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Fedor Bogomolov

Publications and source records attributed to Fedor Bogomolov.

At least 19 recordsLinked to original sources
arXiv2026

Sections of Lagrangian fibrations on holomorphic symplectic manifolds

Let $M$ be a holomorphically symplectic manifold, equipped with a Lagrangian fibration $π:\; M \to X$. A degenerate twistor deformation (sometimes also called ``a Tate-Shafarevich twist'') is a family of holomorphically symplectic structures on $M$ parametrized by $H^{1,1}(X)$. All members of this family are equipped with a holomorphic Lagrangian projection to $X$, and their fibers are isomorphic to the fibers of $π$. Assume that $M$ is a compact hyperkahler manifold of maximal holonomy, and the general fiber of the Lagrangian projection $π$ is primitive (that is, not divisible) in integer homology. We also assume that $π$ has reduced fibers in codimension 1. Then $M$ has a degenerate twistor deformation $M'$ such that the Lagrangian projection $π:\; M' \to X$ admits a meromorphic section.

math.AG↗
arXiv2025

Stabilization of direct images for curves

We discuss phenomena of stabilization for direct images of line bundles over projective curves mapping onto the projective line, for maps of sufficiently big degree.

math.AG↗
arXiv2021

Geometry and automorphisms of non-Kähler holomorphic symplectic manifolds

We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works of D. Guan and the first author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kähler manifold $W_F$ which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction, and prove that the automorphism group of $Q$ satisfies the Jordan property.

math.AG↗
arXiv2020

Sections of Lagrangian fibrations on holomorphically symplectic manifolds and degenerate twistorial deformations

Let $(M,I, Ω)$ be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration $π:\; M \mapsto X$, and $η$ a closed form of Hodge type (1,1)+(2,0) on $X$. We prove that $Ω':=Ω+π^* η$ is again a holomorphically symplectic form, for another complex structure $I'$, which is uniquely determined by $Ω'$. The corresponding deformation of complex structures is called "degenerate twistorial deformation". The map $π$ is holomorphic with respect to this new complex structure, and $X$ and the fibers of $π$ retain the same complex structure as before. Let $s$ be a smooth section of of $π$. We prove that there exists a degenerate twistorial deformation $(M,I', Ω')$ such that $s$ is a holomorphic section.

math.AG↗
arXiv2020

Stable vector bundles on the families of curves

We offer a new approach to proving the Chen-Donaldson-Sun theorem which we demonstrate with a series of examples. We discuss the existence of a construction of a special metric on stable vector bundles over the surfaces formed by a families of curves and its relation to the one-dimensional cycles in the moduli space of stable bundles on curves.

math.AG↗
arXiv2019

On the $\text{PGL}_{2}$-invariant quadruples of torsion points of elliptic curves

Let $E$ be an elliptic curve and $π:E\to\mathbb{P}^{1}$ a standard double cover identifying $\pm P\in E$. It is known that for some torsion points $P_{i}\in E$, $1\leq i\leq4$, the cross ratio of $\{π(P_{i})\}_{i=1}^{4}$ is independent of $E$. In this article, we will give a complete classification of such quadruples.

math.AG↗
arXiv2018

Algebraically hyperbolic manifolds have finite automorphism groups

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.

math.AG↗
arXiv2018

Lagrangian fibrations for IHS fourfolds

In this paper we study the Lagrangian fibrations for projective irreducible symplectic fourfolds and exclude the case of non-smooth base. Our method could be extended to the higher-dimensional cases.

math.AG↗
arXiv2018

Purely noncommuting groups

In this paper we define and investigate a class of groups characterized by a representation-theoretic property we call purely noncommuting or PNC. This property guarantees that the group has an action on a smooth projective variety with mild quotient singularities. It has intrinsic group-theoretic interest as well. The main results are as follows. (i) All supersolvable groups are PNC. (ii) No nonabelian finite simple groups are PNC. (iii) A metabelian group is guaranteed to be PNC if its commutator subgroup's cyclic prime-power-order factors are all distinct, but not in general. We also give a criterion guaranteeing a group is PNC if its nonabelian subgroups are all large, in a suitable sense, and investigate the PNC property for permutations.

math.RT↗
arXiv2018

On stable cohomology of central extensions of elementary abelian groups

We study when kernels of inflation maps associated to extraspecial p-groups in stable group cohomology are generated by their degree two components. This turns out to be true if the prime is large enough compared to the rank of the elementary abelian quotient, but false in general.

math.AG↗
arXiv2017

On Contraction of Algebraic Points

We study contraction of points on $\mathbb{P}^1(\bar{\mathbb{Q}})$ with certain control on local ramification indices, with application to the unramified curve correspondences problem initiated by Bogomolov and Tschinkel.

math.AG↗
arXiv2017

Dominant classes of projective varieties

We give evidence for a uniformization-type conjecture, that any algebraic variety can be altered into a variety endowed with a tower of smooth fibrations of relative dimension one.

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