2001/10/31 by R. F. Goldin
Mathematics · #math.SG
published as Geom. Func. Anal., Vol. 12 (2002) 567-583 · 16 pages, 4 figures, to appear in Geometric and Functional Analysis
arxiv created 2002/05/07 · arxiv updated 2009/11/30
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 κ:HG^*(M)→ H^*(M//G) is generated by a small number of classes α∈ HG^*(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 κ:HG^*(M)→ H^*(M//G) is surjective. We also compute the kernel of the equivariant Kirwan map, generalizing the result due to Tolman and Weitsman 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 \C P2s, quotiented by the diagonal 2-torus action.