Equidistribution results for singular metrics on line bundles

Abstract

Let L 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 associated to the space of square integrable holomorphic sections of the p-th tensor powers of L. Assuming that the singular set of the metric is contained in a compact analytic subset of X and that the logarithm of the Bergman kernel function associated to the p-th tensor power of L (defined outside the singular set) grows like o(p) as p tends to infinity, we prove the following: 1) the k-th power of the Fubini-Study currents converge weakly on the whole X to the k-th power of the curvature current of L. 2) the expectations of the common zeros of a random k-tuple of square integrable holomorphic sections converge weakly in the sense of currents to to the k-th power of the curvature current of L. Here k is so that the codimension of the singular set of the metric is greater or equal as k. Our weak asymptotic condition on the Bergman kernel 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, Kaehler-Einstein metrics on Zariski-open sets, artihmetic quotients) fit into our framework.