Geodesic
Evidence for a generalization of Gieseker's conjecture on stratified bundles in positive characteristic
Documenta mathematica, Tome 18 (2013), pp. 1215-1242
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Let X be a smooth, connected, projective variety over an algebraically closed field of positive characteristic. In [Gie75], Gieseker conjectured that every stratified bundle (i.e. every OX​-coherent DX/k​-module) on X is trivial, if and only if π1eˊt​(X)=0. This was proven by Esnault–Mehta, [EM10]. Building on the classical situation over the complex numbers, we present and motivate a generalization of Gieseker's conjecture, using the notion of regular singular stratified bundles developed in the author's thesis and [Kin12a]. In the main part of this article we establish some important special cases of this generalization; most notably we prove that for not necessarily proper X,π1tame​(X)=0 implies that there are no nontrivial regular singular stratified bundles with abelian monodromy.
DOI : 10.4171/dm/426
Classification : 14E20, 14E22, 14F10
Mots-clés : fundamental group, coverings, stratified bundles, D-modules, tame ramification 
@article{10_4171_dm_426,
     author = {Lars Kindler},
     title = {Evidence for a generalization of {Gieseker's} conjecture on stratified bundles in positive characteristic},
     journal = {Documenta mathematica},
     pages = {1215--1242},
     year = {2013},
     volume = {18},
     doi = {10.4171/dm/426},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/426/}
}
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Lars Kindler. Evidence for a generalization of Gieseker's conjecture on stratified bundles in positive characteristic. Documenta mathematica, Tome 18 (2013), pp. 1215-1242. doi: 10.4171/dm/426

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