Etale covers of Kawamata log terminal spaces and their smooth loci

TL;DR

This paper investigates the relationship between etale covers of Kawamata log terminal (klt) varieties and their smooth loci, establishing finiteness results and constructing coverings that align their etale fundamental groups.

Contribution

It proves finiteness of certain towers of etale covers over klt varieties and constructs coverings aligning the etale fundamental groups of the variety and its smooth locus.

Findings

No infinite towers of etale covers branched over small sets exist over klt varieties.

The difference between etale covers of a klt variety and its smooth locus is finite.

Constructs a finite cover where etale fundamental groups of the cover and its smooth locus agree.

Abstract

Working with a singular variety X, one is often interested in comparing the set of etale covers of X with that of its smooth locus X_reg. More precisely, one may ask: What are the obstructions to extend finite etale covers of X_reg to all of X? How do the etale fundamental groups of X and of its smooth locus compare? For projective varieties with Kawamata log terminal (klt) singularities we answer these questions in part. The main result of this paper asserts that there are no infinite towers of finite morphisms over a klt base variety where all morphisms are etale in codimension one, but branched over a small set. In a certain sense, this result can be seen as saying that the difference between the etale covers of X and those of X_reg is finite if X is klt. As an immediate application, we construct a finite covering Y of X, etale in codimension one, such that the etale fundamental groups of Y and Y_reg agree.