Evidence for a generalization of Gieseker's conjecture on stratified bundles in positive characteristic

Abstract

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.