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.Comment: 40 page
