  Let $X$ be a smooth variety over an algebraically closed field $k$ of
positive characteristic, ${\rm D}_X$ the sheaf of PD-differential operators,
and ${\bar D}_X$ its central reduction, the sheaf of small differential
operators. In this paper we show that if $X$ is a line-hyperplane incidence
variety (a partial flag variety of type $(1,n,n+1)$) or a quadric of arbitrary
dimension (in this case the characteristic is supposed to be odd) then ${\rm
H}^{i}(X,{\bar D}_X)=0$ for $i>0$. Using this vanishing result and the derived
localization theorem for crystalline differential operators (\cite{BMR}) we
show that the Frobenius pushforward of the structure sheaf is a tilting bundle
on these varieties, provided that $p>h$, the Coxeter number of the
corresponding group.
