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 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.
