Publications

For a geometrically rational surface X over a perfect field of characteristic different from 2 that contains all roots of 1, we show that either X is birational to a product of a projective line and a conic, or the group of birational automorphisms of X has bounded finite subgroups. As an auxiliary result, we show boundedness of finite subgroups in anisotropic linear algebraic groups of bounded rank and number of connected components. Also, we provide applications to Jordan property for groups of birational automorphisms.

Given a relatively projective birational morphism f : X → Y of smooth algebraic spaces with dimension of fibers bounded by 1, we construct tilting relative (over Y) generators TX,f and S_X,f in D^b(X). We develop a piece of general theory of strict admissible lattice filtrations in triangulated categories and show that D^b(X) has such a filtration L where the lattice is the set of all birational decompositions f : X −→^g Z  −→^h Y with smooth Z. The t-structures related to T_X,f and S_X,f are proved to be glued via filtrations left and right dual to L. We realise all such Z as the fine moduli spaces of simple quotients of O_X in the heart of the t-structure for which S_X,g is a relative projective generator over Y . This implements the program of interpreting relevant smooth contractions of X in terms of a suitable system of t-structures on D^b(X).

For a simply connected, connected, semisimple complex algebraic group G, we dene two geometric crystals on the A-cluster variety of double Bruhat cell B\cap \ Bw_0B. These crystals are related by the duality . We dene the graded Donaldson-Thomas correspondence as the crystal bijection between these crystals. We show that this correspondence is equal to the composition of the cluster cham- ber Ansatz, the inverse generalized geometric RSK-correspondence, and transposed twist map due to Berenstein and Zelevinsky.

In this article, we will show that for a smooth quasi-projective variety X over complex numbers, and regular function  W on X, the periodic cyclic homology of the DG category of matrix factorizations of W on X is identified (under the Riemann–Hilbert correspondence) with the vanishing cohomology, with the monodromy twisted by a sign. Also, Hochschild homology is identified respectively with the hypercohomology of the complex of differential forms, with a differential being a wedge product with dW.

We prove that the derived category D(C) of a generic curve of genus greater than one embeds into the derived category D(M) of the moduli space M of rank two stable bundles on C with fixed determinant of odd degree.

 

We classify smooth Fano threefolds with infinite automorphism groups.

The construction from [B06], see also [B10], of cohomology classes of compactified moduli spaces of Riemann surfaces, starting from a derivation of associative whose square is nonzero, is generalized to the case of A-infinity algebras. It is shown that the constructed cohomology classes define Cohomological Field Theory.

We give lower bounds for Hodge numbers of smooth well formed Fano weighted complete intersections. In particular, we compute their Hodge complexity, that is, the maximal distance between non-trivial Hodge numbers. This allows us to classify varieties of such type whose Hodge numbers are like that of a projective space, of a curve, or of a Calabi-Yau variety of low dimension.

We consider the conjectures of Katzarkov, Kontsevich, and Pantev  about Landau--Ginzburg Hodge numbers associated to tamely compactifiable Landau--Ginzburg models. We test these conjectures in case of dimension two, verifying some and giving a counterexample to the other.

We describe a new large  class of  Lorentzian Kac--Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices $S$ with the group of 2-reflections of finite volume and with a lattice Weyl vector.  They define the corresponding hyperbolic Kac--Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct Lorentzian Kac--Moody algebras which give their automorphic corrections: they are graded by the S, have the same simple real roots, but their denominator identities are given by automorphic forms with 2-reflective divisors. We give exact constructions of these automorphic forms as Borcherds products and, in some cases, as additive Jacobi liftings.

 

We compare previously found finite-dimensional matrix and integral operator realizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With the help of residue calculus we explicitly show how the integral Bailey lemma can be reduced to its matrix version. As a consequence, we demonstrate that the matrix Bailey lemma can be interpreted as a star-triangle relation, or as a Coxeter relation for a permutation group.  

In this paper we construct the modular Cauchy kernel on the Hilbert modular surface ΞHil,m(z)(z2−z2¯), i.e. the function of two variables, (z1,z2)∈H×H, which is invariant under the action of the Hilbert modular group, with the first order pole on the Hirzebruch-Zagier divisors. The derivative of this function with respect to z2¯ is the function ωm(z1,z2) introduced by Don Zagier in \cite{Za1}. We consider the question of the convergence and the Fourier expansion of the kernel function. The paper generalizes the first part of the results obtained in the preprint \cite{Sa}

We present an example of modified moduli space of special Bohr-Sommerfeld lagrangian submanifolds for the case when the given algebraic variety is the full flag F3 for C3 and the very ample bundle is K^{-1/2}_{F3}

In the previous papers we present a construction of the set U_SBS in the direct product B_S×PΓ(M, L) of the moduli space of Bohr - Sommerfeld lagrangian submanifolds of fixed topological type and the projectivized space of smooth sections of the prequantization bundle L→M over a given compact simply connected symplectic manifold M.  Canonical projections p:U_SBS→PΓ(M, L) and q:U_SBS→ B_S are studied in the present text: first, we show that the differential dp at a given point is an isomorphism, which implies that a natural complex structure can be defined on U_SBS; second, the projection q:U_SBS→ B_S splits as the combination U_SBS→ TB_S→ B_S such that the fibers of the first map are complex subsets in U_SBS. This implies that an appropriate section of the first map should define a complex structure on TB_S; therefore it can be seen as a complexification of the moduli space B_S. The construction can be exploited in the Lagrangian approach to Geometric Quantization.

Let X(n) denote n-th symmetric power of a cubic surface X. We show that X(4)×X is stably birational to X(3)×X, despite examples when X(4) is not stably birational to X(3).

We construct a mininal Lefschetz decomposition of the bounded derived category of coherent sheaves on the isotropic Grassmannian $\IGr(3,7)$. Moreover, we show that $\IGr(3, 7)$ admits a full exceptional collection consisting of equivariant vector bundles.

Perverse schobers are conjectural categorical analogs of perverse sheaves. We show that such structures appear naturally in Homological Minimal Model Program which studies the effect of birational transformations such as flops, on the coherent derived categories. More precisely, the flop data are analogous to hyperbolic stalks of a perverse sheaf. In the first part of the paper we study schober-type diagrams of categories corresponding to flops of relative dimension 1, in particular we determine the categorical analogs of the (compactly supported) cohomology with coefficients in such schobers. In the second part we consider the example of a “web of flops” provided by the Grothendieck resolution associated to a reductive Lie algebra g and study the corresponding schober-type diagram. For g = sl3 we relate this diagram to the classical space of complete triangles studied by Schubert, Semple and others.