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.Comment: 22 page

Derived categories view on rationality problems

We discuss a relation between the structure of derived categories of smooth projective varieties and their birational properties. We suggest a possible definition of a birational invariant, the derived category analogue of the intermediate Jacobian, and discuss its possible applications to the geometry of prime Fano threefolds and cubic fourfolds.Comment: Lecture notes for the CIME-CIRM summer school, Levico Terme, June 22--27, 2015; 26 page

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.Comment: 11 page

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.Comment: 24 pages; the proof of the main theorem rewritten, a section on functoriality is adde

A simple counterexample to the Jordan-H\"older property for derived categories

A counterexample to the Jordan-H\"older property for semiorthogonal decompositions of derived categories of smooth projective varieties was constructed by B\"ohning, Graf von Bothmer and Sosna. In this short note we present a simpler example by realizing Bondal's quiver in the derived category of a blowup of the projective space

Semiorthogonal decompositions in algebraic geometry

In this review we discuss what is known about semiorthogonal decompositions of derived categories of algebraic varieties. We review existing constructions, especially the homological projective duality approach, and discuss some related issues such as categorical resolutions of singularities.Comment: Contribution to the ICM 2014; v2: acknowledgements updated; v3: a reference adde

Exceptional collections on isotropic Grassmannians

We introduce a new construction of exceptional objects in the derived category of coherent sheaves on a compact homogeneous space of a semisimple algebraic group and show that it produces exceptional collections of the length equal to the rank of the Grothendieck group on homogeneous spaces of all classical groups.Comment: v1: 51 page; v2: 55 pages, to appear in JEM

Derived categories of Gushel-Mukai varieties

We study the derived categories of coherent sheaves on Gushel-Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel-Mukai fourfold in more detail. Namely, we show that this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for Gushel-Mukai varieties, which was one of the main motivations for this work.Comment: 43 pages, reorganized and edite

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 $\Gamma$-equivariant Hilbert scheme $X^{\Gamma[n]}$ and the Hilbert scheme $X_\Gamma^{[n]}$ (where X=\C^2, \Gamma\subset SL(\C^2) is a finite subgroup, and $X_\Gamma$ is a minimal resolution of $X/\Gamma$) are quiver varieties for the affine Dynkin graph, corresponding to $\Gamma$ via the McKay correspondence, the same dimension vectors, but different parameters $\zeta$ (for earlier results in this direction see [4, 12, 13]). In particular, it follows that the varieties $X^{\Gamma[n]}$ and $X_\Gamma^{[n]}$ are diffeomorphic. Computing their cohomology (in the case $\Gamma=\Z/d\Z$) 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.Comment: LaTeX, 27 page

On K\"uchle manifolds with Picard number greater than 1

We describe the geometry of K\"uchle varieties (i.e. Fano 4-folds of index 1 contained in the Grassmannians as zero loci of equivariant vector bundles) with Picard number greater than 1 and the structure of their derived categories.Comment: 10 pages, a reference adde