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By use of a natural map introduced recently by the first and third authorsfrom the space of pure-type complex differential forms on a complex manifold tothe corresponding one on the small differentiable deformation of this manifold,we will give a power series proof for Kodaira-Spencer's local stability theoremof K\"ahler structures. We also obtain two new local stability theorems, one ofbalanced structures on an $n$-dimensional balanced manifold with the$(n-1,n)$-th mild $\partial\bar\partial$-lemma by power series method and theother one on $p$-K\"ahler structures with the deformation invariance of$(p,p)$-Bott-Chern numbers.

Power series proofs for local stabilities of K\"ahler and balanced structures with mild $\partial\bar\partial$-lemma