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

Singular hermitian metrics on vector bundles

We introduce a notion of singular hermitian metrics (s.h.m.) for holomorphic vector bundles and define positivity in view of $L^2$-estimates. Associated with a suitably positive s.h.m. there is a (coherent) sheaf 0-th kernel of a certain $d''$-complex. We prove a vanishing theorem for the cohomology of this sheaf. All this generalizes to the case of higher rank known results of Nadel for the case of line bundles. We introduce a new semi-positivity notion, $t$-nefness, for vector bundles, establish some of its basic properties and prove that on curves it coincides with ordinary nefness. We particularize the results on s.h.m. to the case of vector bundles of the form $E=F \otimes L$, where $F$ is a $t$-nef vector bundle and $L$ is a positive (in the sense of currents) line bundle. As applications we generalize to the higher rank case 1) Kawamata-Viehweg Vanishing Theorem, 2) the effective results concerning the global generation of jets for the adjoint to powers of ample line bundles, and 3) Matsusaka Big Theorem made effective.Comment: LaTex (article) 25 pages; revised: minor changes; to appear in Crelle's J; dedicated to Michael Schneide