Hyperpolygon spaces and their cores

Given an nn-tuple of positive real numbers (α1,…,αn)(\alpha _1,\ldots ,\alpha _n), Konno (2000) defines the hyperpolygon space X(α)X(\alpha ), a hyperkähler analogue of the Kähler variety M(α)M(\alpha ) parametrizing polygons in R3\mathbb {R}^3 with edge lengths (α1,…,αn)(\alpha _1,\ldots ,\alpha _n). The polygon space M(α)M(\alpha ) can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, X(α)X(\alpha ) is the hyperkähler quiver variety defined by Nakajima. A quiver variety admits a natural C∗\mathbb {C}^*-action, and the union of the precompact orbits is called the core. We study the components of the core of X(α)X(\alpha ), interpreting each one as a moduli space of pairs of polygons in R3\mathbb {R}^3 with certain properties. Konno gives a presentation of the cohomology ring of X(α)X(\alpha ); we extend this result by computing the C∗\mathbb {C}^*-equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.