  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 K\"ahler 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 $\partial\bar\partial$-lemma by power series method and the
other one on $p$-K\"ahler structures with the deformation invariance of
$(p,p)$-Bott-Chern numbers.
