Einstein Hermitian Metrics of Positive Sectional Curvature

TL;DR

This paper proves that the only compact 4-manifold with an Einstein metric of positive sectional curvature that is also Hermitian is the complex projective plane with its standard Fubini-Study metric.

Contribution

It establishes a uniqueness result for Hermitian Einstein metrics of positive sectional curvature on compact 4-manifolds, identifying CP^2 as the sole example.

Findings

CP^2 with Fubini-Study metric is the unique such manifold.

No other compact 4-manifold admits a Hermitian Einstein metric with positive sectional curvature.

The result narrows the classification of positively curved Einstein Hermitian 4-manifolds.

Abstract

In this paper we will prove that the only compact 4-manifold M with an Einstein metric of positive sectional curvature which is also hermitian with respect to some complex structure on M, is the complex projective plane CP^2, with its Fubini-Study metric.