Extensions of $\overline\partial$-closed forms on compact generalized Hermitian manifolds

TL;DR

This paper develops a criterion for holomorphicity of forms on generalized complex manifolds, extends $ar{ ext{d}}$-closed forms in families, and proves invariance of generalized Hodge numbers under deformations.

Contribution

It introduces a new criterion for holomorphic forms and demonstrates the invariance of generalized Hodge numbers for certain deformations.

Findings

Established a criterion for holomorphic forms on generalized complex manifolds.

Extended $ar{ ext{d}}$-closed forms locally in smooth families.

Proved the invariance of generalized Hodge numbers under deformations with $ ext{d}ar{ ext{d}}$-lemma.

Abstract

In this paper, we first get a criterion formula for whether a differential form is holomorphic with respect to the generalized complex structure induced by $\epsilon$. Next, we get the local extensions of $\overline\partial$-closed forms on a smooth family of compact generalized Hermitian manifolds by using this criterion. Finally, as an application, we use this extension to get the invariance of the generalized Hodge number of the deformations of compact generalized Hermitian manifolds with $\partial\overline\partial$-lemma holds.