Properties of the residual circle action on a toric hyperkahler variety

Abstract

We consider a manifold X obtained by a Kahler reduction of C^n, and we define its hyperkahler analogue M as a hyperkahler reduction of T^*C^n = H^n by the same group. In the case where the group is abelian and X is a smooth toric variety, M is a toric hyperkahler manifold, as defined by Bielawski-Dancer, and further studied by Konno and Hausel-Sturmfels. The manifold M carries a natural action of S^1, induced by the scalar action of S^1 on the fibers of T^*C^n. In this paper we study this action, computing its fixed points and its equivariant cohomology. As an application, we use the associated Z/2 action on the real locus of M to compute a deformation of the Orlik-Solomon algebra of a smooth, generic, real hyperplane arrangement, depending nontrivially on the affine structure of the arrangement. This deformation is given by the Z/2-equivariant cohomology of the complement of the complexification, where Z/2 acts by complex conjugation.