Poincaré-Lelong type formulas and Segre numbers

Let $E$and $F$ be Hermitian vector bundles over a complex manifold $X$ and let $g\text{\hspace{0.17em}⁣}:E\to F$ be a holomorphic morphism. We prove a Poincaré-Lelong type formula with a residue term $M^g$. The currents $M^g$ so obtained have an expected functorial property. We discuss various applications: If $F$ has a trivial holomorphic subbundle of rank $r$ outside the analytic set $Z$, then we get currents with support on $Z$ that represent the Bott-Chern classes ${\widehat{c}}_k(E)$ for $k>\text{rank}E-r$. We also consider Segre and Chern forms associated with certain singular metrics on $F$. The multiplicities (Lelong numbers) of the various components of $M^g$ only depend on the cokernel of the adjoint sheaf morphism $g^{\ast }$. This leads to a notion of distinguished varieties and Segre numbers of an arbitrary coherent sheaf, generalizing these notions, in particular the Hilbert-Samuel multiplicity, in case of an ideal sheaf.