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.

Abstract:
We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety. In particular, let V be a finite-dimensional complex symplectic vector space and G\subset Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V/G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the rational cohomology H^*(X) of any smooth symplectic resolution X \to V/G (multiplicative McKay correspondence). We prove further that if G is an irreducible Weyl group in GL(h) and V=h+ h^* then no smooth symplectic resolution of V/G exists unless G is of types A,B, or C.

Abstract:
We use (non-)additive sheaves to introduce an (absolute) notion of Hochschild cohomology for exact categories as Ext's in a suitable bisheaf category. We compare our approach to various definitions present in the literature.

Abstract:
Moduli spaces of semistable sheaves on a K3 or abelian surface with respect to a general ample divisor are shown to be locally factorial, with the exception of symmetric products of a K3 or abelian surface and the class of moduli spaces found by O'Grady. Consequently, since singular moduli space that do not belong to these exceptional cases have singularities in codimension $\geq4$ they do no admit projective symplectic resolutions.

Abstract:
We apply the technique of formal geometry to give a necessary and sufficient condition for a line bundle supported on a smooth Lagrangian subvariety to deform to a sheaf of modules over a fixed deformation quantization of the structure sheaf of an algebraic symplectic variety.

Abstract:
The article deals with the regionalization of social processes and their specific manifestations in the post-Soviet space. According to the author, regionalization is developing at four levels-international (macroregional), subregional, intraregional and on country levels. The most important one is the international level. Specific features of regionalization at each level of the post-Soviet space are described. The author draws a conclusion that the post-Soviet space is the epicenter of regionalization in contemporary world.

Abstract:
The Kodaira-Nakano Vanishing Theorem has been generalized to the relative setting by A. Sommese. We prove a version of this theorem for non-compact manifolds. As an apllication, we prove that the cohomology of a fiber of a symplectic contraction is trivial in odd degrees and pure Hodge-Tate in even degrees.

Abstract:
The projective coordinate ring of a projective Poisson scheme $X$ does not usually admit a structure of a Poisson algebra. We show that when $H^1(X,O_X)=H^2(X,O_X)=0$, this can be corrected by embedding $X$ into a canonical one-parameter deformation. The scheme $X$ then becomes the Hamiltonian reduction of the spectrum of the deformed projective coordinate ring with respect to $G_m$. The projection into the base of the deformation is the moment map.

Abstract:
This is a write-up of my talk at the Conference on algebraic structures in Montreal, July 2003. I try to give a brief informal introduction to the proof of Y. Ruan's conjecture on orbifold cohomology multiplication for symplectic quotient singularities given in V. Ginzburg and D. Kaledin, math.AG/0212279. Version 2: minor changes, added some references.

Abstract:
We introduce a version of the Cartier isomorphism for de Rham cohomology valid for associative, not necessarily commutative algebras over a field of positive characteristic. Using this, we imitate the well-known argument of P. Deligne and L. Iluusie and prove, in some cases, a conjecture of M. Kontsevich which claims that the Hodge-to-de Rham, a.k.a. Hochschild-to-cyclic spectral sequence degenerates.