Einstein Metrics on Complex Surfaces

TL;DR

This paper classifies compact complex surfaces with Einstein Hermitian metrics that are not Kähler, showing they are mostly CP2 blown up at few points with specific symmetry properties, highlighting the uniqueness of the Page metric.

Contribution

It provides a classification of non-Kähler Einstein Hermitian metrics on complex surfaces, identifying the specific manifolds and symmetry conditions involved.

Findings

Manifolds are CP2 blown up at 1, 2, or 3 points.

The isometry group contains a 2-torus.

The Page metric on CP2#(-CP2) is nearly unique of this type.

Abstract

We consider compact complex surfaces with Hermitian metrics which are Einstein but not Kaehler. It is shown that the manifold must be CP2 blown up at 1,2, or 3 points, and the isometry group of the metric must contain a 2-torus. Thus the Page metric on CP2#(-CP2) is almost the only metric of this type.