Multi-Logarithmic Differential Forms on Complete Intersections

We construct a complex Omega(.)(S)(log C) of sheaves of multi-logarithmic differential forms on a complex analytic manifold S with respect to a reduced complete intersection C subset of S; and define the residue map as a natural morphism from this complex onto the Barlet complex omega(.)(C) of regular meromorphic differential forms on C. 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.