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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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: 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.


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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
PII: S 0002-9947(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