A generalized Poincare-Lelong formula

Journal article, 2007

We prove a generalization of the classical Poincare-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. Then the Chern form c(D-Q) is locally integrable and closed in X and there is a current W such that dd(c)W = c(D-E) - c(D-Q) - 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. We discuss positivity of these currents, and we also reveal a close relation with principal value and residue currents of Cauchy-Fantappie-Leray type.