On the Kawamata-Viehweg vanishing theorem for a surface in positive characteristic

Let X be a nonsingular projective variety defined over an algebraically closed field k. In the characteristic zero case, the Kawamata-Viehweg vanishing theorem for X is an important tool in the adjunction theory and the minimal model program. On the other hand, in positive characteristic cases, an analog of Kodaira vanishing theorem for X was proved by Raynaud under the condition that X has a lifting over W-2(k), the ring of Witt vectors of length two. In this paper we prove the Kawamata-Viehweg vanishing theorem for a nonsingular projective surface defined over an algebraically closed field of characteristic p > 0. In particular, we give an explicit condition that the cohomology groups vanish. This is based on Shepherd-Barren's result on the instability of rank 2 locally free sheaves on a surface in positive characteristic.