On the curvature of Einstein-Hermitian surfaces

摘要

We give a mathematical exposition of the Page metric and introduce anefficient coordinate system for it. We carefully examine the submanifolds ofthe underlying smooth manifold and show that it does not have positiveholomorphic bisectional curvature. We also reprove that a compact complexsurface together with an Einstein-Hermitian metric of positive orthogonalbisectional curvature is biholomorphically isometric to the complex projectiveplane with its Fubini-Study metric up to rescaling. This result relaxes theK\"ahler condition in Berger's theorem, and the positivity condition onsectional curvature in a theorem proved by Koca.