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Localization for nonabelian group actions

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arxiv alg-geom/9307001 v3 pith:HGXQOS5G submitted 1993-07-06 alg-geom math.AG

L.C. Jeffrey , F.C. Kirwan This is my paper · ORCID
classification alg-geommath.AG
keywords xredformulakappaclasscohomologycompactgroupmoment
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abstract

Suppose $X$ is a compact symplectic manifold acted on by a compact Lie group $K$ (which may be nonabelian) in a Hamiltonian fashion, with moment map $\mu: X \to {\rm Lie}(K)^*$ and Marsden-Weinstein reduction $\xred = \mu^{-1}(0)/K$. There is then a natural surjective map $\kappa_0$ from the equivariant cohomology $H^*_K(X) $ of $X$ to the cohomology $H^*(\xred)$. In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of $\xred$ of any $\eta_0 \in H^*(\xred)$ whose degree is the dimension of $\xred$, provided that $0$ is a regular value of the moment map $\mu$ on $X$. This formula is given in terms of any class $\eta \in H^*_K(X)$ for which $\kappa_0(\eta ) = \eta_0$, and involves the restriction of $\eta$ to $K$-orbits $KF$ of components $F \subset X$ of the fixed point set of a chosen maximal torus $T \subset K$. Since $\kappa_0$ is

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Cited by 6 Pith papers

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