Hyperpolygon spaces and their cores

Given an $n$-tuple of positive real numbers $(\alpha_1,\ldots,\alpha_n)$, 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 $(\alpha_1,\ldots,\alpha_n)$. 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 $\mathbb{C} ^*$-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 $\mathbb{C} ^*$-equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.