On a toroidalization for klt singularities

TL;DR

This paper establishes a toroidalization principle for finite group actions on klt singularities, linking geometric properties to algebraic invariants and providing bounds on fundamental group ranks.

Contribution

It introduces a new toroidalization approach for klt singularities and connects the Jordan property with geometric realizations involving toric singularities.

Findings

Jordan property for regional fundamental groups is geometrically realizable.

Rank of fundamental groups is bounded by the singularity's regularity.

Finite actions on dual complexes have controlled behavior.

Abstract

In this article, we prove a toroidalization principle for finite actions on klt singularities. As an application, we prove that the Jordan property for the regional fundamental group of klt singularities can be realized geometrically: by extracting a toric singularity over the klt germ. In the course of the proof, we will prove statements about finite actions on dual complexes and almost fixed points in the fibers of equivariant Fano type morphisms. Furthermore, we will prove that the rank of a fundamental group of the klt singularity is bounded above by its regularity.