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 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].

The first author was supported by a grant from NSERC. The first author’s work was begun during a visit to Harvard University during the spring of 2003, whose support during this period is acknowledged. The second author was supported by the National Science Foundation under agreement DMS-0111298.