Canonical Kähler metrics on classes of Lorentzian $4$-manifolds

Abstract

Conditions for the existence of Kähler-Einstein metrics and central Kähler metrics [MS] along with examples, both old and new, are given on classes of Lorentzian $4$-manifolds with two distinguished vector fields. The results utilize the general construction [AM] of Kähler metrics on such manifolds. The examples include both complete and incomplete metrics, and some reside on Lie groups associated to four types of Lie algebras. An appendix includes a similar construction for scalar-flat Kähler metrics.

arXiv commentsequel to [AM], final version. Various significant changes, in particular with regard to completeness