Hyperpolygon spaces and their cores

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