FROBENIUS MORPHISMS MODULE P(2)

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].