## Illinois Journal of Mathematics

### On the curvature of Einstein–Hermitian surfaces

#### Abstract

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

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

Dates
Revised: 9 October 2018
First available in Project Euclid: 13 March 2019

https://projecteuclid.org/euclid.ijm/1552442655

Digital Object Identifier
doi:10.1215/ijm/1552442655

Mathematical Reviews number (MathSciNet)
MR3922409

Zentralblatt MATH identifier
07036779

#### Citation

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. https://projecteuclid.org/euclid.ijm/1552442655

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