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.

中文翻译：

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洛伦兹四流形类的规范 Kähler 度量

Kähler-Einstein 度量和中心 Kähler 度量（Maschler in Trans Am Math Soc 355:2161–2182, 2003）以及新旧示例的存在条件在具有两个可区分向量的 Lorentzian 4-流形类上给出领域。结果利用了一般构造（Aazami 和 Maschler 在 Kähler 度量中通过第四维的洛伦兹几何，Complex Manifolds 7:36–61 (2020) 在这种流形上的 Kähler 度量。示例包括完整和不完整的度量，有些位于与四种类型的李代数相关的李群。一个附录包括一个类似的标量平坦 Kähler 度量结构。