ON THE CURVATURE OF EINSTEIN-HERMITIAN SURFACES

摘要

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 holo-morphic 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 Kahler condition in Berger's theorem, and the positivity condition on sectional curvature in a theorem proved by the second author.