Intersection homology and torus actions

Abstract

in the Bialynicki-Birula decomposition of X (see 15, 6, 10, 13]). The same formula holds for appropriate integers my when C* is replaced by a torus T=(C*)r. In [9] it is shown that the formula (0.1) is valid even when X is singular, provided that the Bialynicki-Birula decomposition is good. The aim of this paper is to generalize (0.1) to the case when X is singular in a different way, which involves replacing ordinary homology by intersection homology (with respect to the middle perversity). However only rational coefficients are considered. When X is nonsingular its intersection homology and ordinary homology coincide, but when X is singular its intersection homology behaves better in many respects than its ordinary cohomology. It is shown that when X is singular, just as when X is nonsingular, the rational intersection homology of X is determined by the action of T on an arbitrarily small neighborhood of the fixed point set XT . As might be expected, the formula (0.1) does not carry over directly when intersection homology replaces ordinary homology. The terms Hi_2m (F ; Z) appearing in the right-hand side are replaced by hypercohomology groups of certain complexes of sheaves over the F which depend upon how the F meet the singularities of X (see They y orem 2.3). ? 1 of this paper contains a review of a proof of (0.1) for rational coefficients which uses equivariant Morse theory. In ?2 it is shown how this proof can be extended to apply to singular varieties when intersection homology is used. The