Hyperpolygon spaces and their cores
Authors:
Megumi Harada and Nicholas Proudfoot
Journal:
Trans. Amer. Math. Soc. 357 (2005), 14451467
MSC (2000):
Primary 53C26; Secondary 16G20, 14D20
Published electronically:
September 23, 2004
MathSciNet review:
2115372
Fulltext PDF Free Access
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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.
 [AB]
M.
F. Atiyah and R.
Bott, The moment map and equivariant cohomology, Topology
23 (1984), no. 1, 1–28. MR 721448
(85e:58041), http://dx.doi.org/10.1016/00409383(84)900211
 [BD]
Roger
Bielawski and Andrew
S. Dancer, The geometry and topology of toric hyperkähler
manifolds, Comm. Anal. Geom. 8 (2000), no. 4,
727–760. MR 1792372
(2002c:53078)
 [HP]
M. Harada and N. Proudfoot.
Properties of the residual circle action on a hypertoric variety. To appear in Pacific Journal of Mathematics.
 [HS]
T. Hausel and B. Sturmfels.
Toric hyperkähler varieties. math.AG/0203096.
 [HK1]
JeanClaude
Hausmann and Allen
Knutson, Polygon spaces and Grassmannians, Enseign. Math. (2)
43 (1997), no. 12, 173–198. MR 1460127
(98e:58035)
 [HK2]
J.C.
Hausmann and A.
Knutson, The cohomology ring of polygon spaces, Ann. Inst.
Fourier (Grenoble) 48 (1998), no. 1, 281–321
(English, with English and French summaries). MR 1614965
(99a:58027)
 [Ki]
Frances
Clare Kirwan, Cohomology of quotients in symplectic and algebraic
geometry, Mathematical Notes, vol. 31, Princeton University
Press, Princeton, NJ, 1984. MR 766741
(86i:58050)
 [Kl]
Alexander
A. Klyachko, Spatial polygons and stable configurations of points
in the projective line, Algebraic geometry and its applications
(Yaroslavl′, 1992) Aspects Math., E25, Vieweg, Braunschweig, 1994,
pp. 67–84. MR 1282021
(95k:14015)
 [K1]
Hiroshi
Konno, Cohomology rings of toric hyperkähler manifolds,
Internat. J. Math. 11 (2000), no. 8, 1001–1026.
MR
1797675 (2001k:53089), http://dx.doi.org/10.1142/S0129167X00000490
 [K2]
Hiroshi
Konno, On the cohomology ring of the hyperKähler analogue of
the polygon spaces, Integrable systems, topology, and physics (Tokyo,
2000) Contemp. Math., vol. 309, Amer. Math. Soc., Providence, RI,
2002, pp. 129–149. MR 1953356
(2003k:53111), http://dx.doi.org/10.1090/conm/309/05345
 [N1]
Hiraku
Nakajima, Instantons on ALE spaces, quiver varieties, and KacMoody
algebras, Duke Math. J. 76 (1994), no. 2,
365–416. MR 1302318
(95i:53051), http://dx.doi.org/10.1215/S0012709494076138
 [N2]
Hiraku
Nakajima, Varieties associated with quivers, Representation
theory of algebras and related topics (Mexico City, 1994), CMS Conf.
Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996,
pp. 139–157. MR 1388562
(97m:16022)
 [N3]
Hiraku
Nakajima, Quiver varieties and KacMoody algebras, Duke Math.
J. 91 (1998), no. 3, 515–560. MR 1604167
(99b:17033), http://dx.doi.org/10.1215/S0012709498091207
 [N4]
Hiraku
Nakajima, Quiver varieties and
finitedimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), no. 1, 145–238. MR 1808477
(2002i:17023), http://dx.doi.org/10.1090/S0894034700003532
 [AB]
 M. Atiyah and R. Bott.
The moment map and equivariant cohomology. Topology 23 (1984) no. 1, 128. MR 85e:58041
 [BD]
 R. Bielawski and A. Dancer.
The geometry and topology of toric hyperkähler manifolds. Comm. Anal. Geom. 8 (2000), 727760. MR 2002c:53078
 [HP]
 M. Harada and N. Proudfoot.
Properties of the residual circle action on a hypertoric variety. To appear in Pacific Journal of Mathematics.
 [HS]
 T. Hausel and B. Sturmfels.
Toric hyperkähler varieties. math.AG/0203096.
 [HK1]
 JC. Hausmann, A. Knutson.
Polygon spaces and Grassmannians. L'Enseignement Mathématique. 43 (1997), 173198. MR 98e:58035
 [HK2]
 JC. Hausmann, A. Knutson.
The cohomology ring of polygon spaces. Ann. Inst. Fourier, Grenoble 48 (1998), no. 1, 281321. MR 99a:58027
 [Ki]
 F. Kirwan.
Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes 31, Princeton University Press, 1984. MR 86i:58050
 [Kl]
 A. Klyachko.
Spatial polygons and stable configurations of points in the projective line. Algebraic geometry and its applications (Yaroslavl, 1992), Aspects Math., Vieweg, Braunschweig (1994), 6784. MR 95k:14015
 [K1]
 H. Konno.
Cohomology rings of toric hyperkähler manifolds. Internat. J. Math. 11 (2000), no. 8, 10011026. MR 2001k:53089
 [K2]
 H. Konno.
On the cohomology ring of the HyperKähler analogue of the Polygon Spaces. Integrable systems, topology, and physics (Tokyo, 2000), 129149, Contemp. Math., 309, Amer. Math. Soc., Providence, RI, 2002. MR 2003k:53111
 [N1]
 H. Nakajima.
Instantons on ALE spaces, quiver varieties, and KacMoody algebras. Duke Mathematical Journal 72 (1994) no. 2, 365416. MR 95i:53051
 [N2]
 H. Nakajima.
Varieties associated with quivers. Canadian Math. Soc. Conf. Proc. 19 (1996), 139157. MR 97m:16022
 [N3]
 H. Nakajima.
Quiver varieties and KacMoody algebras. Duke Math. J. 91 (1998), no. 3, 515560. MR 99b:17033
 [N4]
 H. Nakajima.
Quiver varieties and finite dimensional representations of quantum affine algebras. J. Amer. Math. Soc. 14 (2001), no. 1, 145238 (electronic). MR 2002i:17023
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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/S0002994704035226
PII:
S 00029947(04)035226
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
