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.
Question. What are examples of compact complex manifolds which are pseudo-Kähler but not Kähler?
I think that the easiest example of compact pseudo-Kähler manifold which does not admit any Kähler metric is the Kodaira-Thurston manifold. See for instance the introduction of
Yamada, Takumi, Ricci flatness of certain compact pseudo-Kähler solvmanifolds, J. Geom. Phys. 62, No. 5, 1338-1345 (2012). ZBL1239.53100.
Another collection of examples come via hyperbolic 4-manifolds. Since hyperbolic metrics are conformally flat, the twistor space $Z$ of an oriented hyperbolic 4-manifold is actually a complex manifold with complex structure $J$ (this is due to Atiyah-Hitchin-Singer, Self duality in 4-dimensional Riemannian geometry, https://www.jstor.org/stable/79638). There is also a natural closed 2-form $\omega$ on the twistor space. In the case of hyperbolic 4-manifolds, this pairs with $\omega$ to give a pseudo-Kähler metric. This is described for $H^4$ itself in a paper I wrote with Dmitri Panov (Section 2.3.2 of this article: https://arxiv.org/pdf/0905.3237.pdf). Now, the twistor space $Z$ of a 4-manifold $M$ has the same fundamental group as $M$. So when $M$ is a compact hyperbolic 4-manifold, $Z$ can't carry any Kähler structure whatsoever, by results on the structure of Kähler fundamental groups (I'm afraid I don't a know a reference off the top of my head here!).
