On small deformations of balanced manifolds

TL;DR

This paper introduces a new property for compact complex manifolds that ensures the stability of balanced metrics under small deformations, expanding understanding beyond the $ ext{d} ext{d}^c$-lemma.

Contribution

It defines a property weaker than the $ ext{d} ext{d}^c$-lemma and characterizes it using the strongly Gauduchon cone and the first $ ext{d} ext{d}^c$-degree.

Findings

Balanced metrics are stable under small deformations for manifolds with this new property.

The property is characterized via the strongly Gauduchon cone.

It relates the first $ ext{d} ext{d}^c$-degree to cohomological differences.

Abstract

We introduce a property of compact complex manifolds under which the existence of balanced metric is stable by small deformations of the complex structure. This property, which is weaker than the $\partial\overline\partial$-Lemma, is characterized in terms of the strongly Gauduchon cone and of the first $\partial\overline\partial$-degree measuring the difference of Aeppli and Bott-Chern cohomologies with respect to the Betti number $b_1$.