On the stability of compact pseudo-K\"ahler and neutral Calabi-Yau manifolds

We study the stability of compact pseudo-K"ahler manifolds, i.e. compact complex manifolds $X$ endowed with a symplectic form compatible with the complex structure of $X$. When the corresponding metric is positive-definite, $X$ is K"ahler and any sufficiently small deformation of $X$ admits a K"ahler metric by a well-known result of Kodaira and Spencer. We prove that compact pseudo-K"ahler surfaces are also stable, but we show that stability fails in every complex dimension $n\ge 3$. Similar results are obtained for compact neutral K"ahler and neutral Calabi-Yau manifolds. Finally, motivated by a question of Streets and Tian in the positive-definite case, we construct compact complex manifolds with pseudo-Hermitian-symplectic structures that do not admit any pseudo-K"ahler metric.