Publications

We explicitly calculate an arithmetic adelic quotient group for a locally free sheaf on an arithmetic surface when the fiber over the infinite point of the base is taken into account. The claculations are presented via a shiort exact sequence. We relate the last term of this short exact sequence with the projective limit of free finitely generated modules over the one-dimensiona real torus such that these modules are connected with first cohomology groups of locally free sheaves on the arithmetic surface.  

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).

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 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.

 

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 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.

Let R be the algebra of functions on a smooth affine irreducible curve S over a field k and let be a smooth and proper DG algebra over R. The relative periodic cyclic homology of over R is equipped with the Hodge filtration and the Gauss-Manin connection (Getzler, in: Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992), Israel mathematics conference proceedings, vol 7, Bar-Ilan University, Ramat Gan, pp 65-78, 1993; Kaledin, in: Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, vol II, pp 23-47, Progress in mathematics, vol 270, Birkhauser Inc., Boston, 2009) satisfying the Griffiths transversality condition. When k is a perfect field of odd characteristic p, we prove that, if the relative Hochschild homology vanishes in degrees , then a lifting of R over and a lifting of over determine the structure of a relative Fontaine-Laffaille module (Faltings, in: Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins University Press, Baltimore, MD, pp 25-80, 1989, A 2 (c); Ogus and Vologodsky in Publ Math Inst Hautes Aetudes Sci No 106:1-138, 2007 A 4.6) on . That is, the inverse Cartier transform of the Higgs R-module is canonically isomorphic to . This is non-commutative counterpart of Faltings' result (1989, Th. 6.2) for the de Rham cohomology of a smooth proper scheme over R. Our result amplifies the non-commutative Deligne-Illusie decomposition proven by Kaledin (Algebra, geometry and physics in the 21st century (Kontsevich Festschrift), Progress in mathematics, vol 324. Birkhauser, pp 99-129, 2017, Th. 5.1). As a corollary, we show that the p-curvature of the Gauss-Manin connection on is nilpotent and, moreover, it can be expressed in terms of the Kodaira-Spencer class [a similar result for the p-curvature of the Gauss-Manin connection on the de Rham cohomology is proven by Katz (Invent Math 18:1-118, 1972)]. As an application of the nilpotency of the p-curvature we prove, using a result from Katz (Inst Hautes Aetudes Sci Publ Math No 39:175-232, 1970), a version of "the local monodromy theorem" of Griffiths-Landman-Grothendieck for the periodic cyclic homology: if , is a smooth compactification of S, then, for any smooth and proper DG algebra over R, the Gauss-Manin connection on the relative periodic cyclic homology has regular singularities, and its monodromy around every point at is quasi-unipotent.

We develop the topological polylogarithm which provides an integral ver- sion of Nori’s Eisenstein cohomology classes for GLn (Z) and yields classes with values in an Iwasawa algebra. This implies directly the integrality properties of spe- cial values of L-functions of totally real fields and a construction of the associated p-adic L-function. Using a result of Graf, we also apply this to prove some integrality and p-adic interpolation results for the Eisenstein cohomology of Hilbert modular varieties.  

We explicitly calculate an adelic quotient group for an excellent Noetherian normal integral two-dimensional separated scheme. An application to an irreducible normal projective algebraic surface over a field is given.

This volume is a tribute to Maxim Kontsevich, one of the most original and influential mathematicians of our time. Maxim’s vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics. Many of his papers have opened completely new directions of research and led to the solutions of many classical problems. This book collects papers by leading experts currently engaged in research on topics close to Maxim’s heart.

We construct a filtration on integrable highest weight module of an affine Lie algebra whose adjoint graded quotient is a direct sum of global Weyl modules. We show that the graded multiplicity of each Weyl module there is given by a corresponding level-restricted Kostka polynomial. This leads to an interpretation of level-restricted Kostka polynomials as the graded dimension of the space of conformal coinvariants. In addition, as an application of the level one case of the main result, we realize global Weyl modules of current algebras of type ADE in terms of Schubert manifolds of thick affine Grassmanian, as predicted by Boris Feigin.

In this paper we prove the smoothness of the moduli space of Landau–Ginzburg models. We formulate and prove a Bogomolov–Tian–Todorov theorem for the deformations of Landau–Ginzburg models, develop the necessary Hodge theory for varieties with potentials, and prove a double degeneration statement needed for the unobstructedness result. We discuss the various definitions of Hodge numbers for non-commutative Hodge structures of Landau–Ginzburg type and the role they play in mirror symmetry. We also interpret the resulting families of de Rham complexes attracted to a potential in terms of mirror symmetry for one parameter families of symplectic Fano manifolds and argue that modulo a natural triviality property the moduli spaces of Landau–Ginzburg models posses canonical special coordinates.

We study two rational Fano threefolds with an action of the icosahedral group 𝔄5. The first one is the famous Burkhardt quartic threefold, and the second one is the double cover of the projective space branched in the Barth sextic surface. We prove that both of them are 𝔄5-Fano varieties that are 𝔄5-birationally superrigid. This gives two new embeddings of the group 𝔄5 into the space Cremona group.

In this paper we prove the indicated conjecture in the last case of known infinite series of theta-blocks of weight 2.

In this paper we study the derived categories of coherent sheaves on Grassmannians Gr(k,n), defined over the ring of integers. We prove that the category Db(Gr(k,n)) has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of GLk. This in particular implies existence of a full exceptional collection, which is a refinement of Kapranov's collection [13], which was constructed over a field of characteristic zero. We also describe the right dual semi-orthogonal decomposition which has a similar form, and its components are full subcategories of the derived category of representations of GLn−k. The resulting equivalences between the components of the two decompositions are given by a version of Koszul duality for strict polynomial functors. We also construct a tilting vector bundle on Gr(k,n). We show that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over Z, and we identify the objects of Db(Gr(k,n)), which correspond to the standard and costandard modules in both structures. All the results automatically extend to the case of arbitrary commutative base ring and the category of perfect complexes on the Grassmannian, by extension of scalars (base change). Similar results over fields of arbitrary characteristic were obtained independently in [7], by different methods.