On the curvature of Einstein-Hermitian surfaces

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 K\"ahler condition in Berger's theorem, and the positivity condition on sectional curvature in a theorem proved by the second author.Comment: 16 pages, Page metric coefficient and Vierbein are fixed, Journal info added. arXiv admin note: text overlap with arXiv:1112.418

Conformally K\"ahler surfaces and orthogonal holomorphic bisectional curvature

We show that a compact complex surface which admits a conformally K\"ahler 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.Comment: 12 pages. Journal information adde