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Working paper

Kamenova L.,

arxiv.org. math. Cornell University, 2012
Added: Feb 6, 2013

Working paper

Val’kov V. V.,

, Aksenov S. arxiv.org. math. Cornell University, 2018
Taking into account an inner structure of the arms of the Aharonov-Bohm ring (AB ring) we have analyzed the transport features related to the Majorana bound states (MBSs) which are induced in a superconducting wire (a SC wire) with strong spin-orbit interaction (SOI). The SC wire acts as a bridge connecting the arms. The in-plane magnetic-field dependence of linear-response conductance obtained using the nonequilibrium Green's functions in the tight-binding approximation revealed the Fano resonances (FRs) if the wire is in the topologically nontrivial phase. The effect is attributed to the presence of bound states in continuum in the AB ring which lifetime is determined by both hopping parameters between subsystems and the SC wire properties. The FRs minima appear at the magnetic fields corresponding to the zero energies of the leads constituting the AB arms. It is established that the FRs shift if the SC pairing reduces or, equivalently, the MBS transforms from the edge to bulk state. In addition, the FR width essentially depends on the MBS spacial distribution. Finally, we demonstrated that the MBS spacial distribution can be also probed in the T-shaped transport scheme where two out of four leads perform as side-coupled ones. The observed possibility to distinguish between the edge- and bulk low-energy states is additionally valuable considering current discussion about the manifestation of Andreev and Majorana excitations in transport properties of the SC wire. The FR properties are analytically studied for effective double-quantum-dot system in one-contact transport regime. The influence of Coulomb interactions in the SC wire on the FR is investigated in the mean-field regime.

Added: Oct 16, 2018

Working paper

We show that G-Fano threefolds are mirror-modular.
1. Mirror maps are inversed reversed Hauptmoduln for moonshine subgroups of SL2(ℝ).
2. Quantum periods, shifted by an integer constant (eigenvalue of quantum operator on primitive cohomology) are expansions of weight 2 modular forms (theta-functions) in terms of inversed Hauptmoduln.
3. Products of inversed Hauptmoduln with some fractional powers of shifted quantum periods are very nice cuspforms (eta-quotients).
The latter cuspforms also appear in work of Mason and others: they are eta-products, related to conjugacy classes of sporadic simple groups, such as Mathieu group M24 and Conway's group of isometries of Leech lattice.
This gives a strange correspondence between deformation classes of G-Fano threefolds and conjugacy classes of Mathieu group M24.

Added: Sep 25, 2018

Working paper

We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis. In the favourable case a basis of the module is parameterized by the lattice points of a normal polytope. The filtrations induce flat degenerations of the corresponding flag variety to its abelianized version and to a toric variety, the special fibres of the degenerations being projectively normal and arithmetically Cohen-Macaulay. The polytope itself can be recovered as a Newton-Okounkov body. We conclude the paper by giving classes of examples for favourable modules.

Added: Jun 24, 2013

Working paper

In this paper we study the geometry of the total space Y of a cotangent bundle to a Kähler manifold N where N is obtained as a Kähler reduction from Cn. Using the hyperkähler reduction we construct a hyperkähler metric on Y and prove that it coincides with the canonical Feix-Kaledin metric. This metric is in general non-complete. We show that the metric completion Y~ of the space Y is equipped with a structure of a stratified hyperkähler space. We give a necessary condition for the Feix-Kaledin metric to be complete using an observation of R.Bielawski. Pick a complex structure J on Y~ induced from quaternions. Suppose that J≠±I where I is the complex structure whose restriction to Y=T∗N is induced by the complex structure on N. We prove that the space Y~J admits an algebraic structure and is an affine variety.

Added: Jul 21, 2020

Working paper

We classify finite groups G in PGL_4(ℂ) such that ℙ^3 is G-birationally rigid.

Added: Oct 21, 2018

Working paper

Let $A$ be an abelian variety over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by the Weil polynomial $f_A$. We assume that $f_A$ is separable. For a given prime number $\ell\neq\ch k$ we give a classification of group schemes $B[\ell]$, where $B$ runs through the isogeny class, in terms of certain Newton polygons associated to $f_A$. As an application we classify zeta functions of Kummer surfaces over $k$.

Added: Dec 17, 2013

Working paper

We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: 𝔄5, PSL2(F7), 𝔄6, SL2(F8), 𝔄7, PSp4(F3), SL2(F7), 2.𝔄5, 2.𝔄6, 3.𝔄6 or 6.𝔄6. All of these groups with a possible exception of 2.𝔄6 and 6.𝔄6 indeed act on some rationally connected threefolds.

Added: Oct 15, 2018

Working paper

We prove (1) the existence of a noncompact smooth four-dimensional manifold M whose diffeomorphism group Diff(M) contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite simple subgroups of Diff(N) for compact manifolds N.

Added: Oct 25, 2013

Working paper

We study the derived categories of varieties X and X^+ connected by a flop. We assume that flopping contractions f: X\to Y, f^+: X^+ \to Y have fibers of relative dimension bounded by one and Y has canonical hypersurface singularities of multiplicity 2. We consider flop functors F: D^b(X) \to \dD^b(X^+), F^+: D^b(X^+) \to D^b(X), which we prove to be equivalences.
The composite F^+F: D(X) \to D(X) is a non-trivial auto-equivalence. When variety Y is affine, we present F^+F (up to shift) as a spherical cotwist associated to two spherical functors. Consider the composite g=if, where i:Y \to {\cal Y} is a divisorial embedding into a smooth variety. The functor Rg_*: D(X) \to D({\cal Y})$ is the first spherical functor. We consider the null category A_f of sheaves on X with derived direct image vanishing. The inclusion of A_f into Coh(X) gives the second spherical functor \Psi: D(A_f) \to D(X).
We construct a spherical pair in a suitable quotient D(W)/K of the fiber product W= X\times_Y X^+.
We show that the subcategory of A_f consisting of sheaves supported on the fiber C= \bigcup C_i over a closed point of Y ind-represents the functor of non-commutative deformations of \oO_{C_i}(-1)'s. For this purpose, we develop a categorical approach to the non-commutative deformation theory of n objects.

Added: Nov 20, 2015

Working paper

Added: Aug 14, 2019

Working paper

We reformulate several basic notions of notions in finite group theory in terms of iterations of the lifting property (orthogonality) with respect to particular morphisms. Our examples include the notions being nilpotent, solvable, perfect, torsion-free; p-groups and prime-to-p-groups; Fitting subgroup, perfect core, p-core, and prime-to-p core. We also reformulate as in similar terms the conjecture that a localisation of a (transfinitely) nilpotent group is (transfinitely) nilpotent.

Added: Oct 29, 2020

Working paper

Entov M.,

arxiv.org. math. Cornell University, 2014
Let M be a closed symplectic manifold of volume V. We say that M admits a full symplectic packing by balls if any collection of symplectic balls of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in terms of Kahler cones of their blow-ups. When M is a Kahler manifold which is not a union of its proper subvarieties (such a manifold is called Campana simple) these Kahler cones can be described explicitly using the Demailly and Paun structure theorem. We prove that any Campana simple Kahler manifold, as well as any manifold which is a limit of Campana simple manifolds, admits a full symplectic packing by balls. This is used to show that all even-dimensional tori equipped with Kahler symplectic forms and all hyperkahler manifolds of maximal holonomy admit full symplectic packings by balls. This generalizes a previous result by Latschev-McDuff-Schlenk. We also consider symplectic packings by other shapes and show using Ratner's orbit closure theorem that any even-dimensional torus equipped with a Kahler form whose cohomology class is not proportional to a rational one admits a full symplectic packing by any number of equal polydisks (and, in particular, by any number of equal cubes).

Added: Feb 5, 2015

Working paper

Braverman A., Dobrovolska G.,

. arxiv.org. math. Cornell University, 2014
Let G be an almost simple simply connected group over complex numbers. For a positive element α of the coroot lattice of G let Z^α denote the space of based maps from the projective line to the flag variety of G of degree α. This space is known to be isomorphic to the space of framed euclidean G-monopoles with maximal symmetry breaking at infinity of charge α. In [Finkelberg-Kuznetsov-Markarian-Mirkovi\'c] a system of (\'etale, rational) coordinates on Z^α is introduced. In this note we compute various known structures on Z^α in terms of the above coordinates. As a byproduct we give a natural interpretation of the Gaiotto-Witten superpotential and relate it to the theory of Whittaker D-modules introduced by D.Gaitsgory.

Added: Feb 3, 2015

Working paper

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the Gamma class Γ_F. When the quantum cohomology of F is semisimple, we say that F satisfies Gamma Conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by Γ_F Ch(E_i) for an exceptional collection {E_i} in the derived category of coherent sheaves D^b_{coh}(F). Gamma Conjecture II refines part (3) of Dubrovin's conjecture. We prove Gamma Conjectures for projective spaces, toric manifolds, certain toric complete intersections and Grassmannians.

Added: May 4, 2014

Working paper

The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class. Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class G_F associated to Euler's Γ-function. We illustrate in the case of toric varieties, toric complete intersections and Grassmannians how this conjecture follows from mirror symmetry. We also prove that Gamma conjecture is compatible with taking hyperplane sections, and give a heuristic argument how the mirror oscillatory integral and the Gamma class for the projective space arise from the polynomial loop space.

Added: Aug 5, 2015

Working paper

The classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the quantum Bruhat graph. We make use of the Orr-Shimozono formula for the nonsymmetric Macdonald polynomials in order to prove that the t=∞ specialization of the nonsymmetric Macdonald polynomials are equal to the characters of the generalized Weyl modules corresponding to the antidominant weights. We also prove a generalization of the t=0 specialization theorem by Ion to the case of the non simply-laced algebras.

Added: Dec 15, 2015

Working paper

We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric q-Whittaker functions. In particular, we show that the series part of the nonsymmetric q-Whittaker function is a generating function for the graded characters of generalized global Weyl modules.

Added: May 6, 2016

Working paper

We introduce the notion of generalized Weyl modules for twisted current algebras. We study their representation-theoretic and combinatorial properties and connection to the theory of nonsymmetric Macdonald polynomials. As an application we compute the dimension of the classical Weyl modules in the remaining unknown case.

Added: Jun 17, 2016

Working paper

In this article, we prove two results. First, we construct a dense subset in the space of polynomial foliations of degree n such that each foliation from this subset has a leaf with at least (n+1)(n+2)/2-4 handles. Next, we prove that for a generic foliation invariant under the map (x, y) ↦ (x, -y) all leaves have infinitely many handles.

Added: Mar 7, 2015

Working paper

For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Generic finite dimensional (not necessarily associative) $\boldsymbol k$-algebras of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent over $\boldsymbol k$ rational functions in the structure constants. (2) There exists an "algebraic normal form", to which the set of structure constants of every such algebra can be uniquely transformed by means of passing to its new basis, namely: there are two finite systems of nonconstant polynomials on the space of structure constants, $\{f_i\}_{i\in I}$ and $\{b_j\}_{j\in J}$, such that the ideal generated by the set $\{f_i\}_{i\in I}$ is prime and, for every tuple $c$ of structure constants satisfying the property $b_j(c)\neq 0$ for all $j\in J$, there exists a unique new basis of this algebra in which the tuple $c'$ of its structure constants satisfies the property $f_i(c')=0$ for all $i\in I$.

Added: Nov 25, 2014