Geometriae Dedicata

, Volume 174, Issue 1, pp 401–408 | Cite as

Conformally Kähler surfaces and orthogonal holomorphic bisectional curvature

  • Mustafa Kalafat
  • Caner Koca
Original Paper


We show that a compact complex surface which admits a conformally Kähler metric \(g\) of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if \(g\) is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which \(g\) becomes a multiple of the Fubini-Study metric.


Einstein metrics Holomorphic curvature Complex surfaces 4-Manifolds 



The authors would like to thank Claude LeBrun for suggesting us this problem, and for his help and encouragement. The authors also thank F. Belgun, Peng Wu for useful inquiries, and the referee for useful remarks which greatly improved the original manuscript. This work is partially supported by the grant #113F159 of Tübitak (Turkish science and research council).


  1. 1.
    Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: Ambitoric geometry I: Einstein metrics and extremal ambikaehler structures. J. Reine Angew. Math. (2014). doi: 10.1515/crelle-2014-0060
  2. 2.
    Andreotti, A.: On the complex structures of a class of simply-connected manifolds. In: Fox, R.H., Spencer, D.C., Tucker, A.W. (eds.) Algebraic Geometry and Topology. A Symposium in Honor of S. Lefschetz, pp. 53–77. Princeton University Press, Princeton, NJ (1957)Google Scholar
  3. 3.
    Berger, M.: Sur quleques variétés riemanniennes compactes d’Einstein. C. R. Acad. Sci. Paris 260, 1554–1557 (1965)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bourguignon, J.-P.: Les variétés de dimension \(4\) à signature non nulle dont la courbure est harmonique sont d’Einstein. Invent. Math. 63(2), 263–286 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces, Volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer, Berlin (1984)Google Scholar
  6. 6.
    Derdziński, A.: Self-dual Kähler manifolds and Einstein manifolds of dimension four. Composit. Math. 49(3), 405–433 (1983)zbMATHGoogle Scholar
  7. 7.
    Goldberg, S.I., Kobayashi, S.: Holomorphic bisectional curvature. J. Differ. Geom. 1, 225–233 (1967)Google Scholar
  8. 8.
    Hawley, N.S.: Constant holomorphic curvature. Can. J. Math. 5, 53–56 (1953)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Igusa, J.: On the structure of a certain class of Kaehler varieties. Am. J. Math. 76, 669–678 (1954)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Kobayashi, S.: On compact Kähler manifolds with positive definite Ricci tensor. Ann. Math. 2(74), 570–574 (1961)CrossRefGoogle Scholar
  11. 11.
    Koca, C.: On Conformal Geometry of Kahler Surfaces. ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)-State University of New York at Stony BrookGoogle Scholar
  12. 12.
    Koca, C.: Einstein Hermitian metrics of positive sectional curvature. Proc. Am. Math. Soc. 142(6), 2119–2122 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    LeBrun, C.: Einstein metrics on complex surfaces. In: Geometry and Physics (Aarhus, 1995), Volume 184 of Lecture Notes in Pure and Applied Mathematical, pp. 167–176. Dekker, New York (1997)Google Scholar
  14. 14.
    Lichnerowicz, A.: Sur les transformations analytiques des variétés kählériennes compactes. C. R. Acad. Sci. Paris 244, 3011–3013 (1957)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Matsushima, Y.: Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne. Nagoya Math. J. 11, 145–150 (1957)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Yum T. S., Shing T.Y.: Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59(2), 189–204 (1980)Google Scholar
  17. 17.
    Synge, J.L.: On the connectivity of spaces of positive curvature. Quart. J. Math. (Oxford series) 7(1), 316–320 (1936)CrossRefGoogle Scholar
  18. 18.
    Viaclovsky, J.: Lecture Notes: Topics in Riemannian Geometry, 2011.
  19. 19.
    Yau, S.T.: On the curvature of compact Hermitian manifolds. Invent. Math. 25, 213–239 (1974)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Tunceli ÜniversitesiAktulukTurkey
  2. 2.Department of MathematicsVanderbilt UniversityNashvilleUSA

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