Stratified bundles and étale fundamental group

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 $[{\pi }_1^{\acute{\mathrm{e}}\mathrm{t}},{\pi }_1^{\acute{\mathrm{e}}\mathrm{t}}]$ of the étale 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 ${\pi }_1^{\acute{\mathrm{e}}\mathrm{t}}$ is abelian without $p$-power quotient. This answers positively a conjecture by Gieseker.

We thank the referee who pointed out that p.7 displayed formula is wrong. We therefore withdraw the article. Lem 3.2 is proven but not Thm 3.1 So in the final Thm (i) (ii) are ok but (iii) has to be replaced by the corresponding statement on the subcat. dual to the abelian quotient of the Tannaka group