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(XSigma) grows like o(p) as p -> infinity, we prove the following:
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