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

TL;DR

This paper generalizes Gieseker's conjecture on stratified bundles in positive characteristic, establishing that the vanishing of the tame fundamental group implies triviality of certain regular singular stratified bundles with abelian monodromy.

Contribution

It introduces a new generalization of Gieseker's conjecture using regular singular stratified bundles and proves special cases linking tame fundamental group vanishing to bundle triviality.

Findings

Vanishing tame fundamental group implies no nontrivial regular singular stratified bundles with abelian monodromy.

Provides evidence for the generalized conjecture in non-proper varieties.

Extends classical results over complex numbers to positive characteristic settings.

Abstract

Let X be a smooth, connected, projective variety over an algebraically closed field of positive characteristic. In "Flat vector bundles and the fundamental group in non-zero characteristics" (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975)), Gieseker conjectured that every stratified bundle (i.e. every O-coherent D-module) on X is trivial, if and only if the \'etale fundamental group of X is trivial. This was proven by Esnault-Mehta in "Simply connected projective manifolds in characteristic p> 0 have no nontrivial stratified bundles" (Invent. Math. 181 (2010)). 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 arXiv:1210.5077. 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, the vanishing of the tame fundamental group of X implies that there are no nontrivial regular singular stratified bundles with abelian monodromy.