On the Bergman kernels of holomorphic vector bundles

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 $⟨,⟩$ on $\mathcal{O}(E)$ functions FS$(⟨,⟩)\in {\mathcal{H}}_{\omega }=\{u\in C^{\infty }(X):\omega +i\partial \overline{\partial }u>0\}$. 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 $\mathcal{O}(E{)}^{\ast }$, and the paper describes some simple properties of this association.

References in corpus (1)

• Griffiths extremality, interpolation of norms, and Kähler quantization

Cited by in corpus (1)

• On Chow stability and balanced embeddings