Let X be a projective normal algebraic variety over a perfect field k
of characteristic p > 0 with a flat lift to a scheme X' over the Witt
vectors W-2 (k) of length two. Let <(Omega)over tilde> be the sheaf of
Zariski i-forms on X. If the absolute Frobenius morphism F : X --> X
lifts to a morphism F' : X' --> X', we prove that H' (X: <(Omega)over
tilde>(X/k) x L) = 0, where L is an ample line bundle on X and i > 0.
When X is a toric variety, Frobenius lifts to W-2 (k) and we get a sim
ple proof of the Bott-Steenbrink-Danilov vanishing theorem and the deg
eneration of the Danilov spectral sequence [2].