The Multiple Residue and the Weight Filtration on the Logarithmic de Rham Complex

We study the multiple residue of logarithmic differential forms with poles along a reducible divisor and compute the kernel and the image of the multiple residue map. As an application we describe the weight filtration on the logarithmic de Rham complex for divisors whose irreducible components are given locally by a regular sequence of holomorphic functions. In particular, this allows us to compute the mixed Hodge structure on the cohomology of the complement of divisors of certain types without the use of theorems on resolution of singularities and the standard reduction to the case of normal crossings.