On the Bergman kernels of holomorphic vector bundles

László Lempert

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arXiv · 2109.08593

On the Bergman kernels of holomorphic vector bundles

László Lempert

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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.

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László Lempert. 2021-09-17. On the Bergman kernels of holomorphic vector bundles. https://arxiv.org/abs/2109.08593

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