Abstract.
Let G be a compact torus acting on a compact symplectic manifold M in a Hamiltonian fashion, and T a subtorus of G. We prove that the kernel of \( \kappa : H^\ast_G(M) \to H^\ast(M//G) \) is generated by a small number of classes \( \alpha \in H^\ast_G(M) \) satisfying very explicit restriction properties. Our main tool is the equivariant Kirwan map, a natural map from the G-equivariant cohomology of M to the G/T-equivariant cohomology of the symplectic reduction of M by T . We show this map is surjective. This is an equivariant version of the well-known result that the (nonequivariant) Kirwan map \( \kappa : H^\ast_G(M) \to H^\ast(M//G) \) is surjective. We also compute the kernel of the equivariant Kirwan map, generalizing the result due to Tolman and Weitsman [TW] in the case T = G and allowing us to apply their methods inductively. This result is new even in the case that dim T = 1. We close with a worked example: the cohomology ring of the product of two \( {\Bbb C}P^2{\rm s} \), quotiented by the diagonal 2-torus action.
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Submitted: September 2001, Revised: December 2001, Revised: February 2002.
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Goldin, R. An effective algorithm for the cohomology ring of symplectic reductions . GAFA, Geom. funct. anal. 12, 567–583 (2002). https://doi.org/10.1007/s00039-002-8257-5
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DOI: https://doi.org/10.1007/s00039-002-8257-5