Hyperpolygon spaces and their cores

Given an n-tuple of positive real numbers, Konno defines an algebraic variety called a hyperpolygon space, a hyperkahler analogue of the Kahler variety parametrizing spacial polygons with fixed edge lengths. The ordinary polygon space can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, the hyperpolygon space is the hyperkahler quiver variety defined by Nakajima. A quiver variety admits a natural action of the nonzero complex numbers, and the union of the precompact orbits is called the core. We study the components of the core of the hyperpolygon space, interpreting each one as a moduli space of pairs of spatial polygons with certain properties. Konno gives a presentation of the cohomology ring of the hyperpolygon space; we extend this result by computing the circle-equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.