I am reading a text which says that if a symplectic manifold is pseudo-Kähler, then there exists a unique symplectic connection on it. Since this a side remark without significance to the core of that text, no definition of "pseudo-Kähler manifold" is given.
I've searched the internet, but I haven't found anything clear. Wikipedia mentions "nearly Kähler manifolds" and "almost Kähler manifolds", and some research papers mention "nearly pseudo-Kähler manifolds" and "generalized pseudo-Kähler structures", but none pseudo-Kähler manifolds.
What is, then, a pseudo-Kähler manifold? What are the prototypical examples?
A pseudo-Riemannian metric on a manifold $M$ is a section $g \in \Gamma(M, S^2(T^*M))$ such that for each $p \in M$, $g_p \in \Gamma(M, S^2(T^*M))$ is non-degenerate. That is, if we view $g_p$ as a symmetric bilinear pairing $g_p : T_pM\times T_pM \to \mathbb{R}$, $g_p(X, Y) = 0$ for all $Y \in T_pM$ if and only if $X = 0$.
Now suppose $M$ is a complex manifold with corresponding almost complex structure $J$. A pseudo-Riemannian metric $g$ on $M$ is called pseudo-hermitian if $g(JX, JY) = g(X, Y)$. If in addition, the two-form $\omega(X, Y) = g(JX, Y)$ is closed (i.e. $d\omega = 0$), $g$ is called a pseudo-Kähler metric.
Note, as a Riemannian metric is a pseudo-Riemannian metric, every Kähler metric is a pseudo-Kähler metric.