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][1].

**Question.** *What are examples of compact complex manifolds which are pseudo-Kähler but not Kähler?*


  [1]: https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold