A generalized Poincaré-Lelong formula

2004/12/22 by Mats Andersson, Andersson, Mats
Mathematics · #32 A 26 #32 C 30 #32 U 40 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:26 #msc:30 #msc:32 #msc:40

paper · pdf · doi:10.48550/arxiv.math/0412446

arxiv created 2010/03/15 · arxiv updated 2010/03/16

Abstract

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 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^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.

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