Bergman kernels and equilibrium measures for ample line bundles

TL;DR

This paper investigates the asymptotic behavior of Bergman kernels for ample line bundles over compact complex manifolds, linking their convergence to equilibrium metrics and Monge-Ampere measures, and establishing regularity properties.

Contribution

It generalizes known results by analyzing Bergman kernel convergence in the large tensor power limit using equilibrium metrics and Monge-Ampere measures.

Findings

Convergence of Bergman kernels to equilibrium measures as tensor power increases

Equilibrium metric possesses Lipschitz continuous first derivatives

Results extend classical cases with positive curvature metrics

Abstract

Let L be an ample holomorphic line bundle over a compact complex Hermitian manifold X. Any fixed smooth Hermitian metric on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k:th tensor power of L. In this paper various convergence results are obtained for the corresponding Bergman kernels. The convergence is studied in the large k limit and is expressed in terms of the equilibrium metric associated to the fixed metric, as well as in terms of the Monge-Ampere measure of the fixed metric itself on a certain support set. It is also shown that the equilibrium metric has Lipschitz continuous first derivatives. These results can be seen as generalizations of well-known results concerning the case when the curvature of the fixed metric is positive (the corresponding equilibrium metric is then simply the fixed metric itself).