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Working paper

We classify smooth Fano weighted complete intersections of large codimension.

Added: Nov 19, 2019

Working paper

Mazzucchi S., Moretti V.,

et al. math. arxive. Cornell University, 2020
Feynman formulas are representations of solutions to initial value problems, for some parabolic and Schrödinger equations, by the limits of integrals over finite Cartesian powers of some spaces. Two versions of these formulas which were suggested by Feynman himself are associated with names of Trotter and Chernoff respectively. These formulas can be interpreted as approximations for path integrals over spaces of functions of a real variable; the corresponding representations of the solutions to the said equations are usually known as Feynman-Kac formulas. This work presents some new Feynman type formulas, related to the Chernoff theorem, on Riemannian manifolds. The used manifolds are of boundend geometry which include all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a generally non-compact manifold of bounded geometry. A construction of Chernoff approximations is presented for those Feller semigroups in terms of shift operators. This provides approximations for solutions to initial-value problems for parabolic equations with variable coefficients on the manifold. It also yields the weak convergence of a sequence of random walks on the manifold to the diffusion process associated with the elliptic generator. For parallelizable manifolds this result is applied to the Brownian motion

Added: Aug 10, 2020

Working paper

We conjecture that the number of components of the fiber over infinity of Landau--Ginzburg model for a smooth Fano variety X equals the dimension of the anticanonical system of X. We verify this conjecture for log Calabi--Yau compactifications of toric Landau--Ginzburg models for smooth Fano threefolds, complete intersections, and some toric varieties.

Added: Aug 19, 2020

Working paper

Given a holomorphic conic bundle without sections, we show that finite groups acting by its fiberwise bimeromorphic transformations are bounded. This provides an analog of a similar result obtained by T.Bandman and Yu.Zarhin for quasi-projective conic bundles.

Added: Nov 19, 2019

Working paper

We consider the class of dissipative reaction-diffusion-convection systems on the circle and obtain conditions under which the final (at large times) phase dynamicsof a system can be described by an ODE with Lipschitz vector field in RN. Precisely in this class, the first example of a parabolic problem of mathematical physics without the indicated property was recently constructed.

Added: Nov 6, 2020

Working paper

We show that automorphism groups of Hopf and Kodaira surfaces have unbounded finite subgroups. For elliptic fibrations on Hopf, Kodaira, bielliptic, and K3 surfaces, we make some observations on finite groups acting along the fibers and on the base of such a fibration.

Added: Nov 19, 2019

Working paper

We classify finite groups that can act by automorphisms and birational automorphisms on non-trivial Severi-Brauer surfaces over fields of characteristic zero.

Added: Aug 19, 2020

Working paper

We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have Jordan property.

Added: Nov 19, 2019

Working paper

We construct a categorification of the maximal commutative subalgebra of the type A Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functor allows one to match the Hochschild homology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. The categorified Jones-Wenzl projectors studied by Abel, Elias and Hogancamp are idempotents in the category of Soergel bimodules, and they correspond to the renormalized Koszul complexes of the torus fixed points on the flag Hilbert scheme. As a consequence, we conjecture that the endomorphism algebras of the categorified projectors correspond to the dg algebras of functions on affine charts of the flag Hilbert schemes. We define a family of differentials d_N on these dg algebras and conjecture that their homology matches that of the gl_N projectors, generalizing earlier conjectures of the first and third authors with Oblomkov and Shende.

Added: Sep 19, 2016

Working paper

The Oeljeklaus-Toma (OT-) manifolds are compact, complex, non-Kahler manifolds constructed by Oeljeklaus and Toma, and generalizing the Inoue surfaces. Their construction uses the number-theoretic data: a number field K and a torsion-free subgroup U in the group of units of the ring of integers of K, with rank of U equal to the number of real embeddings of K. We prove that any complex subvariety of smallest possible positive dimension in an OT-manifold is also flat affine. This is used to show that if all non-trivial elements in U are primitive in K, then X contains no proper complex subvarieties.

Added: Dec 5, 2018

Working paper

Added: Sep 1, 2018

Working paper

We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only constant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety is flexible.

Added: Dec 6, 2018

Working paper

The paper defines a family of nested non-cooperative simultaneous finite games to study coalition structure formation with intra and inter-coalition externalities. Every game has two outcomes - an allocation of players over coalitions and a payoff profile for every player.
Every game in the family has an equilibrium in mixed strategies. The equilibrium can generate more than one coalition with a presence of intra and inter group externalities. These properties make it different from the Shapley value, strong Nash, coalition-proof equilibrium, core, kernel, nucleolus. The paper demonstrates some applications: non-cooperative cooperation, Bayesian game, stochastic games and construction of a non-cooperative criterion of coalition structure stability for studying focal points. An example demonstrates that a payoff profile in the Prisoners' Dilemma is non-informative to deduce a cooperation of players.

Added: Feb 26, 2017

Working paper

The paper defines a non-cooperative simultaneous finite game to study coalition structure formation with intra and inter-coalition externalities. The novelty of the game is that the game definition embeds a \textit{coalition structure formation mechanism}. This mechanism portions a set of strategies of the game into partition-specific strategy domains, what makes every partition to be a non-cooperative game with partition-specific payoffs for every player. The mechanism includes a maximum coalition size, a set of eligible partitions with coalitions sizes no greater than this number (which also serves as a restriction for a maximum number of deviators) and a coalition structure formation rule. The paper defines a family of nested non-cooperative games parametrized by a size of a maximum coalition size. Every game in the family has an equilibrium in mixed strategies. The equilibrium can generate more than one coalition and encompasses intra and inter group externalities, what makes it different from the Shapley value. Presence of individual payoff allocation makes it different from a strong Nash, coalition-proof equilibrium, and some other equilibrium concepts. The accompanying papers demonstrate applications of the proposed toolkit.

Added: Dec 8, 2016

Working paper

The paper uses a non-cooperative simultaneous game for coalition structure formation (Levando, 2016) to demonstrate some applications of the introduced game: a cooperation, a Bayesian game within a coalition with intra-coalition externalities, a stochastic game, where states are coalition structures; self-enforcement properties of a non-cooperative equilibrium and a construction of a non-cooperative stability criterion.

Added: Jan 31, 2017

Working paper

We consider the category of perverse sheaves on a complex vector space smooth with respect to a stratification given by an arrangement of hyperplanes with real equations. As shown in an earlier wotk of two of the authors, this category can be described in terms of certain diagrams of vector spaces labelled by all the faces of the real arrangement (we call such diagrams hyperbolic sheaves). In this paper we calculate, in these terms, several fundamental operations of sheaf theory such as forming the space of vanishing cycles, specialization and the Fourier-Sato transform.

Added: Dec 3, 2018

Working paper

Kroshnin A.,

math. arxive. Cornell University, 2015. No. 1512.08421.
Endow the space P(R) of probability measures on R with a transportation cost J(mu, nu) generated by a translation-invariant convex cost function. For a probability distribution on P(R) we formulate a notion of average with respect to this transportation cost, called here the Fréchet barycenter, prove a version of the law of large numbers for Fréchet barycenters, and discuss the structure of P(R) related to the transportation cost J.

Added: Dec 31, 2015

Working paper

Let (X,C) be a germ of a threefold X with terminal singularities along a connected reduced complete curve C with a contraction f:(X,C)→(Z,o) such that C=f^{−1}(o)_{red} and −K_X is f-ample. Assume that each irreducible component of C contains at most one point of index >2. We prove that a general member D∈|−K_X| is a normal surface with Du Val singularities.

Added: Aug 19, 2020

Working paper

Recent work of the first author, Negut and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov-Rozansky knot homology produces a family of polynomials in q and t labeled by integer sequences. These polynomials can be expressed as equivariant Euler characteristics of certain line bundles on flag Hilbert schemes. The q,t-Catalan numbers and their rational analogues are special cases of this construction. In this paper, we give a purely combinatorial treatment of these polynomials and show that in many cases they have nonnegative integer coefficients.
For sequences of length at most 4, we prove that these coefficients enumerate subdiagrams in a certain fixed Young diagram and give an explicit symmetric chain decomposition of the set of such diagrams. This strengthens results of Lee, Li and Loehr for (4,n) rational q,t-Catalan numbers.

Added: Sep 3, 2019

Working paper

Let G be a connected reductive complex algebraic group with a maximal torus T. We denote by Λ the cocharacter lattice of (T,G). Let Λ^+⊂Λ be the submonoid of dominant coweights. For λ∈Λ+,μ∈Λ,μ⩽λ, in arXiv:1604.03625, authors defined a generalized transversal slice W^λ_μ. This is an algebraic variety of the dimension ⟨2ρ^∨,λ−μ⟩, where 2ρ^∨ is the sum of positive roots of G. We prove that for a minuscule λ and μ appearing as a weight of V^λ (irreducible representation of the Langlands dual group G^∨ with the highest weight λ) the variety W^λ_μ is isomorphic to the affine space 𝔸⟨2ρ^∨,λ−μ⟩ and that in certain coordinates the Poisson structure on it is standard.

Added: May 30, 2019

Working paper

We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works of D. Guan and the first author. Any such manifold Q of dimension 2n−2 is obtained as a finite degree n2 cover of some non-Kähler manifold WF which we call the base of Q. We show that the algebraic reduction of Q and its base is the projective space of dimension n−1. Besides, we give a partial classification of submanifolds in Q, describe the degeneracy locus of its algebraic reduction, and prove that the automorphism group of Q satisfies the Jordan property.

Added: Aug 19, 2020