On the Bergman kernels of holomorphic vector bundles

TL;DR

This paper investigates the properties of the Fubini-Study map for holomorphic line bundles over compact complex manifolds, showing it is an injective immersion and exploring how to extend it to degenerate inner products using Bergman kernels.

Contribution

It proves the Fubini-Study map is an injective immersion and extends it to degenerate inner products via Bergman kernels, providing new insights into the map's range and properties.

Findings

Fubini-Study map is an injective immersion

Its image is generally not closed in the space of metrics

Extension to degenerate inner products involves Bergman kernels

Abstract

Consider a very ample line bundle $ E \to X$ over a compact complex manifold, endowed with a hermitian metric of curvature $-i \omega $, and the space $\mathcal{O}(E)$ of its holomorphic sections. The Fubini--Study map associates with positive definite inner products $\langle \, , \rangle$ on $\mathcal{O}(E)$ functions FS$(\langle \, ,\rangle) \in \mathcal{H}_{\omega}=\{u \in C^{\infty}(X):\omega +i\partial\overline{\partial} u >0\}$. We prove that FS is an injective immersion, but its image in general is not closed in $\mathcal{H}_{\omega}$. To obtain a closed range, FS has to be extended to certain degenerate inner products. This we do by associating Bergman kernels with general inner products on the dual $\mathcal{O}(E)^*$, and the paper describes some simple properties of this association.