By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an n-dimensional balanced manifold with the $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma by power series method and the other one on p-Kähler structures with the deformation invariance of $(p,p)$ -Bott–Chern numbers.

通过使用第一作者和第三作者最近介绍的从复流形上的纯型复微分形式空间到该流形的小可微变形的对应空间的自然映射，我们将给出小平的幂级数证明–Spencer 的 Kähler 结构的局部稳定性定理。我们还获得了两个新的局部稳定性定理，一个n维平衡流形上的平衡结构，通过幂级数方法具有 $(n-1,n)$ th 温和 $\partial \overline {\partial }$ -lemma 和另一个是关于p -Kähler 结构的，具有 $(p,p)$ -Bott-Chern 数的变形不变性。