Curvature of vector bundles and subharmonicity of Bergman kernels

2005/05/23 by Bo Berndtsson, Berndtsson, Bo
Mathematics · #32E #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #math.AG #math.CV #msc:32E

paper · pdf · doi:10.48550/arxiv.math/0505470

arxiv created 2005/05/23 · openalex publication_date 2005/05/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a previous paper, \citeBerndtsson, we have studied a property of subharmonic dependence on a parameter of Bergman kernels for a family of weighted L²-spaces of holomorphic functions. Here we prove a result on the curvature of a vector bundle defined by this family of L²-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\grdet V has an Hermitian metric with curvature strictly positive in the sense of Nakano.

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