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

TL;DR

This paper introduces power series methods to prove local stability theorems for K"ahler and balanced structures on complex manifolds, extending classical results and establishing new stability results under mild conditions.

Contribution

It provides a power series proof of Kodaira-Spencer's stability theorem and introduces two new local stability theorems for balanced and p-K"ahler structures under mild $ar ext{d}ar ext{d}$-lemma conditions.

Findings

Power series proof of Kodaira-Spencer's local stability theorem.

New stability theorem for balanced structures with mild $ar ext{d}ar ext{d}$-lemma.

Stability of p-K"ahler structures with invariance of Bott-Chern numbers.

Abstract

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.