Hyperpolygon spaces and their cores
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- by Megumi Harada and Nicholas Proudfoot PDF
- Trans. Amer. Math. Soc. 357 (2005), 1445-1467 Request permission
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.References
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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
- MR Author ID: 689525
- Email: proudf@math.berkeley.edu
- Received by editor(s): August 23, 2003
- Received by editor(s) in revised form: October 1, 2003
- Published electronically: September 23, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 1445-1467
- MSC (2000): Primary 53C26; Secondary 16G20, 14D20
- DOI: https://doi.org/10.1090/S0002-9947-04-03522-6
- MathSciNet review: 2115372