Geodesic
Evidence for a generalization of Gieseker's conjecture on stratified bundles in positive characteristic
Documenta mathematica, Tome 18 (2013), pp. 1215-1242.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let $X$ be a smooth, connected, projective variety over an algebraically closed field of positive characteristic. In citeGieseker/FlatBundles, Gieseker conjectured that every stratified bundle (i.e. every $O_X$-coherent $\mathscr{D}_{X/k}$-module) on $X$ is trivial, if and only if $\pi_1^{\et}(X)=0$. This was proven by Esnault-Mehta, citeEsnaultMehta/Gieseker. 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 citeKindler/FiniteBundles. 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 $X, \pi_1^{\tame}(X)=0$ implies that there are no nontrivial regular singular stratified bundles with abelian monodromy.
Classification : 14E20, 14E22, 14F10
Keywords: fundamental group, coverings, stratified bundles, D-modules, tame ramification 
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     author = {Kindler, Lars},
     title = {Evidence for a generalization of {Gieseker's} conjecture on stratified bundles in positive characteristic},
     journal = {Documenta mathematica},
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     volume = {18},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2013__18__a12/}
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Kindler, Lars. Evidence for a generalization of Gieseker's conjecture on stratified bundles in positive characteristic. Documenta mathematica, Tome 18 (2013), pp. 1215-1242. http://geodesic.mathdoc.fr/item/DOCMA_2013__18__a12/