Publications

We study the field theory localizing to holomorphic maps from a complex manifold of complex dimension 2 to a toric target (a generalization of A model). Fields are realized as maps to (C*)(N) where one includes special observables supported on (1, 1)-dimensional submanifolds to produce maps to the toric compactification. We study the mirror of this model. It turns out to be a free theory interacting with N-comp topological strings of type A. Here N-comp is the number of compactifying divisors of the toric target. Before the mirror transformation these strings are vortex (actually, holomortex) strings.

We show the boundedness of finite subgroups in any anisotropic reductive group over a perfect field that contains all roots of 1. Also, we provide explicit bounds for orders of finite subgroups of automorphism groups of Severi–Brauer varieties and quadrics over such fields.

A categorical formalism is introduced for studying various features of the symplectic geometry of Lefschetz fibrations and the algebraic geometry of Tyurin degenerations. This approach is informed by homological mirror symmetry, derived noncommutative geometry, and the theory of Fukaya categories with coefficients in a perverse Schober. The main technical results include (i) a comparison between the notion of relative Calabi-Yau structures and a certain refinement of the notion of a spherical functor, (ii) a local-to-global gluing principle for constructing Calabi-Yau structures, and (iii) the construction of shifted symplectic structures and Lagrangian structures on certain derived moduli spaces of branes. Potential applications to a theory of derived hyperkähler geometry are sketched

Given a variety with a finite group action, we compare its equivariant categorical measure, that is, the categorical measure of the corresponding quotient stack, and the categorical measure of the extended quotient. Using weak factorization for orbifolds, we show that for a wide range of cases that these two measures coincide. This implies, in particular, a conjecture of Galkin and Shinder on categorical and motivic zeta-functions of varieties. We provide examples showing that, in general, these two measures are not equal. We also give an example related to a conjecture of Polishchuk and Van den Bergh, showing that a certain condition in this conjecture is indeed necessary.

General reduction of the elliptic hypergeometric equation to the level of complex hypergeometric functions is described. Derived equation is generalized to the Hamiltonian eigenvalue problem for new rational integrable N-body systems emerging from particular degenerations of the elliptic Ruijsenaars and van Diejen models.

This paper is a survey about cylinders in Fano varieties and related problems

We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups.

We introduce a notion of quasi-antisymmetric Higgs G-bundles over curves with marked points. They are endowed with additional structures that replace the parabolic structures at marked points in parabolic Higgs bundles. This means that the coadjoint orbits are attached to the marked points of the curves. The moduli spaces of parabolic Higgs bundles are the phase spaces of complex completely integrable systems. In our case, the coadjoint orbits are replaced by bundles cotangent to some special symmetric spaces in such a way that the moduli space of the modified Higgs bundles are still phase spaces of complex completely integrable systems. We show that the moduli space of parabolic Higgs bundles is the symplectic quotient of the moduli space of the quasi-antisymmetric Higgs bundle with respect to the action of the product of Cartan subgroups. In addition, by changing the symmetric spaces, we introduce quasi-compact and quasi-normal Higgs bundles. The fixed point sets of real involutions acting on their moduli spaces are the phase spaces of real completely integrable systems. Several examples are given including integrable extensions of the SL(2) Euler-Arnold top, two-body elliptic Calogero-Moser system, and the rational SL(2) Gaudin system together with its real reductions.& nbsp;Published under an exclusive license by AIP Publishing

We introduce a notion of a Hodge-proper stack and apply the strategy of Deligne and Illusie to prove the Hodge-to-de Rham degeneration in this setting. In order to reduce the statement in characteristic 0 to characteristic p⁠, we need to find a good integral model of a stack (namely, a Hodge-proper spreading), which, unlike in the case of proper schemes, need not to exist in general. To address this problem, we investigate the property of spreadability in more detail by generalizing standard spreading out results for schemes to higher Artin stacks and showing that all proper and some global quotient stacks are Hodge-properly spreadable. As a corollary, we deduce a (noncanonical) Hodge decomposition of the equivariant cohomology for certain classes of varieties with an algebraic group action.

We study Schlichting's K-theory groups of the Buchweitz-Orlov singularity category D-sg. X/ of a quasiprojective algebraic scheme X= k with applications to algebraic K-theory.

We prove for isolated quotient singularities over an algebraically closed field of characteristic zero that K-0((DX)-X-sg)) is finite torsion, and that K-1(D-sg (X) D-0. One of the main applications is that algebraic varieties with isolated quotient singularities satisfy rational Poincare duality on the level of the Grothendieck group; this allows computing the Grothendieck group of such varieties in terms of their resolution of singularities. Other applications concern the Grothendieck group of perfect complexes supported at a singular point and topological filtration on the Grothendieck groups.

Let Qn denote the space of all n × n skew-symmetric matrices over the complex field ℂ. It is proved that for n = 4, there are no linear maps T : Q4 → Q4 satisfying the condition dχ' (T (A)) = dχ(A) for all matrices A ∈ Q4, where χ, χ' ∈ {1, ∈, [2, 2]} are two distinct irreducible characters of S4. In the case χ = χ' = 1, a complete characterization of the linear maps T : Q4 → Q4 preserving the permanent is obtained. This case is the only one corresponding to equal characters and remaining uninvestigated so far. © 2021, Springer Science+Business Media, LLC, part of Springer Nature.

We investigate necessary conditions for Gorenstein projective varieties to admit semiorthogonal decompositions introduced by Kawamata, with main emphasis on threefolds with isolated compound A(n), singularities. We introduce obstructions coming from Algebraic K-theory and translate them into the concept of maximal nonfactoriality.

Using these obstructions we show that many classes of nodal three-folds do not admit Kawamata type semiorthogonal decompositions. These include nodal hypersurfaces and double solids, with the exception of a nodal quadric, and del Pezzo threefolds of degrees 1 <= d <= 4 with maximal class group rank.

We also investigate when does a blow up of a smooth threefold in a singular curve admit a Kawamata type semiorthogonal decomposition and we give a complete answer to this question when the curve is nodal and has only rational components.

Motivated by results of Thurston, we prove that any autoequivalence of a triangulated category induces a filtration by triangulated subcategories, provided the existence of Bridgeland stability conditions. The filtration is given by the exponential growth rate of masses under iterates of the autoequivalence, and only depends on the choice of a connected component of the stability manifold. We then propose a new definition of pseudo-Anosov autoequivalences, and prove that our definition is more general than the one previously proposed by Dimitrov, Haiden, Katzarkov, and Kontsevich. We construct new examples of pseudo-Anosov autoequivalences on the derived categories of quintic Calabi–Yau threefolds and quiver Calabi–Yau categories. Finally, we prove that certain pseudo-Anosov autoequivalences on quiver 3-Calabi–Yau categories act hyperbolically on the space of Bridgeland stability conditions.

n this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we show that their category is equivalent to a certain category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties. Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure

We show that a Calabi–Yau structure of dimension d on a smooth dg category C induces a symplectic form of degree 2 - d on ‘the moduli space of objects’ MC. We show moreover that a relative Calabi–Yau structure on a dg functor C→ D compatible with the absolute Calabi–Yau structure on C induces a Lagrangian structure on the corresponding map of moduli MD→ MC

A brief description of the relations between the factorization method in quantum mechanics, self-similar potentials, integrable systems and the theory of special functions is given. New coherent states of the harmonic oscillator related to the Fourier transformation are constructed.

We prove that a smooth well-formed Picard rank-one Fano complete intersection of dimension at least 2 in a toric variety is a weighted complete intersection.

We classify triangulated categories that are equivalent to finitely generated thick subcategories $T\subset D^b(cohC)$ for smooth projective curves C over an algebraically closed field.