Regular singular stratified bundles and tame ramification

TL;DR

This paper introduces a new notion of regular singularities for stratified bundles in positive characteristic, establishing an equivalence with tame fundamental group representations without using resolution of singularities.

Contribution

It defines regular singular stratified bundles in positive characteristic and proves their equivalence to tame fundamental group representations, avoiding resolution of singularities.

Findings

Category of regular singular stratified bundles with finite monodromy is equivalent to tame fundamental group representations.

A stratified bundle with finite monodromy is regular singular iff it is along all curves mapping to X.

The approach does not rely on resolution of singularities.

Abstract

Let X be a smooth variety over an algebraically closed field k of positive characteristic. We define and study a general notion of regular singularities for stratified bundles (i.e. O_X-coherent D_X-modules) on X without relying on resolution of singularities. The main result is that the category of regular singular stratified bundles with finite monodromy is equivalent to the category of continuous representations of the tame fundamental group on finite dimensional k-vector spaces. As a corollary we obtain that a stratified bundle with finite monodromy is regular singular if and only if it is regular singular along all curves mapping to X.