Hyperpolygon spaces and their cores

Authors:
Megumi Harada and Nicholas Proudfoot

Journal:
Trans. Amer. Math. Soc. **357** (2005), 1445-1467

MSC (2000):
Primary 53C26; Secondary 16G20, 14D20

Published electronically:
September 23, 2004

MathSciNet review:
2115372

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Abstract | References | Similar Articles | Additional Information

Abstract: Given an -tuple of positive real numbers , Konno (2000) defines the *hyperpolygon space* , a hyperkähler analogue of the Kähler variety parametrizing polygons in with edge lengths . The 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, is the hyperkähler quiver variety defined by Nakajima. A quiver variety admits a natural -action, and the union of the precompact orbits is called the *core*. We study the components of the core of , interpreting each one as a moduli space of pairs of polygons in with certain properties. Konno gives a presentation of the cohomology ring of ; we extend this result by computing the -equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.

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Additional Information

**Megumi Harada**

Affiliation:
Department of Mathematics, University of Toronto, Ontario, Canada M5S 3G3

Email:
megumi@math.toronto.edu

**Nicholas Proudfoot**

Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720

Email:
proudf@math.berkeley.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-04-03522-6

Keywords:
Hyperk\"ahler geometry,
polygon space,
quiver variety,
equivariant cohomology

Received by editor(s):
August 23, 2003

Received by editor(s) in revised form:
October 1, 2003

Published electronically:
September 23, 2004

Article copyright:
© Copyright 2004
American Mathematical Society