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Fedor Bogomolov

Publications and source records attributed to Fedor Bogomolov.

At least 37 records · Page 2Linked to original sources
arXiv2016

On the Kobayashi pseudometric, complex automorphisms and hyperkaehler manifolds

We define the Kobayashi quotient of a complex variety by identifying points with vanishing Kobayashi pseudodistance between them and show that if a compact complex manifold has an automorphism whose order is infinite, then the fibers of this quotient map are nontrivial. We prove that the Kobayashi quotients associated to ergodic complex structures on a compact manifold are isomorphic. We also give a proof of Kobayashi's conjecture on the vanishing of the pseudodistance for hyperkähler manifolds having Lagrangian fibrations without multiple fibers in codimension one. For a hyperbolic automorphism of a hyperkähler manifold, we prove that its cohomology eigenvalues are determined by its Hodge numbers, compute its dynamical degree and show that its cohomological trace grows exponentially, giving estimates on the number of its periodic points.

math.DG↗
arXiv2014

Local structure of closed symmetric 2-differentials

In the authors's previous work on symmetric differentials and their connection to the topological properties of the ambient manifold, a class of symmetric differentials was introduced: closed symmetric differentials ([BoDeO11] and [BoDeO13]). In this article we give a description of the local structure of closed symmetric 2-differentials on complex surfaces, with an emphasis towards the local decompositions as products of 1-differentials. We show that a closed symmetric 2-differential $w$ of rank 2 (i.e. defines two distinct foliations at the general point) has a subvariety $B_w\subset X$ outside of which $w$ is locally the product of closed holomorphic 1-differentials. The main result, theorem 2.6, gives a complete description of a (locally split) closed symmetric 2-differential in a neighborhood of a general point of $B_w$. A key feature of theorem 2.6 is that closed symmetric 2-differentials still have a decomposition as a product of 2 closed 1-differentials (in a generalized sense) even at the points of $B_w$. The (possibly multi-valued) closed 1-differentials can have essential singularities along $B_w$, but one still has a control on these essential singularities. The essential singularities come from exponentials of meromorphic functions acquiring poles along the irreducible components of $B_w$ of order bounded by the order of contact of the 2 foliations defined by the symmetric 2-differential along that irreducible component.

math.AG↗
arXiv2013

Closed symmetric 2-differentials of the 1st kind

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 $\ge 1$.

math.AG↗
arXiv2013

On uniformly rational varieties

We investigate basic properties of uniformly rational varieties, i.e. those smooth varieties for which every point has a Zariski open neighborhood isomorphic to an open subset of A^n. It is an open question of Gromov whether all smooth rational varieties are uniformly rational. We discuss some potential criteria that might allow one to show that they form a proper subclass in the class of all smooth rational varieties. Finally we prove that small algebraic resolutions and big resolutions of nodal cubic threefolds are uniformly rational.

math.AG↗
arXiv2012

Unirationality and existence of infinitely transitive models

We study unirational algebraic varieties and the fields of rational functions on them. We show that after adding a finite number of variables some of these fields admit an infinitely transitive model. The latter is an algebraic variety with the given field of rational functions and an infinitely transitive regular action of a group of algebraic automorphisms generated by unipotent algebraic subgroups. We expect that this property holds for all unirational varieties and in fact is a peculiar one for this class of algebraic varieties among those varieties which are rationally connected.

math.AG↗
arXiv2012

Collineation group as a subgroup of the symmetric group

Let $Ψ$ 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 $\mathfrak{S}_Ψ$ of the set $Ψ$. Suppose that $H$ contains the projective group and an arbitrary self-bijection of $Ψ$ transforming a triple of collinear points to a non-collinear triple. It is well-known from \cite{KantorMcDonough} that if $Ψ$ is finite then $H$ contains the alternating subgroup $\mathfrak{A}_Ψ$ of $\mathfrak{S}_Ψ$. We show in Theorem \ref{density} below that $H=\mathfrak{S}_Ψ$, if $Ψ$ is infinite.

math.GR↗
arXiv2012

Isoclinism and stable cohomology of wreath products

Using the notion of isoclinism introduced by P. Hall for finite p-groups, we show that many important classes of finite p-groups have stable cohomology detected by abelian subgroups, see Theorem 4.4. Moreover, we show that the stable cohomology of the n-fold wreath product of cyclic groups ZZ/p is detected by elementary abelian p-subgroups and we describe the resulting cohomology algebra explicitly. Some applications to the computation of unramified and stable cohomology of finite groups of Lie type are given.

math.AG↗
arXiv2011

Galois theory and projective geometry

We explore connections between birational anabelian geometry and abstract projective geometry. One of the applications is a proof of a version of the birational section conjecture.

math.AG↗
arXiv2011

Linear bounds for levels of stable rationality

Let G be one of the groups SL_n C, Sp_2n C, SO_m C, O_m C, or G_2. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G x P^N is rational. In this paper we improve known bounds for the levels of stable rationality for the quotients V/G. In particular, their growth as functions of the rank of the group is linear for G one of the classical groups.

math.AG↗
arXiv2010

Introduction to birational anabelian geometry

We survey recent developments in the Birational Anabelian Geometry program aimed at the reconstruction of function fields of algebraic varieties over algebraically closed fields from pieces of their absolute Galois groups.

math.AG↗
arXiv2010

Rationality of quotients by linear actions of affine groups

Let G be the (special) affine group, semidirect product of SL_n and C^n. In this paper we study the representation theory of G and in particular the question of rationality for V/G where V is a generically free G-representation. We show that the answer to this question is positive if the dimension of V is sufficiently large and V is indecomposable. We have a more precise theorem if V is a two-step extension 0 -> S -> V -> Q -> 0 with S, Q completely reducible.

math.AG↗
arXiv2010

Remarks on endomorphisms and rational points

Let X be a variety over a number field and let f: X --> X be an "interesting" rational self-map with a fixed point q. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold (originally obtained by Claire Voisin and the first author in 2007).

math.AG↗