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

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