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Working paper

We give an example of two n-by-n chess positions, A and B, such that (1) there is a sequence of legal chess moves leading from A to B; (2) the length of this sequence cannot be less than exp(cn).

Added: Oct 30, 2014

Working paper

We study tunneling of charge carriers in single- and bilayer graphene. We propose an explanation for non-zero "magic angles" with 100% transmission for the case of symmetric potential barrier, as well as for their almost-survival for slightly asymmetric barrier in the bilayer graphene known previously from numerical simulations. Most importantly, we demonstrate that these magic angles are not protected in the case of bilayer and give an explicit example of a barrier with very small electron transmission probability for any angles. This means that one can lock charge carriers by a p-n-p (or n-p-n) junction without opening energy gap. This creates new opportunities for the construction of graphene transistors.

Added: Aug 22, 2015

Working paper

In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum R-matrices. Here we study the simplest case -- the 11-vertex R-matrix and related gl_2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars-Schneider (RS) or the 2-body Calogero-Moser (CM) model depending on its description. We give different descriptions of the integrable tops and use them as building blocks for construction of more complicated integrable systems such as Gaudin models and classical spin chains (periodic and with boundaries). The known relation between the top and CM (or RS) models allows to re-write the Gaudin models (or the spin chains) in the canonical variables. Then they assume the form of n-particle integrable systems with 2n constants. We also describe the generalization of the top to 1+1 field theories. It allows us to get the Landau-Lifshitz type equation. The latter can be treated as non-trivial deformation of the classical continuous Heisenberg model. In a similar way the deformation of the principal chiral model is also described.

Added: Jan 23, 2015

Working paper

We consider the isomonodromy problems for flat $G$-bundles over punctured
elliptic curves $\Sigma_\tau$ with regular singularities of connections at
marked points. The bundles are classified by their characteristic classes.
These classes are elements of the second cohomology group
$H^2(\Sigma_\tau,{\mathcal Z}(G))$, where ${\mathcal Z}(G)$ is the center of
$G$. For any complex simple Lie group $G$ and arbitrary class we define the
moduli space of flat bundles, and in this way construct the monodromy
preserving equations in the Hamiltonian form and their Lax representations. In
particular, they include the Painlev\'e VI equation, its multicomponent
generalizations and elliptic Schlesinger equations. The general construction is
described for punctured curves of arbitrary genus. We extend the
Beilinson-Drinfeld description of the moduli space of Higgs bundles to the case
of flat connections. This local description allows us to establish the
Symplectic Hecke Correspondence for a wide class of the monodromy preserving
equations classified by characteristic classes of underlying bundles. In
particular, the Painlev\'e VI equation can be described in terms of ${\rm
SL}(2, {\mathbb C})$-bundles. Since ${\mathcal Z}({\rm SL}(2, {\mathbb C}))=
{\mathbb Z}_2$, the Painlev\'e VI has two representations related by the Hecke
transformation: 1) as the well-known elliptic form of the Painlev\'e VI (for
the trivial bundles); 2) as the non-autonomous Zhukovsky-Volterra gyrostat (for
non-trivial bundles).

Added: Dec 27, 2013

Working paper

In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine commutative algebraic monoids. In this work, we extend this classification to noncommutative monoid structures on normal affine surfaces. We prove that two-dimensional algebraic monoids are toric. We also show how to find all monoid structures on a normal toric surface. Every such structure is induced by a comultiplication formula involving Demazure roots. We also give descriptions of opposite monoids, quotient monoids, and boundary divisors.

Added: Jun 13, 2021

Working paper

We give a mostly self-contained proof of the classification of non-Kahler surfaces based on Buchdahl-Lamari theorem. We also prove that all non-Kahler surfaces which are not of class VII are locally conformally Kahler.

Added: Oct 18, 2018

Working paper

A closed symmetric differential of the 1st kind is a differential that locally is the product of closed holomorphic 1-forms. We show that closed symmetric 2-differentials of the 1st kind on a projective manifold $X$ come from maps of $X$ to cyclic or dihedral quotients of Abelian varieties and that their presence implies that the fundamental group of $X$ (case of rank 2) or of the complement $X\setminus E$ of a divisor $E$ with negative properties (case of rank 1) contains subgroup of finite index with infinite abelianization. Other results include: i) the identification of the differential operator characterizing closed symmetric 2-differentials on surfaces (which provides in this case a connection to flat Riemannian metrics) and ii) projective manifolds $X$ having symmetric 2-differentials $w$ that are the product of two closed meromorphic 1-forms are irregular, in fact if $w$ is not of the 1st kind (which can happen), then $X$ has a fibration $f:X \to C$ over a curve of genus $\ge 1$.

Added: Oct 9, 2013

Working paper

In this paper the relation between the cluster integrable systems and q-difference equations is extended beyond the Painlevé case. We consider the class of hyperelliptic curves when the Newton polygons contain only four boundary points. The corresponding cluster integrable Toda systems are presented, and their discrete automorphisms are identified with certain reductions of the Hirota difference equation. We also construct non-autonomous versions of these equations and find, that their solutions are expressed in terms of 5d Nekrasov functions with the Chern-Simons contributions, while in the autonomous case these equations are solved in terms of the Riemann theta-functions.

Added: Nov 22, 2018

Working paper

We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all S_6-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman-Edge pencil. As an application, we check that S_6-invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class groups as S_6-representations.

Added: Oct 21, 2018

Working paper

We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations is then identified with the triangulated category of relative singularities, while the similar exotic derived category of locally free matrix factorizations is its full subcategory. The latter category is identified with the kernel of the direct image functor corresponding to the closed embedding of the zero locus and acting between the conventional (absolute) triangulated categories of singularities. Similar results are obtained for matrix factorizations of infinite rank; and two different "large" versions of the triangulated category of relative singularities, corresponding to the approaches of Orlov and Krause, are identified in the case of a Cartier divisor. Contravariant (coherent) and covariant (quasi-coherent) versions of the Serre-Grothendieck duality theorems for matrix factorizations are established, and pull-backs and push-forwards of matrix factorizations are discussed at length. A number of general results about derived categories of the second kind for CDG-modules over quasi-coherent CDG-algebras are proven on the way. Hochschild (co)homology of matrix factorization categories are discussed in an appendix written in collaboration with A.I. Efimov.

Added: Dec 22, 2013

Working paper

A Hermitian symplectic manifold is a complex manifold endowed with a symplectic form ω, for which the bilinear form ω(I·, ·) is positive definite. In this work we prove ddc -lemma for 1- and (1,1)-forms for compact Hermitian symplectic manifolds of dimension 3. This shows that Albanese map for such manifolds is well-defined and allows one to prove Kahlerness if the dimension of the Albanese image of a manifold is maximal.

Added: Nov 19, 2015

Working paper

Let $\Psi$ be the projectivization (i.e., the set of one-dimensional vector
subspaces) of a vector space of dimension $\ge 3$ over a field. Let $H$ be a
closed (in the pointwise convergence topology) subgroup of the permutation
group $\mathfrak{S}_{\Psi}$ of the set $\Psi$. Suppose that $H$ contains the
projective group and an arbitrary self-bijection of $\Psi$ transforming a
triple of collinear points to a non-collinear triple. It is well-known from
\cite{KantorMcDonough} that if $\Psi$ is finite then $H$ contains the
alternating subgroup $\mathfrak{A}_{\Psi}$ of $\mathfrak{S}_{\Psi}$.
We show in Theorem \ref{density} below that $H=\mathfrak{S}_{\Psi}$, if
$\Psi$ is infinite.

Added: Nov 21, 2014

Working paper

Added: Nov 2, 2012

Working paper

Added: Sep 2, 2018

Working paper

Blokh A., Oversteegen L., Ptacek R. et al. arxiv.org. math. Cornell University, 2014

To construct a model for a connectedness locus of polynomials of degree $d\ge
3$ (cf with Thurston's model of the Mandelbrot set), we define \emph{linked}
geolaminations $\mathcal{L}_1$ and $\mathcal{L}_2$. An \emph{accordion} is
defined as the union of a leaf $\ell$ of $\mathcal{L}_1$ and leaves of
$\mathcal{L}_2$ crossing $\ell$. We show that any accordion behaves like a gap
of one lamination and prove that the maximal \emph{perfect} (without isolated
leaves) sublaminations of $\mathcal{L}_1$ and $\mathcal{L}_2$ coincide.
In the cubic case let $\mathcal{D}_3\subset \mathcal{M}_3$ be the set of all
\emph{dendritic} (with only repelling cycles) polynomials. Let $\mathcal{MD}_3$
be the space of all \emph{marked} polynomials $(P, c, w)$, where $P\in
\mathcal{D}_3$ and $c$, $w$ are critical points of $P$ (perhaps, $c=w$). Let
$c^*$ be the \emph{co-critical point} of $c$ (i.e., $P(c^*)=P(c)$ and, if
possible, $c^*\ne c$). By Kiwi, to $P\in \mathcal{D}_3$ one associates its
lamination $\sim_P$ so that each $x\in J(P)$ corresponds to a convex polygon
$G_x$ with vertices in $\mathbb{S}$. We relate to $(P, c, w)\in \mathcal{MD}_3$
its \emph{mixed tag} $\mathrm{Tag}(P, c, w)=G_{c^*}\times G_{P(w)}$ and show
that mixed tags of distinct marked polynomials from $\mathcal{MD}_3$ are
disjoint or coincide. Let $\mathrm{Tag}(\mathcal{MD}_3)^+ =
\bigcup_{\mathcal{D}_3}\mathrm{Tag}(P,c,w)$. The sets $\mathrm{Tag}(P, c, w)$
partition $\mathrm{Tag}(\mathcal{MD}_3)^+$ and generate the corresponding
quotient space $\mathrm{MT}_3$ of $\mathrm{Tag}(\mathcal{MD}_3)^+$. We prove
that $\mathrm{Tag}:\mathcal{MD}_3\to \mathrm{MT}_3$ is continuous so that
$\mathrm{MT}_3$ serves as a model space for $\mathcal{MD}_3$.

Added: Feb 11, 2015

Working paper

As noticed by R. Kulkarni, the conjugacy classes of subgroups of the modular group correspond bijectively to bipartite cuboid graphs. We'll explain how to recover the graph corresponding to a subgroup $G$ of $PSL_2(\mathbb{Z})$ from the combinatorics of the right action of $PSL_2(\mathbb{Z})$ on the right cosets $G\setminus PSL_2(\mathbb{Z})$ This gives a method of constructing nice fundamental domains (which Kulkarni calls "special polygons") for the action of $G$ on the upper half plane. For the classical congruence subgroups $\Gamma(N),\Gamma_0(N),\Gamma_1(N)$ etc. the number of operations the method requires is the index times something that grows not faster than a polynomial in $log(N)$. We also give algorithms to locate a given element of the upper half-plane on the fundamental domain and to write a given element of $G$ as a product of independent generators.

Added: Apr 4, 2014

Working paper

Campana F., Demailly J.,

. arxiv.org. math. Cornell University, 2013
We prove that any compact K\"ahler 3-dimensional manifold which has no non-trivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of 'simple manifolds', central in the bimeromorphic classification of compact K\"ahler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, combined with fundamental results of Siu and of the second author on the Lelong numbers of closed positive (1,1)-currents, and with a version of the hard Lefschetz theorem for pseudo-effective line bundles, due to Takegoshi and Demailly-Peternell-Schneider. In a similar vein, we show that a normal compact and K\"ahler 3-dimensional analytic space with terminal singularities and nef canonical bundle is a cyclic quotient of a simple non-projective torus if it carries no effective divisor. This is a crucial step to complete the bimeromorphic classification of compact K\"ahler 3-folds.

Added: May 13, 2013

Working paper

Ornea L.,

arxiv.org. math. Cornell University, 2015
A locally conformally Kahler manifold is a Hermitian manifold (M,I,ω) satisfying dω=θ∧ω, where θ is a closed 1-form, called the Lee form of M. It is called pluricanonical if ∇θ is of Hodge type (2,0)+(0,2), where ∇ is the Levi-Civita connection, and Vaisman if ∇θ=0. We show that a compact LCK manifold is pluricanonical if and only if the Lee form has constant length and the Kahler form of its covering admits an automorphic potential. Using a degenerate Monge-Ampere equation and the classification of surfaces of Kahler rank one, due to Brunella, Chiose and Toma, we show that any pluricanonical metric on a compact manifold is Vaisman. Several errata to our previous work are given in the last Section.

Added: Dec 5, 2015

Working paper

Blokh A., Oversteegen L., Ptacek R. et al. arxiv.org. math. Cornell University, 2014

We study the closure of the cubic Principal Hyperbolic Domain and its
intersection $\mathcal{P}_\lambda$ with the slice $\mathcal{F}_\lambda$ of the
space of all cubic polynomials with fixed point $0$ defined by the multiplier
$\lambda$ at $0$. We show that any bounded domain $\mathcal{W}$ of
$\mathcal{F}_\lambda\setminus\mathcal{P}_\lambda$ consists of $J$-stable
polynomials $f$ with connected Julia sets $J(f)$ and is either of \emph{Siegel
capture} type (then $f\in \mathcal{W}$ has an invariant Siegel domain $U$
around $0$ and another Fatou domain $V$ such that $f|_V$ is two-to-one and
$f^k(V)=U$ for some $k>0$) or of \emph{queer} type (then at least one critical
point of $f\in \mathcal{W}$ belongs to $J(f)$, the set $J(f)$ has positive
Lebesgue measure, and carries an invariant line field).

Added: Feb 11, 2015

Working paper

Panov T., Ustinovsky Y.,

arxiv.org. math. Cornell University, 2013
Moment-angle manifolds provide a wide class of examples of non-Kaehler compact complex manifolds. A complex moment-angle manifold Z is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In general, a complex moment-angle manifold Z is equipped with a canonical holomorphic foliation F and a C*-torus action transitive in the transverse direction. Examples of moment-angle manifolds include Hopf manifolds of Vaisman type, Calabi-Eckmann manifolds, and their deformations. We construct transversely Kaehler metrics on moment-angle manifolds, under some restriction on the combinatorial data. We prove that all Kaehler submanifolds (or, more generally, Fujiki class C subvarieties) in such a moment-angle manifold lie in a compact complex torus contained in a fibre of the foliation F. For a generic moment-angle manifold Z in its combinatorial class, we prove that all subvarieties are moment-angle manifolds of smaller dimension. This implies, in particular, that the algebraic dimension of Z is zero.

Added: Aug 23, 2013

Working paper

Buff X.,

arxiv.org. math. Cornell University, 2013. No. 1308.3510.
We investigate the notion of complex rotation number which was introduced by V.I.Arnold in 1978. Let f: R/Z \to R/Z be an orientation preserving circle diffeomorphism and let {\omega} \in C/Z be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary components of the annulus {z \in C/Z | 0< Im(z)< Im({\omega})} via the map f+{\omega}. This complex torus is isomorphic to C/(Z+{\tau} Z) for some appropriate {\tau} \in C/Z. According to Moldavskis (2001), if the ordinary rotation number rot (f+\omega_0) is Diophantine and if {\omega} tends to \omega_0 non tangentially to the real axis, then {\tau} tends to rot (f+\omega_0). We show that the Diophantine and non tangential assumptions are unnecessary: if rot (f+\omega_0) is irrational then {\tau} tends to rot (f+\omega_0) as {\omega} tends to \omega_0. This, together with results of N.Goncharuk (2012), motivates us to introduce a new fractal set, given by the limit values of {\tau} as {\omega} tends to the real axis. For the rational values of rot (f+\omega_0), these limits do not necessarily coincide with rot (f+\omega_0)$ and form a countable number of analytic loops in the upper half-plane.

Added: Dec 12, 2013