On isolated singularities with noninvertible finite endomorphism

TL;DR

This paper proves that isolated singularities with a finite, étale-in-codimension-1 endomorphism of degree at least 2 are necessarily $Q$-Gorenstein and log canonical, revealing structural properties of such singularities.

Contribution

It establishes that such singularities must be $Q$-Gorenstein and log canonical, linking endomorphism properties to singularity classifications.

Findings

Singularities with the specified endomorphism are $Q$-Gorenstein.

Such singularities are log canonical.

Endomorphism degree influences singularity structure.

Abstract

We prove that if $\phi:(X,0)\to (X,0)$ is a finite endomorphism of an isolated singularity such that $\operatorname{deg}(\phi)\geq 2$ and $\phi$ is \'etale in codimension 1, then $X$ is $\mathbb{Q}$-Gorenstein and log canonical.