Intersection cohomology of quotients of nonsingular varieties

Abstract

Let M ⊂ P be a nonsingular projective variety acted on by a connected complex reductive group G = KC via a homomorphism G → GL(n + 1) which is the complexification of a homomorphism K → U(n+1). Geometric invariant theory (GIT) gives us a recipe to form a quotient φ : M → M//G of the set of semistable points and many interesting spaces in algebraic geometry are constructed in this manner [MFK94]. But often this quotient is singular and hence intersection cohomology with middle perversity is an important topological invariant. The purpose of this paper is to present a way to compute the middle perversity intersection cohomology of the singular quotients. The choice of an embedding M ⊂ P provides us with a moment map for the K-action and the Morse stratification of M with respect to its norm square is Kequivariantly perfect, with the unique open dense stratum M [Kir84]. Hence one can compute the Betti numbers for the equivariant cohomology H∗ K(M ) as well as the cup product, at least in principle. See also [Kir92]. When G acts locally freely on M, we get an orbifold M//G = M/G and H∗ K(M ) ∼= H∗(M//G) ∼= IH∗(M//G).