Poincare-Lelong Type Formulas and Segre Numbers

Journal article, 2025

Let E and F be Hermitian vector bundles over a complex manifold X and let g : E -> F be a holomorphic morphism. We prove a Poincar & eacute;-Lelong type formula with a residue term M-g. The currents Mg 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 c(k)(<^>)(E) for k>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 Mg only depend on the cokernel of the adjoint sheaf morphism g(& lowast;). 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.