Lisa C. Jeffrey, Augustin-Liviu Mare, Jonathan M. Woolf
The Kirwan map, equivariant Kirwan maps, and their kernels
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
?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
?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(
?K ) and Ker(
?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
?T .
Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter
Print ISSN: 0075-4102
Volume: 2005, 12/2005
Pages: 105 - 127
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