Evidence for a Generalization of Gieseker's Conjecture on Stratified Bundles in Positive Characteristic

Let $X$ be a smooth, connected, projective variety over an algebraically closed field of positive characteristic. In \cite{Gieseker/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, \cite{EsnaultMehta/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 \cite{Kindler/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.

2010 Mathematics Subject Classification: 14E20, 14E22, 14F10

Keywords and Phrases: Fundamental group, coverings, stratified bundles, D-modules, tame ramification

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