The Kirwan Map for Singular Symplectic Quotients

Let M be a Hamiltonian K ‐space with proper moment map μ. The symplectic quotient X = μ −1 (0)/ K is a singular stratified space with a symplectic structure on the strata. In this paper we generalise the Kirwan map, which maps the K equivariant cohomology of μ −1 (0) to the middle perversity intersection cohomology of X , to this symplectic setting. The key technical results which allow us to do this are Meinrenken's and Sjamaar's partial desingularisation of singular symplectic quotients and a decomposition theorem, proved in Section 2 of this paper, exhibiting the intersection cohomology of a ‘symplectic blowup’ of the singular quotient X along a maximal depth stratum as a direct sum of terms including the intersection cohomology of X .