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 [Gie75], Gieseker conjectured that every stratified bundle (i.e. every ${\mathcal{O}}_X$-coherent ${\mathcal{D}}_{X/k}$-module) on $X$ is trivial, if and only if ${\pi }_1^{\acute{\mathrm{e}}\mathrm{t}}(X)=0$. 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 $X,{\pi }_1^{\mathrm{tame}}(X)=0$ implies that there are no nontrivial regular singular stratified bundles with abelian monodromy.