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We introduce a notion of singular hermitian metrics (s.h.m.) for holomorphicvector bundles and define positivity in view of $L^2$-estimates. Associatedwith a suitably positive s.h.m. there is a (coherent) sheaf 0-th kernel of acertain $d''$-complex. We prove a vanishing theorem for the cohomology of thissheaf. All this generalizes to the case of higher rank known results of Nadelfor the case of line bundles. We introduce a new semi-positivity notion,$t$-nefness, for vector bundles, establish some of its basic properties andprove that on curves it coincides with ordinary nefness. We particularize theresults 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 ofcurrents) line bundle. As applications we generalize to the higher rank case 1)Kawamata-Viehweg Vanishing Theorem, 2) the effective results concerning theglobal generation of jets for the adjoint to powers of ample line bundles, and3) Matsusaka Big Theorem made effective.

Singular hermitian metrics on vector bundles