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 Bielawski and Dancer, 2000, and further studied by Konno and by Hausel and Sturmfels. 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 afine 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.

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