Building blocks of amplified endomorphisms of normal projective varieties

TL;DR

This paper generalizes polarized endomorphisms to int-amplified endomorphisms on normal projective varieties, demonstrating that they preserve key properties related to singularities, canonical divisors, and the minimal model program.

Contribution

It introduces and studies int-amplified endomorphisms, extending the class of polarized endomorphisms while maintaining important geometric and algebraic properties.

Findings

Int-amplified endomorphisms preserve singularity types.

They maintain properties of the canonical divisor.

They are compatible with the equivariant minimal model program.

Abstract

Let $X$ be a normal projective variety. A surjective endomorphism $f:X\to X$ is int-amplified if $f^\ast L - L =H$ for some ample Cartier divisors $L$ and $H$. This is a generalization of the so-called polarized endomorphism which requires that $f^*H\sim qH$ for some ample Cartier divisor $H$ and $q>1$. We show that this generalization keeps all nice properties of the polarized case in terms of the singularity, canonical divisor, and equivariant minimal model program.