Curvature of vector bundles and subharmonicity of Bergman kernels

Publication date: 23 May 2005

Abstract: In a previous paper, cite{Berndtsson}, we have studied a property of subharmonic dependence on a parameter of Bergman kernels for a family of weighted

L^{2}

-spaces of holomorphic functions. Here we prove a result on the curvature of a vector bundle defined by this family of

L^{2}

-spaces itself, which has the earlier results on Bergman kernels as a corollary. Applying the same arguments to spaces of holomorphic sections to line bundles over a locally trivial fibration we also prove that if a holomorphic vector bundle,

V

, over a complex manifold is ample in the sense of Hartshorne, then

V g r d e t V

has an Hermitian metric with curvature strictly positive in the sense of Nakano.

Mathematics Subject Classification ID

Holomorphic convexity (32Exx)

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