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 γp associated to the space of L2-holomorphic sections of L⊗p. Assuming that the singular set of the metric is contained in a compact analytic subset Σ ofX and that the logarithm of the Bergman density function of L⊗pjXnΣ grows like o(p) as p ! 1, we prove the following: 1) the currents γkp converge weakly on the whole X to c1(L; h)k, where c1(L; h) is the curvature current of h. 2) the expectations of the common zeros of a random k-tuple of L2-holomorphic sections converge weakly in the sense of currents to c1(L; h)k. Here k is so that codim Σ ≥ 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, K?hler-Einstein metrics on Zariski-open sets, arithmetic quotients) fit into our framework.
|Original language||English (US)|
|Number of pages||40|
|Journal||Annales Scientifiques de l'Ecole Normale Superieure|
|State||Published - May 1 2015|
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