The equivariant cohomology of hypertoric varieties and their real loci

Abstract

Let M be a Hamiltonian T space with a proper moment map, bounded below in some component. In this setting, we give a combinatorial description of the T -equivariant cohomology of M, extending results of Goresky, Kottwitz and MacPherson and techniques of Tolman and Weitsman. Moreover, when M is equipped with an antisymplectic involution σ anticommuting with the action of T , we also extend to this noncompact setting the “mod 2” versions of these results to the real locus Q := M of M. We give applications of these results to the theory of hypertoric varieties.