Curvature of vector bundles and subharmonicity of Bergman kernels

Details Curvature of vector bundles and subharmonicity of Bergman kernels Curvature of vector bundles and subharmonicity of Bergman kernels

2005-05-23

arxiv.org/abs/math/0505470v1

Mathematics

Complex Variables

Scientific paper

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 $V\gr\det V$ has an Hermitian metric with curvature strictly positive in the sense of Nakano.

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