Segre forms of singular metrics on vector bundles and Lelong numbers

Let $E\to X$ be a holomorphic vector bundle. We consider a class of a singular Hermitian metrics on $E$ with analytic singularities that contains all Griffiths negative such metrics. One can define, given a smooth reference metric $h_0$, a current $s(E,h,h_0)$ 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 $s(E,h,h_0)$ is the limit of the Segre forms of a sequence of smooth metrics if the metric is smooth outside the degeneracy locus, and in general as a limit of Segre forms of metrics with empty degeneracy locus. One can also define an associated Chern form $c(E,h,h_0)$. We prove that the Lelong numbers of $s(E,h,h_0)$ and $c(E,h,h_0)$ are integers if the singularities are integral, and non-negative for $s(E,h,h_0)$.