EQUIDISTRIBUTION RESULTS FOR SINGULAR METRICS ON LINE BUNDLES

Let (L, h) be a holomorphic line bundle with a positively curved singular Hermitian metric over a complex manifold X. One can define naturally the sequence of Fubini-Study currents gamma p associated to the space of L-2-holomorphic sections of L-circle times p. Assuming that the singular set of the metric is contained in a compact analytic subset Sigma of X and that the logarithm of the Bergman density function of L-circle times p\(X\Sigma) grows like o(p) as p -> infinity, we prove the following: 1) the currents converge gamma(k)(p) weakly on the whole X to c(1) (L, h)(k), where c(1) (L, h) is the curvature current of h. 2) the expectations of the common zeros of a random k-tuple of L-2-holomorphic sections converge weakly in the sense of currents to c(1) (L,h)(k). Here k is so that codim Sigma >= k. Our weak asymptotic condition on the Bergman density function is known to hold in many cases, as it is a consequence of its asymptotic expansion. We also prove it here in a quite general setting. We then show that many important geometric situations (singular metrics on big line bundles, Kahler-Einstein metrics on Zariski-open sets, arithmetic quotients) fit into our framework.