Properties of the residual circle action on a hypertoric variety

Abstract

We consider an orbifold X obtained by a Khler reduction of C n , and we define its "hyperkhler analogue" M as a hyperkhler reduction of T * C n = H n by the same group. In the case where the group is abelian and X is a toric variety, M is a toric hyperkhler orbifold, as defined in Bielawski and Dancer, 2000, and further studied by Konno and by Hausel and Sturmfels. The variety 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, real hyperplane arrangement H, 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 of H, where Z 2 acts by complex conjugation.