Fundamental groups of proper varieties are finitely presented

TL;DR

This paper extends the recent proof that the étale fundamental group of smooth projective varieties over an algebraically closed field is finitely presented to all connected proper schemes, broadening the class of varieties with this property.

Contribution

It generalizes the finite presentability of étale fundamental groups from smooth projective varieties to all connected proper schemes over an algebraically closed field.

Findings

Étale fundamental groups of all connected proper schemes are finitely presented.

Extension of previous results from smooth projective to proper schemes.

Broadens understanding of fundamental groups in algebraic geometry.

Abstract

It was recently proven by Esnault, Shusterman and the second named author, that the \'etale fundamental group of a connected smooth projective variety over an algebraically closed field $k$ is finitely presented. In this note, we extend this result to all connected proper schemes over $k$.