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Residue forms on singular hypersurfaces

**Andrzej Weber**

Math. Sub. Class.: 14 F10, 14F43, 14C30

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 $X$.

Key words:
Residue differential form, canonical singularities, intersection
cohomology.

Contents:
1. Introduction
2. Residue as a differential form
3. Residues and resolution
4. Vanishing of hidden residues
5. Adjoint ideals
6. $L^2$-cohomology
7. Residues and homology
8. Hodge theory
9. Isolated singularities
10. Quasihomogeneous isolated hypersurface
11. Example: elliptic singularity