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Working paper

The equivariantly closed matrix integrals introduced in [B06], see also [B10], are studied in the case of the graded associative algebras with odd or even scalar product.

Added: Oct 25, 2018

Working paper

Anikin A., Dvurechensky P.,

et al. math. arxive. Cornell University, 2015
Added: Oct 30, 2015

Working paper

Anikin A., Dvurechensky P.,

et al. math. arxive. Cornell University, 2015
Added: Oct 30, 2015

Working paper

We explicitly construct pairs of elliptic curves defined over the algebraic numbers with large intersection of projective torsion points.

Added: Jun 20, 2017

Working paper

We show that the derived category of a general Enriques surface can be realized as a semiorthogonal component in the derived category of a smooth Fano variety with a diagonal Hodge diamond.

Added: Dec 3, 2018

Working paper

Ornea L.,

math. arxive. Cornell University, 2017
An locally conformally Kahler (LCK) manifold with potential is a complex manifold with a cover which admits an automorphic Kahler potential. An LCK manifold with potential can be embedded to a Hopf manifold, if its dimension is at least 3. We give a functional-analytic proof of this result based on Riesz-Schauder theorem and Montel theorem. We give an alternative argument for complex surfaces, deducing embedding theorem from the Spherical Shell Conjecture.

Added: Feb 6, 2017

Working paper

Let B be a simply-connected projective variety such that the first cohomology groups of all line bundles on B are zero. Let E be a vector bundle over B and X=ℙ(E). It is easily seen that a power of any endomorphism of X takes fibers to fibers. We prove that if X admits an endomorphism which is of degree greater than one on the fibers then E splits into a direct sum of line bundles.

Added: Sep 7, 2016

Working paper

Added: Jul 2, 2019

Working paper

Let M be a hyperkahler manifold, Γ its mapping class group, and Teich the Teichmuller space of complex structures of hyperkahler type. After we glue together birationally equivalent points, we obtain the so-called birational Teichmuller space Teichb. Every connected component of Teichb is identified with its period space P by global Torelli theorem. The mapping class group of M acts on P as a finite index subgroup of the group of isometries of the integer cohomology lattice, that is, satisfies assumptions of Ratner theorem. We prove that there are three classes of orbits, closed, dense and the intermediate class which corresponds to varieties with Re(H2,0(M)) containing a given rational vector. The closure of the later orbits is a fixed point set of an anticomplex involution of P. This fixes an error in the paper 1306.1498, where this third class of orbits was overlooked. We explain how this affects the works based on 1306.1498.

Added: Aug 28, 2017

Working paper

We describe the universal target of annular Khovanov-Rozansky link homology functors as the homotopy category of a free symmetric monoidal category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use to characterize the annular invariants of Coxeter braids. Further, we prove the existence of symmetric group actions on the Khovanov-Rozansky invariants of cabled tangles and we introduce spectral sequences that aid in computing the homologies of generalized Hopf links. Finally, we conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter braids in other types.

Added: Sep 3, 2019

Working paper

Let T(n,m) be the set of plane labelled bipartite trees with n white vertices and m -- black. If the number n+m of vertices is even, then the set T(n,m) is a union of two disjoined subsets -- subset of "even" trees and subset of "odd" trees. This partition has a clear geometric meaning.

Added: Nov 7, 2016

Working paper

In this paper we at first consider plane trees with the root vertex and a marked directed edge, outgoing from the root vertex. For such trees we introduce a new characteristic-- the parity, using the bracket code. It turns out that the parity depends only on the root vertex (not on the marked edge). And in the case of an even number of vertices the parity does not depend on the root vertex also. Then we consider rotation groups of bipartite trees, study their properties and prove that in the case of even number of vertices rotation groups of even and odd trees are different.

Added: Nov 27, 2018

Working paper

We study the Monge and Kantorovich transportation problems on R∞ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal transportation problem on the Hilbert space. We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution. The result holds, in particular, under certain analytical assumptions involving log-concavity of the target measure. As a by-product we obtain the following result: any uniformly log-concave exchangeable sequence of random variables is i.i.d.

Added: Feb 23, 2016

Working paper

In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's $r$-spin classes. They are parameterized by a phase space which has one extra dimension and in genus $0$ they correspond to the extended $r$-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while when restricted to the usual smaller phase space, they give in all genera the product of the top Hodge class by the $r$-spin class. They do not form a cohomological field theory, but a more general object which we call F-CohFT, since in genus $0$ it corresponds to a flat F-manifold. For $r=2$ we prove that the partition function of such F-CohFT gives a solution of the discrete KdV hierarchy. Moreover the same integrable system also appears as its double ramification hierarchy.

Added: Oct 5, 2020

Working paper

Added: Mar 13, 2020

Working paper

Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding
K{\"a}hler-Einstein equation $e^{-\Phi} = \det D^2 \Phi$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric
$D^2 \Phi$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2 \Phi$
for arbitrary $K$ is uniformly bounded by $\frac{n-1}{4(n+1)}$ and verify this conjecture in the two-dimensional case.
The general case remains open.

Added: Dec 30, 2017

Working paper

In this paper, we consider the distribution of the supremum of non-stationary Gaussian processes, and present a new theoretical result on the asymptotic behaviour of this distribution. Unlike previously known facts in this field, our main theorem yields the asymptotic representation of the cor- responding distribution function with exponentially decaying remainder term. This result can be efficiently used for studying the projection density estimates, based, for instance, on Legendre polynomials. More precisely, we construct the sequence of accompanying laws, which approximates the distribution of maxi- mal deviation of the considered estimates with polynomial rate. Moreover, we construct the confidence bands for densities, which are honest at polynomial rate to a broad class of densities.

Added: May 25, 2020

Working paper

This paper deals with the extreme value analysis for the triangular arrays, which appear when some parameters of the mixture model vary as the number of observations grow. When the mixing parameter is small, it is natural to associate one of the components with ”an impurity” (in case of regularly varying distribution, ”heavy-tailed impurity”), which ”pollutes” another component. We show that the set of possible limit distributions is much more diverse than in the classical Fisher-Tippett-Gnedenko theorem, and provide the numerical examples showing the efficiency of the proposed model for studying the maximal values of the stock returns.

Added: Apr 11, 2021

Working paper

We introduce a new construction of towers of algebraic curves over finite fields and provide a simple example of an optimal tower.

Added: Oct 23, 2017

Working paper

It is known that the moduli space of smooth Fano-Mukai fourfolds V 18 of genus 10 has dimension one. We show that any such fourfold is a completion of C 4 in two different ways. Up to isomorphism, there is a unique fourfold V s 18 acted upon by SL 2 ( C ). The group Aut( V s 18 ) is a semidirect product GL 2 ( C ) o ( Z / 2 Z ). Furthermore, V s 18 is a GL 2 ( C )-equivariant completion of C 4 , and as well of GL 2 ( C ). The restriction of the GL 2 ( C )-action on V s 18 to C 4 ↪ → V s 18 yields a faithful representation with an open orbit. There is also a unique, up to isomorphism, fourfold V a 18 such that the group Aut( V a 18 ) is a semidirect product ( G a × G m ) o ( Z / 2 Z ). For a Fano-Mukai fourfold V 18 neither isomorphic to V s 18 , nor to V a 18 , one has Aut 0 ( V 18 ) ∼ = ( G m ) 2 , and Aut( V 18 ) is a semidirect product of Aut 0 ( V 18 ) and a finite cyclic group whose order is a factor of 6.

Added: Jun 20, 2017

Working paper

We classify smooth Fano threefolds with infinite automorphism groups.

Added: Nov 19, 2019