Geometriae Dedicata

, Volume 138, Issue 1, pp 151–172 | Cite as

Compact complex surfaces and constant scalar curvature Kähler metrics

Original Paper


In this article, I prove the following statement: Every compact complex surface with even first Betti number is deformation equivalent to one which admits an extremal Kähler metric. In fact, this extremal Kähler metric can even be taken to have constant scalar curvature in all but two cases: the deformation equivalence classes of the blow-up of \({\mathbb {P}_2}\) at one or two points. The explicit construction of compact complex surfaces with constant scalar curvature Kähler metrics in different deformation equivalence classes is given. The main tool repeatedly applied here is the gluing theorem of C. Arezzo and F. Pacard which states that the blow-up/resolution of a compact manifold/orbifold of discrete type, which admits cscK metrics, still admits cscK metrics.


Deformation equivalence Constant scalar curvature Kähler metric 

Mathematics Subject Classification (2000)

53C25 53C55 


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  1. 1.
    Arezzo C., Pacard F.: Blowing up and desingularizing constant scalar curvature Kähler manifolds. Acta Mathematica 196(2), 179–228 (2006)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Arezzo, C., Pacard, F., Singer, M.: Extremal metrics on blow ups. e-print # math.DG/0701028 (2007)Google Scholar
  3. 3.
    Apostolov V., Tønnesen-Friedman C.W.: A remark on Kähler metrics of constant scalar curvature on ruled complex surfaces. Bull. Lond. Math. Soc. 38(3), 494–500 (2006)MATHCrossRefGoogle Scholar
  4. 4.
    Aubin T.: Équations du type de Monge-Ampére sur les variétés Kähleriennes compactes. C.R. Acad. Sci. Paris 283, A119–A121 (1976)MathSciNetGoogle Scholar
  5. 5.
    Barth, W., Peters, C., de Ven, A.V.: Compact Complex Surfaces. Springer-Verlag (1984)Google Scholar
  6. 6.
    Beauville, A.: Complex Algebraic Surfaces. Cambridge University Press (1996)Google Scholar
  7. 7.
    Calabi E.: Extremal Kähler metrics. Ann. Math. Stud. 102, 259–290 (1982)MathSciNetGoogle Scholar
  8. 8.
    Chen, X.X., Tian, G.: Geometry of Kähler metrics and foliations by holomorphic discs. e-print # math.DG/0507148 (2005)Google Scholar
  9. 9.
    Donaldson S.: Scalar curvature and projective embeddings. J. Diff. Geom. 59(3), 479–522 (2001)MathSciNetMATHGoogle Scholar
  10. 10.
    Friedman, R., Morgan, J.W.: Smooth Four-Manifolds and Complex Surfaces. Springer-Verlag (1994)Google Scholar
  11. 11.
    Friedman, R.: Algebraic Surfaces and Holomorphic Vector Bundles. Springer-Verlag, (1998).Google Scholar
  12. 12.
    Futaki, A.: Kähler-Einstein Metrics and Integral Invariants. Springer Verlag (1988)Google Scholar
  13. 13.
    Griffith, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library (1994)Google Scholar
  14. 14.
    Grothendieck, A.: Géométrie formulle et géométrie algébrique, pp. 182-1–182-28. Séminaire Bourbaki 11e Année v.3 (1959)Google Scholar
  15. 15.
    Kobayashi K.: A remark on the Ricci curvature of algebraic surfaces of general type. Tohoku Math. J. 36, 385–399 (1984)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Kodaira K.: On compact analytic surfaces: II. Ann. Math. 77(3), 563–626 (1963)MATHCrossRefGoogle Scholar
  17. 17.
    Kronheimer P.B.: The construction of ALE spaces as hyper-Kähler quotients. J. Diff. Geom. 29(3), 665–683 (1989)MathSciNetMATHGoogle Scholar
  18. 18.
    LeBrun C.: Polarized 4-manifolds, extremal Kähler metrics, and Seiberg-Witten theory. Math. Res. Lett. 2, 653–662 (1995)MathSciNetMATHGoogle Scholar
  19. 19.
    LeBrun C., Simanca S.R.: Extremal Kähler metrics and complex deformation theory. Geom. Funct. Anal. 4, 298–336 (1994)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Lichnérowicz A.: Sur les transformations analytiques des variétiés Kählériennes compactes. C.R. Acad. Sci. Paris 244, 3011–3013 (1957)MathSciNetMATHGoogle Scholar
  21. 21.
    Mabuchi, T.: An Energy-Theoretic Approach to the Hitchin-Kobayashi Correspondence for Manifolds II. e-print # math.DG/0410239 (2004)Google Scholar
  22. 22.
    Matsushima Y.: Sur la structure du goupe d’homéomorphismes analytiques d’une certaine variété Kählériennes. Nag. Math. J. 11, 145–150 (1957)MathSciNetMATHGoogle Scholar
  23. 23.
    Mabuchi T.: Uniqueness of extremal Kähler metrics for an integral kähler class. Int. J. Math. 15, 531–546 (2004)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Maruyama M.: On automorphism groups of ruled surfaces. J. Math. Kyoto Univ. 11, 89–112 (1971)MathSciNetMATHGoogle Scholar
  25. 25.
    Paur K.: The Fenchel-Nielsen coordinates of Techmuller space. MIT UJM 1, 149–154 (1999)Google Scholar
  26. 26.
    Ross J.: Unstable products of smooth curves. Invent. Math. 165(1), 153–162 (2006)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Siu Y.T.: Every K3 surface is Kähler. Invent. Math. 73, 139–150 (1983)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Tian G., Yau S.T.: Kähler-Einstein metrics on complex surfaces with c 1 > 0. Comm. Math. Phys. 112, 175–203 (1987)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Todorov A.N.: Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of K3 surfaces. Invent. Math. 61, 251–265 (1980)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Troyanov M.: Prescribing curvature on compact surfaces with conical singularities. Trans. Am. Math. Soc. 324(2), 793–821 (1991)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Wall C.T.C.: Geometric structures on compact complex analytic surfaces. Topology 25(2), 119–153 (1986)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Yau S.T.: On the Ricci curvature of a compact Kähler manifold and complex Monge-Ampére equation I. Comm. Pure Appl. Math. 31(2), 339–411 (1978)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, Santa BarbaraSanta BarbaraUSA

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