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Real loci of symplectic reductions


Authors: R. F. Goldin and T. S. Holm
Journal: Trans. Amer. Math. Soc. 356 (2004), 4623-4642
MSC (2000): Primary 53D20
DOI: https://doi.org/10.1090/S0002-9947-04-03504-4
Published electronically: April 27, 2004
MathSciNet review: 2067136
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Abstract: Let $M$ be a compact, connected symplectic manifold with a Hamiltonian action of a compact $n$-dimensional torus $T$. Suppose that $M$ is equipped with an anti-symplectic involution $\sigma$ compatible with the $T$-action. The real locus of $M$ is the fixed point set $M^\sigma$ of $\sigma$. Duistermaat introduced real loci, and extended several theorems of symplectic geometry to real loci. In this paper, we extend another classical result of symplectic geometry to real loci: the Kirwan surjectivity theorem. In addition, we compute the kernel of the real Kirwan map. These results are direct consequences of techniques introduced by Tolman and Weitsman. In some examples, these results allow us to show that a symplectic reduction $M/ /T$ has the same ordinary cohomology as its real locus $(M/ /T)^{\sigma_{red}}$, with degrees halved. This extends Duistermaat's original result on real loci to a case in which there is not a natural Hamiltonian torus action.


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Additional Information

R. F. Goldin
Affiliation: Mathematical Sciences, George Mason University, MS 3F2, 4400 University Dr., Fairfax, Virgina 22030
Email: rgoldin@math.gmu.edu

T. S. Holm
Affiliation: Department of Mathematics, University of California Berkeley, 813 Evans Hall, Berkeley, California 94720
Email: tsh@math.berkeley.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03504-4
Keywords: Real locus, symplectic reduction
Received by editor(s): April 4, 2003
Received by editor(s) in revised form: July 24, 2003
Published electronically: April 27, 2004
Additional Notes: The first author was partially supported by NSF grant DMS-0305128. This research was partially conducted during the period when the second author served as a Clay Mathematics Institute Liftoff Fellow. The second author was also partially supported by an NSF postdoctoral fellowship
Article copyright: © Copyright 2004 American Mathematical Society

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