Let $\phi: X \to B$ be a family of complex manifolds. Fix a point $0 \in B$ and $X_0 := \phi^{-1}(0)$. For any $\alpha \in A^{p,q}(X_0) := \{C^\infty (p,q)\text{ forms on }X_0\}$ such that $d \alpha = 0$, can we extend it to $\tilde\alpha \in A^{p,q}(U)$ such that $d\tilde\alpha = 0$ and $\tilde\alpha|_{X_0} = \alpha$, where $U$ is some neighborhood of $X_0$ in $X$?
Of course, we can easily obtain such $d$-closed $\tilde\alpha \in A^{p+q}(U)$ for some neighborhood $U$ by $C^\infty$ local trivialization. And if we assume all fibers of $X$ to be Kahler, we can obtain such $\tilde\alpha$ as a $(p,q)$ form by taking its harmonic part on each fibres. But I don't know how we do it in general, and I think it may be impossible.