Bergman kernels and equilibrium measures for line bundles over projective manifolds
Abstract
Description
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.
35 pages. This paper supersedes "Bergman kernels and equilibrium measures for ample line bundles" (apart from section 4 that will be superseded by a sequal to this paper) v2: two pictures added. Fixed some confusing typos in the proof of lemma 4.4
35 pages. This paper supersedes "Bergman kernels and equilibrium measures for ample line bundles" (apart from section 4 that will be superseded by a sequal to this paper) v2: two pictures added. Fixed some confusing typos in the proof of lemma 4.4