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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 added

A dynamics of a low-dimensional ensemble consisting of connected in a network five discrete phase oscillators is considered. A two-parameter synchronization picture which appears instead of the Arnold tongues with an increase of the system dimension is discussed. An appearance of the Arnol'd resonance web is detected on the "frequency - coupling" parameter plane. The cases of attractive and repulsive interactions are discussed., Comment: 7 figures

Structure of the eigenfrequencies parameter space for three and four dissipatively coupled van der Pol oscillators is discussed. Situations of different codimension relating to the configuration of the full synchronization area as well as a picture of different modes in its neighborhood are revealed. The organization of quasi-periodic areas of different dimensions is considered. The results for the phase model and for the original system are compared., Comment: 16 pages, 9 figures

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.

The conditions are discussed for which an ensemble of interacting oscillators may demonstrate the Landau-Hopf scenario of successive birth of multi-frequency quasi-periodic motions. A model is proposed that is a network of five globally coupled oscillators characterized by controlled degree of activation of individual oscillators. Illustrations are given for successive birth of tori of increasing dimension via quasi-periodic Hopf bifurcations.

We show that the derived category of any singularity over a field of characteristic 0 can be embedded fully and faithfully into a smooth triangulated category which has a semiorthogonal decomposition with components equivalent to derived categories of smooth varieties. This provides a categorical resolution of the singularity., Comment: 71 pages

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 corrected

We introduce the notion of an instanton bundle on a Fano threefold of index 2. For such bundles we give an analogue of a monadic description and discuss the curve of jumping lines. The cases of threefolds of degree 5 and 4 are considered in a greater detail., Comment: 31 page, to appear in CEJM

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 JEMS

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 2011

We consider the class of singular double coverings $X \to \PP^3$ ramified in the degeneration locus $D$ of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such quartic surface $D$ one can associate an Enriques surface $S$ which is the factor of the blowup of $D$ by a natural involution acting without fixed points (such Enriques surfaces are known as nodal Enriques surfaces or Reye congruences). We show that the nontrivial part of the derived category of coherent sheaves on this Enriques surface $S$ is equivalent to the nontrivial part of the derived category of a minimal resolution of singularities of $X$., Comment: 18 pages

Given a generic family $Q$ of 2-dimensional quadrics over a smooth 3-dimensional base $Y$ we consider the relative Fano scheme $M$ of lines of it. The scheme $M$ has a structure of a generically conic bundle $M \to X$ over a double covering $X \to Y$ ramified in the degeneration locus of $Q \to Y$. The double covering $X \to Y$ is singular in a finite number of points (corresponding to the points $y \in Y$ such that the quadric $Q_y$ degenerates to a union of two planes), the fibers of $M$ over such points are unions of two planes intersecting in a point. The main result of the paper is a construction of a semiorthogonal decomposition for the derived category of coherent sheaves on $M$. This decomposition has three components, the first is the derived category of a small resolution $X^+$ of singularities of the double covering $X \to Y$, the second is a twisted resolution of singularities of $X$ (given by the sheaf of even parts of Clifford algebras on $Y$), and the third is generated by a completely orthogonal exceptional collection., Comment: 14 pages

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., Comment: 30 pages

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., Comment: 12 pages

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., Comment: Keywords Gamma rays: bursts

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., Comment: 18 pages

The particular properties of dynamics are discussed for the dissipatively coupled van der Pol oscillators, non-identical in values of parameters controlling the Hopf bifurcation. Possibility of a special synchronization regime in an infinitively long band between oscillation death and quasiperiodic areas is shown for such system. Features of the bifurcation picture are discussed for different values of the control parameters and for the case of additional Duffing-type nonlinearity. Analysis of the abridged equations is presented., Comment: 19 pages, 9 figures

Consider an algebraic variety $X$ over a base scheme $S$ and a faithful base change $T \to S$. Given an admissible subcategory $\CA$ in the bounded derived category of coherent sheaves on $X$, we construct an admissible subcategory in the bounded derived category of coherent sheaves on the fiber product $X\times_S T$, called the base change of $\CA$, in such a way that the following base change theorem holds: if a semiorthogonal decomposition of the bounded derived category of $X$ is given then the base changes of its components form a semiorthogonal decomposition of the bounded derived category of the fiber product. As an intermediate step we construct a compatible system of semiorthogonal decompositions of the unbounded derived category of quasicoherent sheaves on $X$ and of the category of perfect complexes on $X$. As an application we prove that the projection functors of a semiorthogonal decomposition are kernel functors., Comment: 24 pages; derived category of countably-coherent sheaves which appeared in the first version for technical reasons is replaced by the usual quasicoherent category

The neutrino chirality-flip process under the conditions of the supernova core is investigated in detail with the plasma polarization effects in the photon propagator taken into account, in a more consistent way than in earlier publications. It is shown in part that the contribution of the proton fraction of plasma is essential. New upper bounds on the neutrino magnetic moment are obtained: mu_nu < (0.5 - 1.1) 10^{-12} mu_B from the limit on the supernova core luminosity for nu_R emission, and mu_nu < (0.4 - 0.6) 10^{-12} mu_B from the limit on the averaged time of the neutrino spin-flip. The best upper bound on the neutrino magnetic moment from SN1987A is improved by the factor of 3 to 7., Comment: 19 pages, LaTeX, 7 EPS figures, submitted to Journal of Cosmology and Astroparticle Physics

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 pages

In the work a nonlinear Duffing oscillator is considered under impulse excitation with two ways of introduction of the random additive term simulating noise, - with help of amplitude modulation and modulation of period of impulses sequence. The scaling properties both in the Feigenbaum scenario and in the tricritical case are shown., Comment: 14 pages, 8 figures

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., Comment: 36 pages

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 added

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 $n$., Comment: 26 pages; thoroughly revised - a construction of a full exceptional collection for the orthogonal isotropic Grassmannian OGR(2,2n+1) is included

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 pages

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., Comment: 43 pages

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., Comment: 76 pages