Finite presentation of the tame fundamental group

TL;DR

This paper proves that the tame fundamental group of certain algebraic varieties over an algebraically closed field of characteristic p is finitely presented or projective, advancing understanding of their algebraic and topological structure.

Contribution

It establishes finite presentability of tame fundamental groups for smooth affine and quasi-projective varieties over fields of characteristic p, a significant extension of known results.

Findings

Tame fundamental group of smooth affine curves is projective.

Fundamental group of smooth projective varieties is finitely presented.

Tame fundamental group of certain quasi-projective varieties is finitely presented.

Abstract

Let $p$ be a prime number, and let $k$ be an algebraically closed field of characteristic $p$. We show that the tame fundamental group of a smooth affine curve over $k$ is a projective profinite group. We prove that the fundamental group of a smooth projective variety over $k$ is finitely presented. More generally we prove that the tame fundamental group of a smooth quasi-projective variety over $k$ which admits a good compactification is finitely presented. v2: references added. Thank you to all for the friendly and fruitful comments.