Hyperpolygon spaces and their cores

Abstract. Given an n-tuple of positive real numbers (α1, . . . , αn), Konno [K2] defines the hyperpolygon space X(α), a hyperkähler analogue of the Kähler variety M(α) parametrizing polygons in R3 with edge lengths (α1, . . . , αn). The polygon space M(α) 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) is the hyperkähler quiver variety defined by Nakajima [N1, N2]. A quiver variety admits a natural C∗-action, and the union of the precompact orbits is called the core. We study the components of the core of X(α), interpreting each one as a moduli space of pairs of polygons in R3 with certain properties. Konno gives a presentation of the cohomology ring of X(α); we extend this result by computing the C∗-equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.