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Working paper

Generalizing the definition of Cartier, we introduce pn-typical formal group laws over ℤ(p)-algebras. An oriented cohomology theory in the sense of Levin-Morel is called pn-typical if its corresponding formal group law is p^n-typical. The main result of the paper is the construction of 'Chern classes' from the algebraic n-th Morava K-theory to every p^n-typical oriented cohomology theory.
If the coefficient ring of a p^n-typical theory is a free ℤ(p)-module we also prove that these Chern classes freely generate all operations to it. Examples of such theories are algebraic mn-th Morava K-theories K(nm)∗ for all m∈ℕ and CH∗⊗ℤ(p) (operations to Chow groups were studied in a previous paper). The universal pn-typical oriented theory is BP{n}∗=BP∗/(v_j,j∤n) which coefficient ring is also a free ℤ_(p)-module.
Chern classes from the n-th algebraic Morava K-theory K(n)∗ to itself allow us to introduce the gamma filtration on K(n)∗. This is the best approximation to the topological filtration obtained by values of operations and it satisfies properties similar to that of the classical gamma filtration on K0. The major difference from the classical case is that Chern classes from the graded factors gr^i_γ K(n)^∗ to CH^i⊗ℤ_(p) are surjective for i≤p^n. For some projective homogeneous varieties this allows to estimate p-torsion in Chow groups of codimension up to pn.

Added: Dec 6, 2018

Working paper

Motivated by the Chern-Weil theory, we prove that for a given vector bundle E on a smooth scheme X over a field k of any characteristic, the Chern classes of E in the Hodge cohomology can be recovered from the Atiyah class. Although this problem was solved by Illusie in \cite{i}, we present another proof by means of derived algebraic geometry.
Also, for a scheme X over a field k of characteristic p with a vector bundle E we construct elements c^cris_n(E,α(E))∈H^2n_dR(X) using an obstruction α(E) to a lifting of F∗E to a crystal modulo p2 and prove that c^cris_n(E,α(E))=n!⋅c^dR_n(E), where cdRn(E)are the Chern classes of E in the de Rham cohomology and F is the Frobenius map.

Added: Nov 20, 2019

Working paper

We give a new proof of a theorem of Bogomolov, that the only VII_0 surfaces with b_2=0 are those constructed by Hopf and Inoue. The proof follows the strategy of the original one, but it is of purely group-theoretic nature.

Added: Dec 2, 2018

Working paper

The convolution ring K^{GL_n(\OOO)⋊ℂ×}(Gr_{GL_n}) was identified with a quantum unipotent cell of the loop group LSL_2 in [Cautis-Williams, arXiv:1801.08111]. We identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves with the dual canonical basis of the quantum unipotent cell.

Added: Jun 9, 2019

Working paper

Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperkahler geometry

Let M be a hyperk\"ahler manifold with b2(M)≥5. We improve our earlier results on the Morrison-Kawamata cone conjecture by showing that the Beauville-Bogomolov square of the primitive MBM classes (i.e. the classes whose orthogonal hyperplanes bound the K\"ahler cone in the positive cone, or, in other words, the classes of negative extremal rational curves on deformations of M) is bounded in absolute value by a number depending only on the deformation class of M. The proof uses ergodic theory on homogeneous spaces.

Added: Sep 7, 2016

Working paper

In this paper we prove, in a purely combinatorial way, a structural quasi-polynomiality property for the Bousquet-M\'elou--Schaeffer numbers. Conjecturally, this property should follow from the Chekhov-Eynard-Orantin topological recursion for these numbers (or, to be more precise, the Bouchard-Eynard version of the topological recursion for higher order critical points), which we derive in this paper from the recent result of Alexandrov-Chapuy-Eynard-Harnad. To this end, the missing ingredient is a generalization to the case of higher order critical points on the underlying spectral curve of the existing correspondence between the topological recursion and Givental's theory for cohomological field theories.

Added: Oct 8, 2019

Working paper

Added: Dec 15, 2017

Working paper

Compact difference schemes for weakly-nonlinear parabolic and Schrodinger-type equations and systems

The implicit compact finite-difference scheme was developed for evolutionary partial differential parabolic and Schrodinger-type equations and systems with a weak nonlinearity. To make a temporal step of the compact implicit scheme we need to solve a non-linear system. We use for this step a simple explicit difference scheme and then Newton - Raphson iterations, which are implemented by the double-sweep method. Numerical experiments confirm the 4-th order of an algorithm. The Richardson extrapolation improves it up to the 6-th order.

Added: Dec 15, 2017

Working paper

Essential dimension of a family of complex manifolds is the dimension of the image of its base in the Kuranishi space of the fiber. We prove that any family of hyperk\"ahler manifolds over a compact simply connected base has essential dimension not greater than 1. A similar result about families of complex tori is also obtained.

Added: Sep 23, 2016

Working paper

Let M be an irreducible holomorphic symplectic (hyperk¨ahler) manifold. If b2(M) >= 5, we construct a deformation M′ of M which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real (1, 1)-classes is hyperbolic. If b2(M) >= 14, similarly, we construct a deformation which admits a parabolic automorphism.

Added: Apr 13, 2016

Working paper

We prove a partial converse to the main theorem of the author's previous paper "Proper affine actions: a sufficient criterion" (submitted; available at arXiv:1612.08942). More precisely, let G be a semisimple real Lie group with a representation rho on a finite-dimensional real vector space V, that does not satisfy the criterion from the previous paper. Assuming that rho is irreducible and under some additional assumptions on G and rho, we then prove that there does not exist a group of affine transformations acting properly discontinuously on V whose linear part is Zariski-dense in rho(G).

Added: Sep 26, 2018

Working paper

Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the con- tinuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with exist- ing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.

Added: Oct 30, 2020

Working paper

We study the exceptional loci of birational (bimeromorphic) contractions of a hyperkähler manifold M. Such a contraction locus is the union of all minimal rational curves in a collection of cohomology classes which are orthogonal to a wall of the Kähler cone. Homology classes which can possibly be orthogonal to a wall of the Kähler cone of some deformation of M are called MBM classes. We prove that all MBM classes of type (1,1) can be represented by rational curves, called MBM curves. All MBM curves can be contracted on an appropriate birational model of M, unless b_2(M)≤5. When b_2(M)>5, this property can be used as an alternative definition of an MBM class and an MBM curve. Using the results of Bakker and Lehn, we prove that the diffeomorphism type of a contraction locus remains stable under all deformations for which these classes remains of type (1,1), unless the contracted variety has b2≤4. Moreover, these diffeomorphisms preserve the MBM curves, and induce biholomorphic maps on the contraction fibers, if they are normal.

Added: Jun 9, 2019

Working paper

Falaleev A.,

math. arxive. Cornell University, 2019. No. 1910.03862.
In this paper we consider a random walk of a particle in Rd. Convergence of different transformations of trajectories of random flights with Poisson switching moments has been obtained by Davydov and Konakov, as well as diffusion approximation of the process has been built. The goal of this paper is to prove stronger convergence in terms of the Wasserstein distance. Three types of transformations are considered: cases of exponential and super-exponential growth of a switching moment transformation function are quite simple, and the result follows from the fact that the limit processes belong to the unit ball. In the case of the power growth the estimation is more complicated and follows from combinatorial reasoning and properties of the Wasserstein metric.

Added: Oct 10, 2019

Working paper

Braverman A.,

, Nakajima H. math. arxive. Cornell University, 2016
We study Coulomb branches of unframed and framed quiver gauge theories of type ADE. In the unframed case they are isomorphic to the moduli space of based rational maps from ℂP1 to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera, Ben Webster, and Alex Weekes, we identify the quantized Coulomb branch with the truncated shifted Yangian.

Added: Apr 21, 2016

Working paper

These are (somewhat informal) lectures notes for the CIME summer school "Geometric Representation Theory and Gauge Theory" in June 2018. In these notes we review the results and constructions of a series of our joint papers with H.Nakajima where a mathematical definition of Coulomb branches of 3d N=4 quantum gauge theories (of cotangent type) is given. We also discuss some further constructions (such as categories of line operators in the corresponding topologically twisted theories) and briefly discuss a connection to local geometric Langlands correspondence.

Added: Dec 3, 2018

Working paper

We compute the Coulomb branch of a multiloop quiver gauge theory for the quiver with a single vertex, r loops, one-dimensional framing, and dim V = 2. We identify it with a Slodowy slice in the nilpotent cone of the symplectic Lie algebra of rank r. Hence it possesses a symplectic resolution with 2r fixed points with respect to a Hamiltonian torus action. We also idenfity its flavor deformation with a base change of the full Slodowy slice.

Added: Jun 9, 2019

Working paper

We show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A given by generators; (2) as an algebra given by generators and relations; (3) as an algebra of differential-reflection operators preserving some spaces of functions; (4) as equivariant Borel-Moore homology of a certain variety. Also, we define a new q-deformation of this algebra, which we call cyclotomic DAHA. Namely, we give a q-deformation of each of the above four descriptions of the partially spherical rational Cherednik algebra, replacing differential operators with difference operators, degenerate DAHA with DAHA, and homology with K-theory, and show that they give the same algebra. In addition, we show that spherical cyclotomic DAHA are quantizations of certain multiplicative quiver and bow varieties, which may be interpreted as K-theoretic Coulomb branches of a framed quiver gauge theory. Finally, we apply cyclotomic DAHA to prove new flatness results for various kinds of spaces of q-deformed quasiinvariants. In the appendix by H. Nakajima and D. Yamakawa (added in version 2), the authors explain the relations between multiplicative bow varieties and (various versions of) multiplicative quiver varieties for a cyclic quiver.

Added: Apr 10, 2017

Working paper

This paper is a survey about cylinders in Fano varieties and related problems.

Added: Aug 19, 2020

Working paper

We answer a question of Ciro Ciliberto about cylinders in rational surfaces which are obtained by blowing up the plane at points in general position.

Added: Dec 3, 2018

Working paper

Added: Oct 16, 2020