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Intersection forms of toric hyperkähler varieties


Authors: Tamás Hausel and Edward Swartz
Journal: Proc. Amer. Math. Soc. 134 (2006), 2403-2409
MSC (2000): Primary 53C26; Secondary 52C35
DOI: https://doi.org/10.1090/S0002-9939-06-08248-7
Published electronically: February 6, 2006
MathSciNet review: 2213714
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Abstract | References | Similar Articles | Additional Information

Abstract: This note proves combinatorially that the intersection pairing on the middle-dimensional compactly supported cohomology of a toric hyperkähler variety is always definite, providing a large number of non-trivial $ L^2$ harmonic forms for toric hyperkähler metrics on these varieties. This is motivated by a result of Hitchin about the definiteness of the pairing of $ L^2$ harmonic forms on complete hyperkähler manifolds of linear growth.


References [Enhancements On Off] (What's this?)

  • 1. A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G. Ziegler: Oriented Matroids, Cambridge University Press, 1999. MR 1744046 (2000j:52016)
  • 2. R. Bielawski, A. Dancer: The geometry and topology of toric hyperkähler manifolds, Comm. Anal. Geom. 8 (2000), no. 4, 727-760. MR 1792372 (2002c:53078)
  • 3. A. Borel, J-P. Serre:
    Corners and arithmetic groups.
    Comment. Math. Helv., 48:436-491, 1973. MR 0387495 (52:8337)
  • 4. M.A. de Cataldo, L. Migliorini: The Hodge theory of algebraic maps, preprint, arXiv:math.AG/0306030
  • 5. W. Fulton: Intersection theory, Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
  • 6. M. Harada, N. Proudfoot: Properties of the residual circle action on a toric hyperkähler variety, Pac. J. Math. 214 (2004) 263-284. MR 2042933 (2004k:53139)
  • 7. T. Hausel: Vanishing of intersection numbers on the moduli space of Higgs bundles, Adv. Theor. Math. Phys. 2 (1998) 1011-1040. MR 1688480 (2000g:14017)
  • 8. T. Hausel, E. Hunsicker, R. Mazzeo: Hodge cohomology of gravitational instantons, Duke Mathematical Journal 122 no. 3, (2004) 485-548. MR 2057017 (2005d:58039)
  • 9. T. Hausel, B. Sturmfels: Toric hyperkähler varieties, Documenta Mathematica 7 (2002), 495-534. MR 2015052 (2004i:53054)
  • 10. N.J. Hitchin: $ L^2$-cohomology of hyperkähler quotients, Comm. Math. Phys. 211 (2000), no. 1, 153-165. MR 1757010 (2001c:53060)
  • 11. H. Nakajima: Quiver varieties and Kac-Moody algebras, Duke Mathematical Journal 91 (1998), 515-560. MR 1604167 (99b:17033)
  • 12. H. Nakajima: Lectures on Hilbert schemes of points. University Lecture Series, 18. American Mathematical Society, Providence, RI, 1999 MR 1711344 (2001b:14007)
  • 13. G. Segal, A. Selby: Cohomology of the space of magnetic monopoles, Comm. Math. Phys. 177 (1996), 775-787. MR 1385085 (97a:58022)
  • 14. A. Sen: Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and $ SL(2,\mathbb{Z})$ invariance in string theory. Phys.-Lett.-B 329 No.2-3 (1994), 217-221. MR 1281578 (95e:81187)

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Additional Information

Tamás Hausel
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712-1082
Email: hausel@math.utexas.edu

Edward Swartz
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
Email: ebs@math.cornell.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08248-7
Received by editor(s): June 21, 2004
Received by editor(s) in revised form: March 9, 2005
Published electronically: February 6, 2006
Communicated by: Michael Stillman
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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