A vanishing theorem for differential operators in positive characteristic

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 Hⁱ( X,End( sfF_*O_X ) ) = 0 {H^i}left( {X,mathcal{E}ndleft( {{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 sfF_*O_X {sf{F}_*}{mathcal{O}_X} is a tilting bundle on these varieties provided that p > h, the Coxeter number of the corresponding group.