A *pseudo-Kähler manifold* is a complex manifold $(X, I)$ endowed with a non-degenerate closed $(1, 1)$-form $\omega$. In that case, the symmetric tensor $g(\cdot, \cdot) = \omega(\cdot, I \cdot)$ is a [pseudo-Riemannian metric](https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold).



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> **Question.** What are examples of compact complex manifolds which are pseudo-Kähler but not Kähler?
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Non-Kähler pseudo-Kähler manifolds
I think that the easiest example of compact pseudo-Kähler manifold which does not admit any Kähler metric is the [Kodaira-Thurston manifold](https://www.jstor.org/stable/2041749?origin=crossref#metadata_info_tab_contents). See for instance the introduction of


*Yamada, Takumi*, [**Ricci flatness of certain compact pseudo-Kähler solvmanifolds**](https://doi.org/10.1016/j.geomphys.2011.06.006), J. Geom. Phys. 62, No. 5, 1338-1345 (2012). [ZBL1239.53100](https://zbmath.org/?q=an:1239.53100).


