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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
  • 125 Downloads

Abstract

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.

Keywords

Einstein metrics Holomorphic curvature Complex surfaces 4-Manifolds 

Notes

Acknowledgments

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).

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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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