The Kirwan map, equivariant Kirwan maps, and their kernels

DOI10.1515/CRLL.2005.2005.589.105zbMATH Open1092.53061arXivmath/0211297OpenAlexW2070521438MaRDI QIDQ5713251

Augustin-Liviu Mare, Lisa C. Jeffrey, Jonathan Woolf

Publication date: 13 December 2005

Published in: Journal für die reine und angewandte Mathematik (Crelles Journal) (Search for Journal in Brave)

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 $kapp{a}_K$. We start with the case when K is a torus T: we determine the kernel of the equivariant Kirwan map (defined by Goldin in [Go]) corresponding to a generic circle S in T, and show how to recover from this the kernel of $kapp{a}_T$, as described by Tolman and Weitsman. (In the situation when the fixed point set of the torus action is finite, similar results have been obtained in our previous papers [Je], [Je-Ma]). For a compact nonabelian Lie group K we will use the ``non-abelian localization formula of [Je-Ki1] and [Je-Ki2] to establish relationships -- some of them obtained by Tolman and Weitsman in [To-We] -- between $ker(kapp{a}_K)$ and $ker(kapp{a}_T)$, where T is a maximal torus in K. An Appendix generalizes Theorem 1.8 to the case of singular values of $kapp{a}_T$.

Full work available at URL: https://arxiv.org/abs/math/0211297

zbMATH Keywords

symplectic reductionequivariant cohomologyHamiltonian action

Mathematics Subject Classification ID

Momentum maps; symplectic reduction (53D20)

Related Items (2)

The Kernel of the Equivariant Kirwan Map and the Residue Formula ⋮ The residue formula and the Tolman-Weitsman theorem

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