Families of automorphisms of abelian varieties

TL;DR

This paper studies the algebraic dynamics of automorphisms on families of polarized abelian varieties, showing conditions for regularizability and describing orbit closures, with implications for understanding degenerations and translations.

Contribution

It establishes criteria for when automorphisms can be regularized in degenerating families and describes orbit closures, extending previous work on algebraic dynamics of abelian varieties.

Findings

Automorphisms without cyclotomic factors cannot be regularized if the family degenerates.

Families of translations are always regularizable.

Descriptions of orbit closures inspired by Cantat and Amerik-Verbitsky.

Abstract

We consider some algebraic aspects of the dynamics of an automorphism on a family of polarized abelian varieties parameterized by the complex unit disk. When the action on the cohomology of the generic fiber has no cyclotomic factor, we prove that such a map can be made regular only if the family of abelian varieties does not degenerate. As a contrast, we show that families of translations are always regularizable. We further describe the closure of the orbits of such maps, inspired by results of Cantat and Amerik-Verbitsky.