A vanishing theorem for Fano varieties in positive characteristic

We prove a Kodaira-type vanishing theorem for the Witt vector sheaf on a Fano variety over a perfect field of characteristic p. As a corollary, we deduce that the number of rational points on a Fano variety over a finite field with q=p^n elements is congruent to 1 mod q. This refines a result of Esnault which say that the number is congruent to 1 mod p.