Localization theorems by symplectic cuts

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 \kappa_0 the Kirwan map H^*_T(M)-> H^*(M_0). For an equivariant cohomology class \eta\in 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. 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.