  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).
