On Conformally Kaehler, Einstein Manifolds

TL;DR

This paper proves that certain complex surfaces with positive first Chern class can admit Einstein metrics conformally related to Kaehler metrics, including new examples on blow-ups of the complex projective plane.

Contribution

It establishes the existence of conformally Kaehler Einstein metrics on compact complex surfaces with positive first Chern class, notably on blow-ups of the projective plane.

Findings

Existence of conformally Kaehler Einstein metrics on these surfaces

Construction of such metrics on blow-ups of the projective plane

Advancement in understanding Einstein metrics in complex geometry

Abstract

We prove that any compact complex surface with positive first Chern class admits an Einstein metric which is conformally related to a Kaehler metric. The key new ingredient is the existence of such a metric on the blow-up of the complex projective plane at two distinct points.