Einstein Metrics on Complex Surfaces

Abstract

This chapter focuses on Einstein metrics which are Hermitian with respect to some integrable complex structure on a compact complex surface. These will, for the sake of brevity, sometimes be referred to as Einstein Hermitian metrics, so it is worth warning the reader that these are not a priori Hermite-Einstein in the sense of the theory of holomorphic vector bundles. The chapter explores a local result concerning the conformal curvature of Einstein Hermitian metrics. The identity component of the isometry group of an extremal Kahler metric is a maximal compact subgroup of the identity component of the complex automorphism group; and since the maximal compact is unique up to conjugation, a suitable change of homogeneous coordinates will make the extremal Kahler metric g invariant under the torus action. An extremal Kahler metric in the anti-canonical class would have to be a Kahler-Einstein metric because the relevant Futaki invariant vanishes, and so coincide with the metric of Siu.