Illinois Journal of Mathematics

On the curvature of Einstein–Hermitian surfaces

Mustafa Kalafat and Caner Koca

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We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal bundle. We also give another proof of the fact that a compact complex surface together with an Einstein–Hermitian metric of positive orthogonal bisectional curvature is biholomorphically isometric to the complex projective plane with its Fubini–Study metric up to rescaling. This result relaxes the Kähler condition in Berger’s theorem, and the positivity condition on sectional curvature in a theorem proved by the second author.

Article information

Illinois J. Math., Volume 62, Number 1-4 (2018), 25-39.

Received: 4 August 2015
Revised: 9 October 2018
First available in Project Euclid: 13 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]


Kalafat, Mustafa; Koca, Caner. On the curvature of Einstein–Hermitian surfaces. Illinois J. Math. 62 (2018), no. 1-4, 25--39. doi:10.1215/ijm/1552442655.

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