Conformally Kähler surfaces and orthogonal holomorphic bisectional curvature

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.

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Notes

  1. 1.

    Here one should insist on being Kähler with respect to the same complex structure. As an example, the Page metric on \(\mathbb {CP}_2\sharp \, \overline{\mathbb {CP}}_2\) has an orientation-reversing conformal map, consequently has two different conformal rescalings which are Kähler but for different complex structures. Interested reader may consult to [1] and related papers for more regarding this situation.

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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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Correspondence to Mustafa Kalafat.

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Kalafat, M., Koca, C. Conformally Kähler surfaces and orthogonal holomorphic bisectional curvature. Geom Dedicata 174, 401–408 (2015). https://doi.org/10.1007/s10711-014-0025-9

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Keywords

  • Einstein metrics
  • Holomorphic curvature
  • Complex surfaces
  • 4-Manifolds