arXiv subjects

Anna Abasheva

Publications and source records attributed to Anna Abasheva.

5 recordsLinked to original sources

arXiv2025

Shafarevich-Tate groups of holomorphic Lagrangian fibrations II

Let $X$ be a compact hyperkähler manifold with a Lagrangian fibration $π\colon X\to B$. A Shafarevich-Tate twist of $X$ is a holomorphic symplectic manifold with a Lagrangian fibration $π^φ\colon X^φ\to B$ which is isomorphic to $π$ locally over the base. In particular, $π^φ$ has the same fibers as $π$. A twist $X^φ$ corresponds to an element $φ$ in the Shafarevich-Tate group of $X$. We show that $X^φ$ is Kähler when a multiple of $φ$ lies in the connected component of unity of the Shafarevich-Tate group and give a necessary condition for $X^φ$ to be bimeromorphic to a Kähler manifold.

math.AG↗

arXiv2025

Shafarevich-Tate groups of holomorphic Lagrangian fibrations

Consider a Lagrangian fibration $π\colon X\to \mathbb P^n$ on a hyperkähler manifold $X$. There are two ways to construct a holomorphic family of deformations of $π$ over $\mathbb C$. The first one is known under the name Shafarevich-Tate family while the second one is the degenerate twistor family constructed by Verbitsky. We show that both families coincide. We prove that for a very general $X$ all members of the Shafarevich-Tate family are Kähler. There is a related notion of the Shafarevich-Tate group associated to a Lagrangian fibration. Its connected component of unity can be shown to be isomorphic to $\mathbb C/Λ$ where $Λ$ is a finitely generated subgroup of $\mathbb C$ and $\mathbb C$ is thought of as the base of the Shafarevich-Tate family. We show that for a very general $X$, projective deformations in the Shafarevich-Tate family correspond to the torsion points in the connected component of unity of the Shafarevich-Tate group. A sufficient condition for a Lagrangian fibration $X$ to be projective is existence of a holomorphic section. We find sufficient cohomological conditions for existence of a deformation in the Shafarevich-Tate family that admits a section.

math.AG↗

arXiv2023

Complex surfaces with many algebraic structures

We find new examples of complex surfaces with countably many non-isomorphic algebraic structures. Here is one such example: take an elliptic curve $E$ in $\mathbb P^2$ and blow up nine general points on $E$. Then the complement $M$ of the strict transform of $E$ in the blow-up has countably many algebraic structures. Moreover, each algebraic structure comes from an embedding of $M$ into a blow-up of $\mathbb P^2$ in nine points lying on an elliptic curve $F\not\simeq E$. We classify algebraic structures on $M$ using a Hopf transform: a way of constructing a new surface by cutting out an elliptic curve and pasting a different one. Next, we introduce the notion of an analytic K-theory of varieties. Manipulations with the example above lead us to prove that classes of all elliptic curves in this K-theory coincide. To put in another way, all motivic measures on complex algebraic varieties that take equal values on biholomorphic varieties do not distinguish elliptic curves.

math.CV↗

arXiv2021

Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds

A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group equipped with a left-invariant hypercomplex structure. Such a manifold admits a whole 2-dimensional sphere $S^2$ of complex structures induced by quaternions. We prove that for any hypercomplex nilmanifold $M$ and a generic complex structure $L\in S^2$, the complex manifold $(M,L)$ has algebraic dimension 0. A stronger result is proven when the hypercomplex nilmanifold is abelian. Consider the Lie algebra of left-invariant vector fields of Hodge type (1,0) on the corresponding nilpotent Lie group with respect to some complex structure $I\in S^2$. A hypercomplex nilmanifold is called abelian when this Lie algebra is abelian. We prove that all complex subvarieties of $(M,L)$ for generic $L\in S^2$ on a hypercomplex abelian nilmanifold are also hypercomplex nilmanifolds.

math.AG↗

arXiv2021

Feix-Kaledin metric on the total spaces of cotangent bundles to Kähler quotients

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 $\mathbb C^n$. 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 $\tilde Y$ 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 $\tilde Y$ induced from quaternions. Suppose that $J\ne\pm I$ 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 $\tilde{Y}_J$ admits an algebraic structure and is an affine variety.

math.DG↗