An effective algorithm for the cohomology ring of symplectic reductions

Abstract

Let G be a compact torus acting on a compact symplectic manifold M in a Hamiltonian fashion, and T a subtorus of G. We prove that the kernel of $\kappa:H_G^*(M)\to H^*(M//G)$ is generated by a small number of classes $\alpha\in H_G^*(M)$ satisfying very explicit restriction properties. Our main tool is the equivariant Kirwan map, a natural map from the G-equivariant cohomology of M to the G/T-equivariant cohomology of the symplectic reduction of M by T. We show this map is surjective. This is an equivariant version of the well-known result that the (nonequivariant) Kirwan map $\kappa:H_G^*(M)\to H^*(M//G)$ is surjective. We also compute the kernel of the equivariant Kirwan map, generalizing the result due to Tolman and Weitsman in the case T=G and allowing us to apply their methods inductively. This result is new even in the case that dim T = 1. We close with a worked example: the cohomology ring of the product of two $\C P^2$s, quotiented by the diagonal 2-torus action.