# Papers by M. Verbitsky

Recent papers at arxiv.org.

## Three latest papers (in LaTeX 2e):

### Deformations of trianalytic subvarieties.

In the paper where trianalytic subvarieties were first introduced, we proved that a hyperkaehler manifold, if ``generic'' enough, might have only those complex subvarieties which are trianalytic -- i. e., complex analytic with respect to each induced complex structure. Here we study the complex analytic deformations of trianalytic subvarieties. We prove that all deformations of such X are trianalytic and naturally isomorphic to X as complex analytic varieties. We show also that this isomorphism is compatible with the metric induced from the ambient manifold. Also, we prove that the Douady space of complex analytic deformations of X inside its ambient manifold is equipped with a natural hyperk\"ahler structure. There is some discussion of singular hyperkaehler varietieis. A sample definition is given; trianalytic subvarieties and the deformation spaces of holomorphic bundles (as discussed in my paper ``Hyperholomorphic bundles'') all have this ``singular hyperkaehler'' structure. We show that the natural isomorphism of deformations of the trianalytic subvarieties (the isomorphism discussed above) is compatible with the singular hyperkaehler structure.

### Algebraic structures on hyperkaehler manifold.

From abstract:
Let M be a compact hyperkaehler manifold. The hyperkaehler structure equips M with a set R of complex structures parametrized by \$CP^1\$, called "the set of induced complex structures". It was known previously that induced complex structures are non-algebraic, except may be a countable set. We prove that a countable set of induced complex structures is algebraic, and this set is dense in R. A more general version of this theorem was proven by Fujiki.
A note
The paper is 5 pages long, and consists of a rather elementary proof used in the early version on the paper about deformations of trianalytic cycles. After this early version was demolished, I did not use the proof anymore, but it was aesthetically pleasing anyway. So I put it down as a separate paper. After I sent it to alg-geom I learned that my result was superseded by an 180-pages long work of Fujiki (1984). What a mess.
Published in:
Math. Res. Lett. 3 763-767 (1996)

### Non-Hermitian Yang-Mills connections.

A paper which I wrote with my friend and classmate (Moscow school #91) Dima Kaledin (Ph. D. 1995, MIT). I like it very much and appreciate any feedback. Here is the abstract:
We study Yang-Mills connections on holomorphic bundles over complex K\"ahler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isomorphic to the complexification of the space of Hermitian Yang-Mills connections (which is, by Uhlenbeck and Yau, the same as the space of stable bundles). Further, we study the NHYM connections over hyperk\"ahler manifolds. We construct direct and inverse twistor transform from NHYM bundles on a hyperk\"ahler manifold to holomorphic bundles over its twistor space. We study the stability and the modular properties of holomorphic bundles over twistor spaces, and prove that work of Li and Yau, giving the notion of stability for bundles over non-K\"ahler manifolds, can be applied to the twistors. We identify locally the following two spaces: the space of stable holomorphic bundles on a twistor space of a hyperk\"ahler manifold and the space of rational curves in the twistor space of the ``Mukai dual'' hyperk\"ahler manifold.

I wrote a letter (to A. Beilinson, V. Ginzburg, D. Kazhdan, T. Pantev and Carlos Simpson) explaining the matters in more detail. Recommended

# Hyperkaehler geometry

### Mirror Symmetry for hyperkaehler manifolds.

We prove Mirror Conjecture for Calabi-Yau manifolds equipped with holomorphic symplectic form, also known as complex manifolds of hyperkaehler type. We obtain that a complex manifold of hyperkaehler type is mirror dual to itself. The Mirror Conjecture is stated (following Kontsevich, ICM talk) as equivalence of certain algebraic structures related to variations of Hodge structures. We compute the canonical flat coordinates on the moduli space of Calabi-Yau manifolds of hyperkaehler type, introduced to Mirror Symmetry by Bershadsky, Cecotti, Ooguri and Vafa. LaTeX 2e, 65 pages.

### Cohomology of hyperkaehler manifolds.

Let M be a compact simply connected hyperk\"ahler (or holomorphically symplectic) manifold, \dim H^2(M)=n. We prove that the Lie group SO(n) acts by automorphisms on the cohomology ring H^*(M). Under this action, the space H^2(M) is isomorphic to the fundamental representation of SO(n). Let A^r be the subring of H^*(M) generated by H^2(M). We construct an action of the Lie algebra so(n+2) on the space A, which preserves A^r. The space A^r is an irreducible representation of so(n+2). This makes it possible to compute the ring A^r explicitely.

### Cohomology of hyperkaehler manifolds and their applications.

This article contains a compression of results from ``Cohomology of hyperkaehler manifolds.'' with most proofs omitted. We prove that every two points of the connected component of moduli space of holomorphically symplectic manifolds can be connected with so-called ``twistor lines'' -- projective lines holomorphically embedded to the moduli space and corresponding to the hyperk\"ahler structures. This has interesting implications for the geometry of compact hyperk\"ahler manifolds and of holomorphic vector bundles over such manifolds.

This article was written upon demand from GAFA. I am not to be held accountable for its (relative) lack of content: GAFA wanted to publish a short version without proofs instead of the longish Cohomology of hyperkaehler manifolds. Some new results have found a way here, anyway, and it's generally nicer to read short papers. Published in GAFA vol. 6 4 pp. 601--612 (1996)

### Here is an article about hyperholomorphic bundles

In this paper the instanton equation is being generalized to hyperkaehler manifolds of arbitrary dimension. The hyperholomorphic bundle is a stable holomorphic bundle for which this instanton equation can be solved. Its solution is unique, if exist.

Precise equivalence between such instantons and stable holomorphic bundles with first two Chern classes invariant under a natural SU(2)-action in cohomology is obtained, thus generalizing the theorem of Yau and Uhlenbeck. Deformation properties of hyperholomorphic bundles are studied. In particular, we show that deformation space of a hyperholomorphic bundle is again hyperkaehler. As easy calculations imply, on generic hyperkaehler manifold all stable bunles are hyperholomorphic.

I wrote an article, which describes some details on deformations of hyperholomorphic bundles, vaguely motivated by Massey products and homotopical algebra (posted to sci.math.research).

### Here is a paper about what sort of subvarieties a compact holomorphic symplectic manifold has.

If the complex structure on this manifold is "generic enough" in its deformation space (we describe the sufficient conditions explicitly) these subvarieties turn out to be holomorphically symplectic as well. Published in Journ. of Alg. Geom. vol. 5 3 pp. 401-415 (1996)

### Here is a paper, where I prove that all complex analytic subvarieties of a "generic" hyperkaehler manifold are also hyperkaehler.

By Calabi-Yau theorem, all compact holomorphically symplectic manifolds admit a hyperkaehler structure. In this paper I prove that under some genericity conditions, all complex analytic subvarieties of such manifold are also hyperkaehler. Thus we show that the property of being hyperkaehler is very resilient - not only deformation spaces of sheaves or bundles over a hyperkaehler manifold are hyperkaehler, but even complex analytic subvarieties of a hyperkaehler manifolds are hyperkaehler! Amazing. Published in GAFA, January 95.

# Quantum groups.

Here is an article written with David Kazhdan, where we compute cohomology of the restricted quantum algebra. Published in ``Quantum Deformations of Algebras and Their Representations,'' vol. 7 of Isr. Math. Conf. Proc. (1993)

Here are the Postscript files of the papers above. The old links to .DVI files were removed, because our system deletes old .dvi files automatically anyway.