Théorie des résidus de Leray et formes de Barlet sur une intersection complète singulière

Let S be a complex analytic manifold and C⊂S a reduced complete intersection. We construct a complex $\text{Ω}{}_{\mathrm{S}}{}^{\mathrm{•}}\text{(}\text{log}\text{C)}$ of sheaves of the so-called multi-logarithmic differential forms on S with respect to C and define a residue map $\text{res}\text{:Ω}{}_{\mathrm{S}}{}^{\mathrm{•}}\text{(}\text{log}\text{C)→ω}{}_{\mathrm{C}}{}^{\mathrm{•}}$ from this complex onto the Barlet complex ω_C^• of regular meromorphic differential forms on C. The residue map is proved to be a natural morphism between the two complexes; it follows then that sections of the complex ω_C^• may be regarded as a generalization of the residue differential forms defined by Leray. Moreover, we show that the map res can be given explicitly in terms of a certain integration current.