Height of exceptional collections and Hochschild cohomology of quasiphantom categories

We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety $X$ --- a graded vector space which controls the restriction morphism from the Hochschild cohomology of $X$ to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is generated by an exceptional collection, we define its new invariant (the height) and show that the orthogonal to an exceptional collection of height $h$ in the derived category of a smooth projective variety $X$ has the same Hochschild cohomology as $X$ in degrees up to $h - 2$. We use this to describe the second Hochschild cohomology of quasiphantom categories in the derived categories of some surfaces of general type. We also give necessary and sufficient conditions of fullness of an exceptional collection in terms of its height and of its normal Hochschild cohomology.Comment: 23 pages, the construction of the \v{C}ech enhancement is correcte

Derived categories of families of sextic del Pezzo surfaces

We construct a natural semiorthogonal decomposition for the derived category of an arbitrary flat family of sextic del Pezzo surfaces with at worst du Val singularities. This decomposition has three components equivalent to twisted derived categories of finite flat schemes of degrees 1, 3, and 2 over the base of the family. We provide a modular interpretation for these schemes and compute them explicitly in a number of standard families. For two such families the computation is based on a symmetric version of homological projective duality for $\mathbb{P}^2 \times \mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, which we explain in an appendix.Comment: 48 pages; v2: minor improvement

Calabi-Yau and fractional Calabi-Yau categories

We discuss Calabi-Yau and fractional Calabi-Yau semiorthogonal components of derived categories of coherent sheaves on smooth projective varieties. The main result is a general construction of a fractional Calabi-Yau category from a rectangular Lefschetz decomposition and a spherical functor. We give many examples of application of this construction and discuss some general properties of Calabi-Yau categories.Comment: 24 pages, v2: minor correction

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

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

Exceptional collections in surface-like categories

We provide a categorical framework for recent results of Markus Perling on combinatorics of exceptional collections on numerically rational surfaces. Using it we simplify and generalize some of Perling's results as well as Vial's criterion for existence of a numerical exceptional collection.Comment: 23 pages; v2: some references added, the relation to the approach of [dTVdB] discusse

Categorical joins

We introduce the notion of a categorical join, which can be thought of as a categorification of the classical join of two projective varieties. This notion is in the spirit of homological projective duality, which categorifies classical projective duality. Our main theorem says that the homological projective dual category of the categorical join is naturally equivalent to the categorical join of the homological projective dual categories. This categorifies the classical version of this assertion and has many applications, including a nonlinear version of the main theorem of homological projective duality.Comment: 58 pages. Final version, to appear in JAM

Black Hole in Binary System is the Source of Cosmic Gamma-Ray Bursts and Their Counterparts

Discovery of GRB clusters allows us to determine coordinates and characteristics of their sources. The objects radiating GRBs are reliably identified with black hole binaries, including the Galactic binaries. One of the unusual GRB properties, which are determined by black hole, is revealed in that the measured arrival direction of GRB does not coincide with the real location of its source. Just this fact allows us to find the objects radiating GRBs. On the basis of the general relativity theory's effects, observed in the GRB clusters, the technique of the black hole masses measurement is developed. The calculated black hole masses for the majority of known Galactic BH binaries are presented. It is briefly shown how the incorrect interpretations of observational facts result in an erroneous idea of the GRB cosmological origin. In fact, two problems are solved in the paper: the GRB origin and the reality of BH existence.Comment: This paper has been accepted by the Journal of Modern Physics in April 201

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