Conformally K\"ahler surfaces and orthogonal holomorphic bisectional curvature

TL;DR

This paper proves that compact complex surfaces with certain positive curvature conditions are biholomorphic to the complex projective plane, and characterizes Einstein Hermitian metrics as multiples of the Fubini-Study metric.

Contribution

It establishes a classification result linking conformally K"ahler metrics with positive orthogonal holomorphic bisectional curvature to the complex projective plane, and characterizes Einstein Hermitian metrics.

Findings

Compact complex surfaces with positive orthogonal holomorphic bisectional curvature are biholomorphic to  P^2.

Einstein Hermitian metrics under these conditions are multiples of the Fubini-Study metric.

The biholomorphism can be chosen as an isometry in the Einstein case.

Abstract

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.