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The Kirwan Map for Singular Symplectic Quotients
Author(s) -
Kiem Young-Hoon,
Woolf Jonathan
Publication year - 2006
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610705022453
Subject(s) - moment map , symplectic geometry , mathematics , intersection homology , pure mathematics , symplectomorphism , quotient , symplectic representation , cohomology , symplectic manifold , symplectic matrix , equivariant cohomology , symplectic vector space , quantum cohomology , algebra over a field , mathematical analysis
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 .