Properties of the residual circle action on a hypertoric variety

Abstract. We consider an orbifold X obtained by a Kähler reduction of Cn, and we define its “hyperkähler analogue” M as a hyperkähler reduction of T ∗Cn ∼= Hn by the same group. In the case where the group is abelian and X is a toric variety, M is a toric hyperkähler orbifold, as defined in [BD], and further studied in [K1, K2] and [HS]. The variety M carries a natural action of S1, induced by the scalar action of S1 on the fibers of T ∗Cn. In this paper we study this action, computing its fixed points and its equivariant cohomology. As an application, we use the associated Z2 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 Z2-equivariant cohomology of the complement of the complexification of H, where Z2 acts by complex conjugation.