  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.

Differential Geometry