Residue forms on singular hypersurfaces

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The purpose of this paper is to point out a relation between the canonical sheaf and the intersection complex of a singular algebraic variety. We focus on the hypersurface case. Let M be a complex manifold, X⊂ M a singular hypersurface. We study residues of top-dimensional meromorphic forms with poles along X. Applying resolution of singularities sometimes we are able to construct residue classes either in L²-cohomology of X or in the intersection cohomology. The conditions allowing to construct these classes coincide. They can be formulated in terms of the weight filtration. Finally, provided that these conditions hold, we construct in a canonical way a lift of the residue class to cohomology of X.