Periodic subvarieties of a projective variety under the action of a maximal rank abelian group of positive entropy

TL;DR

This paper characterizes positive-dimensional G-periodic subvarieties of a projective variety under a maximal rank abelian group action with positive entropy, shedding light on the structure of such varieties and their quotients.

Contribution

It provides a classification of G-periodic subvarieties under maximal rank abelian groups with positive entropy, advancing understanding of equivariant birational geometry.

Findings

Identification of positive-dimensional G-periodic subvarieties.

Conditions under which X is G-equivariantly birational to a quotient of an abelian variety.

Obstructions to X being a quotient of an abelian variety by a finite group.

Abstract

We determine positive-dimensional G-periodic proper subvarieties of an n-dimensional normal projective variety X under the action of an abelian group G of maximal rank n-1 and of positive entropy. The motivation of the paper is to understand the obstruction for X to be G-equivariant birational to the quotient variety of an abelian variety modulo the action of a finite group.