We study the stability of compact pseudo-Kähler 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 Kähler and any sufficiently small deformation of X admits a Kähler metric by a well-known result of Kodaira and Spencer. We prove that compact pseudo-Kähler surfaces are also stable, but we show that stability fails in every complex dimension $n\ge 3$. Similar results are obtained for compact neutral Kähler 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-Kähler metric.

我们研究紧凑伪凯勒流形，即紧凑复流形的稳定性X赋与的复杂结构兼容的辛形式X。当相应的度量是正定的，X是凯勒和任何足够小的变形X承认一个凯勒通过小平和Spencer的公知的结果度量。我们证明了紧致的伪Kähler曲面也是稳定的，但是我们证明了在每个复杂维度上的稳定性都会失败$ñ\ge 3$。紧凑型中性Kähler和中性Calabi-Yau歧管也获得了类似的结果。最后，受正定情况下的Streets和Tian问题的启发，我们构造了带有准Hermitian辛结构的紧凑复流形，该结构不接受任何拟Kähler度量。