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Working paper

Given two fields k and F, a k-vector space V, an integer r≥1, we study the structure of the F-representation of the projective group PGL(V) in the F-vector space of formal finite linear combinations with coefficients in F of r-dimensional vector subspaces of V.
This gives a series of natural examples of irreducible infinite-dimensional representations of projective groups. These representations are non-smooth when k is a local field.

Added: Dec 5, 2018

Working paper

We develop the formal analogue of the Morse theory for a pair of commuting gradient-like vector fields. The resulting algebraic formalism turns out to be very similar to the algebra of the infrared of Gaiotto, Moore and Witten (see [GMW], [KKS]): from a manifold M with the pair of gradient-like commuting vector fields, subject to some general position conditions we construct an L_∞-algebra and Maurer-Cartan element in it.
We also provide Morse-theoretic examples for the algebra of the infrared data.

Added: Dec 1, 2018

Working paper

We study Calabi-Yau threefolds fibered by abelian surfaces, in particular, their arithmetic properties, e.g., Neron models and Zariski density.

Added: Oct 11, 2016

Working paper

We consider a problem of multiclass classification, where the training sample Sn={(Xi,Yi)}ni=1 is generated from the model ℙ(Y=m|X=x)=ηm(x), 1≤m≤M, and η1(x),…,ηM(x) are unknown α-Holder continuous functions.Given a test point X, our goal is to predict its label. A widely used 𝗄-nearest-neighbors classifier constructs estimates of η1(X),…,ηM(X) and uses a plug-in rule for the prediction. However, it requires a proper choice of the smoothing parameter 𝗄, which may become tricky in some situations. In our solution, we fix several integers n1,…,nK, compute corresponding nk-nearest-neighbor estimates for each m and each nk and apply an aggregation procedure. We study an algorithm, which constructs a convex combination of these estimates such that the aggregated estimate behaves approximately as well as an oracle choice. We also provide a non-asymptotic analysis of the procedure, prove its adaptation to the unknown smoothness parameter α and to the margin and establish rates of convergence under mild assumptions.

Added: May 23, 2018

Working paper

In this work we demonstrate, how the use of polar decomposition allows one to understand metric proporties of non-degenerate linear operators on the plane. We study how the non-isometric operatpr changes the length of vectors and the angles between the vectors.

Added: Mar 10, 2016

Working paper

A projective manifold M is algebraically hyperbolic if there exists a positive constant A such that the degree of any curve of genus g on M is bounded from above by A(g-1). A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.

Added: Nov 17, 2017

Working paper

Fino A., Grantcharov G.,

math. arxive. Cornell University, 2016
Let M be a complex nilmanifold, that is, a compact quotient of a nilpotent Lie group endowed with an invariant complex structure by a discrete lattice. A holomorphic differential on M is a closed, holomorphic 1-form. We show that a(M) ⩽ k, where a(M) is the algebraic dimension a(M) (i.e. the transcendence degree of the field of meromorphic functions) and k is the dimension of the space of holomorphic differentials. We prove a similar result about meromorphic maps to Kahler manifolds.

Added: Mar 16, 2016

Working paper

Kamenova L.,

math. arxive. Cornell University, 2016
A projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. This is a property which follows from Kobayashi hyperbolicity. We prove that hyperk¨ahler manifolds are non algebraically hyperbolic when the Picard rank is at least 3, or if the Picard rank is 2 and the SYZ conjecture on existence of Lagrangian fibrations is true. We also prove that if the automorphism group of a hyperk¨ahler manifold is infinite then it is algebraically non-hyperbolic.

Added: Apr 21, 2016

Working paper

The Robbins-Monro algorithm is a recursive, simulation-based stochastic procedure to approximate
the zeros of a function that can be written as an expectation. It is known that under some technical
assumptions, a Gaussian convergence can be established for the procedure. Here, we are interested in the
local limit theorem, that is, quantifying this convergence on the density of the involved objects. The analysis
relies on a parametrix technique for Markov chains converging to diffusions, where the drift is unbounded.

Added: Oct 25, 2018

Working paper

We present an easy proof of Polya's theorem on random walks: with the probability one a random walk on the two-dimensional lattice returns to the starting point.

Added: Mar 6, 2018

Working paper

We construct an explicit solution for the multimarginal transportation problem on the unit cube [0,1]3 with the cost function xyz and one-dimensional uniform projections. We show that the primal problem is concentrated on a set with non-constant local dimension and admits many solutions, whereas the solution to the corresponding dual problem is unique (up to addition of constants).

Added: Oct 10, 2018

Working paper

In the present article we discuss an approach to cohomological invariants of algebraic groups over fields of characteristic zero based on the Morava K-theories, which are generalized oriented cohomology theories in the sense of Levine--Morel.
We show that the second Morava K-theory detects the triviality of the Rost invariant and, more generally, relate the triviality of cohomological invariants and the splitting of Morava motives.
We describe the Morava K-theory of generalized Rost motives, compute the Morava K-theory of some affine varieties, and characterize the powers of the fundamental ideal of the Witt ring with the help of the Morava K-theory. Besides, we obtain new estimates on torsion in Chow groups of codimensions up to 2^n of quadrics from the (n+2)-nd power of the fundamental ideal of the Witt ring. We compute torsion in Chow groups of K(n)-split varieties with respect to a prime p in all codimensions up to p^{n−1}/(p−1) and provide a combinatorial tool to estimate torsion up to codimension p^n. An important role in the proof is played by the gamma filtration on Morava K-theories, which gives a conceptual explanation of the nature of the torsion.
Furthermore, we show that under some conditions the K(n)-motive of a smooth projective variety splits if and only if its K(m)-motive splits for all m≤n.

Added: Dec 6, 2018

Working paper

In arXiv:1807.09038 we formulated a conjecture describing the derived category D-mod(Gr_GL(n)) of (all) D-modules on the affine Grassmannian of the group GL(n) as the category of ind-coherent sheaves on a certain stack (it is explained in loc. cit. that this conjecture "follows" naturally from some heuristic arguments involving 3-dimensional quantum field theory). In this paper we prove a weaker version of this conjecture for the case n=2.

Added: Dec 3, 2018

Working paper

Brown F.,

math. arxive. Cornell University, 2013. No. 1110.6917 [.
Abstract. We study the de Rham fundamental group of the configuration sp ace of several marked points on a complex elliptic curve, and define multiple elliptic polylogarithms. These are multivalued functions with unipotent monodromy, and are constructed by a general averaging proce dure. We show that all iterated integrals on this configuration space can be expressed in terms of these functions.

Added: Oct 4, 2013

Working paper

The aim of this short note is to give a simple proof of the non-rationality of the double cover of the three-dimensional projective space branched over a sufficiently general quartic.

Added: May 16, 2016

Working paper

We study automorphism groups and birational automorphism groups of compact complex surfaces. We show that the automorphism group of such surface X is always Jordan, and the birational automorphism group is Jordan unless X is birational to a product of an elliptic and a rational curve.

Added: Aug 22, 2017

Working paper

We prove that automorphism groups of Inoue and primary Kodaira surfaces are Jordan.

Added: Jun 8, 2019

Working paper

We show that automorphism groups of Moishezon threefolds are always Jordan.

Added: Jun 8, 2019

Working paper

In 2007, Dubouloz introduced Danielewski varieties. Such varieties general- ize Danielewski surfaces and provide counterexamples to generalized Zariski cancellation problem in arbitrary dimension. In the present work we describe the automorphism group of a Danielewski variety. This result is a generalization of a description of automorphisms of Danielewski surfaces due to Makar-Limanov.

Added: Sep 1, 2018

Working paper

We show that smooth well formed Fano weighted complete intersections of dimension at least 3 have finite automorphism groups, with two obvious exceptions.

Added: May 30, 2019

Working paper

We study groups of bimeromorphic and biholomorphic automorphisms of projective hyperkähler manifolds. Using an action of these groups on some non-positively curved space, we deduce many of their properties, including finite presentation, strong form of Tits' alternative and some structural results about groups consisting of transformations with infinite order.

Added: Dec 2, 2018