1993/07/06 by Jeffrey, L. C., Kirwan, F. C. · 4 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.alg-geom/9307001
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 μ: X → \rm Lie(K)^* and Marsden-Weinstein reduction \xred = μ-1(0)/K. There is then a natural surjective map κ0 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 η0 ∈ H^*(\xred) whose degree is the dimension of \xred, provided that 0 is a regular value of the moment map μ on X. This formula is given in terms of any class η∈ H^*K(X) for which κ0(η) = η0, and involves the restriction of η to K-orbits KF of components F ⊂ X of the fixed point set of a chosen maximal torus T ⊂ K. Since κ0 is