Singular hermitian metrics on vector bundles

Abstract

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 ⊗ 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.

Metrics on semistable and numerically effective Higgs bundles

“…The approaches by Demailly, Peternell and Schneider [9] and by de Cataldo [7] rely on a definition of numerical effectiveness in terms of fibre metrics. In particular, de Cataldo considers metrics on the bundle E, and this approach seems to be well suited to an extension to the case of Higgs bundles, again implementing the idea that the transition from the ordinary to the Higgs case is obtained by replacing the Chern connection with the Hitchin-Simpson connection.…”

“…The guiding idea here is to give a definition of nefness in terms of fibre metrics on the bundles, following Demailly-Peternell-Schneider and de Cataldo [9,7]. One takes the Higgs structure into account by replacing the Chern connection associated with the fibre metrics by a connection (introduced by Hitchin [17] and later extensively used by Simpson [23]) whose definition involves the Higgs field.…”

“…The authors thank M.S. Narasimhan for drawing de Cataldo's paper [7] to their attention, and the referee for helping to improve the exposition. This paper was mostly done during a stay of both authors at the Tata Institute for Fundamental Research in Mumbai, India; they express their warm thanks for the hospitality and support.…”

“…Fixing a C ∞ isomorphism E ≃ F ⊕ G, these metrics induce an Hermitian metric h ξ on E. A simple calculation, which involves the second fundamental form of E, shows that R (E,h ξ ) ≥ 1 −ξω ⊗ h ξ , so that E is 1-H-nef (details in the ordinary case are given in [7]). …”

Remarks on Seshadri constants of vector bundles

“…There is however one technical point, we may no longer assume that ^ (g)7r*f* is ample. Let L be any ample line bundle on P(<f) and 6 > 0 a rational number, then the Q-vector bundle ^^^£^^L 6 is ample. Eventually we take the limit as 6 -> 0, and hence this does not affect the preceding computations.…”