Regular singularities and tame ramification

Abstract: After briefly recalling the notion of regular singularity for vector bundles with flat connections on smooth complex varieties, we present an analogous definition for vector bundles with an action of the ring of differential operators on smooth varieties in positive characteristic, and show how it relates to the notion of tame ramification of coverings. In characteristic 0, the Deligne-Riemann-Hilbert correspondence together with a group theoretic result by Grothendieck-Malcev implies that the category of regular singular flat connections is "controlled" by the étale fundamental group. We discuss an analogous question over a field of positive characteristic.