Bergman kernels and equilibrium measures for line bundles over projective manifolds

Source abstract

Let $L$ be a holomorphic line bundle over a compact complex projective Hermitian manifold $X.$ Any fixed smooth hermitian metric $\phi$ 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 $\phi_{e}$ associated to the fixed metric $\phi,$ as well as in terms of the Monge-Ampere measure of the metric $\phi$ itself on a certain support set. It is also shown that the equilibrium metric is ${\cal C}^{1,1}$ 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 $\phi$ is globally positive (then $\phi_{e}=\phi$). In general, the results can be seen as local metrized versions of Fujita's approximation theorem for the volume of $L$.