Residues of holomorphic sections and lelong currents

LetZ be the zero set of a holomorphic sectionf of a Hermitian vector bundle. It is proved that the current of integration over the irreducible components ofZ of top degree, counted with multiplicities, is a product of a residue factorR ^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 powersk. The author was partially supported by the Swedish Research Council.