Shadow forms of Brasselet-Goresky-MacPherson

Abstract:

Brasselet, Goresky and MacPherson constructed an explicit morphism, providing a De Rham isomorphism between the intersection homology of a singular variety $X$ and the cohomology of some complex of differential forms, called "shadow forms" and generalizing Whitney forms, on the smooth part of $X$. The coefficients of shadow forms are integrals of Dirichlet type. We find an explicit formula for them; from that follows an alternative proof of Brasselet, Goresky and MacPherson’s theorem. Next, we give a duality formula and a product formula for shadow forms and construct the correct algebra structure, for which shadow forms yield a morphism.