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

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.