FROBENIUS MORPHISMS OVER Z/p AND BOTT VANISHING

when 0 < k ≤ p. We will say that a smooth projective variety X has Bott vanishing if for every ample line bundle L, i > 0 and j H(X,ΩjX ⊗ L) = 0 The purpose of this paper is to show that Bott vanishing is a simple consequence of a very specific condition on the Frobenius morphism in prime characteristic p. Recall that the absolute Frobenius morphism F : X → X on X, where X is a variety over Z/p is the identity on point spaces and the p-th power map locally on functions. Assume that there is a flat scheme X over Z/p, such that X ∼= X ×Z/p2 Z/p. The condition on F is that there should be a morphism F (2) : X → X which gives F by reduction mod p. In this case we will say that the Frobenius morphism lifts to Z/p. It is known that a lift of the Frobenius morphism to Z/p leads to a quasi-isomorphism