Segre forms of singular metrics on vector bundles and Lelong numbers

Let E→ X be a holomorphic vector bundle with a singular Griffiths negative Hermitian metric h with analytic singularities. One can define, given a smooth reference metric h₀, a current s(E,h,h₀) that is called the associated Segre form, which defines the expected Bott-Chern class and coincides with the usual Segre form of h where it is smooth. We prove that s(E,h,h₀) is the limit of the Segre forms of a sequence of smooth metrics if the metric is smooth outside the unbounded locus, and in general as a limit of Segre forms of metrics with empty unbounded locus. One can also define an associated Chern form c(E,h,h₀) and we prove that the Lelong numbers of s(E,h,h₀) and c(E,h,h₀) are integers if the singularities are integral, and non-negative for s(E,h,h₀). We also extend these results to a slightly larger class of singular metrics with analytic singularities.