Properties of the residual circle action on a hypertoric variety

We consider an orbifold X obtained by a Kähler reduction of Cn, and we define its “hyperkähler analogue ” M as a hy-perkähler 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 Sturm-fels. The varietyM 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 equivari-ant cohomology. As an application, we use the associated Z2 action on the real locus ofM to compute a deformation of the Orlik-Solomon algebra of a smooth, real hyperplane arrange-ment 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. In order to construct a toric variety as a Kähler quotient of Cn by a torus, one begins with the combinatorial data of an arrangement H of n cooriented, rational, affine hyperplanes in Rd. The normal vectors to these hyperplanes determine a subtorus T k ⊆ Tn (k = n − d), and the affine structure determines a value α ∈ (tk) ∗ at which to reduce, so that we may define X = Cn//αT k. Using the same combinatorial data, one can also con-struct a hypertoric variety,1 which is defined as the hyperkähler quotient M = Hn///(α,0)T k of Hn ∼ = T ∗Cn by the induced action of the same subtorus T k ⊆ Tn [BD]. It is well-known that the toric variety X does not retain all of the information of H; indeed, it depends only on the polyhedron ∆ obtained by intersecting the half-spaces associated to each of the cooriented hyperplanes. Thus it is always possible to add an extra hyperplane to H without changing X. In contrast, the hypertoric variety M remembers the number of hyperplanes in H, but its equivariant diffeomorphism type de-pends neither on the coorientations nor on the affine structure of H (se