Singularities of varieties admitting an endomorphism

TL;DR

This paper investigates the relationship between the singularities of normal varieties with Q-Cartier canonical divisors and the dynamics of finite surjective endomorphisms, showing that polarized endomorphisms imply at most log-canonical singularities.

Contribution

It establishes a connection between the non-log-canonical locus of singularities and the dynamics of endomorphisms, proving that polarized endomorphisms restrict singularities to be at most log-canonical.

Findings

The non-log-canonical locus relates closely to endomorphism dynamics.

Polarized endomorphisms on projective varieties limit singularities to log-canonical.

Varieties with such endomorphisms cannot have worse singularities than log-canonical.

Abstract

Let X be a normal variety such that $K_X$ is Q-Cartier, and let $f: X \rightarrow X$ be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that are not log-canonical and the dynamics of the endomorphism f. As a consequence we prove that if X is projective and f polarised, then X has at most log-canonical singularities.