham complex introduced in Subsection 7.1. We have also introduced in Subsection 7.1 Hodge bundle subalgebras Λ_{o}, Λ_{l}, Λ_{r} ⊂ Λ_{tot} in the total de Rham complex Λ_{tot} and ideals Λ_{ll} ⊂ Λ_{l}, Λ_{rr} ⊂ Λ_{r} in the algebras Λ_{l}, Λ_{r}. Let

B_{o}^{k} = S ⊗ Λ_{o}^{k} ⊂ B_{tot}^{k}
B_{l}^{k} = S ⊗ Λ_{l}^{k} ⊂ B_{tot}^{k}
B_{r}^{k} = S ⊗ Λ_{r}^{k} ⊂ B_{tot}^{k}

be the associated subalgebras in the total Weil algebra B_{tot} and let

B_{ll}^{k} = S ⊗ Λ_{ll}^{k} ⊂ B_{l}^{k}
B_{rr}^{k} = S ⊗ Λ_{rr}^{k} ⊂ B_{r}^{k}

be the corresponding ideals in the Hodge bundle algebras B_{l}, B_{r}.

By 7.1.10 we have bundle isomorphisms Λ_{tot} = Λ_{l}+Λ_{r}and Λ_{o}= Λ_{l}∩Λ_{r}, and the direct sum decomposition Λ_{tot} ≅ Λ_{ll}⊕ Λ_{o}⊕ Λ_{rr}. Therefore we also have

\begin{aligned}B_{tot} & = B_{l}+ B_{r}= B_{ll}⊕ B_{o}⊕ B_{rr} \\ B_{o} & = B_{l}⋂ B_{r}⊂ B_{tot} \end{aligned}  (7.3)

Moreover, the algebra Λ_{o} is isomorphic to the usual de Rham complex Λ , therefore the subalgebra B_{o}⊂ B_{tot} is isomorphic to the usual Weil algebra B . These isomorphisms are not weakly Hodge. 7.2.6. The total Weil algebra carries a canonical weight 0 Hodge bundle structure, and we will denote the corresponding Hodge type grading by upper indices: B_{tot} = ⊕_{p} (B_{tot})^{p,−p}. The augmentation bigrading on the Weil algebra introduced in 7.2.2 extends to a bigrading of the total Weil algebra, which we will denote by lower indices. In general, both these grading and the direct sum decomposition (7.3) are independent, so that, in general, for every i ≥ 0 we have a decomposition

B_{tot}^{i} = ⨁_{n,p,q} (B_{ll}^{i})_{p,q}^{n,−n} ⊕ (B_{o}^{i})_{p,q}^{n,−n} ⊕ (B_{rr}^{i})_{p,q}^{n,−n} .

We would like to note, however, that some terms in this decomposition vanish when i = 0, 1. Namely, we have the following fact.

Lemma. Let n, k be arbitrary integers such that k ≥ 0.

(i) If n + k is odd, then (B_{tot}^{0})_{k}^{n,−n} = 0.

(ii) If n + k is even, then (B_{ll}^{1})_{k}^{n,−n} = (B_{rr}^{1})_{k}^{n,−n} = 0, while if n + k is odd, then (B_{o}^{1})_{k}^{n,−n} = 0.

Proof.

(i) The bundle B_{tot}^{0} by definition coincides with B^{0}, and it is generated by the subbundles S^{1,−1}, S^{−1,1} ⊂ B^{0}. Both these subbundles have augmentation degree 1 and Hodge degree ±1, so that the sum n + k of the Hodge degree with the augmentation degree is even. Since both gradings are multiplicative, for all non-zero components B_{k}^{n,−n} ⊂ B^{0} the sum n + k must also be even.

(ii) By definition we have B_{tot}^{1} = B^{0} ⊗ Λ_{tot}^{1}. The subbundle Λ_{tot}^{1} ⊂ B_{tot}^{1} has augmentation degree 1, and it decomposes

Λ_{tot}^{1} = Λ_{o}^{1} ⊕ Λ_{ll}^{1} ⊕ Λ_{rr}^{1}.

By 7.1.5 we have Λ_{o}^{1} ≅ S^{1} = S^{1,−1} ⊕ S^{−1,1} as Hodge bundles, so that the Hodge degrees on Λ_{o}^{1} ⊂ Λ_{tot}^{1} are odd. On the other hand, the subbundles Λ_{ll}^{1}, Λ_{rr}^{1} ⊂ Λ_{tot}^{1} are by 7.1.6 of Hodge bidegree (0, 0). Therefore the sum n + k of the Hodge and the augmentation degrees is even for Λ_{o}^{1} and odd for Λ_{ll}^{1} and Λ_{rr}^{1}. Together with (i) this proves the claim. □

7.3. Derivations of the Weil algebra

7.3.1. We will now introduce certain canonical derivations of the Weil algebra B (M, ℂ) which will play an important part in the rest of the paper. First of all, to simplify notation, for any two linear maps a, b let

{a, b} = a ∘ b + b ∘ a

be their anticommutator, and for any linear map a : B → B^{+i} let a = ∑_{p+q=i} a^{p,q} be the Hodge type decomposition. The following fact is wellknown, but we have included a proof for the sake of completeness.

Lemma. For every two odd derivations P, Q of a graded-commutative algebra A, their anticommutator {P, Q} is an even derivation of the algebra A.

Proof. Indeed, for every a, b ∈ A we have

{P, Q}(ab) = P (Q(ab)) + Q(P (ab))

= P (Q(a)b + (−1)^{deg a}aQ(b)) + Q(P (a)b + (−1)^{deg a}aP (b))

= P (Q(a))b + (−1)^{deg Q(a)}Q(a)P (b) + (−1)^{deg a}P (a)Q(b)
+ aP (Q(b)) + Q(P (a))b + (−1)^{deg P (a)}P (a)Q(b)
+ (−1)^{deg a}Q(a)P (b) + aQ(P (b))
= P (Q(a))b − (−1)^{deg a}Q(a)P (b) + (−1)^{deg a}P (a)Q(b)
+ aP (Q(b)) + Q(P (a))b − (−1)^{deg a}P (a)Q(b)
+ (−1)^{deg a}Q(a)P (b) + aQ(P (b))
= P (Q(a))b + aP (Q(b)) + Q(P (a))b + aQ(P (b))
= {P, Q}(a)b + a{P, Q}(b).
□

7.3.2. Let C : S^{1} → Λ^{1} be the canonical weakly Hodge map introduced in 6.3.8. Extend C to an algebra derivation C : B → B^{+1} by setting C = 0 on Λ^{1} ⊂ B^{1}. By 6.3.8 the derivation C is weakly Hodge. The composition

C ∘ C = \frac{1}{2}\{C, C\} : B → B^{+2}

is also an algebra derivation, and it obviously vanishes on the generator subbundles S^{1}, Λ^{1} ⊂ B . Therefore C ∘ C = 0 everywhere.

Let also σ : B^{+1} → B be the derivation introduced in 6.3.7. The derivation σ is not weakly Hodge; however, it is real and admits a decomposition σ = σ^{−1,0} +σ^{0,−1} into components of Hodge types (−1, 0) and (0, −1). Both these components are algebra derivations of the Weil algebra B . We obviously have σ∘σ = σ^{−1,0}∘σ^{−1,0} = σ^{0,−1}∘σ^{0,−1} = 0 on generators S^{1}, Λ^{1} ⊂ B , and, therefore, on the whole Weil algebra. Remark. Up to a sign the derivations C, σ and their Hodge bidegree components coincide with the so-called Koszul differentials on the Weil algebra B = S ⊗ Λ . 7.3.3. The derivation C : B → B^{+1} is by definition weakly Hodge. Applying the functor Γ to it, we obtain a derivation C : B_{tot} → B_{tot}^{+1} of the total Weil algebra B_{tot} preserving the weight 0 Hodge bundle structure on B_{tot}. The canonical identification B ≅ B_{o} ⊂ B_{tot} is compatible with the derivation C : B_{tot} → B_{tot}^{+1}. Moreover, by 7.3.2 this derivation satisfies C ∘ C = 0 : B_{tot} → B_{tot}^{+2}. Therefore the total Weil algebra B_{tot} equipped with the derivation C is a complex of Hodge bundles of weight 0.

The crucial linear algebraic property of the total Weil algebra B_{tot} of the manifold M which will allow us to classify flat extended connections on M is the following.

Proposition 7.1 Consider the subbundle

⨁_{p,q≥1} (B_{tot})_{p,q} ⊂ B_{tot}  (7.4)

of the total Weil algebra B_{tot} consisting of the components of augmentation bidegrees (p, q) with p, q ≥ 1. This subbundle equipped with the differential C : (B_{tot})_{,}→ (B_{tot}^{+1})_{,}is an acyclic complex of Hodge bundles of weight 0 on M.

7.3.4. We sketch a more or less simple and conceptual proof of Proposition 7.1 in the Appendix. However, in order to be able to study in Section 10 the analytic properties of our formal constructions, we will need an explicit contracting homotopy for the complex (7.4), which we now introduce.

The restriction of the derivation C : B_{tot}^{0} → B_{tot}^{1} to the subbundle S^{1} ⊂ B_{0} ≅ B_{tot}^{0} induces a Hodge bundle isomorphism

C : S^{1} → Λ_{o}^{1} ⊂ Λ_{tot}^{1} ⊂ B_{tot}^{1}.

Define a map σ_{tot} : Λ_{tot}^{1} → S^{1} by

σ_{tot} = \begin{cases}0 & on Λ_{ll}^{1}, Λ_{rr}^{1} ⊂ Λ_{tot}^{1}, \\ C^{−1} & on Λ_{o}^{1} ⊂ Λ_{tot}^{1}. \end{cases}  (7.5)

The map σ_{tot} : Λ_{tot}^{1} → S^{1} preserves the Hodge bundle structures of weight 0 on both sides. Moreover, its restriction to the subbundle Λ^{1} ≅ Λ_{o}^{1} ⊂ Λ_{tot}^{1} coincides with the canonical map σ : Λ^{1} → S^{1} introduced in 7.3.2. 7.3.5. Unfortunately, unlike σ : Λ^{1} → S^{1}, the map σ_{tot} : Λ_{tot}^{1} → S^{1} does not admit an extension to a derivation B_{tot}^{+1} → B_{tot} of the total Weil algebra B_{tot}. We will extend it to a bundle map σ_{tot} : B_{tot}^{+1} → B_{tot} in a somewhat roundabout way. To do this, define a map σ_{l} : Λ_{l}^{1} → S^{0} ⊂ B^{0} by

σ_{l} = \begin{cases}0 & on Λ_{ll}^{1} ⊂ Λ_{l}^{1} and on (Λ_{o}^{1})^{−1,1} ⊂ Λ_{tot}^{1}, \\ C^{−1} & on (Λ_{o}^{1})^{1,−1} ⊂ Λ_{tot}^{1}. \end{cases}

and set σ_{l} = 0 on S^{1}. By 7.1.7 we have

Λ_{l}^{1} = Λ^{1,0} ⊕ (Λ^{0,1} ⊗ W_{1}^{∗}) .

The map σ_{l} : Λ_{l}^{1} → S^{1} vanishes on the second summand in this direct sum, and it equals C^{−1} : Λ^{1,0} → S^{1,−1} ⊂ S^{1} on the first summand. The restriction of the map σ_{l} to the subbundle

Λ^{1,0} ⊕ Λ^{0,1} = Λ^{1} ≅ Λ_{o}^{1} ⊂ Λ_{l}^{1}

vanishes on Λ^{0,1} and equals C^{−1} on Λ^{1,0}. Thus it is equal to the Hodge type-(0, −1) component σ^{0,−1} : Λ^{1} → S^{1} of the canonical map σ : Λ^{1} → S^{1}.

7.3.6. By 7.1.8 the algebra B_{l}is generated by the bundles S^{1} and Λ_{l}^{1}, and the ideal of relations is generated by the subbundle

S^{2} (Λ^{0,1}) ⊗ Λ^{2} (W_{1}^{∗}) ⊂ Λ^{2} (Λ_{l}^{1}) .  (7.6)

Since the map σ_{l} : Λ_{l}^{1} → S^{1} vanishes on Λ^{0,1} ⊗ W_{1}^{∗} ⊂ Λ_{l}^{1}, it extends to an algebra derivation σ_{l} : B_{l}^{+1} → B_{l}. The restriction of the derivation σ_{l} to the subalgebra B ≅ B_{0}⊂ B_{tot} coincides with the (0, −1)-component σ^{0,−1} of the derivation σ : B^{+1} → B .

Analogously, the (−1, 0)-component σ^{−1,0} of the derivation σ : B^{+1} → B extends to an algebra derivation σ_{r} : B_{r}^{+1} → B_{r} of the subalgebra B_{r} ⊂ B_{tot}. By definition, the derivation σ_{l} preserves the decomposition B_{l} = B_{ll} ⊕ B_{o}, while the derivation σ_{r} preserves the decomposition B_{r} = B_{rr} ⊕ B_{o}. Both these derivations vanish on B_{tot}^{0}, therefore both are maps of B_{tot}^{0}-modules. In addition, the compositions σ_{l} ∘ σ_{l} and σ_{r} ∘ σ_{r} vanish on generator and, therefore, vanish identically. 7.3.7. Extend both σ_{l} and σ_{r} to the whole B_{tot} by setting

σ_{l} = 0 on B_{r} σ_{r} = 0 on B_{l},  (7.7)

and let

σ_{tot} = σ_{l} + σ_{r} : B_{tot}^{+1} → B_{tot}.

On Λ_{tot}^{1} ⊂ B_{tot}^{1} this is the same map as in (7.5). The bundle map σ_{tot} : B_{tot}^{+1} → B_{tot} preserves the direct sum decomposition (7.3), and its restriction to B_{o} ⊂ B_{tot} coincides with the derivation σ. Note that neither of the maps σ_{l}, σ_{r}, σ_{tot} is a derivation of the total Weil algebra B_{tot}. However, all these maps are linear with respect to the B_{tot}^{0}-module structure on B_{tot} and preserve the decomposition (7.3). The map σ_{tot} : B_{tot}^{+1} → B_{tot} is equal to σ_{l}

on B_{ll} ⊂ B_{tot}, to σ_{r} on B_{rr} and to σ : B^{+1} → B on B ≅ B_{o}⊂ B_{tot}. Since

σ_{l} ∘ σ_{l} = σ_{r} ∘ σ_{r} = σ ∘ σ = 0, we have σ_{tot} ∘ σ_{tot} = 0. 7.3.8. The commutator

h = {C, σ_{tot}} : B_{tot} → B_{tot}

of the maps C and σ_{tot} also preserves the decomposition (7.3), and we have the following.

Lemma. The map h acts as multiplication by p on (B_{ll})_{p,q}, as multiplication by q on (B_{rr})_{p,q} and as multiplication by (p + q) on (B_{o})_{p,q}.

Proof. It suffices to prove the claim separately on each term in the decomposition (7.3). By definition σ_{tot} = σ_{l} +σ_{r}, and h = h_{l} +h_{r}, where h_{l} = {σ_{l}, C} and h_{r} = {σ_{r}, C}. Moreover, h_{l} vanishes on B_{rr} and h_{r} vanishes on B_{ll}. Therefore it suffices to prove that h_{l} = pid on (B_{l})_{p,q} and that h_{r} = qid on (B_{r})_{p,q}. The proofs of these two identities are completely symmetrical, and we will only give a proof for h_{l}.

The algebra B_{l} is generated by the subbundles S^{1} ⊂ B_{l}^{0} and Λ_{l}^{1} ⊂ B_{l}^{1}. The augmentation bidegree decomposition of S^{1} is by definition given by

S_{1,0}^{1} = S^{1,−1} S_{0,1}^{1} = S^{−1,1},

while the augmentation bidegree decomposition of Λ_{l}^{1} is given by

(Λ_{l}^{1})_{1,0} = Λ^{1,0} (Λ_{l}^{1})_{0,1} = Λ^{0,1} ⊗ W_{1}^{∗}.

By the definition of the map σ_{l} : Λ_{l}^{1} → S^{1} (see 7.3.5) we have h_{l} = {C, σ_{l}} = id on Λ^{1,0} and S^{1,−1}, and h_{l} = 0 on Λ^{0,1} ⊗ W_{1}^{∗} and on S^{−1,1}. Therefore for every p, q ≥ 0 we have h_{l} = pid on the generator subbundles S_{p,q}^{1} and on (Λ_{l}^{1})_{p,q}. Since the map h_{l} is a derivation and the augmentation bidegree is multiplicative, the same holds on the whole algebra B_{l}= ⊕ (B_{l})_{p,q}. □

Lemma 7.3.8 shows that the map σ_{tot} is a homotopy, contracting the subcomplex (7.4) in the total Weil algebra B_{tot}, which immediately implies Proposition 7.1. Remark. In fact, in our classification of flat extended connections given in Section 8 it will be more convenient for us to use Lemma 7.3.8 directly rather than refer to Proposition 7.1. 7.3.9. We finish this section with the following corollary of Lemma 7.2.6 and Lemma 7.3.8, which we will need in Section 10.

Lemma. Let n = ±1. If the integer k ≥ 1 is odd, then the map h : B_{tot} → B_{tot} acts on (B_{tot}^{1})_{k}^{n,−n} by multiplication by k. If k = 2m ≥ 1 is even, then the endomorphism h : (B_{tot}^{1})_{k}^{n,−n} → (B_{tot}^{1})_{k}^{n,−n} is diagonalizable, and its only eigenvalues are m and m − 1.

Proof. If k is odd, then by Lemma 7.2.6 we have (B_{tot}^{1})_{k}^{n,−n} = (B_{o}^{1})_{k}^{n,−n}, and Lemma 7.3.8 immediately implies the claim. Assume that the integer k = 2m is even. By Lemma 7.2.6 we have

(B_{tot})_{k}^{n,−n} = (B_{ll})_{k}^{n,−n} ⊕ (B_{rr})_{k}^{n,−n} = B_{k−1}^{n,−n} ⊗ (Λ_{ll}^{1} ⊕ Λ_{rr}^{1}) .

The bundle B^{0} is generated by subbundles S^{1,−1} and S^{−1,1}. The first of these subbundles has augmentation bidegree (1, 0), while the second one has augmentation bidegree (0, 1). Therefore for every augmentation bidegree component B_{p,q}^{n,−n} ⊂ B_{k−1}^{n,−n} we have p−q = n and p+q = k −1. This implies that B_{k−1}^{n,−n} = B_{p,q}^{n,−n} with p = m − (1 − n)/2 and q = m − (1 + n)/2.

By definition the augmentation bidegrees of the bundles Λ_{ll}^{1} and Λ_{rr}^{1} are, respectively, (0, 1) and (1, 0). Lemma 7.3.8 shows that the only eigenvalue of the map h on (B_{ll}^{1})_{k+1} is p = (m − (1 − n)/2)), while its only eigenvalue on (B_{rr}^{1})_{k+1} is q = (m − (1 + n)/2). Since n = ±1, one of these numbers equals m and the other one equals m − 1. □

8. Classification of flat extended connections

8.1. Kählerian connections

8.1.1. Let M be a complex manifold. In Section 6 we have shown that formal Hodge manifold structures on the tangent bundle \overline{T} M are in one-to-one correspondence with linear flat extended connections on the manifold M (see 6.4.1–6.4.6 for the definitions). It turns out that flat linear extended connections on M are, in turn, in natural one-to-one correspondence with differential operators of a much simpler type, namely, connections on the cotangent bundle Λ^{1,0}(M) satisfying certain vanishing conditions (Theorem 8.1). We call such connections Kählerian. In this section we use the results of Section 7 establish the correspondence between extended connections on M and Kählerian connections on Λ^{1,0}(M). 8.1.2. We first give the definition of Kählerian connections. Assume that the manifold M is equipped with a connection

∇ : Λ^{1}(M) → Λ^{1}(M) ⊗ Λ^{1}(M)

on its cotangent bundle Λ^{1}(M). Let

T = Alt ∘∇ − d_{M} : Λ^{1}(M) → Λ^{2}(M)
R = Alt ∇ ∘ ∇ : Λ^{1}(M) → Λ^{1}(M) ⊗ Λ^{2}(M)

be its torsion and curvature, and let R = R^{2,0} + R^{1,1} + R^{0,2} be the decomposition of the curvature according to the Hodge type. Definition. The connection ∇ is called Kählerian if

T = 0  (i)
R^{2,0} = R^{0,2} = 0  (ii)

Example. The Levi-Civita connection on a Kähler manifold is Kählerian. Remark. The condition T = 0 implies, in particular, that the component

∇^{0,1} : Λ^{1,0}(M) → Λ^{1,1}(M)

of the connection ∇ coincides with the Dolbeault differential. Therefore a Kählerian connection is always holomorphic . 8.1.3. Recall that in 6.4.2 we have associated to any extended connection D on M a connection ∇ on the cotangent bundle Λ^{1}(M, ℂ) called the reduction of D. We can now formulate the main result of this section.

Theorem 8.1 (i) If an extended connection D on M is flat and linear, then its reduction ∇ is Kählerian.

(ii) Every Kählerian connection ∇ on the cotangent bundle Λ^{1}(M, ℂ) is the reduction of a unique linear flat extended connection D on M.

The rest of this section is taken up with the proof of Theorem 8.1. To make it more accessible, we first give an informal outline. The actual proof starts with Subsection 8.2, and it is independent from the rest of this subsection. 8.1.4. Assume given a Kählerian connection ∇ on the manifold M. To prove Theorem 8.1, we have to construct a flat linear extended connection D on M with reduction ∇. Every extended connection decomposes into a series D = ∑_{k≥0} D_{k} as in (6.2), and, since ∇ is the reduction of D, we must have D_{1} = ∇. We begin by checking in Lemma 8.2.1 that if D is linear, then D_{0} = C, where C : S^{1}(M, ℂ) → Λ^{1}(M, ℂ) is as in 6.3.8. The sum C + ∇ is already a linear extended connection on M. By 6.4.5 it extends to a derivation D_{≤1} of the Weil algebra B (M, ℂ) of the manifold M, but this derivation does not necessarily satisfy D_{≤1} ∘ D_{≤1} = 0, thus the extended connection D_{≤1} is not necessarily flat.

We have to show that one can add the “correction terms” D_{k}, k ≥ 2 to D_{≤1} so that D = ∑_{k} D_{k} satisfies all the conditions of Theorem 8.1. To do this, we introduce in 8.3.3 a certain quotient B̃(M, ℂ) of the Weil algebra B (M, ℂ), called the reduced Weil algebra. The reduced Weil algebra is defined in such a way that for every extended connection D the associated derivation D : B (M, ℂ) → B^{+1}(M, ℂ) preserves the kernel of the surjection B → B̃, thus inducing a derivation D̃ : B̃(M, ℂ) → B̃^{+1}(M, ℂ). Moreover, the algebra B̃(M, ℂ) has the following two properties:

(i) The derivation D̃ : B̃(M, ℂ) → B̃^{+1}(M, ℂ) satisfies D̃ ∘ D̃ = 0 if and only if the connection D_{1} is Kählerian.

(ii) Let D̃ be the weakly Hodge derivation of the quotient algebra B̃(M, ℂ) induced by an arbitrary linear extended connection D_{≤1} and such that D̃ ∘ D̃ = 0. Then the derivation D̃ lifts uniquely to a weakly Hodge derivation D of the Weil algebra B (M, ℂ) such that D ∘ D = 0, and the derivation also D comes from a linear extended connection on M (see Proposition 8.1 for a precise formulation of this statement).

8.1.5. The property (i) is relatively easy to check, and we do it in the end of the proof, in Subsection 8.4. The rest is taken up with establishing the property (ii). The actual proof of this statement is contained in Proposition 8.1, and Subsection 8.2 contains the necessary preliminaries.

Recall that we have introduced in 7.2.2 a new grading on the Weil algebra B (M, ℂ), called the augmentation grading, so that the component D_{k} in the decomposition D = ∑_{k} D_{k} is of augmentation degree k. In order to lift D̃ to a derivation D so that D ∘ D = 0, we begin with the given lifting D_{≤1} and then add components D_{k}, k ≥ 2, one by one, so that on each step for D_{≤k} = D_{≤1} + ∑_{2≤p≤k} the composition D_{≤k} ∘ D_{≤k} is zero in augmentation degrees from 0 to k. In order to do it, we must find for each k a solution to the equation

D_{0} ∘ D_{k} = −R_{k},  (8.1)

where R_{k} is the component of augmentation degree k in the composition D_{≤k−1} ∘ D_{≤k−1}. This solution must be weakly Hodge, and the extended connection D_{≤k} = D_{≤k−1} + D_{k} must be linear.

We prove in Lemma 8.2.1 that since D_{≤0} is linear, we may assume that D_{0} = C. In addition, since D̃ ∘D̃ = 0, we may assume by induction that the image of R_{k} lies in the kernel I of the quotient map B (M, ℂ) → B̃(M, ℂ). 8.1.6. In order to analyze weakly Hodge maps from S^{1}(M, ℂ) to the Weil algebra B (M, ℂ), we apply the functor Γ : WHodge_{≥0}(M) → WHodge_{0}(M) constructed in 2.1.4 to the bundle B (M, ℂ) to obtain the total Weil algebra B_{tot}(M, ℂ) = Γ(B (M, ℂ)) of weight 0, which we studied in Subsection 7.2. The Hodge bundle S^{1}(M, ℂ) on the manifold M is of weight 0, and, by the universal property of the functor Γ, weakly Hodge maps from S^{1}(M, ℂ) to B (M, ℂ) are in one-to-one correspondence with Hodge bundle maps from S^{1}(M, ℂ) to the total Weil algebra B_{tot}(M, ℂ). The canonical map C : S^{1}(M, ℂ) → B^{1}(M, ℂ) extends to a derivation C : B_{tot}(M, ℂ) → B_{tot}^{+1}(M, ℂ). Moreover, the weakly Hodge map R_{k} : S^{1}(M, ℂ) → B^{2}(M, ℂ) defines a Hodge bundle map R_{k}^{tot} : S^{1}(M, ℂ) → B_{tot}^{2}(M, ℂ), and solving (8.1) is equivalent to finding a Hodge bundle map D_{k} : S^{1}(M, ℂ) → B_{tot}^{1}(M, ℂ) such that

C ∘ D_{k} = −R_{k}.  (8.2)

8.1.7. Recall that by 7.3.2 the derivation

C : B_{tot}(M, ℂ) → B_{tot}^{+1}(M, ℂ)

satisfies C ∘C = 0, so that the total Weil algebra B_{tot}(M, ℂ) becomes a complex with differential C. The crucial part of the proof of Theorem 8.1 consists in noticing that the subcomplex I_{tot}(M, ℂ) = Γ(I (M, ℂ)) ⊂ B_{tot}(M, ℂ) of the total Weil algebra B_{tot}(M, ℂ) corresponding to the kernel I (M, ℂ) ⊂ B (M, ℂ) of the quotient map B (M, ℂ) → B̃(M, ℂ) is canonically contractible. This statement is analogous to Proposition 7.1, and we prove it in the same way. Namely, we check that the subcomplex I_{tot} ⊂ B_{tot} is preserved by the bundle map σ_{tot} : B_{tot}^{+1} → B_{tot} constructed in 7.3.7, and that the anticommutator h = {σ_{tot}, C} : B_{tot}(M, ℂ) → B_{tot}(M, ℂ) is invertible on the subcomplex I_{tot}(M, ℂ) ⊂ B_{tot}(M, ℂ) (Corollary 8.3.4 of Lemma 7.3.8). We also check that C ∘ R_{k}^{tot} = 0, which implies that the Hodge bundle map

D_{k} = −h^{−1} ∘ σ_{tot} ∘ R_{k}^{tot} : S^{1}(M, ℂ) → B_{tot}^{1}(M, ℂ)  (8.3)

provides a solution to the equation (8.2). 8.1.8. To establish the property (ii), we have to insure additionally that the extended connection D = D_{≤k} is linear, and and we have to show that the solution D_{k} of (8.2) with this property is unique. This turns out to be pretty straightforward. We show in Lemma 8.2.1 that D_{≤k} is linear if and only if

σ_{tot} ∘ D_{k} = 0.  (8.4)

Moreover, we show that the homotopy σ_{tot} : B_{tot}^{+1}(M, ℂ) → B_{tot}(M, ℂ) satisfies σ_{tot} ∘ σ_{tot} = 0. Therefore the solution D_{k} to (8.2) given by (8.3) satisfies (8.4) automatically.

The uniqueness of such a solution D_{k} follows from the invertibility of h = C ∘ σ_{tot} + σ_{tot} ∘ C. Indeed, for every two solutions D_{k}, D′_{k} to (8.1), both satisfying (8.4), their difference P = D_{k} − D′_{k} satisfies C ∘ P = 0. If, in addition, both D_{k} and D′_{k} were to satisfy (8.4), we would have had σ_{tot} ∘ P = 0. Therefore h ∘ P = 0, and P has to vanish. 8.1.9. These are the main ideas of the proof of Theorem 8.1. The proof itself begins in the next subsection, and it is organized as follows. In Subsection 8.2 we express the linearity condition on an extended connection D in terms of the associated derivation D_{tot} : B_{C} → B_{tot}^{+1} of the total Weil algebra B_{tot} of the manifold M. After that, we introduce in Subsection 8.3 the reduced Weil algebra B̃(M, ℂ) and prove Proposition 8.1, thus reducing Theorem 8.1 to a statement about derivations of the reduced Weil algebra. Finally, in Subsection 8.4 we prove this statement. Remark. In the Appendix we give, following Deligne and Simpson, a more geometric description of the functor Γ : WHodge_{≥0} → WHodge and of the total Weil algebra B_{tot}(M, ℂ), which allows to give a simpler and more conceptual proof for the key parts of Theorem 8.1.

8.2. Linearity and the total Weil algebra

8.2.1. Assume given an extended connection D : S^{1} → B^{1} on the manifold M, and extend it to a derivation D : B → B^{+1} of the Weil algebra as in 6.4.5. Let D = ∑_{k≥0} D_{k} be the augmentation degree decomposition. The derivation D is weakly Hodge and defines therefore a derivation D = ∑_{k≥0} D_{k} : B_{tot} → B_{tot}^{+1} of the total Weil algebra B_{tot}.

Before we begin the proof of Theorem 8.1, we give the following rewriting of the linearity condition 6.4.4 on the extended connection D in terms of the total Weil algebra.

Lemma. The extended connection D is linear if and only if D_{0} = C and σ_{tot} ∘ D_{k} = 0 on S^{1} ⊂ B_{tot}^{0} for every k ≥ 0.

Proof. Indeed, by Lemma 7.2.6 for odd integers k and n = ±1 the subbundle (B_{o}^{1})_{k+1}^{n,−n} ⊂ B_{tot}^{1} vanishes. Therefore the map D_{k} : S^{1} → B_{tot}^{1} factors through B_{ll}^{1} ⊕ B_{rr}^{1}. Since by definition (7.3.7) we have σ_{tot} = 0 on both B_{ll}^{1} and B_{rr}^{1}, for odd k we have σ_{tot} ∘ D_{k} = 0 on S^{1} regardless of the extended connection D. On the other hand, for even k we have (B_{tot}^{1})_{k+1}^{n,−n} = (B_{o}^{1})_{k+1}^{n,−n}. Therefore on S^{1} we have σ_{tot} ∘ D_{k} = σ ∘ D_{k} (where σ : B^{+1} → B is as in 7.3.2). Moreover, since σ : Λ^{1} → S^{1} is an isomorphism, D_{0} = C is equivalent to σ ∘ D_{0} = id : S^{1} → Λ^{1} → S^{1}. Therefore the condition of the lemma is equivalent to the following

σ ∘ D_{k} = \begin{cases}id, & for k = 0 \\ 0, & for even integers k > 0. \end{cases}  (8.5)

Let now ι^{∗} : B → B be the operator given by the action of the canonical involution ι : \overline{T} M → \overline{T} M, as in 6.3.6, and let D^{ι} = ∑_{k≥0} D_{k}^{ι} = ι^{∗}∘D∘(ι^{∗})^{−1} be the operator ι^{∗}-conjugate to the derivation D. The operator ι^{∗} acts as −id on S^{1} ⊂ B^{0} and as id on Λ^{1} ⊂ B^{1}. Since it is an algebra automorphism, it acts as (−1)^{i+k} on B_{k}^{i} ⊂ B . Therefore D_{k}^{ι} = (−1)^{k+1}D_{k}, and (8.5) is equivalent to

σ ∘ \frac{1}{2}(D − D^{ι}) = id : S^{1} → B^{1} → S^{1},

which is precisely the definition of a linear extended connection. □

8.3. The reduced Weil algebra

8.3.1. We now begin the proof of Theorem 8.1. Our first step is to reduce the classification of linear flat extended connections D : S^{1} → B^{1} on the manifold M to the study of derivations of a certain quotient B̃of the Weil algebra B . We introduce this quotient in this subsection under the name of reduced Weil algebra. We then show that every extended connection D on M induces a derivation D̃ : B̃→ B̃^{+1} of the reduced Weil algebra, and that a linear flat extended connection D on M is completely defined by the derivation D̃. 8.3.2. By Lemma 7.3.8 the anticommutator h = {C, σ_{tot}} : B_{tot} → B_{tot} of the canonical bundle endomorphisms C, σ_{tot} of the total Weil algebra B_{tot} is invertible on every component (B_{tot})_{p,q} of augmentation bidegree (p, q) with p, q ≥ 1.

The direct sum ⊕_{p,q≥1} (B_{tot})_{p,q} is an ideal in the total Weil algebra B_{tot}, and it is obtained by applying the functor Γ to the ideal ⊕_{p,q≥1}B_{p,q} in the Weil algebra B . For technical reasons, it will be more convenient for us to consider the smaller subbundle

I = ⨁_{p≥2,q≥1} B_{p,q} + ⨁_{p≥1,q≥2} B_{p,q} ⊂ B

of the Weil algebra B . The subbundle I is a Hodge subbundle in B , and it is an ideal with respect to the multiplication in B . 8.3.3. Definition. The reduced Weil algebra B̃= B̃(M, ℂ) of the manifold M is the quotient

B̃= B /I

of the full Weil algebra B by the ideal I .

The reduced Weil algebra decomposes as

B̃= B_{1,1} ⊕ ⨁_{p≥0} B_{p,0} ⊕ ⨁_{q≥0} B_{0,q}

with respect to the augmentation bigrading on the Weil algebra B . The two summands on the right are equal to

⨁_{p≥0} B_{p,0} = ⨁_{pgeq0} S^{p} (S^{1,−1}) ⊗ Λ^{,0},
⨁_{q≥0} B_{0,q} = ⨁_{qgeq0} S^{q} (S^{−1,1}) ⊗ Λ^{0,},

8.3.4. Since I is a Hodge subbundle, the reduced Weil algebra carries a canonical Hodge bundle structure compatible with the multiplication. It also obviously inherits the augmentation bigrading, and defines an ideal I_{tot} = Γ(I ) ⊂ B_{tot} in the total Weil algebra B_{tot}. Lemma 7.3.8 immediately implies the following fact.

Corollary. The map h = {C, σ_{tot}} : B_{tot} → B_{tot} is invertible on I_{tot} ⊂ B_{tot}.

8.3.5. Let now D : B → B be the derivation associated to the extended connection D as in 6.4.5. The derivation D does not increase the augmentation bidegree, it preserves the ideal I ⊂ B and defines therefore a weakly Hodge derivation of the reduced Weil algebra B̃, which we denote by D̃. If the extended connection D is flat, then the derivation D̃ satisfies D̃ ∘ D̃ = 0.

We now prove that every derivation D̃ : B̃→ B̃^{+1} of this type comes from a linear flat extended connection D, and that the connection D is completely defined by D̃. More precisely, we have the following.

Proposition 8.1 Let D : S^{1} → B^{1} be a linear but not necessarily flat extended connection on M, and let D̃ : B̃→ B̃^{+1} be the associated weakly Hodge derivation of the reduced Weil algebra B̃. Assume that D̃ ∘ D̃ = 0.

There exists a unique weakly Hodge bundle map P : S^{1} → I^{1} such that the extended connection D′ = D + P : S^{1} → B^{1} is linear and flat.

8.3.6. Proof. Assume given a linear extended connection D satisfying the condition of Proposition 8.1. To prove the proposition, we have to construct a weakly Hodge map P : S^{1} → I^{1} such that the extended connection D + P is linear and flat. We do it by induction on the augmentation degree, that is, we construct one-by-one the terms P_{k} in the augmentation degree decomposition P = ∑_{k} P_{k}. The identity D̃ ∘ D̃ = 0 is the base of the induction, and the induction step is given by applying the following lemma to D + ∑_{i=2}^{k} P_{k}, for each k ≥ 1 in turn.

Lemma. Assume given a linear extended connection D : S^{1} → B^{+1} on M and let D : B → B^{+1} also denote the associated derivation. Assume also that the composition D ∘ D : B → B^{+2} maps S^{1} into I_{>k}^{2} = ⊕_{p>k}I_{p}^{2}.

There exists a unique weakly Hodge bundle map P_{k} : S^{1} → I_{k+1}^{1} such that the extended connection D′ = D + P_{k} : S^{1} → B^{1} is linear, and for the associated derivation D′ : B → B^{+1} the composition D′ ∘ D′ maps S^{1} into I_{>k+1}^{2} = ⊕_{p>k+1}I_{p}^{2}.

Proof. Let D : B_{tot} → B_{tot} be the derivation of the total Weil algebra associated to the extended connection D, and let

R : (B_{tot}) → (B_{tot}^{+2})_{+k}

be the component of augmentation degree k of the composition D∘D : S^{1} → B_{tot}^{2}. Note that by 6.4.5 the map R vanishes on the subbundle Λ_{tot}^{1} ⊂ (B_{tot}^{1})_{1}. Moreover, the composition C ∘ R : B_{tot} → B^{+3} of the map R with the canonical derivation C : B_{tot} → B_{tot}^{+1} vanishes on the subbundle S^{1} ⊂ B_{tot}^{0}. Indeed, since C maps S^{1} into Λ_{tot}^{1}, the composition C ∘ R is equal to the commutator [C, R] : S^{1} → B_{tot}^{3}. Since the extended connection D is by assumption linear, we have D_{0} = C, and

C ∘ R = [C, R] = ∑_{0≤p≤k}[C, D_{p} ∘ D_{k−p}] =

= [C, {C, D_{k}}] + ∑_{1≤p≤k−1}[C, D_{p} ∘ D_{k−p}].

Since C ∘ C = 0, the first term in the right hand side vanishes. Let Θ = ∑_{1≤p≤k−1} D_{p} : B_{tot} → B_{tot}^{+1}. Then the second term is the component of augmentation degree k in the commutator [C, Θ ∘ Θ] : S^{1} → B_{tot}^{3}. By assumption {D, D} = 0 in augmentation degrees < k. Therefore we have {C, Θ} = −{Θ, Θ} in augmentation degrees < k. Since Θ increases the augmentation degree, this implies that in augmentation degree k

[C, Θ ∘ Θ] = {C, Θ} ∘ Θ − Θ ∘ {C, Θ} = [Θ, {Θ, Θ}],

which vanishes tautologically.

The set of all weakly Hodge maps P : S^{1} → I_{k}^{1} coincides with the set of all maps P : S^{1} → (I_{tot}^{1})_{k} preserving the Hodge bundle structures. Let P be such a map, and let D′ : B_{tot} → B_{tot}^{+1} be the derivation associated to the extended connection D′ = D + P.

Since the extended connection D is by assumption linear, by Lemma 8.2.1 the extended connection D′ is linear if and only if σ_{tot} ∘ P = 0. Moreover, since the augmentation degree-0 component of the derivation D equals C, the augmentation degree-k component Q : S^{1} → B_{tot}^{2} in the composition D′ ∘ D′ is equal to

Q = R + {C, (D′ − D)}.

By definition D′ − D : B_{tot} → B_{tot}^{+1} equals P on S^{1} ⊂ B_{tot}^{0} and vanishes on Λ_{tot}^{1} ⊂ B_{tot}^{1}. Since C maps S^{1} into Λ_{tot}^{1} ⊂ B_{tot}^{1}, we have Q = R + C ∘ P. Thus, a map P satisfies the condition of the lemma if and only if

\begin{cases}C ∘ P = −R \\ σ_{tot} ∘ P = 0 \end{cases}

To prove that such a map P is unique, note that these equations imply

h ∘ P = (σ_{tot} ∘ C + C ∘ σ_{tot}) ∘ P = σ_{tot} ∘ R,

and h is invertible by Corollary 8.3.4. To prove that such a map P exists, define P by

P = −h^{−1} ∘ σ_{tot} ∘ R : S^{1} → I_{k+1}^{1}.

The map h = {C, σ_{tot}} and its inverse h^{−1} commute with C and with σ_{tot}. Since σ_{tot} ∘ σ_{tot} = C ∘ C0, we have σ_{tot} ∘ P = h^{−1} ∘ σ_{tot} ∘ σ_{tot} ∘ R = 0. On the other hand, C ∘ R = 0. Therefore

C ∘ P = −C ∘ h^{−1} ∘ σ_{C} ∘ R = −h^{−1} ∘ C ∘ σ_{tot} ∘ R =
= h^{−1} ∘ σ_{tot} ∘ C ∘ R − h^{−1} ∘ h ∘ R = −R.

This finishes the proof of the lemma and of Proposition 8.1. □

8.4. Reduction of extended connections

8.4.1. We now complete the proof of Theorem 8.1. First we will need to identify explicitly the low Hodge bidegree components of the reduced Weil algebra B̃. The following is easily checked by direct inspection.

Lemma. We have

B̃^{2,−1} ⊕ B̃^{1,0} ⊕ B̃^{0,1} ⊕ B̃^{−1,2} = Λ^{1} ⊕ (S^{1} ⊗ Λ^{1}) ⊂ B̃^{1}

B̃^{3,−2} ⊕ B̃^{2,−1} ⊕ B̃^{1,1} ⊕ B̃^{−1,2} ⊕ B̃^{−2,3} =

= (S^{1,−1} ⊗ Λ^{2,0}) ⊕ (S^{−1,1} ⊗ Λ^{0,2}) ⊕ Λ^{2} ⊂ B̃^{2}

8.4.2. Let now ∇ : S^{1} → S^{1} ⊗ Λ^{1} be an arbitrary real connection on the bundle S^{1}. The operator

D = C + ∇ : S^{1} → Λ^{1} ⊕ (Λ^{1} ⊗ S^{1}) ⊂ B^{1}

is then automatically weakly Hodge and defines therefore an extended connection on M. This connection is linear by Lemma 8.2.1. Extend D to a derivation D : B → B^{+1} as in 6.4.5, and let D̃ : B̃ → B̃^{+1} be the associated derivation of the reduced Weil algebra.

Lemma. The derivation D̃ satisfies D̃ ∘ D̃ = 0 if and only if the connection ∇ is Kählerian.

Proof. Indeed, the operator D̃ ∘ D̃ is weakly Hodge, hence factors through a bundle map

D̃ ∘ D̃ : S^{1} → B̃^{3,−1} ⊕ B̃^{2,0} ⊕ B̃^{1,1} ⊕ B̃^{0,2} ⊕ B̃^{−1,3} =

= (S^{1,−1} ⊗ Λ^{2,0}) ⊕ (S^{−1,1} ⊗ Λ^{0,2}) ⊕ Λ^{2} ⊂ B^{2}.

By definition we have

D̃ ∘ D̃ = (C + ∇) ∘ (C + ∇) = {C, ∇} + {∇, ∇}.

An easy inspection shows that the sum is direct, and the first summand equals

{C, ∇} = T ∘ C : S^{1} → Λ^{2},

where T is the torsion of the connection ∇, while the second summand equals

{∇, ∇} = R^{2,0} ⊕ R^{0,2} : S^{1,−1} ⊕ S^{−1,1} → (S^{1,−1} ⊗ Λ^{2,0}) ⊕ (S^{−1,1} ⊗ Λ^{0,2}) ,

where R^{2,0}, R^{0,2} are the Hodge type components of the curvature of the connection ∇. Hence D̃ ∘ D̃ = 0 if and only if R^{2,0} = R^{0,2} = T = 0, which proves the lemma and finishes the proof of Theorem 8.1. □ 8.4.3. We finish this section with the following corollary of Theorem 8.1 which gives an explicit expression for the augmentation degree-2 component D_{2} of a flat linear extended connection D on the manifold M. We will need this expression in Section 9.

Corollary. Let D = ∑_{k≥0} D_{k} : S^{1} → B^{1} be a flat linear extended connection on M, so that D_{0} = C and D_{1} is a Kählerian connection on M. We have

D_{2} = \frac{1}{3}σ ∘ R,

where σ : B^{+1} → B is the canonical derivation introduced in 6.3.7, and R = D_{1} ∘ D_{1} : S^{1} → S^{1} ⊗ Λ^{1,1} ⊂ B^{2} is the curvature of the Kählerian connection D_{1}.

Proof. Extend the connection D to a derivation D_{C} = ∑_{k≥0} D_{k}^{tot} : B_{tot} → B_{tot}^{+1} of the total Weil algebra. By the construction used in the proof of

Lemma 8.3.6 we have D_{2}^{tot} = h^{−1}∘σ_{tot}∘R_{tot} : S^{1} → (B_{tot}^{1})_{3}, where h : B_{tot} →

B_{tot} is as in Lemma 7.3.8, the map σ_{tot} : B_{tot}^{+1} → B_{tot} is the canonical map constructed in 7.3.7, and R^{tot} : S^{1} → (B_{tot}^{2})_{3} is the square R_{tot} = D_{1}^{tot} ∘ D_{1}^{tot} of the derivation D_{1}^{tot} : B_{tot} → B_{tot}^{+1}. By Lemma 7.3.9 the map h acts on (B_{tot}^{1})_{3} by multiplication by 3. Moreover, it is easy to check that

(B_{tot}^{2})_{3} = (S^{1} ⊗ Λ^{2}) ⊕ (S^{−1,1} ⊗ Λ^{2,0}) ⊕ (S^{1,−1} ⊗ Λ^{0,2}) ,

and the map R^{tot} : S^{1} → (B_{tot}^{2})_{3} sends S^{1} into the first summand in this decomposition and coincides with the curvature R : S^{1} → S^{1} ⊗Λ^{2} ⊂ (B_{tot}^{2})_{3} of the Kählerian connection D_{1}. Therefore σ_{tot} ∘ R^{tot} = σ ∘ R, which proves the claim. □

9. Metrics

9.1. Hyperkähler metrics on Hodge manifolds

9.1.1. Let M be a complex manifold equipped with a Kählerian connection ∇, and consider the associated linear formal Hodge manifold structure on the tangent bundle \overline{T} M. In this section we construct a natural bijection between the set of all polarizations on the Hodge manifold \overline{T} M in the sense of Subsection 3.3 and the set of all Kähler metrics on M compatible with the given connection ∇. 9.1.2. Let h be a hyperkähler metric on \overline{T} M, or, more generally, a formal germ of such a metric in the neighborhood of the zero section M ⊂ \overline{T} M. Assume that the metric h is compatible with the given hypercomplex structure and Hermitian-Hodge in the sense of 1.5.2, and let ω_{I} be the Kähler form associated to h in the preferred complex structure \overline{T} M_{I} on \overline{T} M.

Let h_{M} be the restriction of the metric h to the zero section M ⊂ \overline{T} M, and let ω ∈ C^{∞}(M, Λ^{1,1}(M)) be the associated real (1, 1)-form on the complex manifold M. Since the embedding M ⊂ \overline{T} M_{I} is holomorphic, the form ω is the restriction onto M of the form ω_{I}. In particular, it is closed, and the metric h_{M} is therefore Kähler. 9.1.3. The main result of this section is the following.

Theorem 9.1 Restriction onto the zero section M ⊂ \overline{T} M defines a one-to-one correspondence between

(i) Kähler metrics on M compatible with the Kählerian connection ∇, and

(ii) formal germs in the neighborhood on M ⊂ \overline{T} M of Hermitian-Hodge hyperkähler metrics on \overline{T} M compatible with the given formal Hodge manifold structure.

The rest of this section is devoted to the proof of this theorem. 9.1.4. In order to prove Theorem 9.1, we reformulate it in terms of polarizations rather than metrics. Recall (see 3.3.3) that a polarization of the formal Hodge manifold \overline{T} M is by definition a (2, 0)-form Ω ∈ C_{M}^{∞}(\overline{T} M, Λ^{2,0}(\overline{T} M_{J})) for the complementary complex structure \overline{T} M_{J} which is holomorphic, real and of H-type (1, 1) with respect to the canonical Hodge bundle structure on Λ^{2,0}(\overline{T} M_{J}), and satisfies a certain positivity condition (3.3). 9.1.5. By Lemma 3.3.4 Hermitian-Hodge hyperkähler metrics on \overline{T} M are in one-to-one correspondence with polarizations. Let h be a Kähler metric on M, and let ω_{I} and ω be the Kähler forms for h on \overline{T} M_{I} and on M ⊂ \overline{T} M_{I}. The corresponding polarization Ω ∈ C^{∞}(\overline{T} M, Λ^{2,0}(\overline{T} M_{J})) satisfies by (1.4)

ω_{I} = \frac{1}{2} (Ω + ν(Ω)) ∈ Λ^{2}(\overline{T} M, ℂ),  (9.1)

where ν : Λ^{2}(\overline{T} M, ℂ) → \overline{Λ^{2}(\overline{T} M, ℂ)} is the usual complex conjugation.

9.1.6. Let ρ : \overline{T} M → M be the natural projection, and let

Res : ρ_{∗}Λ^{,0}(\overline{T} M_{J}) → Λ (M, ℂ)

be the map given by the restriction onto the zero section M ⊂ \overline{T} M. Both bundles are naturally Hodge bundles of the same weight on M in the sense of 2.1.2, and the bundle map Res preserves the Hodge bundle structures. Since Ω is of H-type (1, 1), the form Res Ω ∈ C^{∞}(M, Λ^{2}(M, ℂ)) is real and of Hodge type (1, 1). By (9.1)

Res Ω = \frac{1}{2} (Res Ω + \overline{Res Ω}) = \frac{1}{2}(Ω + ν(Ω))|_{M⊂\overline{T} M} =

= ω_{I}|_{M⊂\overline{T} M} = ω ∈ Λ^{1,1}(M).

Therefore to prove Theorem 9.1, it suffices to prove the following.

• For every polarization Ω of the formal Hodge manifold \overline{T} M the restriction ω = Res Ω ∈ C^{∞}(Λ^{1,1}(M)) is compatible with the connection ∇, that is, ∇ω = 0. Vice versa, every real positive (1, 1)-form ω ∈ C^{∞}(Λ^{1,1}(M)) satisfying ∇ω = 0 extends to a polarization Ω of \overline{T} M, and such an extension is unique.

This is what we will actually prove.

9.2. Preliminaries

9.2.1. We begin with introducing a convenient model for the holomorphic de Rham algebra Λ^{,0}(\overline{T} M_{J}) of the complex manifold \overline{T} M_{J}, which would be independent of the Hodge manifold structure on \overline{T} M. To construct such a model, consider the relative de Rham complex Λ (\overline{T} M/M, ℂ) of \overline{T} M over M (see 5.2 for a reminder of its definition and main properties). Let π : Λ (\overline{T} M, ℂ) → Λ (\overline{T} M/M, ℂ) be the canonical projection. Recall that the bundle Λ^{i}(\overline{T} M/M, ℂ) of relative i-forms on \overline{T} M over M carries a natural structure of a Hodge bundle of weight i. Moreover, we have introduced in (5.5) a Hodge bundle isomorphism

η : ρ^{∗}Λ (M, ℂ) → Λ (\overline{T} M/M, ℂ)

between Λ^{i}(\overline{T} M/M, ℂ) and the pullback ρ^{∗}Λ (M, ℂ) of the bundle of ℂvalued i-forms on M.

Lemma. (i) The projection π induces an algebra isomorphism

π : Λ^{,0}(\overline{T} M_{J}) → Λ (\overline{T} M/M, ℂ)

compatible with the natural Hodge bundle structures.

(ii) Let α ∈ C_{M}^{∞}(\overline{T} M, Λ^{i,0}(\overline{T} M_{J})) be a smooth (i, 0)-form on \overline{T} M_{J}, and consider the smooth i-form

β = η^{−1}π(α) ∈ C_{M}^{∞}(\overline{T} M, Λ^{i}(\overline{T} M, ℂ))

on \overline{T} M. The forms α and β have the same restriction to the zero section M ⊂ \overline{T} M.

Proof. Since η, π and the restriction map are compatible with the algebra structure on Λ (M, ℂ), it suffices to prove both claims for Λ^{1}(M, ℂ). For every bundle E on \overline{T} M denote by E|_{M⊂\overline{T} M} the restriction of E to the zero section M ⊂ \overline{T} M. Consider the bundle map

χ = η ∘ Res : Λ^{1,0}(\overline{T} M_{J})|_{M⊂\overline{T} M} → Λ^{1}(M, ℂ) → Λ^{1}(\overline{T} M/M, ℂ)|_{M⊂\overline{T} M}.

The second claim of the lemma is then equivalent to the identity χ = π. Moreover, note that the contraction with the canonical vector field φ on \overline{T} M defines an injective map i_{φ} : C^{∞}(M, Λ^{1}(\overline{T} M/M, ℂ)|_{M}) → C^{∞}(\overline{T} M, ℂ). Therefore it suffices to prove that i_{φ} ∘ χ = i_{φ} ∘ π.

Every smooth section s of the bundle Λ^{1,0}(\overline{T} M_{J})|_{M⊂\overline{T} M} is of the form

s = (ρ^{∗}α + \sqrt{−1}jρ^{∗}α)|_{M⊂\overline{T} M},

where α ∈ C^{∞}(M, Λ^{1}(M, ℂ)) is a smooth 1-form on M, and

j : Λ^{1}(\overline{T} M, ℂ) → \overline{Λ^{1}(\overline{T} M, ℂ)}

is the map induced by the quaternionic structure on \overline{T} M. For such a section s we have Res(s) = α, and by (5.2.8) i_{φ}(χ(s)) = \sqrt{−1}τ(α), where τ : C^{∞}(M, Λ^{1}(M, ℂ)) → C^{∞}(\overline{T} M, ℂ) is the tautological map introduced in 4.3.2. On the other hand, since π ∘ ρ^{∗} = 0, we have

i_{φ}(π(s)) = i_{φ}(π(\sqrt{−1}jρ^{∗}α)) = i_{φ}(\sqrt{−1}jρ^{∗}α).

Since the Hodge manifold structure on \overline{T} M is linear, this equals

i_{φ}(π(s)) = \sqrt{−1}i_{φ}(j(ρ^{∗}α)) = \sqrt{−1}τ(α) = χ(s),

which proves the second claim of the lemma. Moreover, it shows that the restriction of the map π to the zero section M ⊂ \overline{T} M is an isomorphism. As in the proof of Lemma 5.1.9, this implies that the map π is an isomorphism on the whole \overline{T} M, which proves the first claim and finishes the proof of the lemma. □

9.2.2. Lemma 9.2.1 (i) allows to define the bundle isomorphism

π^{−1} ∘ η : ρ^{∗}Λ (M, ℂ) → Λ (\overline{T} M/M, ℂ) → Λ^{,0}(\overline{T} M_{J}),

between ρ^{∗}Λ (M, ℂ) and Λ^{,0}(\overline{T} M_{J}), and it induces an isomorphism

ρ_{∗}(η ∘ π^{−1}) : ρ_{∗}ρ^{∗}Λ (M, ℂ) ≅ ρ_{∗}Λ^{,0}(\overline{T} M_{J})

between the direct images of these bundles under the canonical projection ρ : \overline{T} M → M.

On the other hand, by adjunction we have the canonical embedding

Λ (M, ℂ) ↪ ρ_{∗}ρ^{∗}Λ (M, ℂ),

and by the projection formula it extends to an isomorphism

ρ_{∗}ρ^{∗}Λ (M, ℂ) ≅ Λ (M, ℂ) ⊗ B^{0},

where B^{0} = ρ_{∗}Λ^{0}(\overline{T} M, ℂ) is the 0-th component of the Weil algebra B of M. All these isomorphisms are compatible with the Hodge bundle structures and with the multiplication. 9.2.3. It will be convenient to denote the image ρ_{∗}(η ∘ π^{−1}) (Λ (M, ℂ)) ⊂ ρ_{∗}Λ^{,0}(\overline{T} M_{J}) by L (M, ℂ) or, to simplify the notation, by L . (The algebra L (M, ℂ) is, of course, canonically isomorphic to Λ (M, ℂ).) We then have the identification

L ⊗ B^{0} ≅ ρ_{∗}ρ^{∗}Λ (M, ℂ) ≅ ρ_{∗}Λ^{,0}(\overline{T} M_{J}).  (9.2)

This identification is independent of the Hodge manifold structure on \overline{T} M. Moreover, by Lemma 9.2.1 (ii) the restriction map Res : ρ_{∗}Λ^{,0}(\overline{T} M_{J}) → Λ (M, ℂ) is identified under (9.2) with the canonical projection L ⊗ B^{0} → L ⊗ B_{0}^{0} ≅ L .

By Lemma 5.1.9 we also have the identification ρ_{∗}Λ^{0,}(\overline{T} M_{J}) ≅ B . Therefore (9.2) extends to an algebra isomorphism

ρ_{∗}Λ^{,}(\overline{T} M_{J}) ≅ ρ_{∗}Λ (\overline{T} M/M, ℂ) ⊗ Λ (M, ℂ) ≅ L ⊗ B .  (9.3)

This isomorphism is also compatible with the Hodge bundle structures on both sides.

9.3. The Dolbeult differential on \overline{T} M_{J}

9.3.1. Our next goal is to express the Dolbeult differential ¯∂_{J} of the complex manifold \overline{T} M_{J} in terms of the model for the de Rham complex Λ^{,}(\overline{T} M_{J}) given by (9.3). For every k ≥ 0 denote by

D : L^{k} ⊗ B → L^{k} ⊗ B^{+1}.

the differential operator induced by ¯∂_{J} : Λ^{,}(\overline{T} M_{J}) → Λ^{, +1}(\overline{T} M_{J}) under (9.3). The operator D is weakly Hodge. It satisfies the Leibnitz rule with respect to the algebra structure on L ⊗ B , and we have D ∘ D = 0. By definition for k = 0 it coincides with the derivation D : B → B^{+1} defined by the Hodge manifold structure on \overline{T} M. For k > 0 the complex ⟨L^{k}⊗B , D⟩ is a free differential graded module over the Weil algebra ⟨B , D⟩. 9.3.2. The relative de Rham differential d^{r} (see Subsection 5.2) induces under the isomorphism (9.3) an algebra derivation

d^{r} : L ⊗ B → L^{+1} ⊗ B .

The derivation d^{r} also is weakly Hodge, and we have the following.

Lemma. The derivations D and d^{r} commute, that is,

{D, d^{r}} = 0 : L ⊗ B → L^{+1} ⊗ B^{+1}.

Proof. The operator {D, d^{r}} satisfies the Leibnitz rule, so it suffices to prove that it vanishes on B^{0}, B^{1} and L^{1} ⊗ B^{0}. Moreover, the B^{0}-modules B^{1} and L^{1} ⊗ B^{0} are generated, respectively, by local sections of the form Df and d^{r}f, f ∈ B^{0}. Since {D, d^{r}} commutes with D and d^{r}, it suffices to prove that it vanishes on B^{0}. Finally, {D, d^{r}} is continuous in the adic topology on B^{0}. Since the subspace

{fg|f, g ∈ B^{0}, Df = d^{r}g = 0} ⊂ B^{0}

is dense in this topology, it suffices to prove that for a local section f ∈ B^{0} we have {D, d^{r}}f = 0 if either d^{r}f = 0 of Df = 0.

It is easy to see that for every local section α ∈ B we have d^{r}α = 0 if and only if α ∈ B_{0} is of augmentation degree 0. By definition the derivation D preserves the component B_{0}⊂ B of augmentation degree 0 in B^{0}. Therefore d^{r}f = 0 implies d^{r}Df = 0 and consequently {D, d^{r}}f = 0. This handles the case d^{r}f = 0. To finish the proof, assume given a local section f ∈ B^{0} such that Df = 0. Such a section by definition comes from a germ at M ⊂ \overline{T} M of a holomorphic function f̂on \overline{T} M_{J}. Since f̂is holomorphic, we have ¯∂_{J} f̂ = 0 and df̂ = ∂_{J} f̂. Therefore d^{r}f = π(df̂) = π(∂_{J}f), and

Dd^{r}f = π(¯∂_{J}∂_{J} f̂) = −π(∂_{J} ¯∂_{J} f̂) = 0,

which, again, implies {D, d^{r}}f = 0. □ 9.3.3. Let now ∇ = D_{1} : S^{1} → S^{1} ⊗ Λ^{1} be the reduction of the extended connection D. It induces a connection on the bundle L^{1} ≅ S^{1}, and this connection extends by the Leibnitz rule to a connection on the exterior algebra L of the bundle L^{1}, which we will also denote by ∇.

Denote by R = ∇ ∘ ∇ : L → L ⊗ Λ^{2} the curvature of the connection ∇. Since ∇ ∘ ∇ = \frac{1}{2}\{∇, ∇\}, the operator R also satisfies the Leibnitz rule with respect to the multiplication in L . 9.3.4. Introduce the augmentation grading on the bundle L ⊗ B by setting deg L = 0. The derivation D : L ⊗ B → L ⊗ B^{+1} obviously does not increase the augmentation degree, and we have the decomposition D = ∑_{k≥0} D_{k}. On the other hand, the derivation d^{r} preserves the augmentation degree. Therefore Lemma 9.3.2 implies that for every k ≥ 0 we have {D_{k}, d^{r}} = 0. This in turn implies that D_{0} = 0 on L^{p} for p > 0, and therefore D_{0} = id ⊗ C : L^{p} ⊗ B → L^{p} ⊗ B^{+1}. Moreover, this allows to identify explicitly the components D_{1} and D_{2} of the derivation D : L → L ⊗B^{1}. Namely, we have the following.

Lemma. We have

D_{1} = ∇ : L → L ⊗ B_{1}^{1} = L ⊗ Λ^{1}
D_{2} = \frac{1}{3}σ ∘ R : L → L ⊗ B_{2}^{1},

where σ = id ⊗ σ : L ⊗ B^{+1} → L ⊗ B is as in 7.3.2.

Proof. Since both sides of these identities satisfy the Leibnitz rule with respect to the multiplication in L , it suffices to prove them for L^{1}. But d^{r} : B^{0} → L^{1} ⊗ B^{0} restricted to S^{1} ⊂ B^{0} becomes an isomorphism d^{r} : S^{1} → L^{1}. Since {D_{1}, d^{r}} = {D_{2}, d^{r}} = 0, it suffices to prove the identities with L^{1} replaced with S^{1}. The first one then becomes the definition of ∇, and the second one is Corollary 8.4.3. □

9.4. The proof of Theorem 9.1

9.4.1. We can now prove Theorem 9.1 in the form 9.1.6. We begin with the following corollary of Lemma 9.3.4.

Corollary. Let I ⊂ B be the ideal introduced in 8.3.3. An arbitrary smooth section α ∈ C^{∞}(M, L ) satisfies

Dα ∈ C^{∞}(M, L ⊗ I^{1}) ⊂ C^{∞}(M, L ⊗ B^{1})  (9.4)

if and only if ∇α = 0.

Proof. Again, both the identity (9.4) and the equality ∇α are compatible with the Leibnitz rule with respect to the multiplication in α. Therefore it suffices to prove that they are equivalent for every α ∈ L^{1}. By definition of the ideal I the equality (9.4) holds if and only if D_{1}α = D_{2}α = 0. By Lemma 9.3.4 this is equivalent to ∇α = σ ∘ R(α) = 0. But since R = ∇ ∘ ∇, ∇α = 0 implies σ ∘ R(α) = 0, which proves the claim. □ 9.4.2. Let now Ω ∈ C^{∞}(M, ρ_{∗}Λ^{2,0}(\overline{T} M_{J}) ≅ L^{2} ⊗B^{0} be a polarization of the Hodge manifold \overline{T} M_{J}, so that Ω is of Hodge type (1, 1) and DΩ = 0. Let ω = Res Ω ∈ C^{∞}(M, Λ^{1,1}(M, ℂ)) be its restriction, and let Ω = ∑_{k≥0} Ω_{k} be its augmentation degree decomposition.

As noted in 9.2.3, the restriction map Res : ρ_{∗}Λ^{,0}(\overline{T} M_{J}) → Λ (M, ℂ) is identified under the isomorphism (9.3) with the projection L ⊗ B^{0} → L onto the component of augmentation degree 0. Therefore ω = Ω_{0}. Since the augmentation degree-1 component (L^{2} ⊗ B_{tot}^{0})_{1}^{1,1} = 0 and DΩ = 0, we have ∇ω = D_{1}Ω_{0} = 0, which proves the first claim of Theorem 9.1. 9.4.3. To prove the second claim of the theorem, let ω be a Kähler form on M compatible with the connection ∇, so that ∇ω = 0. We have to show that there exists a unique section Ω = ∑_{k≥0} Ω_{k} ∈ C^{∞}(M, L^{2} ⊗ B^{0}) which is of Hodge type (1, 1) and satisfies DΩ = 0 and Ω_{0} = ω. As in the proof of Theorem 8.1, we will use induction on k. Since Ω is of Hodge type (1, 1), we must have Ω_{1} = 0, and by Corollary 9.4.1 we have D(Ω_{0} + Ω_{1}) ∈ C^{∞}(M, L^{2} ⊗ I^{1}), which gives the base of our induction. The induction step is given by applying the following proposition to ∑_{0≥p≥k} Ω_{k} for each k ≥ 1 in turn.

Proposition 9.1 Assume given integers p, q, k; p, q ≥ 0, k ≥ 1 and assume given a section α ∈ C^{∞}(M, L^{p+q} ⊗ B^{0}) of Hodge type (p, q) such that

Dα ∈ C^{∞}(M, L^{p+q} ⊗ I_{≥k}^{1}),

where I_{≥k} = ⊕_{m≥k}I_{m}. Then there exists a unique section

β ∈ C^{∞}(M, L^{p+q} ⊗ B_{k})

of the same Hodge type (p, q) and such that

D(α + β) ∈ C^{∞}(M, L^{p+q} ⊗ I_{≥k+1}^{1}).

Proof. Let B_{tot} be the total Weil algebra introduced in 7.2.4, and consider the free module L^{p+q} ⊗ B_{tot} over B_{tot} generated by the Hodge bundle L^{p+q}. This module carries a canonical Hodge bundle structure of weight p + q. Consider the maps C : B_{tot} → B_{tot}^{+1}, σ_{tot} : B^{+1} → B_{tot} introduced in 7.3.3 and 7.3.7, and let C = id ⊗ C, σ_{tot} = id ⊗ σ_{tot} : L^{p+q} ⊗ B_{tot} → L^{p+q} ⊗ B_{tot} be the associated endomorphisms of the free module L^{p+q} ⊗ B_{tot}.

The maps C and σ_{tot} preserve the Hodge bundle structure. The commutator h = {C, σ_{tot}} : B_{tot} → B_{tot} is invertible on I_{tot} ⊂ B_{tot} by Corollary 8.3.4 and acts as kid on B_{k}^{0} ⊂ B_{tot}. Therefore the endomorphism

h = id ⊗ h = {C, σ_{tot}} : L^{p+q} ⊗ B_{tot} → L^{p+q} ⊗ B_{tot}

is invertible on L^{p+q} ⊗ I_{tot} and acts as kid on L^{p+q} ⊗ B_{k}^{0}.

Since the derivation D : L^{p+q} ⊗ B → L^{p+q} ⊗ B^{+1} is weakly Hodge, it

induces a map D^{tot} : L^{p+q} ⊗ B_{tot} → L^{p+q} ⊗ B_{tot}^{+1}, and Dα ∈ L^{p+q} ⊗ I_{≥k}^{1}

if and only if the same holds for D^{tot}α. To prove uniqueness, note that D_{0} = C is injective on L^{p+q} ⊗ B_{k}^{0}. If there are two sections β, β′ satisfying the conditions of the proposition, then D_{0}(β − β′) = 0, hence β = β′.

To prove existence, let γ = (D^{tot}α)_{k} be the component of the section D^{tot}α of augmentation degree k. Since D^{tot} ∘ D^{tot} = 0, we have Cγ = D_{0}^{tot}γ = 0 and C ∘ σ_{tot}γ = hγ. Let β = −\frac{1}{k} ∘ σ_{tot}(γ). The section β is of Hodge type (p, q) and of augmentation degree k. Moreover, it satisfies

D_{0}^{tot}β = Cβ = −Ch^{−1}σ_{tot}(γ) = −h^{−1} ∘ C ∘ σ_{tot}γ = −h^{−1} ∘ hγ = −γ.

Therefore D_{tot}(α + β) is indeed a section of L^{p+q} ⊗ (I_{tot}^{1})_{≥k+1}, which proves the proposition and finishes the proof of Theorem 9.1. □

9.5. The cotangent bundle

9.5.1. For every Kähler manifold M Theorem 9.1 provides a canonical formal hyperkähler structure on the total space \overline{T} M of the complex-conjugate to the tangent bundle to M. In particular, we have a closed holomorphic 2-form Ω_{I} for the preferred complex structure \overline{T} M_{I} on M.

Let T^{∗}M be the total space to the cotangent bundle to M equipped with the canonical holomorphic symplectic form Ω. To obtain a hyperkähler metric of the formal neighborhood of the zero section M ⊂ T^{∗}M, one can apply an appropriate version of the Darboux-Weinstein Theorem, which gives a local symplectic isomorphism κ : \overline{T} M → T^{∗}M in a neighborhood of the zero section. However, this theorem is not quite standard in the holomorphic and formal situations. For the sake of completeness, we finish this section with a sketch of a construction of such an isomorphism κ : \overline{T} M → T^{∗}M which can be used to obtain a hyperkähler metric on T^{∗}M rather than on \overline{T} M. 9.5.2. We begin with the following preliminary fact on the holomorphic de Rham complex of the manifold \overline{T} M_{I}.

Lemma. Assume given either a formal Hodge manifold structure on the U(1)-manifold \overline{T} M along the zero section M ⊂ \overline{T} M, or an actual Hodge manifold structure on an open neighborhood U ⊂ \overline{T} M of the zero section.

(i) For every point m ∈ M there exists an open neighborhood U ⊂ \overline{T} M_{I} such that the spaces Ω (U) of holomorphic forms on the complex manifold \overline{T} M_{I} (formally completed along M ⊂ \overline{T} M if necessary) equipped with the holomorphic de Rham differential ∂_{I} : Ω (U) → Ω^{+1}(U) form an exact complex.

(ii) If the subset U ⊂ \overline{T} M is invariant under the U(1)-action on the manifold \overline{T} M, then the same is true for the subspaces Ω_{k}(U) ⊂ Ω (U) of forms of weight k with respect to the U(1)-action.

(iii) Assume further that the canonical projection ρ : \overline{T} M_{I} → M is holomorphic for the preferred complex structure \overline{T} M_{I} on \overline{T} M. Then both these claims also hold for the spaces Ω (U/M) of relative holomorphic forms on U over M.

Proof. The claim (i) is standard. To prove (ii), note that, both in the formal and in the analytic situation, the spaces Ω (U) are equipped with a natural topology. Both this topology and the U(1)-action are preserved by the holomorphic Dolbeult differential ∂_{I}.

The subspaces Ω_{fin}(U) ⊂ Ω (U) of U(1)-finite vectors are dense in the natural topology. Therefore the complex ⟨Ω_{fin}(U), ∂_{I}⟩ is also exact. Since the group U(1) is compact, we have

Ω_{fin}(U) = ⨁_{k} Ω_{k}(U),

which proves (ii). The claim (iii) is, again, standard. □ 9.5.3. We can now formulate and prove the main result of this subsection.

Proposition 9.2 Assume given a formal polarized Hodge manifold structure on the manifold \overline{T} M along the zero section M ⊂ \overline{T} M such that the canonical projection ρ : \overline{T} M_{I} → M is holomorphic for the preferred complex structure \overline{T} M_{I} on \overline{T} M. Let Ω_{I} be the associated formal holomorphic 2-form on \overline{T} M_{I}. Let T^{∗}M be the total space of the cotangent bundle to the manifold M equipped with a canonical holomorphic symplectic form Ω. There exists a unique U(1)-equivariant holomorphic map κ : \overline{T} M_{I} → T^{∗}M, defined in a formal neighborhood of the zero section, which commutes with the canonical projections onto M and satisfies Ω_{I} = κ^{∗}Ω. Moreover, if the polarized Hodge manifold structure on \overline{T} M_{I} is defined in an open neighborhood U ⊂ \overline{T} M of the zero section M ⊂ \overline{T} M, then the map κ is also defined in a (possibly smaller) open neighborhood of the zero section.

Proof. By virtue of the uniqueness, the claim is local on M, so that we can assume that the whole M is contained in a U(1)-invariant neighborhood U ⊂ \overline{T} M_{I} satisfying the conditions of Lemma 9.5.2. Holomorphic maps κ : U → T^{∗}M which commute with the canonical projections onto M are in a natural one-to-one correspondence with holomorphic sections α of the bundle ρ^{∗}Λ^{1,0}(M) on \overline{T} M_{I}. Such a map κ is U(1)-equivariant if and only if the corresponding 1-form α ∈ Ω^{1}(U) is of weight 1 with respect to the U(1)- action. Moreover, it satisfies κ^{∗}Ω = Ω_{I} if and only if ∂_{I}α = Ω_{I}. Therefore to prove the formal resp. analytic parts of the proposition it suffices to prove that there exists a unique holomorphic formal resp. analytic section α ∈ C^{∞}(U, ρ^{∗}Λ^{1,0}(M) which is of weight 1 with respect to the U(1)-action and satisfies ∂_{I}α = Ω_{I}.

The proof of this fact is the same in the formal and in the analytic situations. By definition of the polarized Hodge manifold the 2-form Ω_{I} ∈ Ω^{2}(U) is of weight 1 with respect to the U(1)-action. Therefore by Lemma 9.5.2 (ii) there exists a holomorphic 1-form α ∈ Ω^{1}(U) of weight 1 with respect to the U(1)-action and such that ∂_{I}α = Ω_{I}. Moreover, the image of the form Ω_{I} under the canonical projection Ω^{2}(U) → Ω^{2}(U/M) is zero. Therefore by Lemma 9.5.2 (iii) we can arrange so that the image of the form α under the projection Ω^{1}(U) → Ω^{1}(U/M) is also zero, so that α is in fact a section of the bundle ρ^{∗}Λ^{1,0}(M). This proves the existence part. To prove uniqueness, note that every two such 1-forms must differ by a form of the type ∂_{i}f for a certain holomorphic function f ∈ Ω^{0}(U). Moreover, by Lemma 9.5.2 (ii) we can assume that the function f is of weight 1 with respect to the U(1)-action. On the other hand, by Lemma 9.5.2 (iii) we can assume that the function f is constant along the fibers of the canonical projection ρ : \overline{T} M_{I} → M. Therefore we have f = 0 identically on the whole U. □

10. Convergence

10.1. Preliminaries

10.1.1. Let M be a complex manifold. By Theorem 8.1 every Kählerian connection ∇ : Λ^{1}(M, ℂ) → Λ^{1}(M, ℂ) ⊗ Λ^{1}(M, ℂ) on the cotangent bundle Λ^{1}(M, ℂ) to the manifold M defines a flat linear extended connection D : S^{1}(M, ℂ) → B^{1}(M, ℂ) on M and therefore a formal Hodge connection D on the total space \overline{T} M of the complex-conjugate to the tangent bundle to M. By Proposition 5.2, this formal Hodge connection defines, in turn, a formal Hodge manifold structure on \overline{T} M in the formal neighborhood of the zero section M ⊂ \overline{T} M.

In this section we show that if the Kählerian connection ∇ is realanalytic, then the corresponding formal Hodge manifold structure on \overline{T} M is the completion of an actual Hodge manifold structure on an open neighborhood U ⊂ \overline{T} M of the zero section M ⊂ \overline{T} M. We also show that if the connection ∇ comes from a Kähler metric ω on M, then the corresponding polarization Ω of the formal Hodge manifold \overline{T} M defined in Theorem 9.1 converges in a neighborhood U′ ⊂ U ⊂ \overline{T} M of the zero section M ⊂ \overline{T} M to a polarization of the Hodge manifold structure on U′. Here is the precise formulation of these results.

Theorem 10.1 Let M be a complex manifold equipped with a real-analytic Kählerian connection ∇ : Λ^{1}(M, ℂ) → Λ^{1}(M, ℂ)⊗Λ^{1}(M, ℂ) on its cotangent bundle Λ^{1}(M, ℂ). There exists an open neighborhood U ⊂ \overline{T} M of the zero section M ⊂ \overline{T} M in the total space \overline{T} M of the complex-conjugate to the tangent bundle to M and a Hodge manifold structure on U ⊂ \overline{T} M such that its completion along the zero section M ⊂ \overline{T} M defines a linear flat extended connection D on M with reduction ∇.

Moreover, assume that M is equipped with a Kähler metric ω such that ∇ω = 0, and let Ω ∈ C_{M}^{∞}(\overline{T} M, Λ^{2}(\overline{T} M, ℂ)) be the formal polarization of the Hodge manifold structure on U ⊂ \overline{T} M along M ⊂ \overline{T} M. Then there exists an open neighborhood U′ ⊂ U of M ⊂ U such that Ω ∈ C^{∞}(U′, Λ^{2}(\overline{T} M, ℂ)) ⊂ C_{M}^{∞}(\overline{T} M, Λ^{2}(\overline{T} M, ℂ)).

10.1.2. We begin with some preliminary observations. First of all, the question is local on M, therefore we may assume that M is an open neighborhood of 0 in the complex vector space V = ℂ^{n}. Fix once and for all a real structure and an Hermitian metrics on the vector space V, so that it is isomorphic to its dual V ≅ V^{∗}.

The subspace 𝔍 ⊂ C^{∞}(M, ℂ) of functions vanishing at 0 ∈ M is an ideal in the algebra C^{∞}(M, ℂ), and 𝔍-adic topology on C^{∞}(M, ℂ) extends canonically to the de Rham algebra Λ (M, ℂ) of the manifold M and, further, to the Weil algebra B (M, ℂ) of M introduced in 6.3.5. Instead of working with bundle algebra B (M, ℂ) on M, it will be convenient for us to consider the vector space

B = C_{𝔍}^{∞}(M, B (M, ℂ)),

which is by definition the 𝔍-adic completion of the space C^{∞}(M, B (M, ℂ)) of global sections of the Weil algebra. This vector space is canonically a (pro-)algebra over ℂ. Moreover, the Hodge bundle structure on B (M, ℂ) induces an ℝ-Hodge structure on the algebra B . 10.1.3. The 𝔍-adic completion C_{𝔍}^{∞}(M, ℂ) of the space of smooth functions on M is canonically isomorphic to the completion Ŝ(V) of the symmetric algebra of the vector space V ≅ V^{∗}. The cotangent bundle Λ^{1}(M, ℂ) is isomorphic to the trivial bundle V with fiber V over M, and the completed de Rham algebra C_{𝔍}^{∞}(M, Λ (M, ℂ)) is isomorphic to the product

C_{𝔍}^{∞}(M, Λ (M, ℂ)) ≅ S (V) ⊗ Λ (V).

This is a free graded-commutative algebra generated by two copies of the vector space V, which we denote by V_{1} = V ⊂ Λ^{1}(V) and by V_{2} = V ⊂ S^{1}(V). It is convenient to choose the trivialization Λ^{1}(M, ℂ) ≅ V in such a way that the de Rham differential d_{M} : Λ (M, ℂ) → Λ^{+1}(M, ℂ) induces an identity map d_{M} : V_{2} → V_{1} ⊂ C_{𝔍}^{∞}(M, Λ^{1}(M, ℂ)). 10.1.4. The complex vector bundle S^{1}(M, ℂ) on M is also isomorphic to the trivial bundle V. Choose a trivialization S^{1}(M, ℂ) ≅ V in such a way that the canonical map C : S^{1}(M, ℂ) → Λ^{1}(M, ℂ) is the identity map. Denote by

V_{3} = V ⊂ C^{∞}(M, S^{1}(M, ℂ)) ⊂ B^{0}

the subset of constant sections in S^{1}(M, ℂ) ≅ V. Then the Weil algebra B becomes isomorphic to the product

B^{i} ≅ Ŝ(V_{2} ⊕ V_{3}) ⊗ Λ^{i}(V_{1})

of the completed symmetric algebra Ŝ(V_{2} ⊕ V_{3}) of the sum V_{2} ⊕ V_{3} of two copies of the vector space V and the exterior algebra Λ (V_{1}) of the third copy of the vector space V. 10.1.5. Recall that we have introduced in 7.2.2 a grading on the Weil algebra B (M, ℂ) which we call the augmentation grading. It induces a grading on the the Weil algebra B . The augmentation grading on B is multiplicative, and it is obtained by assigning degree 1 to the generator subspaces V_{1}, V_{3} ⊂ B and degree 0 to the generator subspace V_{2} ⊂ B . As in 7.2.2, we will denote the augmentation grading on B by lower indices.

We will now introduce yet another grading on the algebra B which we will call the total grading. It is by definition the multiplicative grading obtained by assigning degree 1 to all the generators V_{1}, V_{2}, V_{3} ⊂ B of the Weil algebra B We will denote by B_{k,n} ⊂ B the component of augmentation degree k and total degree n. Note that by definition n, k ≥ 0 and, moreover, n ≥ k. Remark. In 7.2.2 we have also defined a finer augmentation bigrading on the Weil algebra B (M, ℂ) and it this bigrading that was denoted by double lower indices throughout Section 8. We will now longer need the augmentation bigrading, so there is no danger of confusion. 10.1.6. The trivialization of the cotangent bundle to M defines an isomorphism \overline{T} M ≅ M ×V and a constant Hodge connection on the pair ⟨\overline{T} M, M⟩. The corresponding extended connection D^{const} : S^{1}(M, ℂ) → B^{1}(M, ℂ) is the sum of the trivial connection

∇_{1}^{const} : S^{1}(M, ℂ) → S^{1}(M, ℂ) ⊗ Λ^{1}(M, ℂ) ⊂ B^{1}(M, ℂ)

on S^{1}(M, ℂ) ≅ V and the canonical isomorphism

C = id : S^{1}(M, ℂ) → Λ^{1}(M, ℂ) ⊂ B^{1}(M, ℂ).

The derivation D^{const} : B → B^{+1} of the Weil algebra associated to the extended connection D^{const} by 6.4.5 is equal to

D^{const} = C = id : V_{3} → V_{1}
D^{const} = d_{M} = id : V_{2} → V_{1}
D^{const} = d_{M} = 0 on V_{1}

on the generator spaces V_{1}, V_{2}, V_{3} ⊂ B . In particular, the derivation D^{const} preserves the total degree. 10.1.7. Let now D : S^{1}(M, ℂ) → B^{1}(M, ℂ) be the an arbitrary linear extended connection on the manifold M, and let

D = ∑_{k≥0} D_{k} : B → B^{+1}

be the derivation of the Weil algebra B associated to the extended connection D by 6.4.5. The derivation D admits a finer decomposition

D = ∑_{k,n≥0} D_{k,n} : B → B^{+1}

according to both the augmentation and the total degree on B . The summand D_{k,n} by definition raises the augmentation degree by k and the total degree by n. 10.1.8. Since the extended connection D : S^{1}(M, ℂ) → B^{1}(M, ℂ) is linear, its component D_{0} : S^{1}(M, ℂ) → Λ^{1}(M, ℂ) of augmentation degree 0 coincides with the canonical isomorphism C : S^{1}(M, ℂ) → Λ^{1}(M, ℂ). Therefore the restriction of the derivation D : B → B^{+1} to the generator subspace V_{3} ⊂ S^{1}(M, ℂ) ⊂ B^{0} satisfies

D_{0} = C = D_{0}^{const} = D_{0,0}^{const} : V_{3} → V_{1} ⊂ B^{1}.

In particular, all the components D_{0,n} except for D_{0,0} vanish on the subspace V_{3} ⊂ B .

The restriction of the derivation D to the subspace Λ (M, ℂ) ⊂ B (M, ℂ) by definition coincides with the de Rham differential d_{M} : Λ (M, ℂ) → Λ^{+1}(M, ℂ). Therefore on the generator subspaces V_{1}, V_{2} ⊂ B we have D = d_{M} = D^{const}. In particular, all the components D_{k,n} except for D_{1,0} vanish on the subspaces V_{1}, V_{2} ⊂ B . 10.1.9. The fixed Hermitian metric on the generator spaces V_{1} = V_{2} = V_{3} = V extends uniquely to a metric on the whole Weil algebra such that the multiplication map B ⊗ B → B is an isometry. We call this metric the standard metric on B . We finish our preliminary observations with the following fact which we will use to deduce Theorem 10.1 from estimates on the components D_{n,k} of the derivation D : B → B^{+1}.

Lemma. Let D = ∑_{n,k} D_{n,k} : B → B^{+1} be a derivation associated to an extended connection D on the manifold M. Consider the norms ∥D_{k,n}∥ of the restrictions D_{k,n} : V_{3} → B_{n+1,k+1}^{1} of the derivations D_{k,n} : B → B^{+1} to the generator subspace V_{3} ⊂ B^{0} taken with respect to the standard metric on the Weil algebra B . If for certain constants C, ε > 0 and for every natural n ≥ k ≥ 0 we have

∥D_{k,n}∥ < Cε^{n},  (10.1)

then the formal Hodge connection on \overline{T} M along M associated to D converges to an actual real-analytic Hodge connection on the open ball of radius ε in \overline{T} M with center at 0 ∈ M ⊂ \overline{T} M. Conversely, if the extended connection D comes from a real-analytic Hodge connection on an open neighborhood U ⊂ \overline{T} M, and if the Taylor series for this Hodge connection converge in the closed ball of radius ε with center at 0 ∈ M ⊂ \overline{T} M, then there exists a constant C > 0 such that (10.1) holds for every n, k ≥ 0.

Proof. The constant Hodge connection D^{const} is obviously defined on the whole \overline{T} M, and every other formal Hodge connection on \overline{T} M is of the form

D = D^{const} + d^{r} ∘ Θ : Λ^{0}(\overline{T} M, ℂ) → ρ^{∗}Λ^{1}(M, ℂ),

where d^{r} : Λ^{0}(\overline{T} M, ℂ) → Λ^{1}(\overline{T} M/M, ℂ) is the relative de Rham differential, and Θ ∈ C_{M}^{∞}(\overline{T} M, Λ^{1}(\overline{T} M/M, ℂ) ⊗ ρ^{∗}Λ^{1}(M, ℂ)) is a certain relative 1-form on the formal neighborhood of M ⊂ \overline{T} M with values in the bundle ρ^{∗}Λ^{1}(M, ℂ). Both bundles Λ^{1}(\overline{T} M/M, ℂ) and ρ^{∗}Λ^{1}(M, ℂ) are canonically isomorphic to the trivial bundle V with fiber V on \overline{T} M. Therefore we can treat the 1-form Θ as a formal germ of a End(V)-valued function on \overline{T} M along M. The Hodge connection D converges on a subset U ⊂ \overline{T} M if and only if this formal germ comes from a real-analytic End(V)-valued function Θ on U.

The space of all formal Taylor series for End(V)-valued functions on \overline{T} M at 0 ∈ M ⊂ \overline{T} M is by definition equal to End(V) ⊗ B^{0}. Moreover, for every n ≥ 0 the component Θ_{n} ∈ B_{n}^{0} ⊗ End(V) = Hom(V, B_{n}^{0} ⊗ V) of total degree n of the formal power series for the function Θ at 0 ∈ \overline{T} M is equal to the derivation

∑_{0≤k≤n} D_{k,n} : V = V_{3} → V = V_{1} ⊗ ⨁_{0≤k≤n} B_{k,n}^{0}.

Every point x ∈ \overline{T} M defines the “evaluation at x” map

ev_{x} : C^{∞}(\overline{T} M, End(V)) → ℂ,

and the formal Taylor series for Θ ∈ End(V) ⊗ B^{0} converges at the point x ∈ \overline{T} M if and only if the series

Θ(x) = ∑_{n≥0} ev_{x}(Θ_{n}) ∈ End(V)

converges. But we have

∥ ev_{x}(Θ_{n})∥ = ∥ ∑_{0≤k≤n} D_{k,n}∥ |x|^{n},

where |x| is the distance from the point x to 0 ∈ \overline{T} M. Now the application of standard criteria of convergence finishes the proof of the lemma. □

10.2. Combinatorics

10.2.1. We now derive some purely combinatorial facts needed to obtain estimates for the components D_{k,n} of the extended connection D. First, let a_{n} be the Catalan numbers, that is, the numbers defined by the recurrence relation

a_{n} = ∑_{1≤k≤n−1} a_{k}a_{n−k}

and the initial conditions a_{1} = 1, a_{n} = 0 for n ≤ 0. As is well-known, the generating function f(z) = ∑_{k≥0} a_{k}z^{k} for the Catalan numbers satisfies the equation f(z) = f(z)^{2} + z and equals therefore

f(z) = \frac{1}{2} − \sqrt{\frac{1}{4} − z}.

The Taylor series for this function at z = 0 converges for 4|z| < 1, which implies that

a_{k} < C(4 + ε)^{k}

for some positive constant C > 0 and every ε > 0. 10.2.2. We will need a more complicated sequence of integers, numbered by two natural indices, which we denote by b_{k,n}. The sequence b_{k,n} is defined by the recurrence relation

b_{k,n} = ∑_{p,q;1≤p≤k−1} \frac{q + 1}{k}b_{p,q}b_{k−p,n−q}

and the initial conditions

b_{k,n} = 0 for k ≤ 0,
b_{k,n} = 0 for k = 1, n < 0,
b_{k,n} = 1 for k = 1, n ≥ 0,

which imply, in particular, that if n < 0, then b_{k,n} = 0 for every k. For every k ≥ 1 let g_{k}(z) = ∑_{n≥0} b_{k,n}z^{n} be the generating function for the numbers b_{k,n}. The recurrence relations on b_{k,n} give

g_{k}(z) = \frac{1}{k} ∑_{1≤p≤k−1} g_{k−p}(z) (1 + z \frac{∂}{∂z} ) (g_{p}(z))

= \frac{1}{2k} ∑_{1≤p≤k−1} (2 + z \frac{∂}{∂z} ) (g_{p}(z)g_{k−p}(z)),

and the initial conditions give

g_{1}(z) = \frac{1}{1 − z} .

10.2.3. Say that a formal series f(z) in the variable z is non-negative if all the terms in the series are non-negative real numbers. The sum and product of two non-negative series and the derivative of a non-negative series is also obviously non-negative. For two formal series s(z), t(z) write s(z) ≪ t(z) if the difference t(z) − s(z) is a non-negative power series.

Our main estimate for the generating functions g_{k}(z) is the following.

Lemma. For every k ≥ 1 we have

g_{k}(z) ≪ a_{k} \frac{1}{(1 − z)^{2k−1}} ,

where a_{k} are the Catalan numbers.

Proof. Use induction on k. For k = 1 we have g_{1}(z) = \frac{1}{1−z} and a_{1} = 1, which gives the base for induction. Assume that the claim is proved for all p < k. Since all the g_{n}(z) are non-negative power series, this implies that for every p, 1 ≤ p ≤ k − 1 we have

g_{p}(z)g_{k−p}(z) ≪ a_{p}a_{k−p} \frac{1}{(1 − z)^{2p−1}} \frac{1}{(1 − z)^{2k−2p−1}} = a_{p}a_{k−p} \frac{1}{(1 − z)^{2k−2}} .

Therefore

(2 + z \frac{∂}{∂z} ) (g_{p}(z)g_{k−p}(z)) ≪ a_{p}a_{k−p} (2 + z \frac{∂}{∂z} ) \frac{1}{(1 − z)^{2k−2}}

= a_{p}a_{k−p} (\frac{2}{(1 − z)^{2k−2}} + \frac{(2k − 2)z}{(1 − z)^{2k−1}} )

= a_{p}a_{k−p} (\frac{2k − 2}{(1 − z)^{2k−1}} − \frac{2k − 4}{(1 − z)^{2k−2}} )

≪ (2k − 2)a_{p}a_{k−p} \frac{1}{(1 − z)^{2k−1}} .

Hence

g_{k}(z) = \frac{1}{2k} ∑_{1≤p≤k−1} (2 + z \frac{∂}{∂z} ) (g_{p}(z)g_{k−p}(z))

≪ \frac{2k − 2}{2k} ∑_{1≤p≤k−1} a_{p}a_{k−p} \frac{1}{(1 − z)^{2k−1}}

≪ \frac{1}{(1 − z)^{2k−1}} ∑_{1≤p≤k−1} a_{p}a_{k−p} = a_{k} \frac{1}{(1 − z)^{2k−1}} ,

which proves the lemma. □ 10.2.4. This estimate yields the following estimate for the numbers b_{k,n}.

Corollary. The power series

g(z) = ∑_{k,n} b_{k,n}z^{n+k} = ∑_{k≥1} g_{k}(z)z^{k}

converges for z < 3−\sqrt{8}. Consequently, for every C_{2} such that (3−\sqrt{8})C_{2} > 1 there exists a positive constant C > 0 such that

b_{n,k} < CC_{2}^{n+k}

for every n and k. (One can take, for example, C_{2} = 6.)

Proof. Indeed, we have

g(z) ≪ ∑_{k≥1} a_{k}z^{k} \frac{1}{(1 − z)^{2k−1}} = (1 − z)f (\frac{z}{(1 − z)^{2}} ) ,  (10.2)

where f(z) = \frac{1}{2} −\sqrt{\frac{1}{4} − z} is the generating function for the Catalan numbers. Therefore

g(z) ≪ (1 − z) (\frac{1}{2} − \sqrt{\frac{1}{4} − \frac{z}{(1 − z)^{2}}} ) ,

and the right hand side converges absolutely when

\frac{|z|}{(1 − z)^{2}} < \frac{1}{4}.

Since 3 − \sqrt{8} is the root of the quadratic equation (1 − z)^{2} = 4z, this inequality holds for every z such that |z| < 3 − \sqrt{8}. □ 10.2.5. To study polarizations of Hodge manifold structures on \overline{T} M, we will need yet another recursive sequence of integers, which we denote by b_{k,n}^{m}. This sequence is defined by the recurrence relation

b_{k,n}^{m} = ∑_{p,q;1≤p≤k−1} \frac{q + m(k − p)}{k}b_{p,q}^{m}b_{k−p,n−q}

and the initial conditions

b_{k,n}^{m} = 0 unless k, m ≤ 0,
b_{k,n}^{m} = 0 for k = 1, n < 0,
b_{k,n}^{m} = 1 for k = 1, n ≥ 0.

10.2.6. To estimate the numbers b_{k,n}^{m}, consider the auxiliary sequence c_{k,n} defined by setting

c_{k,n} = ∑_{p,q;1≤p≤k−1} c_{p.q}b_{k−p,n−q} k ≥ 2,

and c_{k,n} = b_{k,n} for k ≤ 1. The generating series c(z) = ∑_{k,n≥0} c_{k,n}z^{n+k} satisfies

c(z) = c(z)g(z) + \frac{z}{1 − z} ,

so that we have c(z) = \frac{z}{(1−z)(1−g(z))}, which is non-singular when |z|, |g(z)| < 1 and g(z) is non-singular. By (10.2) the latter inequality holds if

|(1 − z) (\frac{1}{2} − \sqrt{\frac{1}{4} − \frac{z}{(1 − z)^{2}}} )| < 1,

which holds in the whole disc where g(z) converges, that is, for |z| < 3−\sqrt{8}. Therefore, as in Corollary 10.2.4, we have

c_{k,n} < C6^{n+k}  (10.3)

for some positive constant C. 10.2.7. We can now estimate the numbers b_{k,n}^{m}.

Lemma. For every m, k, n we have

b_{k,n}^{m} ≤ (2m)^{k−1}c_{k,n}b_{k,n},  (10.4)

where c_{k,n} are as in 10.2.6 and b_{k,n} are the numbers introduced in 10.2.2. Consequently, we have

b_{k,n}^{m} < C(72m)^{n+k+m}

for some positive constant C > 0.

Proof. Use induction on k. The case k = 1 follows from the initial conditions. Assume the estimate (10.4) proved for all b_{p,q}^{m} with p < k. Note that by the recurrence relations we have b_{p,q} ≤ b_{k,n} and c_{p,q} ≤ c_{k,n} whenever p < k.

Therefore

b_{k,n}^{m} = ∑_{p,q;1≤p≤k−1} \frac{q + m(k − p)}{k}b_{p,q}^{m}b_{k−p,n−q}

≤ ∑_{p,q;1≤p≤k−1} \frac{q}{k} b_{p,q}^{m}b_{k−p,n−q} + ∑_{p,q;1≤p≤k−1} b_{p,q}^{m}b_{k−p,n−q}

≤ ∑_{p,q;1≤p≤k−1} 2^{p−1}c_{p,q} \frac{q}{k} b_{p,q}b_{k−p,n−q} + ∑_{p,q;1≤p≤k−1} 2^{p−1}b_{p,q}c_{p,q}b_{k−p,n−q}

≤ 2^{k−2}c_{k,n} ∑_{p,q;1≤p≤k−1} \frac{q + 1}{k}b_{p,q}b_{k−p,n−q}

+ 2^{k−2}b_{k,n} ∑_{p,q;1≤p≤k−1} c_{p,q}b_{k−p,n−q}

= 2^{k−2}c_{k,n}b_{k,n} + 2^{k−2}c_{k,n}b_{k,n} = 2^{k−1}c_{k,n}b_{k,n},

which proves (10.4) for b_{k,n}^{m}. The second estimate of the lemma now follows from (10.3) and Corollary 10.2.4. □

10.3. The main estimate

10.3.1. Let now D = ∑_{k,n} D_{n,k} : B → B^{+1} be a derivation of the Weil algebra B associated to a flat linear extended connection on M. Consider the restriction D_{k,n} : B_{p,q} → B p + k, q + n of the derivation D_{k,n} to the component B_{p,q} ⊂ B of augmentation degree p and total degree q. Since both B_{p,q} and B p + k, q + n are finite-dimensional vector spaces, the norm of this restriction with respect to the standard metric on B is well-defined. Denote this norm by ∥D_{k,n}∥_{p,q}.

By Lemma 10.1.9 the convergence of the Hodge manifold structure on \overline{T} M corresponding to the extended connection D is related to the growth of the norms ∥D_{k,n}∥_{1,1}. Our main estimate on the norms ∥D_{k,n}∥_{1,1} is the following.

Proposition 10.1 Assume that there exist a positive constant C_{0} such that for every n the norms ∥D_{1,n}∥_{1,1} and ∥D_{1,n}∥_{0,1} satisfy

∥D_{1,n}∥_{1,1}, ∥D_{1,n}∥_{1,0} < C_{0}^{n}.

Then there exists a positive constant C_{1} such that for every n, k the norm ∥D_{k,n}∥_{1,1} satisfies

∥D_{k,n}∥_{1,1} < C_{1}^{n}.

10.3.2. In order to prove Proposition 10.1, we need some preliminary facts. Recall that we have introduced in 7.2.4 the total Weil algebra B_{tot}(M, ℂ) of the manifold M, and let

B_{tot} = C_{𝔍}^{∞}(M, B_{tot}(M, ℂ))

be the algebra of its smooth sections completed at 0 ⊂ M. By definition for every k ≥ 0 we have B_{tot}^{k} = B^{k} ⊗ W_{k}^{∗}, where W_{k} is the ℝ-Hodge structure of weight k universal for weakly Hodge maps, as in 1.4.5. There exists a unique Hermitian metric on W_{k} such that all the Hodge components W^{p,q} ⊂ W_{k} are orthogonal and all the Hodge degree components w_{k}^{p,q} of the universal weakly Hodge map w_{k} : ℝ(0) → W_{k} are isometries. This metric defines a canonical Hermitian metric on W_{k}^{∗}. Definition. The standard metric on the total Weil algebra B_{tot} is the product of the canonical metric and the standard metric on B . 10.3.3. By Lemma 7.1.4 the total Weil algebra B_{tot} is generated by the subspaces V_{2}, V_{3} ⊂ B^{0} = B_{tot}^{0} and the subspace V_{1} ⊗ W_{1}^{∗} ⊂ B_{tot}^{1}, which we denote by V_{1}^{tot}. The ideal of relations for the algebra B_{tot} is the ideal in S (V_{2} ⊕ V_{3}) ⊗ Λ (V_{1}^{tot}) generated by S^{2}(V_{1}) ⊗ Λ^{2}(W_{1}^{∗}) ⊂ Λ^{2}(V_{1}^{tot}).

The direct sum decomposition (7.3) induces a direct sum decomposition

V_{1}^{tot} = V_{1}^{ll} ⊕ V_{1}^{o} ⊕ V_{1}^{rr}

of the generator subspace V_{1}^{tot} ⊂ B_{tot}^{1}. The subspaces V_{1}^{o} ⊂ V_{1}^{tot} and V_{1}^{ll} ⊕ V_{1}^{rr} ⊂ V_{1}^{tot} are both isomorphic to the vector space V_{1}. More precisely, the universal weakly Hodge map w_{1} : ℝ(0) → W_{1} defines a projection

P : V_{1}^{tot} = V_{1} ⊗ W_{1}^{∗} → V_{1},

and the restriction of the projection P to either of the subspaces V_{1}^{o}, V_{1}^{ll} ⊕ V_{1}^{rr} ⊂ V_{1}^{tot} is an isomorphism. Moreover, either of these restrictions is an isometry with respect to the standard metrics. 10.3.4. The multiplication in B_{tot} is not an isometry with respect to this metric. However, for every b_{1}, b_{2} ⊂ B_{tot} we have the inequality

∥b_{1}b_{2}∥ ≤ ∥b_{1}∥ ∥b_{2}∥.

Moreover, this inequality becomes an equality when b_{1} ⊂ B_{tot}^{0}. In particular, if we extend the map P : V_{1}^{tot} → V_{1} to a B^{0}-module map

P : B_{tot}^{1} → B^{1},

then the restriction of the map P to either of the subspaces B_{o}^{1}, B_{ll}^{1} ⊕ B_{rr}^{1} ⊂ B_{tot}^{1} is an isometry with respect to the standard metric. Therefore the norm of the projection P : B_{tot}^{1} → B^{1} is at most 2. 10.3.5. The total and augmentation gradings on the Weil algebra B extend to gradings on the total Weil algebra B_{tot}, also denoted by lower indices. The extended connection D on M induces a derivation D^{tot} = ∑_{n,k} D_{k,n}^{tot} : B_{tot} → B_{tot}^{+1}. As in 10.3.1, denote by ∥D_{k,n}^{tot}∥_{p,q} the norm of the map D_{k,n}^{tot} : (B_{tot})_{p,q} → (B_{tot})_{p+k,q+n} with respect to the standard metric on B_{tot}. The derivations D_{k,n}^{tot} are related to D_{k,n} by

D_{k,n} = P ∘ D_{k,n}^{tot} : B^{0} → B^{1},

and we have the following.

Lemma. For every k, n and p = 0, 1 we have

∥D_{k,n}∥_{p,1} = ∥D_{k,n}^{tot}∥_{p,1}.

Proof. By definition we have B_{0,1} ⊕ B_{1,1} = V_{1} ⊕ V_{2} ⊕ V_{3}. Moreover, the derivation D_{k,n} vanishes on V_{1}, hence D_{k,n}^{tot} vanishes on V_{1}^{tot}. Therefore it suffices to compare their norms on V_{2} ⊕ V_{3} ⊂ B^{0} = B_{tot}^{0}.

Since on S (V_{2}) ⊂ B^{0} the derivation D_{k,n} coincides with the de Rham differential, the derivation D_{k,n}^{tot} maps the subspace V_{2} into B_{ll}^{1} ⊕B_{rr}^{1}. Moreover, by Lemma 7.2.6 the derivation D_{k,n}^{tot} maps V_{3} either into B_{o}^{1} or into B_{ll}^{1} ⊕B_{rr}^{1}, depending on the parity of the number k. Since D_{k,n} = P ∘ D_{k,n}^{tot} and the map P : B_{tot}^{1} → B^{1} is an isometry on both B_{o}^{1} ⊂ B_{tot}^{1} and B_{ll}^{1} ⊕ B_{rr}^{1} ⊂ B_{tot}^{1}, we have ∥D_{k,n}∥ = ∥D_{k,n}^{tot}∥ on both V_{2} ⊂ B^{0} and V_{3} ⊂ B^{0}, which proves the lemma. □

This lemma allows to replace the derivations D_{k,n} in Proposition 10.1 with associated derivations D_{k,n}^{tot} of the total Weil algebra B_{tot}. 10.3.6. Since the extended connection D is linear and flat, the construction used in the proof of Lemma 8.3.6 shows that

D_{k}^{tot} = h^{−1} ∘ σ_{tot} ∘ ∑_{1≤p≤k−1} D_{p}^{tot} ∘ D_{k−p}^{tot} : V_{3} → (B_{tot}^{1})_{k+1} ,  (10.5)

where σ_{tot} : B_{tot}^{+1} → B_{tot} is the canonical map constructed in 7.3.7, and h : B_{tot} → B_{tot} is as in Lemma 7.3.8. Both σ_{tot} and h preserve the augmentation degree. In order to obtain estimates on ∥D_{k,n}∥_{1,1} = ∥D_{k,n}^{tot}∥_{1,1}, we have to estimate the norms ∥h^{−1}∥ and ∥σ_{tot}∥ of the restrictions of maps h^{−1} and

σ_{tot} on the subspace (B_{tot})_{k+1} ⊂ B_{tot}.

By Lemma 7.3.9 the map h : (B )_{k+1} → (B )_{k+1} is diagonalizable, with eigenvalues k + 1 if k is even and (k + 1)/2, (k − 1)/2 if k is odd. Since m ≥ 2, in any case on (B )_{k+1} we have

∥h^{−1}∥ < \frac{3}{k} .  (10.6)

10.3.7. To estimate σ_{tot} : B_{tot}^{2} → B_{tot}^{2}, recall that, as noted in 7.3.7, the map σ_{tot} is a map of B^{0}-modules. The space B_{tot}^{2} = B^{0} ⊗ (B_{tot}^{2})_{2} is a free B^{0}-module generated by a finite-dimensional vector space (B_{tot}^{2})_{2}. The map σ_{tot} preserves the augmentation degree, hence it maps (B_{tot}^{2})_{2} into the finitedimensional vector space (B_{tot}^{1})_{2}. Therefore there exists a constant K such that

∥σ_{tot}∥ ≤ K  (10.7)

on (B_{tot}^{2})_{2}. Since we have ∥b_{1}b_{2}∥ = ∥b_{1}∥ ∥b_{2}∥ for every b_{1} ∈ B^{0}, b_{2} ∈ B_{tot}^{2}, and the map σ_{tot} is B^{0}-linear, the estimate (10.7) holds on the whole B_{tot}^{2} = B^{0} ⊗ (B_{tot}^{2})_{2}. We can assume, in addition, that K ≥ 1. Remark. In fact K = 2, but we will not need this. 10.3.8. Let now b_{k,n} be the numbers defined recursively in 10.2.2. Our estimate for ∥D_{k,n}^{tot}∥_{p,q} is the following.

Lemma. In the assumptions and notations of Proposition 10.1, we have

∥D_{k,n}^{tot}∥_{p,q} < q(3K)^{k−1}C_{0}^{n}b_{k,n}

for every k, n.

Proof. Use induction on k. The base of induction is the case k = 1, when the inequality holds by assumption. Assume that for some k we have proved the inequality for all ∥D_{m,n}^{tot}∥_{p,q} with m < k, and fix a number n ≥ k.

Consider first the restriction of D_{k,n}^{tot} onto the generator subspace V_{3} ⊂ B^{0}. Taking into account the total degree, we can rewrite (10.5) as

D_{k,n}^{tot} = h^{−1} ∘ σ_{tot} ∘ ∑_{1≤p≤k−1} ∑_{q} D_{k−p,n−q}^{tot} ∘ D_{p,q}^{tot} : V_{3} → (B_{tot}^{1})_{k+1} .

Therefore the norm of the map D_{k,n}^{tot} : V_{3} → B_{tot}^{1} satisfies

∥D_{k,n}^{tot}} ≤ ∥h^{−1}∥ ∥σ_{tot}∥ ∑_{1≤p≤k−1} ∑_{q} ∥D_{k−p,n−q}^{tot}∥_{p+1,q+1} ∥D_{p,q}^{tot}∥_{1,1}.

Substituting into this the estimates (10.6), (10.7) and the inductive assumption, we get

∥D_{k,n}^{tot}∥ < \frac{3}{k} K ∑_{1≤p≤k−1} ∑_{q}(q + 1)(3K)^{k−2}C_{0}^{n}b_{k−p,n−q}b_{p,q} = (3K)^{k−1}C_{0}^{n}b_{k,n}.

Since by definition D_{k,n}^{tot} vanishes on V_{2} ⊂ B^{0} and on V_{1}^{tot} ⊂ B_{tot}^{1}, this proves that

∥D_{k,n}^{tot}∥_{p,1} < (3K)^{k−1}C_{0}^{n}b_{k,n}

when p = 0, 1. Since the map D_{k,n}^{tot} : B_{tot} → B_{tot}^{+1} is a derivation, the Leibnitz rule and the triangle inequality show that for every p, q

∥D_{k,n}^{tot}∥_{p,q} < q(3K)^{k−1}C_{0}^{n}b_{k,n},

which proves the lemma. □ 10.3.9. Proof of Proposition 10.1. By Lemma 10.3.5 we have ∥D_{k,n}∥_{p,1} = ∥D_{k,n}^{tot}∥_{p,1}, and by Lemma 10.3.8 we have

∥D_{k,n}∥_{p,1} = ∥D_{k,n}^{tot}∥_{p,1} < (3K)^{k−1}C_{0}^{n}b_{n,k}.

Since k ≤ n and K ≥ 1, this estimate together with Corollary 10.2.4 implies that

∥D_{k,n}∥_{p,1} < C(3K)^{k−1}C_{0}^{n}6^{2n} < C(108KC_{0})^{n}

for some positive constant C > 0, which proves the proposition. □ 10.3.10. Proposition 10.1 gives estimates for the derivation D : B → B^{+1} or, equivalently, for the Dolbeault differential

¯∂_{J} : Λ^{0,}(\overline{T} M_{J}) → Λ^{0, +1}(\overline{T} M_{J})

for the complementary complex structure \overline{T} M_{J} on \overline{T} M associated to the extended connection D on M. To prove the second part of Theorem 10.1, we will need to obtain estimates on the Dolbeult differential ¯∂_{J} : Λ^{p,0}(\overline{T} M_{J}) → Λ^{p,1}(\overline{T} M_{J}) with p > 0. To do this, we use the model for the de Rham complex Λ^{,}(\overline{T} M_{J}) constructed in Subsection 9.2.

Recall that in 9.2.3 we have identified the direct image ρ_{∗}Λ^{,}(\overline{T} M_{J}) of the de Rham algebra of the manifold \overline{T} M with the free module over the Weil algebra B (M, ℂ) generated by a graded algebra bundle L (M, ℂ) on M. The Dolbeult differential ¯∂_{J} for the complementary complex structure \overline{T} M_{J} induces an algebra derivation D : L (M, ℂ)⊗B (M, ℂ) → L (M, ℂ)⊗ B^{+1}(M, ℂ), so that the free module ρ_{∗}Λ^{,}(\overline{T} M_{J}) ≅ L (M, ℂ) ⊗ B (M, ℂ) becomes a differential graded module over the Weil algebra.

10.3.11. The algebra bundle L (M, ℂ) is isomorphic to the de Rham algebra Λ (M, ℂ). In particular, the bundle L^{1}(M, ℂ) is isomorphic to the trivial bundle V with fiber V over M. By 9.3.2 the relative de Rham differential

d^{r} : Λ (\overline{T} M/M, ℂ) → Λ^{+1}(\overline{T} M/M, ℂ)

induces a derivation

d^{r} : L (M, ℂ) ⊗ B (M, ℂ) → L^{+1}(M, ℂ) ⊗ B (M, ℂ),

and we can choose the trivialization L^{1}(M, ℂ) ≅ V in such a way that d^{r} identifies the generator subspace V_{3} ⊂ B^{0} with the subspace of constant sections of V ≅ L^{1}(M, ℂ) ⊂ L^{1}(M, ℂ) ⊗ B^{0}(M, ℂ). 10.3.12. Denote by

L^{,}= C_{𝔍}^{∞}(L (M, ℂ) ⊗ B_{tot}(M, ℂ))

the 𝔍-adic completion of the space of smooth sections of the algebra bundle L (M, ℂ)⊗B_{tot}(M, ℂ) on M. The space L^{,}is a bigraded algebra equipped with the derivations d^{r} : L^{,}→ L^{+1,}, D^{tot} : L^{,}→ L^{, +1}, which commute by Lemma 9.3.2. The algebra L^{,}is the free graded-commutative algebra generated by the subspaces V_{1}, V_{2}, V_{3} ⊂ L^{0,}= B and the subspace V = d^{r}(V_{3}) ⊂ L^{1,0}, which we denote by V_{4}. 10.3.13. As in 10.3.2, the given metric on the generator subspaces V_{1} = V_{2} = V_{3} = V_{4} = V extends uniquely to a multiplicative metric on the algebra L^{,}, which we call the standard metric. For every k > 0, introduce the total and augmentation gradings on the free B -module L^{k,}= Λ^{k}(V_{4}) ⊗ B by setting deg Λ^{k}(V_{4}) = (0, 0). Let D = D_{k,n} be the decomposition of the derivation D : L^{,}→ L^{, +1} with respect to the total and the augmentation degrees. Denote by ∥D_{k,n}∥_{q}^{p} the norm with respect to the standard metric of the restriction of the derivation D_{k,n} : L^{p,0} → L^{p,1} to the component in L^{p,0} of total degree q. 10.3.14. Let now L_{tot}^{,}= Λ (V_{4})⊗B_{tot} be the product of the exterior algebra Λ (V_{4}) with the total Weil algebra B_{tot}. We have the canonical identification L_{tot}^{p,q} = L^{p,q} ⊗ W_{q}^{∗}, and the canonical projection P : L_{tot}^{,}→ L^{,}, identical on L_{tot}^{,0} = L^{,0}. As in 10.3.4, the norm of the projection P on L_{tot}^{,1} is at most 2.

The derivation D : L^{,0} → L^{,1} induces a derivation D^{tot} : L_{tot}^{,0} → L_{tot}^{,1}, related to D by D = P ∘ D^{tot}. The gradings and the metric on L^{,}extend to L_{tot}^{,}, in particular, we have the decomposition D^{tot} = ∑_{k,n} D_{k,n}^{tot} with respect to the total and the augmentation degrees. Denote by ∥D_{k,n}^{tot}∥_{q}^{p} the norm with respect to the standard metric of the restriction of the derivation D_{k,n}^{tot} : L_{tot}^{p,0} → L_{C}^{p,1} to the component in L^{p,0} of total degree q. Since ∥P ∥ ≤ 2 on L_{tot}^{,1}, we have

∥D_{k,n}^{tot}∥_{q}^{p} ≤ 2 ∥D_{k,n}∥_{q}^{p}.  (10.8)

10.3.15. The estimate on the norms ∥D_{k,n}^{tot}∥_{p,q}^{L} that we will need is the following.

Lemma. In the notation of Lemma 10.3.8, we have

∥D_{k,n}^{tot}∥_{q}^{p} < 2(q + pk)(3K)^{k−1}C_{0}^{n}b_{k,n},

for every p, q, k, n.

Proof. Since D_{k,n} satisfies the Leibnitz rule, it suffices to prove the estimate for the restriction of the derivation D_{k,n}^{tot} to the generator subspaces V_{2}, V_{3}, V_{4} ⊂ L^{,0}. By (10.8) it suffices to prove that on V_{2}, V_{3}, V_{4} we have

∥D_{k,n}∥_{q}^{p} < (q + pk)(3K)^{k−1}C_{0}^{n}b_{k,n}.

On the generator subspaces V_{2}, V_{3} ⊂ B^{0} we have p = 0, and this equality is the claim of Lemma 10.3.8. Therefore it suffices to consider the restriction of the derivation D_{k,n} to the subspace V_{4} = d^{r}(V_{3}) ⊂ L^{1,0}. We have d^{r} ∘D_{k,n} = D_{k,n} ∘ d^{r} : V_{3} → L^{1,1}. Moreover, d^{r} = id : V_{3} → V_{4} is an isometry. Since the operator d^{r} : B → L^{1,}satisfies the Leibnitz rule and vanishes on the generators V_{1}, V_{2} ⊂ B , the norm ∥d^{r}∥_{k} of its restriction to the subspace B_{k} of augmentation degree k does not exceed k. Therefore

∥D_{k,n}∥_{0}^{1} = ∥D_{k,n}|_{V_{4}}∥ = ∥D_{k,n} ∘ d^{r}|_{V_{3}}∥ = ∥d^{r} ∘ D_{k,n}|_{V_{3}}∥
≤ ∥D_{k,n}|_{V_{3}}| ∥d_{B_{k}^{1}}^{r} ∥ < k(3K)^{k−1}C_{0}^{n}b_{k,n},

which proves the lemma. □

10.4. The proof of Theorem 10.1

10.4.1. We can now prove Theorem 10.1. Let D : S^{1}(M, ℂ) → B^{1}(M, ℂ) be a flat linear extended connection on M. Assume that its reduction D_{1} = ∇ : S^{1}(M, ℂ) → S^{1}(M, ℂ) ⊗ Λ^{1}(M, ℂ) is a real-analytic connection on the bundle S^{1}(M, ℂ).

The operator D_{1} : S^{1}(M, ℂ) → S^{1}(M, ℂ) ⊗ Λ^{1}(M, ℂ) ⊂ B^{1}(M, ℂ) considered as an extended connection on M defines a Hodge connection D_{1} : Λ^{0}(\overline{T} M, ℂ) → ρ^{∗}Λ^{1}(M, ℂ) on the pair ⟨\overline{T} M, M⟩, and this Hodge connection is also real-analytic. Assume further that the Taylor series at 0 ⊂ M ⊂ \overline{T} M for the Hodge connection D_{1} converge in the closed ball of radius ε > 0. 10.4.2. Let D = ∑_{n,k} D_{k,n} : B → B^{+1} be the derivation of the Weil algebra B associated to the extended connection D. Applying Lemma 10.1.9 to the Hodge connection D_{1} proves that there exists a constant C > 0 such that for every n ≥ 0 the norm ∥D_{1,n}∥_{1,1} of the restriction of the derivation D_{1,n} to the generator subspace V_{3} = B_{1,1}^{0} ⊂ B^{0} satisfies

∥D_{1,n}∥_{1,1} < CC_{0}^{n},

where C_{0} = 1/ε.

By definition the derivation D_{1,n} vanishes on the generator subspace V_{1} ⊂ B^{1}. If n > 0, then it also vanishes on the generator subspace V_{2} = B_{0,1}^{0} ⊂ B^{0}. If n = 0, then its restriction to V_{2} = B_{0,1}^{0} ⊂ B^{0} is the identity isomorphism D_{1,0} = id : V_{2} → V_{1}. In any case, we have ∥D_{1,n}∥_{0,1} ≤ 1. Increasing if necessary the constant C_{0}, we can assume that for any n and for p = 0, 1 we have

∥D_{1,n}∥_{p,1} < C_{0}^{n}.

10.4.3. We can now apply our main estimate, Proposition 10.1. It shows that there exists a constant C_{1} > 0 such that for every k, n we have

∥D_{k,n}∥_{1,1} < C_{1}^{n}.

Together with Lemma 10.1.9 this estimate implies that the formal Hodge connection D on \overline{T} M along M ⊂ \overline{T} M corresponding to the extended connection D converges to a real-analytic Hodge connection on an open neighborhood U ⊂ \overline{T} M of the zero section M ⊂ \overline{T} M. This in turn implies the first claim of Theorem 10.1. 10.4.4. To prove the second claim of Theorem 10.1, assume that the manifold M is equipped with a Kähler form ω compatible with the Kählerian connection ∇, so that ∇ω = 0. The differential operator ∇ : Λ^{1,1}(M) → Λ^{1,1}(M) ⊗ Λ^{1}(M, ℂ) is elliptic and real-analytic. Since ∇ω = 0, the Kähler form ω is also real-analytic.

For every p, q ≥ 0 we have introduced in 10.3.12 the space L^{p,q}, which coincides with the space of formal germs at 0 ∈ M ⊂ \overline{T} M of smooth forms on \overline{T} M of type (p, q) with respect to the complementary complex structure \overline{T} M_{J}. The spaces L^{,}carry the total and the augmentation gradings. Consider ω as an element of the vector space L_{0}^{2,0}, and let ω = ∑_{n} ω_{n} be the total degree decomposition. The decomposition ω = ∑_{n} ω_{n} is the Taylor series decomposition for the form ω at 0 ∈ M. Since the form ω is real-analytic, there exists a constant C_{2} such that

∥ω_{n}∥ < C_{2}^{n}  (10.9)

for every n. 10.4.5. Let Ω = ∑_{k} Ω_{k} = ∑_{k,n} Ω_{k,n} ⊂ L^{2,0} be the formal polarization of the Hodge manifold \overline{T} M at M ⊂ \overline{T} M corresponding to the Kähler form ω by Theorem 9.1. By definition we have ω = Ω_{0}. Moreover, by construction used in the proof of Proposition 9.1 we have

Ω_{k} = −\frac{1}{k} ∑_{1≤p≤k−1} σ_{tot}(D_{k−p}^{tot}Ω_{p}),  (10.10)

where D^{tot} : L^{2,0} → L^{2,1} is the derivation associated to the extended connection D on M and σ_{tot} : L^{2, +1} → L^{2,}is the extension to L^{2,}= L_{0}^{2} ⊗ B_{tot} of the canonical endomorphism of the total Weil algebra B_{tot} constructed in the proof of Proposition 9.1. 10.4.6. The map σ_{tot} : L^{2,1} → L^{2,0} is a map of B_{tot}^{0}-modules. Therefore, as in 10.3.7, there exists a constant K_{1} > 0 such that

∥σ_{tot}∥ < K_{1}  (10.11)

on L^{2,1}. We can assume that K_{1} > 3K, where K is as in (10.7). Together with the recursive formula (10.10), this estimate implies the following estimate on the norms ∥Ω_{k,n}∥ of the components Ω_{k,n} of the formal polarization Ω taken with respect to the standard metric.

Lemma. For every k, n we have

∥Ω_{k,n}∥ < (2K_{1})^{k−1}C^{n}b_{k,n}^{2},

where C = max(C_{0}, C_{2}) is the bigger of the constants C_{0}, C_{2}, and b_{k,n}^{2} are the numbers defined recursively in 10.2.5.

Proof. Use induction on k. The base of the induction is the case k = 1, where the estimate holds by (10.9). Assume the estimate proved for all Ω_{p,n} with p < k, and fix a number n. By (10.10) we have

Ω_{k,n} = −\frac{1}{k} ∑_{1≤p≤k−1} ∑_{q} σ_{tot}(D_{k−p,n−q}^{tot}Ω_{p,q}).

Substituting the estimate (10.11) together with the inductive assumption and the estimate on ∥D_{k−p,n−q}^{tot}∥_{q}^{2} obtained in Lemma 10.3.15, we get

∥Ω_{k,n}∥ < \frac{1}{k} ∑_{1≤p≤k−1} ∑_{q} ∥σ_{tot}∥ ∥D_{k−p,n−q}^{tot}∥_{q}^{2} ∥Ω_{p,q}∥

< \frac{1}{k} ∑_{1≤p≤k−1} ∑_{q} K_{1} 2(q + 2(k − p))(3K)^{k−p−1}C_{0}^{n−q}b_{k−p,n−q}

(2K_{1})^{p−1}C^{q}b_{p,q}

< \frac{1}{k} (2K_{1})^{k−1}C^{n} ∑_{1≤p≤k−1} ∑_{q}(q + 2(k − p))b_{k−p,n−q}b_{p,q}^{2}

= (2K_{1})^{k−1}C^{n}b_{k,n}^{2},

which proves the lemma. □ 10.4.7. This estimate immediately implies the last claim of Theorem 10.1. Indeed, together with Lemma 10.2.7 it implies that for every k, n

∥Ω_{k,n}∥ < (C_{3})^{n}

for some constant C_{3} > 0. But Ω = ∑_{n} ∑_{0≤k≤n} Ω_{k,n} is the Taylor series decomposition for the formal polarization Ω at 0 ⊂ M. The standard convergence criterion shows that this series converges in an open ball of radius 1/C_{3} > 0. Therefore the polarization Ω is indeed real-analytic in a neighborhood of 0 ⊂ M ⊂ \overline{T} M. □

Appendix

A.1.1. In this appendix we describe a well-known Borel-Weyl type localization construction for quaternionic vector spaces (see, e.g. [HKLR]) which provides a different and somewhat more geometric approach to many facts in the theory of Hodge manifolds. In particular, we establish, following Deligne and Simpson ([D2], [S1]), a relation between Hodge manifolds and the theory of mixed ℝ-Hodge structures. For the sake of simplicity, we consider only Hodge manifold structures on the formal neighborhood of 0 ∈ ℝ^{4n} instead of actual Hodge manifolds, as in Section 10. To save the space all proofs are either omitted or only sketched. A.1.2. Let SB be the Severi-Brauer variety associated to the algebra ℍ, that is, the real algebraic variety of minimal right ideals in ℍ. The variety SB is a twisted ℝ-form of the complex projective line ℂP^{1}.

For every algebra map I : ℂ → ℍ let the algebra ℍ ⊗_{ℝ} ℂ act on the 2-dimensional complex vector space ℍ_{I} by left multiplication, and let Î ⊂ ℍ ⊗_{ℝ} ℂ be the annihilator of the subspace I(ℂ) ⊂ ℍ_{I} with respect to this action. The subspace Î is a minimal right ideal in ℍ ⊗_{ℝ} ℂ. Therefore it defines a ℂ-valued point Î ⊂ SB(ℂ) of the real algebraic variety SB. This establishes a bijection between the set SB(ℂ) and the set of algebra maps from ℂ to ℍ. A.1.3. Let Shv(SB) be the category of flat coherent sheaves on SB. Say that a sheaf E ∈ Ob Shv(SB) is of weight p if the sheaf E ⊗ ℂ on ℂP^{1} = SB ⊗ ℂ is a sum of several copies of the sheaf O(p).

Consider a quaternionic vector space V. Let I ∈ ℍ ⊗ O_{SB} be the tautological minimal left ideal in the algebra sheaf ℍ ⊗ O_{SB}, and let V_{loc} ∈ Ob Shv(SB) be the sheaf defined by

V_{loc} = V ⊗ O_{SB}/I V ⊗ O_{SB}.

The correspondence V ↦ V_{loc} defines a functor from quaternionic vector spaces to Shv(SB). It is easy to check that this functor is a full embedding, and its essential image is the subcategory of sheaves of weight 1. Call V_{loc} the localization of the quaternionic vector space V. For every algebra map I : ℂ → ℍ the fiber V_{loc}|Î of the localization V_{loc} over the point Î ⊂ SB(ℂ) corresponding to the map i : ℂ → ℍ is canonically isomorphic to the real vector space V with the complex structure V_{I}. A.1.4. The compact Lie group U(1) carries a canonical structure of a real algebraic group. Fix an algebra embedding I : ℂ → ℍ and let the group U(1) act on the algebra ℍ as in 1.1.2. This action is algebraic and induces therefore an algebraic action of the group U(1) on the Severi-Brauer variety SB. The point Î: Spec ℂ ⊂ SB is preserved by the U(1)-action. The action of the group U(1) on the complement SB \ Î(Spec ℂ) ⊂ SB is free, so that the variety SB consists of two U(1)-orbits. The corresponding orbits of the complexified group ℂ^{∗} = U(1) × Spec ℂ on the complexification SB × Spec ℂ ≅ ℂP are the pair of points 0, ∞ ⊂ ℂP and the open complement ℂP \ {0, ∞} ≅ ℂ^{∗} ⊂ ℂP.

Let Shv^{U(1)}(SB) be the category of U(1)-equivariant flat coherent sheaves on the variety SB. The localization construction immediately extends to give the equivalence V ↦ V_{loc} between the category of equivariant quaternionic vector spaces and the full subcategory in Shv^{U(1)}(SB) consisting of sheaves of weight 1. For an equivariant quaternionic vector space V, the fibers of the sheaf V_{loc} over the point Î ⊂ SB(ℂ) and over the complement SB\Î(Spec ℂ) are isomorphic to the space V equipped, respectively, with the preferred and the complementary complex structures V_{I} and V_{J}.

A.1.5. The category of U(1)-equivariant flat coherent sheaves on the variety SB admits the following beautiful description, due to Deligne.

Lemma ([D2],[S1]). (i) For every integer n the full subcategory

Shv_{n}^{U(1)}(SB) ⊂ Shv^{U(1)}(SB)

of sheaves of weight n is equivalent to the category of pure ℝ-Hodge structures of weight n.

(ii) The category of pairs ⟨E, W ⟩ of a flat U(1)-equivariant sheaf

E ∈ Ob Shv^{U(1)}(SB)

and an increasing filtration W on E such that for every integer n

W_{n}E/W_{n−1}E is a sheaf of weight n on SB  (A.1)

is equivalent to the category of mixed ℝ-Hodge structures. (In particular, it is abelian.)

A.1.6. For every pure ℝ-Hodge structure V call the corresponding U(1)- equivariant flat coherent sheaf on the variety SB the localization of V and denote it by V_{loc}. For the trivial ℝ-Hodge structure ℝ(0) of weight 0 the sheaf ℝ(0)_{loc} coincides with the structure sheaf O on SB. If V, W are two pure ℝ-Hodge structures, then the space Hom(V_{loc}, W_{loc}) of U(1)-equivariant maps between the corresponding sheaves coincides with the space of weakly Hodge maps from V to W in the sense of Subsection 1.4.

For every pure ℝ-Hodge structure V the space Γ(SB, V_{loc}) of the global sections of the sheaf V_{loc} is equipped with an action of the group U(1) and carries therefore a canonical ℝ-Hodge structure of weight 0. This ℝ-Hodge structure is the same as the universal ℝ-Hodge structure Γ(V) of weight 0 constructed in Lemma 1.4.6. This explains our notation for the functor Γ : WHodge_{≥0} → WHodge_{0}. A.1.7. Assume given a complex vector space V and let M be the formal neighborhood of 0 ∈ V. Let B be the Weil algebra of the manifold M, as in 10.1.2. For every n ≥ 0 the vector space B^{n} is equipped with an ℝ-Hodge structure of weight n, so that we can consider the localization B_{loc}^{n}. The sheaf ⊕B_{loc} is a commutative algebra in the tensor category Shv^{U(1)}(SB). We will call the localized Weil algebra.

The augmentation grading on B defined in 7.2.2 is compatible with the ℝ-Hodge structures. Therefore it defines an augmentation grading on the localized Weil algebra B_{loc}. The finer augmentation bigrading on B does not define a bigrading on B_{loc}. However, it does define a bigrading on the complexified algebra B_{loc} ⊗ ℂ of ℂ^{∗}-equivariant sheaves on the manifold SB ⊗ ℂ ≅ ℂP. A.1.8. Assume now given a flat extended connection D : B → B^{+1} on M. Since the derivation D is weakly Hodge, it corresponds to a derivation D : B_{loc} → B_{loc}^{+1} of the localized Weil algebra B_{loc}. It is easy to check that the complex ⟨B_{loc}, D⟩ is acyclic in all degrees but 0. Denote by H^{0} the 0-th cohomology sheaf H^{0}(B_{loc}). The sheaf H^{0} carries a canonical algebra structure. Moreover, while the derivation D does not preserve the augmentation grading on B_{loc}, it preserves the decreasing augmentation filtration (B_{loc})_{≥}. Therefore we have a canonical decreasing filtration on the algebra H^{0}, which we also call the augmentation filtration. A.1.9. It turns out that the associated graded quotient algebra gr H^{0} with respect to the augmentation filtration does not depend on the extended connection D. To describe it, introduce the ℝ-Hodge structure W of weight −1 by setting W = V as a real vector space and

\begin{aligned}W^{−1,0} & = V ⊂ V ⊗_{ℝ} ℂ, \\ W^{0,−1} & = \overline{V} ⊂ V ⊗_{ℝ} ℂ. \end{aligned}  (A.2)

The k-th graded piece gr_{k} H^{0} with respect to the augmentation filtration is then isomorphic to the symmetric power S^{k}(W_{loc}) of the localization W_{loc} of the ℝ-Hodge structure W. In particular, it is a sheaf of weight −n, so that up to a change of numbering the augmentation filtration on H^{0} satisfies the condition (A.1). The extension data between these graded pieces depend on the extended connection D. The whole associated graded algebra gr H^{0} is isomorphic to the completed symmetric algebra Ŝ(W_{loc}). A.1.10. Using standard deformation theory, one can show that the algebra map H^{0} → B_{loc}^{0} is the universal map from the algebra H^{0} to a complete commutative pro-algebra in the tensor category of U(1)-equivariant flat coherent sheaves of weight 0 on SB. Moreover, the localized Weil algebra B_{loc} coincides with the relative de Rham complex of B_{loc}^{0} over H^{0}. Therefore one can recover, up to an isomorphism, the whole algebra B_{loc} and, consequently, the extended connection D, solely from the algebra H^{0} in Shv^{U(1)}(SB). Together with Lemma A.1.5 (ii) this gives the following, due also to Deligne (in a different form).

Proposition A.1 The correspondence ⟨B , D⟩ ↦ H^{0} is a bijection between the set of all isomorphism classes of flat extended connections on M and the set of all algebras H^{0} in the tensor category of mixed ℝ-Hodge structures equipped with an isomorphism gr_{−1}^{W} H^{0} ≅ W between the −1-th associated graded piece of the weight filtration on H^{0} and the pure ℝ-Hodge structure W defined in (A.2) which induces for every n ≥ 0 an isomorphism gr_{−n}^{W} H^{0} ≅ S^{n}W.

Remark. The scheme Spec H^{0} over SB coincides with the so-called twistor space of the manifold \overline{T} M with the hypercomplex structure given by the extended connection D (see [HKLR] for the definition). Deligne’s and Simpson’s ([D2], [S1]) approach differs from ours in that they use the language of twistor spaces to describe the relation between U(1)-equivariant hypercomplex manifolds and mixed ℝ-Hodge structures. Since this requires some additional machinery, we have avoided introducing twistor spaces in this paper. A.1.11. We will now try to use the localization construction to eludicate some of the complicated linear algebra used in Section 8 to prove our main theorem. As we have already noted, the category WHodge of pure ℝ-Hodge structures with weakly Hodge maps as morphisms is identified by localization with the category Shv^{U(1)}(SB) of U(1)-equivariant flat coherent sheaves on SB. Moreover, the functor Γ : WHodge_{≥0} → WHodge_{0} introduced in Lemma 1.4.6 is simply the functor of global sections Γ(SB, ). A.1.12. Consider the localized Weil algebra B_{loc} with the derivation C : B_{loc} → B_{loc}^{+1} associated to the canonical weakly Hodge derivation introduced in 7.3.2. The differential graded algebra ⟨B_{loc}, C⟩ is canonically an algebra over the completed symmetric algebra Ŝ(V) generated by the constant sheaf on SB with the fiber V. Moreover, it is a free commutative algebra generated by the complex

V ⟶ V (1)  (A.3)

placed in degrees 0 and 1, where V (1) is the U(1)-equivariant sheaf of weight 1 on SB corresponding to the ℝ-Hodge structure given by V (1)^{1,0} = V and V (1)^{0,1} = \overline{V}. A.1.13. The homology sheaves of the complex (A.3) are non-trivial only in degree 1. This non-trivial homology sheaf is a skyscraper sheaf concentrated in the point Î(Spec ℂ) ⊂ SB with fiber V. The associated sheaf on the complexification SB ⊗ ℂ ≅ ℂP splits into the sum of skyscraper sheaf with fiber V concentrated at 0 ∈ ℂP and the skyscraper sheaf with fiber \overline{V} concentrated at ∞ ∈ ℂP. This splitting corresponds to the splitting of the complex (A.3) itself into the components of augmentation bidegrees (1, 0) and (0, 1).

A.1.14. Let now I = B_{≥0,≥0} be the sum of the components in the Weil algebra B of augmentation bidegree greater or equal than (1, 1). The subspace I ⊂ B is compatible with the ℝ-Hodge structure. The crucial point in the proof of Theorem 8.1 is Proposition 7.1, which claim the acyclycity of the complex ⟨Γ(I ), C⟩. It is this fact that becomes almost obvious from the point of view of the localization construction. To show it, we first prove the following.

Lemma. The complex

⟨I_{loc}, C⟩

of U(1)-equivariant sheaves on SB is acyclic.

Proof. It suffices to prove that the complex I_{loc}⊗ℂ of sheaves on SB⊗ℂ ≅ ℂP is acyclic. To prove it, let p, q ≥ 1 be arbitrary integer, and consider the component (B_{loc})_{p,q} of augmentation bidegree (p, q) in the localized Weil algebra B_{loc}. By definition we have

(B_{loc})_{p,q} = (B_{loc})_{p,0} ⊗ (B_{loc})_{0,q}

Since the complex (B_{loc})_{p,0} = S^{p} (B_{loc})_{1,0} has homology concentrated at 0 ∈ ℂP, while the complex (B_{loc})_{0,q} = S^{q} (B_{loc})_{0,1} has homology concentrated at ∞ ∈ ℂP, their product is indeed acyclic. □

Now, we have Γ(I ) = Γ(SB, I_{loc}), and the functor Γ(SB, ) is exact on the full subcategory in Shv^{U(1)}(SB) consisting of sheaves of positive weight . Therefore the complex ⟨Γ(I ), C⟩ is also acyclic, which gives an alternative proof of Proposition 7.1. A.1.15. We would like to finish the paper with the following observation. Proposition A.1 can be extended to the following claim.

Proposition A.2 Let M be a complex manifold. There exists a natural bijection between the set of isomorphism classes of germs of Hodge manifold structures on \overline{T} M in the neighborhood of the zero section M ⊂ \overline{T} M and the set of multiplicative filtrations F on the sheaf O_{ℝ}(M) ⊗ ℂ of ℂ-valued real-analytic functions on M satisfying the following condition:

• For every point m ∈ M let Ô_{m} be the formal completion of the local ring O_{m} of germs of real-analytic functions on M in a neighborhood of m with respect to the maximal ideal. Consider the filtration F on Ô_{m} ⊗ ℂ induced by the filtration f on the sheaf O_{ℝ}(M) ⊗ ℂ, and for every k ≥ 0 let W_{−k}Ô_{m} ⊂ Ô_{m} be the k-th power of the maximal ideal in Ô_{m}. Then the triple ⟨Ô_{m}, F , W ⟩ is a mixed ℝ-Hodge structure. (In particular, F^{k} = 0 for k > 0.)

If the Hodge manifold structure on \overline{T} M is such that the projection ρ : \overline{T} M_{I} → M is holomorphic for the preferred complex structure \overline{T} M_{I} on \overline{T} M, then it is easy to see that the first non-trivial piece F^{0}O_{ℝ}(M) ⊗ ℂ of the filtration F on the sheaf O_{ℝ}(M) ⊗ ℂ coincides with the subsheaf O(M) ⊂ O_{ℝ}(M) ⊗ ℂ of holomorphic functions on M. Moreover, since the filtration F is multiplicative, it is completely defined by the subsheaf F^{−1}O_{ℝ}(M) ⊗ ℂ ⊂ O_{ℝ}(M) ⊗ ℂ. It would be very interesting to find an explicit description of this subsheaf in terms of the Kählerian connection ∇ on M which corresponds to the Hodge manifold structure on \overline{T} M.

[B] A. Besse, Einstein manifolds, Springer-Verlag, 1987.

[Beau] A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Diff. Geom. 19 (1983), 755–782.

[BG] O. Biquard and P. Gauduchon, Hyperkähler metrics on cotangent bundles of Hermitian symmetric spaces, in ”Proc. of the Special Session an Geometry and Physics held at Aarhus University and of the Summer School at Odense University, Denmark, 1995”, Lecture Notes in Pure and Applied Mathematics, volume 184, Marcel Dekker Inc. 1996, 768 pages.

[B] Charles P. Boyer, A note on hyper-Hermitian four-manifolds, Proc. Amer. Math. Soc. 102 (1988), 157–164.

[C] E. Calabi, Métriques kähleriennes et fibrés holomorphes, Ann. Ecol. Norm. Sup. 12 (1979), 269–294.

[DS] A. Dancer and A. Swann, The structure of quaternionic kähler quotients, in ”Proc. of the Special Session an Geometry and Physics held at Aarhus University and of the Summer School at Odense University, Denmark, 1995”, Lecture Notes in Pure and Applied Mathematics, volume 184, Marcel Dekker Inc. 1996, 768 pages.

[D1] P. Deligne, Cohomologie a support propre et construction du foncteur f^{!}, Appendix in: Robin Hartshorne, Residues and Duality, Lecture Notes in Math. vol. 20, Springer - Verlag, 1966.

[D2] P. Deligne, a letter to Simpson, quoted in [S1].

[H1] N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59–126.

[H2] N.J. Hitchin, Metrics on moduli spaces, Proc. Lefschetz Centennial Conference (Mexico City 1984), Contemp. Math. 58, Part I, AMS, Providence, RI, 1986.

[HKLR] N.J. Hitchin, A. Karlhede, U. Lindström, M. Roček, Hyperkähler metrics and supersymmetry, Comm. Math. Phys (1987).

[K] D. Kaledin, Integrability of the twistor space for a hypercomplex manifold, alg-geom preprint 9612016.

[Kr1] P.B. Kronheimer, A hyper-Kählerian structure on coadjoint orbits of a semisimple complex group, J. of LMS, 42 (1990), 193–208.

[Kr2] P.B. Kronheimer, Instantons and the nilpotent variety, J. Diff. Geom. 32 (1990), 473–490.

[N] H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), 365–416.

[S1] C. Simpson, Mixed twistor structures, alg-geom preprint, 1997.

[S2] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety, I: Publ. Math. I.H.E.S. 79 (1994), 47–129; II: Publ. Math. I.H.E.S. 80 (1994), 5–79.

[S3] C. Simpson, Higgs bundles and local systems, Publ. Math. I.H.E.S. 75 (1992), 5–95.

Independent University of Moscow B. Vlassievsky, 11 Moscow, Russia, 121002 E-mail: kaledin@@balthi.dnttm.rssi.ru

INDEX OF TERMS AND NOTATION 247

Index of terms and notation

B_{2}-bundles, 85, 90, 92 and ℂ^{∗}-equivariant holomorphic bundles, 27, 91–93 and Yang–Mills connections, 86 definition of, 24, 84 moduli of, 86 twistor transform for, 85

B_{2}-connections, 85 B_{2}-forms, 84, 85, 91 C-restricted complex structures, 20–22, 26, 27, 49, 58, 110, 116–119, 122

and coherent sheaves, 67

are dense, 67 definition of, 57 existence of, 58 utility of, 23

S-equivalence of coherent sheaves, 108 ℂ^{∗}-action on hyperkähler manifolds, 20, 27, 69, 75, 78, 95, 96

construction of, 71 ℂ^{∗}-action on twistor spaces, 89, 92, 93, 97, 99 ℂ^{∗}-equivariant holomorphic bundles on twistor spaces, 91–93, 95, 97, 99, 102 ℂ^{∗}-equivariant sheaves, 69, 77, 78

construction of, 77

desingularization of, 27, 78 ℂ^{∗}-equivariant structure, 78 Tw(M), 36

adic completion, see formal completion augmentation bidegree, see bidegree, augmentation augmentation bigrading, see bigrading, augmentation augmentation degree, see degree, augmentation augmentation grading, see grading, augmentation autodual bundles, 79

direct and inverse twistor transform for, 79–81

Hermitian structures on, 80, 82

bidegree augmentation, 192–242 Hodge, 192, 194, 195, 207

bigrading augmentation, 192, 193, 205, 222, 240 Hodge type, 130, 138–142, 145, 146, 153, 154, 156, 171, 181, 188, 192–195, 197,

200, 208, 211, 216, 217, 236 Bogomolov’s theorem on decomposition, 30 on period mapping, 65, 66, 121

Bogomolov–Beauville pairing, 52, 53, 55, 62, 65, 66, 115

CA-type (cohomology classes of), 26, 56, 57, 68

definition, 56 Calabi–Yau Theorem, 29, 66, 114, 115, 119, 127, 130 Cartan Homotopy Formula, 170 coherent sheaves

hyperholomorphic, 15 ℂ^{∗}-equivariant, 20, 72 a family of, 102 basic properties of, 26 category of, 48 connections on, 23, 27, 42, 44 definition of, 22, 40 deformations of, 67, 109 desingularization of, 20, 68 flat deformations of, 20 moduli of, 102 over generic complex structures, 20 polystable, 44 semistable, 20 singularities of, 23, 40 slope of, 48 stable, 20 tensor category of, 47 with isolated singularities, 20, 27

normal, 103 polystable

definition of, 38 reflexive, 19, 20, 23, 26, 27, 38–40, 42, 43, 47, 48, 56, 69, 73, 75, 78, 98, 99,

101–103 are normal, 103 category of, 47 hyperholomorphic connections on, 44 push-forward of, 107 singularities of, 43, 69

semistable, 23, 26, 40, 67, 103–106 S-equivalent, 108 a family of, 102 definition of, 35 Gieseker–Maruyama semistable, 107 Jordan-Hölder filtration on, 108

stable, 20, 47 admissible Yang–Mills metric on, 39 Chern classes of, 56 connections on, 19 definition of, 35 Gieseker–Maruyama stable, 107 on generic hyperkähler manifolds, 111 singularities of, 19 tensor product of, 47

Wirtinger’s inequality for, 26, 47 complementary complex structure, 137–139, 148, 150–153, 155, 156, 173, 178, 210,

233, 236, 239 complex manifold of hyperkähler type, 28 complexification, 76, 100

definition of, 69 properties of, 69 the space of twistor lines as, 70

connections ℂ-valued, 164–166, 172 autodual, 79 compatible with a holomorphic structure, 33 extended, see extended connection Hodge, see Hodge connection holomorphic, 131, 200 hyperholomorphic, 18, 23, 69, 72, 73, 75, 78, 79

are Yang-Mills, 36 definition of, 23 existence of, 23, 36, 44 on coherent sheaves, 26 singular set of, 42 twistor transform for, 24

integrable, 33 Kählerian, 131, 132, 135, 199–201, 208–210, 219, 220, 236, 243 Levi-Civita, 129, 200 on a fibration, 163, 164 on a vector bundle, 183, 200, 208, 211, 214, 216, 235 Yang–Mills, 15, 19, 34–36

admissible, 20, 39, 46, 48 and stability, 35 definition, 35 hyperholomorphic, 44 in coherent sheaves, 19, 26 non-Hermitian, 126 on Li–Yau manifolds, 37 on twistor spaces of quaternionic-Kähler manifolds, 101 with Λ∇^{2} = 0, 39 Darboux–Weinstein Theorem, 130, 133, 217 de Rham complex, 134, 152–154, 167, 168, 187, 190, 193, 213, 220, 221, 233

holomorphic, 211, 217 relative, 167, 168, 211, 241 total, 134, 187–193

degree

augmentation, 192, 194, 201, 203, 205–207, 209, 214–217, 222, 223, 229, 231,

234, 235 for different induced complex structures, 58, 59 Hodge, 194, 229 of a bundle obtained as a twistor transform, 86 of a form, 37, 41, 44, 51, 54 of a sheaf, 34, 39, 47 of a subvariety, 31 total, 222, 224, 229, 231, 232, 234, 236

Euclidean metric, see metric, Euclidean extended connection, 180, 183–187, 199–201, 203–208, 222–224, 229, 233, 235, 236,

240, 241 derivation associated to, 185, 205–207, 222, 223, 230, 235, 236 flat, 185, 196, 198–200, 204, 205, 209, 219, 220, 229, 231, 240, 241 linear, 184, 199–209, 219, 220, 222, 223, 229, 231, 235 principal part of, 183 reduction of, 183, 200, 207, 214, 235

formal completion, 177, 178, 180, 221, 233, 243 formal neighborhood, 130–132, 134, 177, 178, 180, 217, 218, 220, 224, 238, 240 Fourier–Mukai transform, 122 Frobenius Theorem, 164

generic polarization, 114 grading

augmentation, 192, 201, 215, 221, 230, 234, 236, 240

total, 221, 230, 234, 236

Hermitian metric, see metric, Hermitian Hermitian-Hodge metric, see metric, Hermitian-Hodge Higgs bundle, 190, 246 Hitchin’s compatibility condition, 132, 133 Hitchin-Simpson moduli space of flat connections, 133 Hodge bidegree, see bidegree, Hodge Hodge bundle, 133, 144–153, 155, 156, 163, 165–167, 169, 171, 172, 178, 181–183,

187–196, 201, 202, 205, 207, 210, 211, 213, 216, 217, 221 S^{1}(M, ℂ), 171–173, 176, 181–185, 188–192, 194–198, 200–209, 214, 215, 219,

221–223, 235 Hodge connection, 134, 163, 165, 167, 171–174, 176, 184–186, 222–224, 235, 236 formal, 134, 179, 180, 184, 185, 219, 220, 223, 236 holonomic, 171, 172, 174, 176 linear, 176, 179, 180, 184 principal part of, 172, 174

Hodge decomposition, 33 Hodge degree, see degree, Hodge Hodge manifold, 132–134, 151–155, 157, 161–163, 165–167, 173–175, 178, 209, 211–

214, 216, 218–220, 227, 229, 236, 238, 243 formal, 177–180, 186, 199, 209–211, 218, 220 linear, 160, 162, 163, 175, 176, 179, 186, 209, 212 polarized, 133 regular, 133, 134, 156, 157, 159, 160, 162, 163, 177, 180

ℝ-Hodge structure, 132, 137–146, 172, 187, 191, 221, 229, 238–242

mixed, 239, 241, 243 Hodge type bigrading, see bigrading, Hodge type H-type, 145, 146, 149, 150, 152, 153, 156, 210, 211 holomorphic symplectic structure, 29, 32, 46, 120, 121, 126, 127 holonomic derivation, 150, 151, 153, 154, 165–167, 171, 172, 178 holonomy group, 83, 88 hypercomplex manifold, 132, 133, 151, 155, 210, 241 hypercomplex structure, 132, 241 hyperkähler desingularization, 17, 20, 21, 27, 32, 43, 109, 119–121 hyperkähler manifold, 129–133, 155, 156

U(1)-equivariant, 132, 133 definition of, 28

hyperkähler metric, see metric, hyperkähler hyperkähler reduction, 132 hyperkähler structures, 17, 29, 31, 32, 34, 52, 54, 64, 112, 114, 115, 117, 130, 131,

144 𝔰𝔬(1, 4)-action associated with, 53 admitting C-restricted complex structures, 26, 67

definition of, 58 Bogomolov’s theorem for, 66 compatible with a complex structure, 28 corresponding to generic polarizations, 117, 119 formal, 217 generic, 21, 22, 55, 109, 118, 122 moduli of, 64 on a Hilbert scheme of points, 32 period map for, 65 twistor curves associated with, 66 twistor curves corresponding to

definition of, 112

induced complex structure of general type

definition of, 31 induced complex structures, 16

C-restricted, 57 definition of, 28

involution ι, 145, 146, 148, 149, 155, 169, 171, 173–176, 179, 182, 184, 188, 204 involution ζ, 169, 170, 176

Jordan-Hölder filtration, 108

Kähler metric, see metric, Kähler Kählerian connection, see connections, Kählerian Koszul differential, 195

Leibnitz rule, 154, 213–215, 232, 234, 235 Li–Yau

Yang–Mills metric, in the sense of, 42 Li–Yau condition, 37, 39 linearity condition, 167, 173–175, 179, 184, 186, 203 linearization, 133, 134, 157, 159–163

metric Euclidean, 143 formal hyperkähler, 131 Hermitian, 143, 144, 155, 190, 220, 223, 229 Hermitian-Hodge, 143, 144, 155, 156, 210 hyperkähler, 130, 131, 134, 155, 156, 209, 210, 217 Kähler, 130, 131, 156, 209, 210, 220 Quaternionic-Hermitian, 143, 154, 155 real-analytic, 131 Ricci-flat, 130 Riemannian, 154, 155 Yang–Mills, 19, 40

admissible, 19, 39 definition, 35 in the sense of Li–Yau, 42 on a polystable sheaf, 39

moduli space coarse, 108 of coherent sheaves, 20 of complex structures, 111, 119, 120

definition of, 64

twistor curves in, 111 of deformations of tangent bundle, 21 of Hermitian B_{2}-bundles, 86 of hyperkähler structures, 23, 26, 67

definition of, 64 of Maruyama, 107, 109, 110 of semistable bundles, 109 of stable hyperholomorphic bundles, 15, 21, 22, 113, 116–119, 121

Newlander–Nirenberg Theorem, 178

polarization, 133, 135, 143, 144, 155, 209–211, 216, 220, 227, 237

formal, 220, 236, 237 preferred complex structure, 135, 138, 139, 143, 144, 148, 151–154, 157, 161, 165–

167, 176, 178, 210, 217, 218, 239, 243

quaternion algebra, 129, 135, 147 quaternionic bundle, 148, 151 quaternionic manifold, 133, 144, 147, 148, 151

equivariant, 147, 148, 151, 152, 154 quaternionic projective space, 25, 83 quaternionic structure, 133, 147–151, 159, 161, 162, 173, 175, 212

formal, 178, 179 formal equivariant, 178

quaternionic vector space, 132, 135, 143, 238 equivariant, 136, 137, 143, 144, 148, 166, 239 regular equivariant, 137–139, 160

Quaternionic-Hermitian metric, see metric, Quaternionic-Hermitian quaternionic-Kähler manifolds, 21, 24, 27, 83, 86, 91, 96

definition of, 24, 83 holonomy group of, 88 hyperkähler manifolds associated with, 25, 89, 90, 93 is Einstein, 83 of positive scalar curvature, 84, 86, 90 twistor transform for, 24, 85 twistors of, 84 with non-trivial scalar curvature, 24

reflexization, 18–20, 22, 38, 40, 102–107, 109, 110 regular fixed point, 157 regular stable point, 157 Riemannian metric, see metric, Riemannian

slope of a coherent sheaf, 34, 102, 104 of a hyperholomorphic sheaf, 40, 48, 68

Swann’s formalism, 25, 88, 90 for vector bundles, 21, 25, 27, 90, 92, 102

Tannakian categories, 48, 49, 125 total degree, see degree, total total grading, see grading, total trianalytic subvarieties, 18, 23, 27, 30–33, 40–43, 58, 73, 109, 110, 116–118, 120–122,

127 definition of, 30 desingularization of, 32

twistor curves, 112, 113, 115, 120 definition of, 21, 112 hyperkähler structures associated to, 112

twistor lines, 70, 71, 73, 76, 80, 98 definition of, 70 horizontal

definition of, 70 space of, 70

twistor paths, 21, 113, 117

admissible, 113 definition of, 112 existence, 114, 115

consisting of a single twistor curve, 113 definition of, 112 edges of, 112 vertices of, 112

twistor projection, 82 twistor space, 48, 70, 79, 81, 156, 241

ℂ^{∗}-equivariant, 92, 93 involution of, 70 of a flat manifold, 92 of a hypercomplex manifold, 86 of a hyperkähler manifold, 24, 25, 27

definition of, 37 of a quaternionic-Kähler manifold, 24, 84, 86 of the quaternionic projective space, 86, 102

twistor transform, 21, 27, 42, 79, 90, 92–94, 97, 99, 123 direct, 24 for B_{2}-bundles, 24, 25, 85 for for bundles with a semilinear pairing, 82 for hypercomplex manifolds, 87 inverse, 24 vector bundles compatible with, 80

Uhlenbeck–Yau Theorem, 17, 19, 34, 35, 37, 39, 126

vector bundles compatible with twistor transform, 80, 83 hyperholomorphic, 19, 24, 109, 112, 119

category of, 113, 115, 117 definition of, 17 deformations of, 17, 27 examples of, 34 moduli of, 17, 19, 21, 27, 102, 110, 116, 118, 120 over higher-dimensional hyperkähler manifolds, 18 tangent bundle considered as, 119 tensor powers of, 34

polystable, 86 semistable, 21

deformations of, 108, 116, 118 moduli of, 16, 27, 109, 110

stable, 15, 17, 18, 21, 113, 119, 125, 126 category of, 27 deformations of, 18, 108 hyperholomorphic connections on, 36 moduli of, 27, 116, 117, 119–121 obtained as twistor transform of B_{2}-bundles, 24 on Li–Yau manifolds, 37 Yang–Mills connections on, 35, 36

Yang–Mills, 17, 24, 36, 86, 91 definition, 35 existence of, 35

weakly Hodge map, 133, 135, 140–142, 145, 146, 149–153, 165, 169, 170, 172, 174, 182, 183, 185–187, 193, 195, 201, 203, 205–208, 213, 214, 217, 229, 230, 240, 241

universal, 141, 229 weight

of a Hodge bundle, 145–148, 150–153, 155, 156, 165–167, 169, 171, 178, 181,

187, 190–193, 195, 196, 201, 210, 211, 216 of a Hodge structure, 138–144, 146, 147, 172, 229, 239, 240 of a sheaf on ℂP^{1}, 238–242 of an U(1)-action, 133, 136, 137, 147, 155, 170, 188, 218, 219

weight filtration, 241 Weil algebra, 134, 180, 182, 185, 187, 191–195, 200, 201, 203–205, 213, 214, 220–223,

229, 230, 233, 235, 240, 242 localized, 240–242 reduced, 201, 203–205, 207, 208 standard metric on, 223, 229, 230, 234, 237 total, 134, 187, 191–193, 195–198, 201–206, 209, 216, 229–231, 234, 236

Weil operator, 139, 143

Authors index

Atiyah, M., 33

Bando, S., 37, 38, 47, 125 Bartocci, C., 125 Beauville, A., 3, 7, 16, 18, 26, 118, 125, 245 Beilinson, A., 135 Berard Bergery, L., 24 Berger, M., 2, 7 Besse, A., 16, 125, 245 Biquard, O., 7, 245 Bochner, S., 3, 4, 7 Bogomolov, F. A., 26, 30, 65, 66, 68, 121, 125 Bott, R., 33 Boyer, C. P., 245 Bruzzo, U., 125

Calabi, E., 2–4, 7, 16, 28, 29, 125, 129, 130, 245

Dancer, A., 7, 245 Deligne, P., 49, 123, 125, 132, 135, 203, 238, 239, 241, 245 Donaldson, S., 5, 7 Douady, A., 70, 125

Fujiki, A., 3, 7, 49, 126

Gaudichon, P., 7, 245 Gieseker, D., 107, 125 Grothendieck, A., 111, 126

Hernandez Ruiperez, D., 125 Hitchin, N. J., 4, 5, 7, 8, 70, 126, 131–133, 245, 246 Huybrechts, D., 8, 126

Karlhede, A., 8, 70, 126, 246 Kazhdan, D., 123, 135 Kobayashi, S., 8, 126 Kodaira, K., 6, 8 Kronheimer, P. B., 5, 8, 246

Levin, A., 135 Li, J., 37, 39, 126 Lindström, U., 8, 246

Maruyama, M., 27, 107, 109, 126

AUTHORS INDEX 257

Mukai, S., 3, 4, 8, 122

Nakajima, H., 5, 8, 132, 246 Nitta, T., 24, 27, 84, 86, 126

O’Grady, K., 8, 16, 125 Okonek, C., 126 Orlov, D., 126

Pantev, T., 123, 135 Penrose, R., 4 Posicelsky, L., 135

Roček, M., 8, 246

Salamon, S., 24, 126 Schmid, W., 5, 8 Schneider, M., 126 Shen, A., 135 Simpson, C., 5, 8, 132–135, 190, 191, 203, 238, 241, 245, 246 Siu, Y.-T., 19, 37, 38, 125 Spencer, D., 6, 8 Spindler, H., 126 Swann, A., 7, 21, 25, 27, 79, 88–90, 102, 126, 245

Tyurin, A., 135

Uhlenbeck, K., 86, 102, 126

Weinstein, A., 133, 217

Yau, S.-T., 3, 4, 17, 19, 29, 34, 35, 37, 39, 42, 66, 68, 86, 102, 115, 126, 127, 130