Derived equivalences by quantization

We assume given a smooth symplectic (in the algebraic sense) resolution $X$ of an affine algebraic variety $Y$, and we prove that, possibly after replacing $Y$ with an etale neighborhood of a point, the derived category of coherent sheaves on $X$ is equivalent to the dervied category of finitely generated left modules over a non-commutative algebra $R$, a non-commutative resolution of $Y$ in a sense close to that of M. Van den Bergh. We also prove some applications, such as: two resolutions are derived-equivalent; every resolution $X$ admits a "resolution of the diagonal"; the cohomology groups of the fibers of the map $X \to Y$ are spanned by fundamental classes of algebraic cycles.Comment: Latex 2e, 39 pages. Added a dedication (to J. Bernstein

Symplectic resolutions: deformations and birational maps

Unfortunately, some proofs in the first version of this paper were incorrect. In this revised version, some minor gaps are fixed, one serious mistake found. The main theorem is now claimed only under a restrictive technical assumption. This invalidates the application to quotient singularities by the Weyl group of type $G_2$. Everything else still stands (in particular, the claim that every symplectic resolution is semismall).Comment: 34 pages, LaTeX2

Multiplicative McKay correspondence in the symplectic case

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.Comment: Latex2e, 19 page

Non-commutative Cartier operator and Hodge-to-de Rham degeneration

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.Comment: 53 pages, LaTeX2

Bokstein homomorphism as a universal object

We give a simple construction of the correspondence between square-zero extensions $R'$ of a ring $R$ by an $R$-bimodule $M$ and second MacLane cohomology classes of $R$ with coefficients in $M$ (the simplest non-trivial case of the construction is $R=M=Z/p$, $R'=Z/p^2$, thus the Bokstein homomorphism of the title). Following Jibladze and Pirashvili, we treat MacLane cohomology as cohomology of non-additive endofunctors of the category of projective $R$-modules. We explain how to describe liftings of $R$-modules and complexes of $R$-modules to $R'$ in terms of data purely over $R$. We show that if $R$ is commutative, then commutative square-zero extensions $R'$ correspond to multiplicative extensions of endofunctors. We then explore in detail one particular multiplicative non-additive endofunctor constructed from cyclic powers of a module $V$ over a commutative ring $R$ annihilated by a prime $p$. In this case, $R'$ is the second Witt vectors ring $W_2(R)$ considered as a square-zero extension of $R$ by the Frobenius twist $R^{(1)}$.Comment: LaTeX2e, 63 pages (updates references

Sommese Vanishing for non-compact manifolds

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.Comment: No changes. Replacement is done just to add comments, which are: the preprint is left intact for historical reasons (there are references to it in other papers); however, as H. Esnault and E. Viehweg kindly indicated to me, the main result is actually an easy corollary of an old result of theirs. The application to symplectic manifolds, which is new, is now a part of my paper math.AG/031018

Normalisation of a Poisson algebra is Poisson

We prove that the integral closure of a Poisson algebra $A$ over a field of characteristic 0 is again a Poisson algebra.Comment: 4 pages, LaTeX2

Motivic structures in non-commutative geometry

We review some recent results and conjectures saying that, roughly speaking, periodic cyclic homology of a smooth non-commutative algebraic variety should carry all the additional "motivic" structures possessed by the usual de Rham cohomology of a smooth algebraic variety (specifically, an R-Hodge structure for varieties over R, and a filtered Dieudonne module structure for varieties over Z_p). To appear in Proc. ICM 2010.Comment: LaTeX 2e, 24 pages

A canonical hyperkaehler metric on the total space of a cotangent bundle

A canonical hyperkaehler metric on the total space $T^*M$ of a cotangent bundle to a complex manifold $M$ has been constructed recently by the author (see alg-geom/9710026). This paper presents the results of alg-geom/9710026 in a streamlined and simplified form. The only new result is an explicit formula obtained for the case when $M$ is an Hermitian symmetric space.Comment: 44 pages, 2 eps figures, LaTeX2e. A talk at the Second Quaternionic Meeting, Rome, 199

On the coordinate ring of a projective Poisson scheme

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.Comment: Final version, slightly expanded; to appear in MR