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Equivariant localization and stationary phase. (English) Zbl 1033.57016
Mladenov, Ivaïlo 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 (ISBN 954-90618-4-1/pbk). 88-124 (2003).
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 ${H}_{G}^{*}\left(M\right)$ 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 ${\int }_{M}\alpha \left(\xi \right)$ for any $G$-equivariantly closed differential form $\alpha$ on $M$ and any nondegenerate element $\xi \in 𝔤$ for which the associated vector field ${\xi }^{#}$ has only isolated zeros, where $𝔤$ 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 ${\int }_{M}{e}^{i\mu \left(\xi \right)}{\nu }_{\omega }$ for any compact symplectic manifold $\left(M,\omega \right)$ of dimension $2k$ with a Hamiltonian action of $G$ and corresponding symplectic moments given by $\mu :𝔤\to {C}^{\infty }\left(M\right)$, where $M$ is oriented with the Liouville form ${\nu }_{\omega }=\frac{1}{k!}{\omega }^{k}$.
##### MSC:
 57R91 Equivariant algebraic topology of manifolds 53D05 Symplectic manifolds, general 53D35 Global theory of symplectic and contact manifolds 55N91 Equivariant homology and cohomology (algebraic topology) 55P60 Localization and completion (homotopy theory)