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. everyOX-coherent DX~k- module) on X is trivial, if and only if π ´ 1 (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, π tame (X)= 0 implies that there are no nontrivial regular singular