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Zbl 1033.57016
Naber, Gregory L.
Equivariant localization and stationary phase. (English)
[A] Mladenov, Iva\"ilo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6--15, 2002. Sofia: Coral Press Scientific Publishing. 88-124 (2003). ISBN 954-90618-4-1/pbk

The author discusses Hamiltonian actions on symplectic manifolds and gives a self-contained introduction to the Cartan model of equivariant cohomology. He relates the results to the Cartan theorem asserting that the $G$-equivariant cohomology algebra $H^*_G(M)$ is isomorphic to the de Rham cohomology with complex coefficients of the orbit manifold $M/G$, where $G$ is a compact connected Lie group acting smoothly and freely on a smooth manifold $M$. Then the author proves the major result of the paper, the equivariant localization theorem about computing $\int_M\alpha(\xi)$ for any $G$-equivariantly closed differential form $\alpha$ on $M$ and any nondegenerate element $\xi\in{\germ g}$ for which the associated vector field $\xi^{\#}$ has only isolated zeros, where ${\germ g}$ is the Lie algebra of a compact Lie group $G$ acting smoothly on a compact oriented manifold $M$ of dimension $2k$. As an application of the theorem, the author derives the generalized Duistermaat-Heckman theorem about computing $\int_M e^{i\mu(\xi)}\nu_\omega$ for any compact symplectic manifold $(M,\omega)$ of dimension $2k$ with a Hamiltonian action of $G$ and corresponding symplectic moments given by $\mu:{\germ g}\to C^\infty(M)$, where $M$ is oriented with the Liouville form $\nu_\omega= {1\over k!} \omega^k$.
[Krzysztof Pawałowski (Poznań)]

MSC 2000:: *57R91 Equivariant algebraic topology of manifolds
53D05 Symplectic manifolds, general
53D35 Global theory of symplectic and contact manifolds
55N91 Equivariant homology and cohomology
55P60 Localization and completion

Keywords: compact Lie group; Lie algebra; Hamiltonian action; symplectic manifold; equivariant cohomology; equivariant localization

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