Vanishing via lifting to second Witt vectors and a proof of an   isotriviality result
  A proof based on reduction to finite fields of Esnault-Viehweg's stronger version of Sommese Vanishing Theorem for $k$-ample line bundles is given. This result is used to give different proofs of isotriviality results of A. Parshin and L. Migliorini. 
Mark Andrea A. de Cataldo