A vanishing theorem for differential operators in positive characteristic

Abstract

Let X be a smooth variety over an algebraically closed field k of characteristic p, and let F: X → X be the Frobenius morphism. We prove that if X is an incidence variety (a partial flag variety in type A n ) or a smooth quadric (in this case p is supposed to be odd) then \( {H^i}\left( {X,\mathcal{E}nd\left( {{\sf{F}_*}{\mathcal{O}_X}} \right)} \right) = 0 \) for i > 0. Using this vanishing result and the derived localization theorem for crystalline differential operators [3], we show that the Frobenius direct image \( {\sf{F}_*}{\mathcal{O}_X} \) is a tilting bundle on these varieties provided that p > h, the Coxeter number of the corresponding group.