Residue forms on singular hypersurfaces

Details Residue forms on singular hypersurfaces Residue forms on singular hypersurfaces

2003-01-27

arxiv.org/abs/math/0301313v4

Mathematics

Algebraic Geometry

Final version, lots of remarks added, to appear in Michigan Math. J

Scientific paper

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, $X\subset M$ 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 $L^2$-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 $X$.

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