Properties of the residual circle action on a toric hyperkahler variety

TL;DR

This paper investigates the properties of a residual circle action on toric hyperkähler varieties, focusing on fixed points, equivariant cohomology, and applications to hyperplane arrangements.

Contribution

It computes fixed points and equivariant cohomology of the circle action on toric hyperkähler manifolds and applies these results to deform the Orlik-Solomon algebra.

Findings

Identified fixed points of the S^1 action on M

Computed the equivariant cohomology of M

Deformed the Orlik-Solomon algebra using Z/2-equivariant cohomology

Abstract

We consider a manifold X obtained by a Kahler reduction of C^n, and we define its hyperkahler analogue M as a hyperkahler reduction of T^*C^n = H^n by the same group. In the case where the group is abelian and X is a smooth toric variety, M is a toric hyperkahler manifold, as defined by Bielawski-Dancer, and further studied by Konno and Hausel-Sturmfels. The manifold 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, generic, real hyperplane arrangement, 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, where Z/2 acts by complex conjugation.