The Kirwan map, equivariant Kirwan maps, and their kernels

Abstract

Consider a Hamiltonian action of a compact Lie group K on a compact symplectic manifold. We find descriptions of the kernel of the Kirwan map corresponding to a regular value of the moment map κ K . We start with the case when K is a torus T : we determine the kernel of the equivariant Kirwan map (defined by Goldin in [ R. F. Goldin , An effective algorithm for the cohomology ring of symplectic reductions, Geom. Func. Anal. 12 (2002), 567–583]) corresponding to a generic circle S ⊂ T , and show how to recover from this the kernel of κ T , as described by Tolman and Weitsman in [ S. Tolman and J. Weitsman , The cohomology rings of symplectic quotients, Comm. Anal. Geom. 11 No. 4 (2003), 751–773]. (In the situation when the fixed point set of the torus action is finite, similar results have been obtained in our previous papers [ L. C. Jeffrey , The residue formula and the Tolman-Weitsman theorem, J. reine angew. Math. 562 (2003), 51–58], [ L. C. Jeffrey and A.-L. Mare , The kernel of the equivariant Kirwan map and the residue formula, Quart. J. Math. Oxford 54 (2004), 435–444].) For a compact nonabelian Lie group K we will use the ‘‘non-abelian localization formula’’ of [ L. C. Jeffrey and F. C. Kirwan , Localization for nonabelian group actions, Topology 34 (1995), 291–327] and [ L. C. Jeffrey and F. C. Kirwan , Localization and the quantization conjecture, Topology 36 (1995), 647–693] to establish relationships—some of them obtained by Tolman and Weitsman in [ S. Tolman and J. Weitsman , The cohomology rings of symplectic quotients, Comm. Anal. Geom. 11 No. 4 (2003), 751–773]—between Ker( κ K ) and Ker( κ T ), where T ⊂ K is a maximal torus. In the appendix we prove that the same relationships remain true in the case when 0 is no longer a regular value of μ T .