# Non-Kähler pseudo-Kähler manifolds

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?

• Do you want (a) compact complex manifolds with a pseudo-Kähler metric which is not Kähler, or (b) compact complex manifolds which admit a pseudo-Kähler metric but do not admit a Kähler metric? (a) is easy, (b) is harder. Commented Nov 7, 2022 at 14:45
• @FrancoisZiegler (b) Commented Nov 7, 2022 at 15:00

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!).
• A reference for the final statement is Theorem $6.22$ of Fundamental Groups of Compact Kähler Manifolds by Amorós, Burger, Corlette, Kotschick, & Toledo. Commented Nov 24, 2022 at 18:36