Local cohomology and D-affinity in positive characteristic

TL;DR

This paper provides a counterexample showing that D-affinity and Beilinson-Bernstein equivalence do not hold in positive characteristic for certain flag varieties, contrasting with the zero characteristic case.

Contribution

It constructs a specific D-module on a Grassmannian in positive characteristic with non-vanishing cohomology, challenging existing assumptions.

Findings

Counterexample to D-affinity in positive characteristic

Non-vanishing first cohomology group for a D-module

Failure of Beilinson-Bernstein equivalence in this setting

Abstract

Comparing vanishing of local cohomology in zero and positive characteristic, we give an example of a D-module on a Grassmann variety in positive characteristic with non-vanishing first cohomology group. This is a counterexample to D-affinity and the Beilinson-Bernstein equivalence for flag manifolds in positive characteristic.