A generalized Poincaré-Lelong formula - Andersson, Mats (MATHEMATICA SCANDINAVICA, 2007)
We prove a 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 XZ 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^cW=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-Fantappiè-Leray type.