A Generalized Poincaré-lelong Formula and Explicit Green Currents

We prove the following generalization of the classical Poincaré-Lelong formula. Given a holomorphic section f , with zero set Z, of a Hermitian vector bundle E → X , let S be the line bundle over X \Z spanned by f and let Q = E/S. We prove that the Chern form c(DQ) is locally integrable and closed in X and that there is a current W such that ddW = c(DE)− c(DQ)− M, where M is a current with support on Z. In particular, the top Bott-Chern class is represented by a current with support on Z. By means of this formula we make a new construction of explicit Green currents of algebraic cycles.