Stratified bundles and \'etale fundamental group (new version)

TL;DR

This paper proves a conjecture by Gieseker relating the structure of stratified bundles on smooth projective varieties over algebraically closed fields of characteristic p to properties of their étale fundamental group, specifically its commutator subgroup and abelian quotients.

Contribution

It establishes a precise correspondence between the rank and simplicity of stratified bundles and the algebraic structure of the étale fundamental group, resolving a conjecture by Gieseker.

Findings

All irreducible stratified bundles have rank 1 iff the commutator of π₁ is a pro-p-group.

The category of stratified bundles is semi-simple with rank 1 irreducibles iff π₁ is abelian without p-power quotients.

The results confirm conjectures relating fundamental group properties to stratified bundle structures.

Abstract

This submission replaces the arXiv:1012.5381 submission with the same title, which had been withdrawn as it contained a mistake, repaired in this submission: on $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that all irreducible stratified bundles have rank 1 if and only if the commutator $[\pi_1, \pi_1]$ of the \'etale fundamental group $\pi_1$ 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 $ is abelian without $p$-power quotient. This answers positively a conjecture by Gieseker.