Stratified bundles and \'etale fundamental group

TL;DR

This paper characterizes the structure of stratified bundles on smooth projective varieties over algebraically closed fields of characteristic p, linking their properties to the structure of the étale fundamental group and confirming a conjecture by Gieseker.

Contribution

It establishes a precise relationship between the rank of irreducible stratified bundles and the pro-p nature of the commutator subgroup of the étale fundamental group, confirming Gieseker's conjecture.

Findings

Irreducible stratified bundles have rank 1 iff the commutator of the étale fundamental group is pro-p.

The category of stratified bundles is semi-simple with rank 1 irreducibles iff the étale fundamental group is abelian without p-power quotients.

Confirmed Gieseker's conjecture relating stratified bundles and fundamental group structure.

Abstract

v2: A few typos corrected, a few formulations improved. On $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that irreducible stratified bundles have rank 1 if and only if the commutator $[\pi_1^{{\rm \acute{e}t}}, \pi_1^{{\rm \acute{e}t}}]$ of the \'etale fundamental group is a pro-$p$-group, and we show that the category of stratified bundles is semi-simple with irreducible objects of rank 1 if and only if $ \pi_1^{{\rm \acute{e}t}}$ is abelian without $p$-power quotient. This answers positively a conjecture by Gieseker.