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
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 $[π_1^{{\rm \acute{e}t}}, π_1^{{\rm \acute{e}t}}]$ 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 $ π_1^{{\rm \acute{e}t}}$ is abelian without $p$-power quotient. This answers positively a conjecture by Gieseker.
Evidence for a generalization of Gieseker's conjecture on stratified bundles in positive characteristic
设X是正特征的代数闭域上的光滑连通射影簇。在“Flat向量丛and the基本群in non-zero characteristics”（Ann. Scuola Norm.辅助核算比萨海峡Sci. (4)2（1975）），Gieseker证明了X上的每个层丛（即每个O-凝聚D-模）是平凡的，当且仅当X的代数基本群是平凡的。Esnault-Mehta在“Simplified connected projective manifolds in characteristic p> 0 have no nontrivial stratified bundles”（Invent. 181（2010））。在复数的经典情况的基础上，我们提出和激励的Gieseker猜想的推广使用的概念，经常奇异分层丛中开发的作者的论文和arXiv：1210.5077。在本文的主要部分，我们建立了一些重要的特殊情况下，这种推广，最值得注意的是，我们证明了不一定是正确的X，驯服的基本群的X的消失意味着没有非平凡的正则奇异层丛的阿贝尔单值。
The variation of the monodromy group in families of stratified bundles in positive characteristic
本文研究具有正特征的层丛的光滑族及其单值群的变化，特别是加强Esnault和Langer在“On a positive equicharacteristic variant of the $p$ -curvature conjecture”（Doc. 18（2013））。主要结果是，如果基域是不可数的，那么强形式成立。在基域可数的情况下，我们对可能的推广给出了肯定和否定的答案。
