Singularities of non-$\mathbb{Q}$-Gorenstein varieties admitting a polarized endomorphism

TL;DR

This paper extends the concept of log canonical singularities to non-$\mathbb{Q}$-Gorenstein varieties and proves that varieties with polarized endomorphisms have these singularities, confirming a related conjecture.

Contribution

It introduces a generalized notion of log canonical singularities for non-$\mathbb{Q}$-Gorenstein varieties and proves that polarized endomorphisms imply these singularities, confirming a conjecture.

Findings

Varieties with polarized endomorphisms have generalized log canonical singularities.

Affirmative resolution of Broustet and H"{o}ring's conjecture.

Extension of singularity theory to non-$\mathbb{Q}$-Gorenstein cases.

Abstract

In this paper, we discuss a generalization of log canonical singularities in the non-$\mathbb{Q}$-Gorenstein setting. We prove that if a normal complex projective variety has a non-invertible polarized endomorphism, then it has log canonical singularities in our sense. As a corollary, we give an affirmative answer to a conjecture of Broustet and H\"{o}ring.