Simple abelian varieties and primitive automorphisms of null entropy of surfaces

TL;DR

This paper characterizes simple abelian varieties and surfaces through translation automorphisms and classifies primitive birational automorphisms with null entropy on surfaces over algebraically closed fields.

Contribution

It provides a new characterization of simple abelian varieties and surfaces and classifies primitive automorphisms with trivial dynamical degree on surfaces.

Findings

Characterization of simple abelian varieties and surfaces

Classification of primitive birational automorphisms with null entropy

Applicable over any characteristic field

Abstract

We characterize simple complex abelian varieties and simple abelian surfaces in terms of primitivity of translation automorphisms. Applying this together with a result due to Diller and Favre, we then classify all primitive birational automorphisms with trivial first dynamical degree of smooth projective surfaces over an algebraically closed field of any characteristic.