Newsgroups: sci.math.research

Subject: Re: This Week's Finds in Mathematical Physics (Week 35)

[ From Grothendieck's letter, which was posted to sci.math.research by John Baez: ]

"I write you under the assumption that you have not entirely lost interest for those foundational questions you were looking at more than fifteen years ago. One thing which strikes me, is that (as far as I know) there has not been any substantial progress since - it looks to me that understanding of the basic structures underlying homotopy theory, or even homological algebra only, is still lacking - probably because the few people who have a wide enough background and perspective enabling them to feel the main questions, are devoting their energies to things which seem more directly rewarding. Maybe even a wind of disrepute for any foundational matters whatever is blowing nowadays! In this respect, what seems to me even more striking than the lack of proper foundations for homological and homotopical algebra, is the absence I daresay of proper foundations for topology itself! I am thinking here mainly of the development of a context of "tame" topology, which (I am convinced) would have on the everyday technique of geometric topology (I use this expression in contrast to the topology of use for analysts) a comparable impact or even a greater one, than the introduction of the point of view of schemes had on algebraic geometry.

[ my comments: ]

In fact, there was quite a progress in homotopical algebra after Quillen's works. Or, rather, his ideas were applied elsewhere in mathematics. First of all, the homotopical algebra was applied to the theory of the deformation spaces. The basic concepts can be found in Illusie's thesis (?), Complexe cotangent et deformations, Lecture Notes 239.

Take any object of algebraic geometry, like holomorphic vector bundle or scheme. Locally, the deformation space G of this object X is a closed subspace in H^1(End X), where by End (X) we understand the sheaf of endomorphisms if X is a bundle, or the sheaf of vector fields if X is a scheme. The theory of the cotangent complex describes the deformations in terms of homotopical algebra.

I will roughly sketch this construction for a bundle X. Consider the space of smooth differential forms with coefficients in the vector bundle End (X). This space is equipped with the differential (\bar \partial) which maps (p,q)-forms to (p,q+1)-forms. Since End (X) is an algebra, the complex K of forms with coefficients in End(X) is equipped with the structure of differential graded (DG) algebra. Using Quillen's formalism of homotopy algebra, we may define the cohomology, homotopy and all other topological constructs for a DG-algebra. Of course, the cohomology of DG-algebra are identified with its cohomology as a complex. To determine G what we need is Massey's products on the cohomology H^i(K). It is trivial to check that H^i(K) as a vector space coincides with H^i(End(X). Quillen's machinery gives us these Massey's products. Now, G is locally isomorphic to the set of all vectors x\in H^1(End(X)) such that all Massey's products vanish on x and Yoneda product vanish on x. By definition, Yoneda product is equal to the symmetrization of the standard product H^1(End(X))\times H^1(End(X)) -> H^2(End(X)).

In other words, the obstructions to the deformation of the bundle (or a scheme, whatever) are Yoneda product and Massey products in the corresponding DG-algebra.

Another (and most famous) application of the homotopic algebra is the topology of Kaehler manifolds. Deligne, Griffiths, Morgan and Sullivan proved in their pioneering paper that there are no Massey's products on the cohomology of the Kaehler manifold. In particular, this implies that the rational homotopy type (homotopy up to torsion) is uniquely defined by the multiplicative structure in the cohomology. This result has crucial importance for the motive geometry, as well as algebraic geometry as whole. For example, Sullivan proved, using this result, that the group of diffeomorphisms up to isotopy of an algebraic manifold of dimension more than 2 is ariphmetic (up to torsion (?)). In patricular, we obtain that the homotopy group of a moduli spaces of (some of) algebraic manifolds is an arithmetic group. The DGMS theorem is proved miraculously easy: they apply a lemma (called d d^c-lemma) which is due to the Hodge decomposition on the space of differential forms to Quillen's formalism, and obtain the vanishing of Massey's products by formal reasons.

Coupling these two approaches to homotopical algebra, I was be able to prove that the obstruction to deformations vanish for the bundles over the hyperkaehler manifolds. The hyperkaehler manifold M is a Riemannian manifold which is equipped with the parallel quaternion action. Each quaternion which has a square -1 defines a complex structure on such manifold. Hence, the hyperkaehler manifold can be considered as a complex one. Taking a holomorphic bundle over this manifold, we can consider the Hodge decomposition on the differential forms with coefficients in this bundle. We twist this decomposition by another quaternion. We obtain the bigraduated DG-algebra, akin to the algebra of differential forms over a Kaehler manifold. It is easy to prove that the (variation of) d d^c-lemma is formally implied by certain identities on differential forms over this manifold. Hence, we obtain that all Massey products in the cotangent algebra K (defined above) vanish! This shows that the deformation space of such a bundle is also hyperkaehler, and generally nice. However, I cannot prove anything about its singularities, except that these singularities are quadratic. It would be nice to prove that they are for example Cohen-Macaylay, using the formality of the corresponding DG-algebra. I pestered several specialists in Koszul duality and quadratic rings (including Yu. I. Manin), and none helped. Still, it seems likely that formality of cotangential complex implies much information about the deformation space.

S.-T. Yau in his list of problems in differential geometry mentioned the following problem. Take the cotangent complex associated with the Calabi-Yau or hyperkaehler manifold M. It is proven (by Bogomolov for the hyperkaehler case, and by Tian-Todorov for Calabi-Yau) that there are no obstructions to the deformations of M. In other words, the local deformation space of M is isomorphic to an open set in H^1(TM), where TM is a sheaf of holomorphic vector fields. Yau asked if there are obstructions to the deformation of bundles, sheaves and other geometrical objects defined over a compact Kaehler manifold with first Chern class 0 (Calabi-Yau or hyperkaehler). Using homotopical algebra as I did, we may answer his question for (some) bundles and sheaves over the hyperkaehler manifold. Still, it is completely mysterious if there are Massey products in the similar situation for Calabi-Yau manifolds. Personally, I am sure that there are none, and that this is very much connected to the mirror symmetry. Still, this question is open and I don't see an easy way to approach it.