Publications

The EA-matrix integrals, introduced in Barannikov (Comptes Rendus Math 348:359–362, hal-00102085 (2006)), are studied in the case of graded associative algebras with odd or even scalar product. I prove that the EA-matrix integrals for associative algebras with scalar product are integrals of equivariantly closed differential forms with respect to the Lie algebra   𝑔𝑙(𝐴) .

In this paper we construct the modular Cauchy kernel $\Xi_N(z_1, z_2)$, i.e. the modular invariant function of two variables, $(z_1, z_2) \in \mathbb{H} \times \mathbb{H}$, with the first order pole on the curve $$D_N=\left\{(z_1, z_2) \in \mathbb{H} \times \mathbb{H}|~ z_2=\gamma z_1, ~\gamma \in \Gamma_0(N)  \right\}.$$ 

 

The  function $\Xi_N(z_1, z_2)$ is used in two cases and for two different purposes. Firstly,  we prove generalization of the Zagier theorem (\cite{La}, \cite{Za3}) for the Hecke subgroups $\Gamma_0(N)$ of genus $g>0$. Namely, we obtain a kind of ``kernel function'' for the Hecke operator $T_N(m)$ on the space of the weight 2 cusp forms for $\Gamma_0(N)$, which is the analogue of the Zagier series $\omega_{m, N}(z_1,\bar{z_2}, 2)$. Secondly, we consider an elementary proof of the formula for the infinite Borcherds product of the difference of two normalized Hauptmoduls, ~$J_{\Gamma_0(N)}(z_1)-J_{\Gamma_0(N)}(z_2)$, for genus zero congruence subgroup $\Gamma_0(N)$.

https://arxiv.org/abs/1803.11549

I describe a combinatorial construction of the cohomology classes in compactified moduli spaces of curves ZˆI∈H∗(barM_g,n) starting from the following data: an odd derivation I, whose square is non-zero in general, I2≠0, acting on a ℤ/2ℤ-graded associative algebra with odd scalar product. The constructed cocycles were first described in the theorem 2 in the author's paper "Noncommmutative Batalin-Vilkovisky geometry and Matrix integrals". Comptes Rendus Mathematique, 348, pp. 359-362, arXiv:0912.5484 , preprint HAL-00102085 (09/2006). By the theorem 3 from loc.cit. the family of the cohomology classes obtained in the case of the algebra Q(N) and the derivation I=[Λ,⋅] coincided with the generating function of products of ψ−classes. This was the first nontrivial computation of categorical Gromov-Witten invariants of higher genus. The result matched with the mirror symmetry prediction, i.e. with the classical (non-categorical) Gromov-Witten descendent invariants of a point for all genus. As a byproduct of that computation a new combinatorial formula for products of ψ-classes ψi=c1(T∗pi) in the cohomology H∗(barM_g,n) is written out.

The goal of this work is to present an alternative way of calculating  the values of the Green's function associated with the Hirzebruch-Zagier divisor at the points of another Hirzebruch-Zagier divisor $T_m$.

We study the rationality problem for nodal quartic double solids. In particular, we prove that nodal quartic double solids with at most six singular points are irrational and nodal quartic double solids with at least eleven singular points are rational.

A Calogero–Sutherland system with two types of interacting spin variables has been described using the Hitchin approach and quasicompact structure. Complete integrability has been established by means of the Lax equation specified on a singular curve and the classical r-matrix depending on the spectral parameter. Generalized Toda systems have also been considered. Their phase portraits have been described.

We study the integral Bailey lemma associated with the A_n-root system and identities for elliptic hypergeometric integrals generated thereby. Interpreting integrals as superconformal indices of four-dimensional N = 1 quiver gauge theories with the gauge groups being products of SU(n + 1), we provide evidence for various new dualities. Further con rmation is achieved by explicitly checking that the `t Hooft anomaly matching conditions holds. We discuss a flavour symmetry breaking phenomenon for supersymmetric quantum chromodynamics (SQCD), and by making use of the Bailey lemma we indicate its manifestation in a web of linear quivers dual to SQCD that exhibits full s-confinement.

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.

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.

In this paper we use the theory of formal moduli problems developed by Lurie in order to study the space of formal deformations of a k-linear 1-category for a eld k. Our main result states that if C is a k-linear 1-category which has a compact generator whose groups of self-extensions vanish for suciently high positive degrees, then every formal deformation of C has zero curvature and moreover admits a compact generator.

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 verify Katzarkov-Kontsevich-Pantev conjecture for Landau-Ginzburg models of smooth Fano threefolds.

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.