Dan Coman and George Marinescu
Abstract: 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 _p associated to the space of L^2-holomorphic sections of L^p. Assuming that the singular set of the metric is contained in a compact analytic subset of X and that the logarithm of the Bergman density function of L^p_X grows like o(p) as p, we prove the following: 1) the currents _p^k converge 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 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, Kähler-Einstein metrics on Zariski-open sets, arithmetic quotients) fit into our framework.
Journal: Annales scientifiques de l'ENS 48, fascicule 3 (2015), 497-536