Localization theorems by symplectic cuts
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Abstract
Given a compact symplectic manifold M with the Hamiltonian action of a torus T, let zero be a regular value of the moment map, and M 0 the symplectic reduction at zero. Denote by κ 0 the Kirwan map H T * (M) → H*(M 0). For an equivariant cohomology class η ∊ H T * (M) we present new localization formulas which express $$ \int_{M_0 } \kappa 0(\eta ) $$ as sums of certain integrals over the connected components of the fixed point set M T. To produce such a formula we apply a residue operation to the Atiyah-Bott-Berline-Vergne localization formula for an equivariant form on the symplectic cut of M with respect to a certain cone, and then, if necessary, iterate this process using other cones. When all cones used to produce the formula are one-dimensional we recover, as a special case, the localization formula of Guillemin and Kalkman [GK]. Using similar ideas, for a special choice of the cone (whose dimension is equal to that of T) we give a new proof of the Jeffrey-Kirwan localization formula [JK1].
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