On endomorphisms of projective varieties with numerically trivial canonical divisors

TL;DR

This paper investigates the properties of endomorphisms on projective varieties with trivial canonical divisors, establishing conditions under which they are amplified or quasi-amplified, and exploring their dynamical behavior and periodic points.

Contribution

It demonstrates that quasi-amplified endomorphisms can be transformed into amplified ones through iteration and birational contractions, and characterizes their dynamics on Hyperk"ahler and abelian varieties.

Findings

Quasi-amplified endomorphisms descend to amplified ones after certain transformations.

On Hyperk"ahler varieties, quasi-amplified endomorphisms have positive entropy.

Endomorphisms on abelian varieties have dense periodic points under specific conditions.

Abstract

Let $X$ be a klt projective variety with numerically trivial canonical divisor. A surjective endomorphism $f:X\to X$ is amplified (resp.~quasi-amplified) if $f^*D-D$ is ample (resp.~big) for some Cartier divisor $D$. We show that after iteration and equivariant birational contractions, an quasi-amplified endomorphism will descend to an amplified endomorphism. As an application, when $X$ is Hyperk\"ahler, $f$ is quasi-amplified if and only if it is of positive entropy. In both cases, $f$ has Zariski dense periodic points. When $X$ is an abelian variety, we give and compare several cohomological and geometric criteria of amplified endomorphisms and endomorphisms with countable and Zariski dense periodic points (after an uncountable field extension).