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D. Kaledin

Publications and source records attributed to D. Kaledin.

52 records · Page 3Linked to original sources
arXiv2006

Symplectic singularities from the Poisson point of view

We consider symplectic singularities in the sense of A. Beauville as examples of Poisson schemes. Using Poisson methods, we prove that a symplectic singularity admits a finite stratification with smooth symplectic strata. We also prove that in the formal neighborhood of a closed point in some stratum, the singularity is a product of the stratum and a transversal slice. The transversal slice is also a symplectic singularity, and the product decomposition is compatible with natural Poisson structures. Moreover, we prove that the transversal slice admits a $C^*$-action dilating the symplectic form.

math.AG↗
arXiv2005

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.

math.AG↗
arXiv2005

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.

math.AG↗
arXiv2005

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.

math.AG↗
arXiv2004

Fedosov quantization in algebraic context

We consider the problem of quantization of smooth symplectic varieties in the algebro-geometric setting. We show that, under appropriate cohomological assumptions, the Fedosov quantization procedure goes through with minimal changes. The assumptions are satisfied, for example, for affine and for projective varieties. We also give a classification of all possible quantizations.

math.AG↗
arXiv2003

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.

math.AG↗
arXiv2001

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).

math.AG↗
arXiv2000

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.

math.AG↗
arXiv2000

Period map for non-compact holomorphically symplectic manifolds

We study the deformations of a holomorphic symplectic manifold $M$, not necessarily compact, over a formal ring. We show (under some additional, but mild, assumptions on $M$) that the coarse deformation space exists and is smooth, finite-dimensional and naturally embedded into $H^2(M)$. For a holomorphic symplectic manifold $M$ which satisfies $H^1(M,{\cal O}_M) = H^2(M,{\cal O}_M)=0$, the coarse moduli of formal deformations is isomorphic to $\Spec\C[[t_1, ..., t_n]]$, where $t_1$, ... $t_n$ are coordinates in $H^2(M)$. This revised version contains one minor improvement: exposition in Subsection 5.1 has been made more detailed and rigourous.

math.AG↗
arXiv1999

McKay correspondence for symplectic quotient singularities

We consider the quotients $X = V/G$ of a symplectic complex vector space $V$ by a finite subgroup $G \subset Sp(V)$ which admit a smooth crepant resolution $Y \to X$. For such quotients, we prove the homological McKay correspondence conjectured by M. Reid. Namely, we construct a natural basis in the homology space $H_\cdot(Y,\Q)$ whose elements are numbered by the conjugacy classes in the finite group $G$.

math.AG↗
arXiv1999

Dynkin diagrams and crepant resolutions of quotient singularities

Let $V$ be a complex vector space on which a finite group $G$ acts by linear transformations. Let $W = V \oplus V^*$ be the sum of $V$ with its dual $V^*$. We prove that if the quotient $W/G$ admits a smooth crepant resolution, then the subgroup $G \subset Aut V$ is generated by complex reflections. We also obtain some results on the structure of smooth crepant resolutions of the quotients $W/G$, where $W$ is a symplectic vector space, and $G \subset Aut W$ is a finite group of symplectic linear transformations of the vector space $W$.

math.AG↗
arXiv1998

Partial resolutions of Hilbert type, Dynkin diagrams, and generalized Kummer varieties

We study the partial resolutions of singularities related to Hilbert schemes of points on an affine space. Consider a quotient of a vector space $V$ by an action of a finite group $G$ of linear transforms. Under some additional assumptions, we prove that the partial desingularization of Hilbert type is smooth only if the action of $G$ is generated by complex reflections. This is used to study the subvarieties of a Hilbert scheme of a complex torus. We show that any subvariety of a generic deformation of a Hilbert scheme of a torus is birational to a quotient of another torus by an action of a Weyl group of some semisimple Lie algebra. In Appendix, we produce counterexamples to a false theorem stated in our preprint math.AG/9801038.

math.AG↗
arXiv1998

Trianalytic subvarieties of generalized Kummer varieties

Let $X$ be a hyperkaehler manifold. Trianalytic subvarieties of $X$ are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkaehler structure. Given a 2-dimensional complex torus $T$, the Hilbert scheme $T^{[n]}$ classifying zero-dimensional subschemes of $T$ admits a hyperkaehler structure. A finite cover of $T^{[n]}$ is a product of $T$ and a simply connected hyperkaehler manifold $K^{[n-1]}$, called generalized Kummer variety. We show that for $T$ generic, the corresponding generalized Kummer variety has no trianalytic subvarieties. This implies that a generic deformation of the generalized Kummer variety has no proper complex subvarieties.

math.AG↗
arXiv1997

Hyperkaehler structures on total spaces of holomorphic cotangent bundles

Let $M$ be a Kaehler manifold, and consider the total space $T^*M$ of the cotangent bundle to $M$. We show that in the formal neighborhood of the zero section $M \subset T^*M$ the space $T^*M$ admits a canonical hyperkaehler structure, compatible with the complex and holomorphic symplectic structures on $T^*M$. The associated hyperkaehler metric $h$ coincides with the given Kaehler metric on the zero section $M \subset T^*M$. Moreover, $h$ is invariant under the canonical circle action on $T^*M$ by dilatations along the fibers of $T^*M$ over $M$. We show that a hyperkaehler structure with these properties is unique. When the Kaehler metric on $M$ is real-analytic, we show that this formal hyperkaehler structure can be extended to an open neighborhood of the zero section. We also prove a hyperkaehler analog of the Darboux-Weinstein Theorem. To prove these results, we use the machinery of $R$-Hodge structures, following Deligne and Simpson.

alg-geom↗
arXiv1996

Integrability of the twistor space for a hypercomplex manifold

A hypercomplex manifold is by definition a smooth manifold equipped with two anticommuting integrable almost complex structures. For example, every hyperkaehler manifold is canonically hypercomplex (the converse is not true). For every hypercomplex manifold M, the two almost complex structures define a smooth action of the algebra of quaternions on the tangent bundle to M. This allows to associate to every hypercomplex manifold M of dimension 4n a certain almost complex manifold X of dimension 4n+2, called the twistor space of M. When M is hyperkaehler, X is well-known to be integrable. We show that for an arbitrary hypercomplex manifold its twistor space is also integrable.

alg-geom↗