Let E and F be Hermitian vector bundles over a complex manifold X and let \(g:E\rightarrow F\) 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 \({\hat{c}}_k(E)\) for \(k >\mathrm{rank \,}E-r\) . We also consider Segre and Chern forms associated with certain singular metrics on E. The multiplicities (Lelong numbers) of the various components of \(M^g\) only depend on the cokernel of the adjoint sheaf morphism \(g^*\) . 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.
