Finiteness of fundamental groups

TL;DR

This paper establishes a link between the finiteness of fundamental groups of certain algebraic varieties and singularities, proving finiteness in specific three-dimensional cases.

Contribution

It demonstrates that finiteness of fundamental groups in lower-dimensional log Fano pairs implies finiteness of local fundamental groups of klt singularities, with new results for three-dimensional cases.

Findings

Finiteness of fundamental groups of smooth loci of lower-dimensional log Fano pairs implies finiteness of local fundamental groups of klt singularities.

The local fundamental group of three-dimensional klt singularities is finite.

The fundamental group of the smooth locus of three-dimensional Fano varieties with canonical singularities is finite.

Abstract

We show that the finiteness of the fundamental groups of the smooth locus of lower dimensional log Fano pairs would imply the finiteness of the local fundamental group of klt singularities. As an application, we verify that the local fundamental group of a three dimensional klt singularity and the fundamental group of the smooth locus of a three dimensional Fano variety with canonical singularities are always finite.