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Working paper

For every smooth del Pezzo surface $S$, smooth curve $C\in|-K_{S}|$ and $\beta\in(0,1]$, we compute the $\alpha$-invariant of Tian $\alpha(S,(1-\beta)C)$ and prove the existence of K\"ahler--Einstein metrics on $S$ with edge singularities along $C$ of angle $2\pi\beta$ for $\beta$ in certain interval. In particular we give lower bounds for the invariant $R(S,C)$, introduced by Donaldson as the supremum of all $\beta\in(0,1]$ for which such a metric exists.

Added: Feb 5, 2015

Working paper

We investigate the Brenier map \nabla \Phi between the uniform measures on two convex domains in \mathbb{R}^n or more generally, between two log-concave probability measures on \mathbb{R}^n. We show that the eigenvalues of the Hessian matrix D^2 \Phi exhibit remarkable concentration properties on a multiplicative scale, regardless of the choice of the two measures or the dimension n.

Added: Mar 12, 2014

Working paper

For each integer n>0, we construct a series of irreducible algebraic varieties X, for which the automorphism group Aut(X) contains as a subgroup the automorphism group Aut(F_n) of a free group F_n of rank n. For n > 1, such groups Aut(X) are nonamenable, and for n > 2, they are nonlinear and contain the braid group B_n. Some of these varieties X are affine, and among affine, some are rational and some are not, some are smooth and some are singular. The byproduct is that for n > 2, each Cremona group of rank > 3n-1 contains Aut(F_n) and the braid group B_n.

Added: Jun 7, 2021

Working paper

Buchstaber V.,

arxiv.org. math. Cornell University, 2018. No. 1808.08851.
We introduce the notions of algebraic and geometric direct families of polytopes and develop a theory of such families. The theory is then applied to the problem of existence of nontrivial higher Massey products in cohomology of moment-angle-complexes.

Added: Sep 29, 2019

Working paper

Added: Feb 21, 2013

Working paper

Fang and Fourier defined the symplectic Dellac configurations in order to parametrize the torus fixed points of the symplectic degenerated flag varieties, and conjectured that their numbers are the elements of a sequence of integers (1, 2, 10, 98, 1594, ...) which appears in the study by Randrianarivony and Zeng of the median Euler numbers. In this paper, we prove the conjecture by considering a combinatorial interpretation of the latter integers in terms of the surjective pistols (which form a well-known combinatorial model of the Genocchi numbers), and constructing an appropriate surjection from the symplectic Dellac configurations to the surjective pistols.

Added: May 11, 2017

Working paper

Let M be a compact complex manifold. The corresponding Teichmuller space $\Teich$ is a space of all complex structures on M up to the action of the group of isotopies. The group Γ of connected components of the diffeomorphism group (known as the mapping class group) acts on $\Teich$ in a natural way. An ergodic complex structure is the one with a Γ-orbit dense in $\Teich$. Let M be a complex torus of complex dimension ≥2 or a hyperkahler manifold with b_2>3. We prove that M is ergodic, unless M has maximal Picard rank (there is a countable number of such M). This is used to show that all hyperkahler manifolds are Kobayashi non-hyperbolic.

Added: Dec 27, 2013

Working paper

For a continuous semicascade on a metrizable compact set Ω, we consider the weak* convergence of generalized operator ergodic means in End C*(Ω). We discuss conditions on the dynamical system under which: (a) every ergodic net contains a convergent subsequence; (b) all ergodic nets converge; (c) all ergodic sequences converge. We study the relationships between the convergence of ergodic means and the properties of transitivity of the proximality relation on Ω, minimality of supports of ergodic measures, and uniqueness of minimal sets in the closure of trajectories of a semicascade. These problems are solved in terms of three algebraic-topological objects associated with the dynamical system: the Ellis enveloping semigroup, the Kohler operator semigroup Г, and the semigroup G that is the weak* closure of the convex hull of Г in End C*(Ω). The main results are stated for ordinary semicascades (whose Ellis semigroup is metrizable) and tame semicascades. For a dynamics, being ordinary is equivalent to being “nonchaotic” in an appropriate sense. We present a classification of compact dynamical systems in terms of topological properties of the above-mentioned semigroups.

Added: Nov 19, 2013

Working paper

We study the problem on the weak-star decomposability of a topological N0-dynamical system (Ω, '), where ' is an endomorphism of a metric compact set Ω, into ergodic components in terms of the associated enveloping semigroups. In the tame case (where the Ellis semigroup E(Ω,') consists of B1-transformations Ω → Ω), we show that (i) the desired decomposition exists for an appropriate choice of the generalized sequential averaging method; (ii) every sequence of weighted ergodic means for the shift operator x → x ◦ ', x ∈ C(Ω), contains a pointwise convergent subsequence. We also discuss the relationship between the statistical properties of (Ω, ') and the mutual structure of minimal sets and ergodic measures.

Added: Jul 3, 2018

Working paper

We compare the notions of essential dimension and stable cohomological dimension of a finite group G, prove that the latter is bounded by the length of any normal series with cyclic quotients for G, and show that, however, this bound is not sharp by showing that the stable cohomological dimension of the finite Heisenberg groups H_p, p any prime, is equal to two.

Added: Nov 21, 2014

Working paper

Let f:X->X be a morphism of a variety over a number field K. We consider local conditions and a "Bruaer-Manin" condition, defined by Hsia and Silverman, for the orbit of a point P in X(K) to be disjoint from a subvariety V of X, i.e., the intersection of the orbit of P with V is empty. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection of the orbit of P with V; this evidence stems from a probabilistic argument as well as unconditional results in the case of etale maps.

Added: Jun 25, 2013

Working paper

We look at how evolution method deforms, when one considers Khovanov polynomials instead of Jones polynomials. We do this for the figure-eight-like knots (also known as 'double braid' knots, see arXiv:1306.3197) -- a two-parametric family of knots which "grows" from the figure-eight knot and contains both two-strand torus knots and twist knots. We prove that parameter space splits into four chambers, each with its own evolution, and two isolated points. Remarkably, the evolution in the Khovanov case features an extra eigenvalue, which drops out in the Jones (t -> -1) limit.

Added: Feb 20, 2019

Working paper

The problem of the exact bounded control of oscillations of the two-dimensional membrane is considered. Control force is applied to the boundary of the membrane, which is located in a domain on a plane. The goal of the control is to drive the system to rest in a finite time.

Added: Oct 19, 2016

Working paper

We will consider the exact controllability of the distributed system governed by wave equation with memory. It will be proved that this mechanical system can be driven to an equilibrium point in a finite time, the absolute value of the distributed control function being bounded. In this case the memory kernel is a linear combination of exponentials.

Added: Mar 21, 2015

Working paper

We construct 4 different families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form Z x A1, where Z is a quasiprojective variety. The affi ne cones over such a fourfold admit eff ective Ga-actions. Similar constructions of cylindrical Fano threefolds were done previously in our papers jointly with Takashi Kishimoto.

Added: Aug 18, 2014

Working paper

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface S. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on S. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections are spires of two helices.

Added: Sep 14, 2013

Working paper

We introduce a new construction of exceptional objects in the derived category of coherent sheaves on a compact homogeneous space of a semisimple algebraic group and show that it produces exceptional collections of the length equal to the rank of the Grothendieck group on homogeneous spaces of all classical groups.

Added: Oct 4, 2013

Working paper

Lee K.,

arxiv.org. math. Cornell University, 2014
We construct exceptional collections of maximal length on four families of
surfaces of general type with $p_g=q=0$ which are isogenous to a product of
curves. From these constructions we obtain new examples of quasiphantom
categories as their orthogonal complements.

Added: Oct 17, 2014

Working paper

A hypercomplex manifold M is a manifold equipped with three complex structures satisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called Obata connection. A quaternionic Hermitian metric is a Riemannian metric on which is invariant with respect to unitary quaternions. Such a metric is called HKT if it is locally obtained as a Hessian of a function averaged with quaternions. HKT metric is a natural analogue of a Kahler metric on a complex manifold. We push this analogy further, proving a quaternionic analogue of Buchdahl-Lamari's theorem for complex surfaces. Buchdahl and Lamari have shown that a complex surface M admits a Kahler structure iff b1(M) is even. We show that a hypercomplex manifold M with Obata holonomy SL(2,H) admits an HKT structure iff H0,1(M)=H1(OM) is even.

Added: Sep 19, 2014

Working paper

Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials, also called E-polynomails, in the limit t=infinity and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q-Whittaker function. These coefficients are pure q-powers and their degrees are expected to coincide in the untwisted setting with the extremal degrees of the so-called PBW-filtration in the corresponding finite-dimensional irreducible representations of the simple Lie algebra for any root systems. This is a particular case of a general conjecture in terms of the level-one Demazure modules. We prove this coincidence for all Lie algebras of classical type and for G_2, and also establish the relations of our extremal degrees to minimal q-degrees of the extremal terms of the Kostant q-partition function; they coincide with the latter only for some root systems.

Added: Jun 24, 2013

Working paper

Thesis of the author.

Added: Dec 16, 2019