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Bergman kernels and equilibrium measures for line bundles over projective manifolds
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 131, Number 5, October 2009
- pp. 1485-1524
- 10.1353/ajm.0.0077
- Article
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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 $L$.