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?

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?