Siberian Federal University

We construct a complex of sheaves of multi-logarithmic differential forms on a complex analytic manifold with respect to a reduced complete intersection; and define the residue map as a natural morphism from this complex onto the Barlet complex of regular meromorphic differential forms: It follows then that sections of the Barlet complex can be regarded as a generalization of the residue differential forms defined by Leray. Moreover, we show that the residue map can be described explicitly in terms of certain integration current.