arXiv subjects

Alexander Kuznetsov

Publications and source records attributed to Alexander Kuznetsov.

At least 91 records · Page 5Linked to original sources

arXiv2009

Hochschild homology and semiorthogonal decompositions

We investigate Hochschild cohomology and homology of admissible subcategories of derived categories of coherent sheaves on smooth projective varieties. We show that the Hochschild cohomology of an admissible subcategory is isomorphic to the derived endomorphisms of the kernel giving the corresponding projection functor, and the Hochschild homology is isomorphic to derived morphisms from this kernel to its convolution with the kernel of the Serre functor. We investigate some basic properties of Hochschild homology and cohomology of admissible subcategories. In particular, we check that the Hochschild homology is additive with respect to semiorthogonal decompositions and construct some long exact sequences relating the Hochschild cohomology of a category and its semiorthogonal components. We also compute Hochschild homology and cohomology of some interesting admissible subcategories, in particular of the nontrivial components of derived categories of some Fano threefolds and of the nontrivial components of the derived categories of conic bundles.

math.AG↗

arXiv2008

Semiinfinite Flags. II. Local and Global Intersection Cohomology of Quasimaps' Spaces

For a simple algebraic group $G$ we study the space $Q$ of Quasimaps from the projective line $C$ to the flag variety of $G$. We prove that the global Intersection Cohomology of $Q$ carries a natural pure Tate Hodge structure, and compute its generating function. We define an action of the Langlands dual Lie algebra $g^L$ on this cohomology. We present a new geometric construction of the universal enveloping algebra $U(n^L_+)$ of the nilpotent subalgebra of $g^L$. It is realized in the Ext-groups of certain perverse sheaves on Quasimaps' spaces, and it is equipped with a canonical basis numbered by the irreducible components of certain algebraic cycles, isomorphic to the intersections of semiinfinite orbits in the affine Grassmannian of $G$. We compute the stalks of the IC-sheaves on the Schubert strata closures in $Q$. They carry a natural pure Tate Hodge structure, and their generating functions are given by the generic affine Kazhdan-Lusztig polynomials. In the Appendix we prove that Kontsevich's space of stable maps provides a natural resolution of $Q$.

alg-geom↗

arXiv2008

Derived categories of Fano threefolds

We consider the structure of the derived categories of coherent sheaves on Fano threefolds with Picard number 1 and describe a strange relation between derived categories of different threefolds. In the Appendix we discuss how the ring of algebraic cycles of a smooth projective variety is related to the Grothendieck group of its derived category.

math.AG↗

arXiv2008

Derived categories of cubic fourfolds

We discuss the structure of the derived category of coherent sheaves on cubic fourfolds of three types: Pfaffian cubics, cubics containing a plane and singular cubics, and discuss its relation to the rationality of these cubics.

math.AG↗

arXiv2008

United classification of cosmic gamma-ray bursts and their counterparts

United classification of gamma-ray bursts and their counterparts is established on the basis of measured characteristics: photon energy E and emission duration T. The founded interrelation between the mentioned characteristics of events consists in that, as the energy increases, the duration decreases (and vice versa). The given interrelation reflects the nature of the phenomenon and forms the E-T diagram, which represents a natural classification of all observed events in the energy range from 10E9 to 10E-6 eV and in the corresponding interval of durations from about 10E-2 up to 10E8 s. The proposed classification results in the consequences, which are principal for the theory and practical study of the phenomenon.

astro-ph↗

arXiv2007

Boundary behaviour of Loewner Chains

In paper found conditions that guarantee that solution of Loewner-Kufarev equation maps unit disc onto domain with quasiconformal rectifiable boundary, or it has continuation on closed unit disc, or it's inverse function has continuation on closure of domain.

math.CV↗

arXiv2006

Exceptional collections for Grassmannians of isotropic lines

We construct a full exceptional collection of vector bundles in the derived category of coherent sheaves on the Grassmannian of isotropic two-dimensional subspaces in a symplectic vector space of dimension $2n$ and in the derived category of coherent sheaves on the Grassmannian of isotropic two-dimensional subspaces in an orthogonal vector space of dimension $2n + 1$ for all $n$.

math.AG↗

arXiv2006

Lefschetz decompositions and Categorical resolutions of singularities

Let $Y$ be a singular algebraic variety and let $\TY$ be a resolution of singularities of $Y$. Assume that the exceptional locus of $\TY$ over $Y$ is an irreducible divisor $\TZ$ in $\TY$. For every Lefschetz decomposition of $\TZ$ we construct a triangulated subcategory $\TD \subset \D^b(\TY)$ which gives a desingularization of $\D^b(Y)$. If the Lefschetz decomposition is generated by a vector bundle tilting over $Y$ then $\TD$ is a noncommutative resolution, and if the Lefschetz decomposition is rectangular, then $\TD$ is a crepant resolution.

math.AG↗

arXiv2006

Homological projective duality for Grassmannians of lines

We show that homologically projectively dual varieties for Grassmannians Gr(2,6) and Gr(2,7) are given by certain noncommutative resolutions of singularities of the corresponding Pfaffian varieties. As an application we describe the derived categories of linear sections of these Grassmannians and Pfaffians. In particular, we show that (1) the derived category of a Pfaffian cubic 4-fold admits a semiorthogonal decompositions consisting of 3 exceptional line bundles, and of the derived category of a K3-surface; (2) mutually orthogonal Calabi-Yau linear sections of Gr(2,7) and of the corresponding Pfaffian variety are derived equivalent. We also conjecture a rationality criterion for cubic 4-folds in terms of their derived categories.

math.AG↗

arXiv2005

Derived Categories of Quadric Fibrations and Intersections of Quadrics

We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algebras on the base corresponding to this quadric fibration, generalizing the Kapranov's description of the derived category of a single quadric. As an application we verify that the noncommutative algebraic variety $(\PP(S^2W^*),\CB_0)$, where $\CB_0$ is the universal sheaf of even parts of Clifford algebras, is Homologically Projectively Dual to the projective space $\PP(W)$ in the double Veronese embedding $\PP(W) \to \PP(S^2W)$. Using the properties of the Homological Projective Duality we obtain a description of the derived category of coherent sheaves on a complete intersection of any number of quadrics.

math.AG↗

arXiv2005

Homological Projective Duality

We introduce a notion of Homological Projective Duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties $X$ and $Y$ in dual projective spaces are Homologically Projectively Dual, then we prove that the orthogonal linear sections of $X$ and $Y$ admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these sections are equivalent. We also investigate Homological Projective Duality for projectivizations of vector bundles.

math.AG↗

arXiv2005

Hyperplane sections and derived categories

We give a generalization of the theorem of Bondal and Orlov about the derived categories of coherent sheaves on intersections of quadrics revealing its relation to projective duality. As an application we describe the derived categories of coherent sheaves on Fano 3-folds of index 1 and degrees 12, 16 and 18.

math.AG↗

arXiv2003

Derived category of V_{12} Fano threefolds

A V_{12} Fano threefold is a smooth Fano threefold X of index 1 with Pic X = Z and (-K_X)^3=12. We show that the bounded derived category of coherent sheaves on any V_{12} threefold X admits a semiorthogonal decomposition consisting of two exceptional bundles and of the derived category of a curve of genus 7. As an application we show that the Fano surface of X (the surface parameterizing conics on X) is canonically isomorphic to the symmetric square of the associated genus 7 curve.

math.AG↗

arXiv2003

Derived categories of cubic and V14 threefolds

We show that the projectivization of the exceptional rank 2 vector bundle on an arbitrary smooth V14 Fano threefold after a certain natural flop turns into the projectivization of an instanton vector bundle on a smooth cubic threefold. And vice versa, starting from a smooth cubic threefold with an instanton vector bundle of charge 2 on it we reconstruct V14 threefold. Relying on the geometric properties of the above correspondence we prove that the orthogonals to the exceptional pairs in the bounded derived categories of coherent sheaves on a smooth V14 threefold and on the corresponding cubic threefold are equivalent as triangulated categories.

math.AG↗

arXiv2002

Wilson's grassmannian and a noncommutative Quadric

Let the group G of m-th roots of unity act on the complex line by multiplication, inducing an action on the algebra, Diff, of polynomial differential operators on the line. Following Crawley-Boevey and Holland, we introduce a multiparameter deformation, D_c, of the smash-product (Diff # G). Our main result provides natural bijections between (roughly speaking) the following spaces: (1) G-equivariant version of Wilson's adelic Grassmannian of rank r; (2) Rank r projective D_c-modules (equipped with generic trivialization); (3) Rank r torsion-free sheaves on a `noncommutative quadric'; (4) Disjoint union of Nakajima quiver varieties for the cyclic quiver with m vertices. The bijection between (1) and (2) is provided by a version of Riemann-Hilbert correspondence between D-modules and sheaves. The bijections between (2), (3) and (4) were motivated by our previous work math.AG/0103068. The resulting bijection between (1) and (4) reduces, in the very special case: r=1 and G=1, to the partition of (rank 1) adelic Grassmannian into a union of Calogero-Moser spaces, discovered by Wilson. This gives, in particular, a natural and purely algebraic approach to Wilson's result.

math.AG↗

arXiv2002

Uhlenbeck spaces for $A^2$ and affine Lie algebra $\hat{sl}_n$

We introduce an Uhlenbeck closure of the space of based maps from projective line to the Kashiwara flag scheme of an untwisted affine Lie algebra. For the algebra $\hat{sl}_n$ this space of based maps is isomorphic to the moduli space of locally free parabolic sheaves on $P^1\times P^1$ trivialized at infinity. The Uhlenbeck closure admits a resolution of singularities: the moduli space of torsion free parabolic sheaves on $P^1\times P^1$ trivialized at infinity. We compute the Intersection Cohomology sheaf of the Uhlenbeck space using this resolution of singularities. The moduli spaces of parabolic sheaves of various degrees are connected by certain Hecke correspondences. We prove that these correspondences define an action of $\hat{sl}_n$ in the cohomology of the above moduli spaces.

math.AG↗

arXiv2001

Quiver varieties and Hilbert schemes

In this note we give an explicit geometric description of some of the Nakajima's quiver varieties. More precisely, we show that the $Γ$-equivariant Hilbert scheme $X^{Γ[n]}$ and the Hilbert scheme $X_Γ^{[n]}$ (where $X=\C^2$, $Γ\subset SL(\C^2)$ is a finite subgroup, and $X_Γ$ is a minimal resolution of $X/Γ$) are quiver varieties for the affine Dynkin graph, corresponding to $Γ$ via the McKay correspondence, the same dimension vectors, but different parameters $ζ$ (for earlier results in this direction see [4, 12, 13]). In particular, it follows that the varieties $X^{Γ[n]}$ and $X_Γ^{[n]}$ are diffeomorphic. Computing their cohomology (in the case $Γ=\Z/d\Z$) via the fixed points of $(\C^*\times\C^*)$-action we deduce the following combinatorial identity: the number $UCY(n,d)$ of uniformly coloured in d colours Young diagrams consisting of nd boxes coincides with the number $CY(n,d)$ of collections of d Young diagrams with the total number of boxes equal to n.

math.AG↗

arXiv2001

Quiver varieties and a non-commutative P^2

To any finite group G in SL_2(C), and each `t' in the center of the group algebra of G, we associate a category, Coh_t. It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category Coh_t should be thought of as the category of coherent sheaves on a `noncommutative projective 2-space', equipped with a framing at the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of Coh_t and the Nakajima quiver variety arising from G via the McKay correspodence. We apply the above isomorphism to deduce generalized Crawley-Boevey & Holland conjecture, saying that the moduli space of `rank 1' projective modules over the deformed preprojective algebra is isomorphic to a particular quiver variety. This reduces, for G=1, to the recently obtained parametrisation of the isomorphism classes of right ideals in the first Weyl algebra, A_1, by points of the Calogero-Moser space, due to Cannings-Holland and Berest-Wilson. Our approach is algebraic and is based on a monadic description of torsion free sheaves on the noncommutative projective 2-space. It is totally different from the one used by Berest-Wilson, involving $τ$-functions.

math.AG↗