Canonical Kähler metrics on classes of Lorentzian 4-manifolds

Conditions for the existence of Kähler–Einstein metrics and central Kähler metrics (Maschler in Trans Am Math Soc 355:2161–2182, 2003) 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 (Aazami and Maschler in Kähler metrics via Lorentzian geometry in dimension four, Complex Manifolds 7:36–61 (2020) of Kähler metrics on such manifolds. The examples include both complete and incomplete metrics, and some reside on Lie groups associated with four types of Lie algebras. An appendix includes a similar construction for scalar-flat Kähler metrics.