Bergman kernels and equilibrium measures for ample line bundles

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).