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All the GIT quotients at once
Author:
Nicholas Proudfoot
Journal:
Trans. Amer. Math. Soc. 363 (2011), 1687-1698
MSC (2010):
Primary 14L24; Secondary 14L30, 53D20
Posted:
November 2, 2010
MathSciNet review:
2746660
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Abstract: Let be an algebraic torus acting on a smooth variety . We study the relationship between the various GIT quotients of and the symplectic quotient of the cotangent bundle of .
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Sternberg, The coefficients of the Duistermaat-Heckman polynomial
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Hiraku
Nakajima, Quiver varieties and
finite-dimensional 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/S0894-0347-00-00353-2
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Nicholas
Proudfoot, Geometric invariant theory and projective toric
varieties, Snowbird lectures in algebraic geometry, Contemp. Math.,
vol. 388, Amer. Math. Soc., Providence, RI, 2005,
pp. 161–167. MR 2182896
(2006j:14065), http://dx.doi.org/10.1090/conm/388/07263
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Nicholas
Proudfoot and Benjamin
Webster, Intersection cohomology of hypertoric
varieties, J. Algebraic Geom.
16 (2007), no. 1,
39–63. MR
2257319 (2008m:14037), http://dx.doi.org/10.1090/S1056-3911-06-00448-6
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Francisco
Santos, Triangulations of oriented matroids, Mem. Amer. Math.
Soc. 156 (2002), no. 741, viii+80. MR 1880595
(2003b:52013)
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Richard
P. Stanley, The number of faces of a simplicial convex
polytope, Adv. in Math. 35 (1980), no. 3,
236–238. MR
563925 (81f:52014), http://dx.doi.org/10.1016/0001-8708(80)90050-X
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Michael
Thaddeus, Geometric invariant theory and
flips, J. Amer. Math. Soc.
9 (1996), no. 3,
691–723. MR 1333296
(96m:14017), http://dx.doi.org/10.1090/S0894-0347-96-00204-4
- [BD]
- R. Bielawski and A. Dancer.
The geometry and topology of toric hyperkähler manifolds. Comm. Anal. Geom. 8 (2000), 727-760. MR 1792372 (2002c:53078)
- [CB]
- W. Crawley-Boevey.
Geometry of the moment map for representations of quivers. Compositio Math. 126 (2001), no. 3, 257-293. MR 1834739 (2002g:16021)
- [CBVdB]
- W. Crawley-Boevey and M. Van den Bergh.
Absolutely indecomposable representations and Kac-Moody Lie algebras. With an appendix by Hiraku Nakajima. Invent. Math. 155 (2004), no. 3, 537-559. MR 2038196 (2004m:17032)
- [DH]
- I. Dolgachev and Y. Hu.
Variation of geometric invariant theory quotients. With an appendix by Nicolas Ressayre. Inst. Hautes Études Sci. Publ. Math. No. 87, (1998), 5-56. MR 1659282 (2000b:14060)
- [GS]
- V. Guillemin and S. Sternberg.
The coefficients of the Duistermaat-Heckman polynomial and the cohomology ring of reduced spaces. Geometry, topology, & physics, 202-213, Conf. Proc. Lecture Notes Geom. Topology, IV, Internat. Press, Cambridge, MA, 1995. MR 1358618 (96k:58082)
- [HP]
- M. Harada and N. Proudfoot.
Hyperpolygon spaces and their cores. Trans. A.M.S. 357 (2005), 1445-1467. MR 2115372 (2006f:53064)
- [Ha]
- T. Hausel.
Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform. Proc. of the Nat. Acad. Sci. 103 (2006), no. 16, 6120-6124 (electronic). MR 2221039 (2007m:53109)
- [HS]
- T. Hausel and B. Sturmfels.
Toric hyperkähler varieties. Doc. Math. 7 (2002), 495-534 (electronic). MR 2015052 (2004i:53054)
- [HK]
- J.C. Hausmann and A. Knutson.
Polygon spaces and Grassmannians. L'Enseignement Mathematique 43 (1997), no. 1-2, 173-198. MR 1460127 (98e:58035)
- [Ki]
- F. Kirwan.
Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes 31, Princeton University Press, 1984. MR 766741 (86i:58050)
- [KP]
- A. Khovanskii and A. Pukhlikov.
The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes. St. Petersburg Math. J. 4 (1993), 789-812. MR 1190788 (94c:14044)
- [K1]
- H. Konno.
Cohomology rings of toric hyperkähler manifolds. Int. J. of Math. 11 (2000), no. 8, 1001-1026. MR 1797675 (2001k:53089)
- [K2]
- H. Konno.
On the cohomology ring of the HyperKähler analogue of the Polygon Spaces. Integrable systems, topology, and physics (Tokyo, 2000), 129-149, Contemp. Math., 309, Amer. Math. Soc., Providence, RI, 2002. MR 1953356 (2003k:53111)
- [MFK]
- D. Mumford, J. Fogarty, and F. Kirwan.
Geometric Invariant Theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34. Springer-Verlag, 1994. MR 1304906 (95m:14012)
- [N1]
- H. Nakajima.
Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76 (1994), no. 2, 365-416. MR 1302318 (95i:53051)
- [N2]
- H. Nakajima.
Quiver varieties and Kac-Moody algebras. Duke Math. J. 91 (1998), no. 3, 515-560. MR 1604167 (99b:17033)
- [N3]
- H. Nakajima.
Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Amer. Math. Soc. 14 (2001), no. 1, 145-238. MR 1808477 (2002i:17023)
- [Pr]
- N. Proudfoot.
Geometric invariant theory and projective toric varieties. Snowbird lectures in algebraic geometry, 161-167, Contemp. Math. 388, Amer. Math. Soc., Providence, RI, 2005. MR 2182896 (2006j:14065)
- [PW]
- N. Proudfoot and B. Webster.
Intersection cohomology of hypertoric varieties. J. Algebraic Geom. 16 (2007), no. 1, 39-63. MR 2257319 (2008m:14037)
- [Sa]
- F. Santos.
Triangulations of oriented matroids. Mem. Amer. Math. Soc. 156 (2002), no. 741. MR 1880595 (2003b:52013)
- [St]
- R. Stanley.
The number of faces of a simplicial convex polytope. Adv. in Math. 35 (1980), no. 3, 236-238. MR 563925 (81f:52014)
- [Th]
- M. Thaddeus.
Geometric invariant theory and flips. J. Amer. Math. Soc. 9 (1996), no. 3, 691-723. MR 1333296 (96m:14017)
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Additional Information
Nicholas Proudfoot
Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403
DOI:
http://dx.doi.org/10.1090/S0002-9947-2010-05483-2
PII:
S 0002-9947(2010)05483-2
Received by editor(s):
February 17, 2006
Received by editor(s) in revised form:
March 23, 2006
Posted:
November 2, 2010
Additional Notes:
This research was partially supported by a National Science Foundation Postdoctoral Research Fellowship
Article copyright:
© Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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