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.Comment: Latex, 7 page

Effective nonvanishing, effective global generation

We prove a multiple-points higher-jets nonvanishing theorem by the use of local Seshadri constants. Applications are given to effectivity problems such as constructing rational and birational maps into Grassmannians, and the global generation of vector bundles.Comment: LaTex (article) 13 pages; revised: one section added; to appear in Ann. Inst. Fourie