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Annals of Global Analysis and Geometry

, Volume 41, Issue 1, pp 109–123 | Cite as

ALE Ricci-flat Kähler metrics and deformations of quotient surface singularities

  • Ioana ŞuvainaEmail author
Original Paper

Abstract

Let \({N_0={\mathbb C}^2/H}\) be an isolated quotient singularity with \({H\subset U(2)}\) a finite subgroup. We show that for any \({\mathbb Q}\) -Gorenstein smoothings of N 0 a nearby fiber admits ALE Ricci-flat Kähler metrics in any Kähler class. Moreover, we generalize Kronheimer’s results on hyperkähler 4-manifolds (J Differ Geom 29(3):685–697, 1989), by giving an explicit classification of the ALE Ricci-flat Kähler surfaces. We construct ALF Ricci-flat Kähler metrics on the above non-simply connected manifolds. These provide new examples of ALF Ricci-flat Kähler 4-manifolds, with cubic volume growth and cyclic fundamental group at infinity.

Keywords

ALE Ricci-flat manifolds Deformations of quotient singularities 

Mathematics Subject Classification (1991)

Primary 53C25 53C55 Secondary 32G05 14B07 32M05 

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References

  1. 1.
    Anderson M., Kronheimer P., Le Brun C.: Complete Ricci-flat Kähler manifolds of infinite topological type. Comm. Math. Phys. 125(4), 637–642 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Atiyah M.F., Hitchin N.J., Singer I. M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. Ser. A 362(1711), 425–461 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Besse A.: Einstein manifolds. Springer-Verlag, Berlin (1987)zbMATHGoogle Scholar
  4. 4.
    Calderbank D., Singer M.: Einstein metrics and complex singularities. Invent. Math. 156(2), 405–443 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Catanese F.: Automorphisms of rational double points and moduli spaces of surfaces of general type. Compos. Math. 61(1), 81–102 (1987)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Eguchi T., Hanson A.: Asymptotically flat solution to Euclidean gravity. Phys. Lett. 74(B), 249–251 (1978)Google Scholar
  7. 7.
    Gibbons G., Hawking S.: Gravitational multi-instantons. Phys. Lett. 78, 430–432 (1978)CrossRefGoogle Scholar
  8. 8.
    Hitchin N.J.: Polygons and gravitons. Math. Proc. Cambridge Philos. Soc. 85(3), 465–476 (1979)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hitchin N.J., Karlhede A., Lindström U., Roçek M.: Hyper-Kähler metrics and supersymmetry. Comm. Math. Phys. 108(4), 535–589 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Joyce D.: Compact manifolds with special holonomy, Oxford mathematical monographs. Oxford University Press, Oxford (2000)Google Scholar
  11. 11.
    Kollár J., Shepherd-Barron N.I.: Threefolds and deformations of surface singularities. Invent. Math. 91(2), 299–338 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kronheimer P. B.: The construction of ALE spaces as hyper-Kähler quotients. J. Differ. Geom. 29(3), 665–683 (1989)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kronheimer P. B.: A Torelli-type theorem for gravitational instantons. J. Differ. Geom. 29(3), 685–697 (1989)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Manetti M.: Q-Gorenstein smoothings of quotient singularities. Scuola Normale Superiore, Pisa (1990)Google Scholar
  15. 15.
    Minerbe, V.: Rigidity for Multi-Taub-NUT metrics. arXiv:0910.5792Google Scholar
  16. 16.
    Petersen P.: Riemannian geometry, 2nd edn. Graduate Texts in Mathematics, vol. 171. Springer, New York (2006)Google Scholar
  17. 17.
    Schlessinger M.: Rigidity of quotient singularities. Invent. Math. 14, 17–26 (1971)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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