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Quick and simple...

Is it possible to define complex structures on Lorentzian manifolds? If so, Can you point me to some example(s)?

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    $\begingroup$ If you want just an example, you can take $T^2$ (or $T^{2n}$), it is Lorentzian (with the metric $dx^2-dy^2$) and has a complex structure. But generally one should not expect to have a complex structure on Lorentzian manifolds. $\endgroup$ – Dmitri Panov Dec 12 '18 at 10:00
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    $\begingroup$ There are some papers on holomorphic metrics, i.e. the metric tensor is a holomorphic tensor on a complex manifold. The expert in this field is Sorin Dumitrescu. Such tensors cannot be Lorentzian or Euclidean signature, as they have no signature, since complex bilinear symmetric tensors have no signature invariant, but Sorin requires that they be nondegenerate. $\endgroup$ – Ben McKay Dec 12 '18 at 10:57
  • $\begingroup$ And what if we take a complex $n$-manifold with a $\mathbb{C}$-sesquilinear metric of signature $(n-1, 1)$? Have this been studied? This may seem to be the complex analog, doesn't it? $\endgroup$ – Ivan Solonenko Dec 12 '18 at 12:42

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