  Let L be a holomorphic line bundle over a compact complex projective
Hermitian manifold X. Any fixed smooth hermitian metric h 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 (i.e. orthogonal projection
kernels). The convergence is studied in the large k limit and is expressed in
terms of the equilibrium metric h_e associated to h, as well as in terms of the
Monge-Ampere measure of h on a certain support set. It is also shown that the
equilibrium metric h_e is in the class C^{1,1} on the complement of the
augmented base locus of L. For L ample these results give generalizations of
well-known results concerning the case when the curvature of h is globally
positive (then h_e=h). In general, the results can be seen as local metrized
versions of Fujita's approximation theorem for the volume of L.
