Communications in Mathematical Physics

, Volume 301, Issue 3, pp 749–770 | Cite as

Degenerations of LeBrun Twistor Spaces



We investigate various limits of the twistor spaces associated to the self-dual metrics on \({n \mathbb{CP}^2}\), the connected sum of the complex projective planes, constructed by C. LeBrun. In particular, we explicitly present the following 3 kinds of degenerations whose limits of the corresponding metrics are: (a) LeBrun metrics on \({(n-1) \mathbb{CP}^2}\), (b) (another) LeBrun metrics on the total space of the line bundle \({\fancyscript O(-n)}\) over \({\mathbb{CP}^1}\), (c) the hyper-Kähler metrics on the small resolution of rational double points of type An-1, constructed by G.W. Gibbons and S.W. Hawking.


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  1. 1.
    Atiyah M., Hitchin N., Singer I.: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London, Ser. A 362, 425–461 (1978)MATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Donaldson S.K., Friedman R.: Connected sums of self-dual manifolds and deformations of singular spaces. Nonlinearity 2, 197–239 (1989)MATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Gibbons G.W., Hawking S.W.: Gravitational multi-instantons. Phys. Lett. 78B, 430–432 (1978)ADSGoogle Scholar
  4. 4.
    Hitchin N.: Polygons and Gravitons. Math. Proc. Cambridge Philos. Soc. 85, 465–476 (1979)CrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Honda, N., Viaclovsky, J.: Conformal symmetries of self-dual hyperbolic monopole metrics. http://arxiv./org/abs/0902.2019v1 [math.DG], 2009
  6. 6.
    LeBrun C.: Counter-examples to the generalized positive action conjecture. Commun. Math. Phys. 118, 591–596 (1988)MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    LeBrun C.: Explicit self-dual metrics on \({{\mathbb{CP}}^2 \cdots {\mathbb{CP}}^2}\). J. Diff. Geom. 34, 223–253 (1991)MATHMathSciNetGoogle Scholar
  8. 8.
    LeBrun C.: Twistors, Kähler manifolds, and bimeromorphic geometry I. J. Amer. Math. Soc. 5, 289–316 (1992)MATHMathSciNetGoogle Scholar
  9. 9.
    Pontecorvo M.: On twistor spaces of anti-self-dual Hermitian surfaces. Tran. Amer. Math. Soc. 331, 653–661 (1992)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Poon Y.S.: Compact self-dual manifolds of positive scalar curvature. J. Diff. Geom. 24, 97–132 (1986)MATHMathSciNetGoogle Scholar
  11. 11.
    Tian G., Viaclovsky J.: Moduli spaces of critical Riemannian metrics in dimension four. Adv. Math. 196, 346–372 (2005)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Viaclovsky, J.: Yamabe invariants and limits of self-dual hyperbolic monopole metrics. Ann. Inst. Fourier (Grenoble) (accepted) 2010Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsTokyo Institute of TechnologyMeguro, TokyoJapan
  2. 2.Mathematical Institute, Graduate School of ScienceTohoku UniversitySendaiJapan

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