Curvature of vector bundles and subharmonicity of Bergman kernels

TL;DR

This paper investigates the curvature properties of vector bundles derived from Bergman kernels and establishes conditions under which certain holomorphic vector bundles exhibit strictly positive Nakano curvature.

Contribution

It proves a new curvature result for vector bundles associated with Bergman kernels and links ampleness of vector bundles to positive curvature of their determinants.

Findings

Curvature of vector bundles from Bergman kernels is positive under certain conditions.

Holomorphic vector bundles that are ample have Hermitian metrics with strictly positive Nakano curvature.

Results generalize previous subharmonicity properties of Bergman kernels.

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