Residues of holomorphic sections and lelong currents

Abstract

LetZ be the zero set of a holomorphic section f of a Hermitian vector bundle. It is proved that the current of integration over the irreducible components of Z of top degree, counted with multiplicities, is a product of a residue factor R^f and a “Jacobian factor”. There is also a relation to the Monge-Ampère expressions (dd^c log|f|)^k, which we define for all positive powers k.