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


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.


ALE Ricci-flat manifolds Deformations of quotient singularities 

Mathematics Subject Classification (1991)

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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Personalised recommendations