On the curvature of Einstein-Hermitian surfaces

TL;DR

This paper analyzes the curvature properties of Einstein-Hermitian surfaces, providing new insights into their geometric structure and curvature constraints, including a detailed examination of the Page metric and related curvature conditions.

Contribution

It introduces an efficient coordinate system for the Page metric, demonstrates that it lacks positive holomorphic bisectional curvature, and extends curvature classification results beyond Kähler surfaces.

Findings

Page metric does not have positive holomorphic bisectional curvature

A holomorphic subsurface with flat normal bundle is exhibited

A classification result for Einstein-Hermitian surfaces with positive orthogonal bisectional curvature

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 K\"ahler condition in Berger's theorem, and the positivity condition on sectional curvature in a theorem proved by the second author.