Suppose we have a fibration over the unit disc $\Delta$ of $\mathbb{C}$, \begin{equation} \pi: X \rightarrow \Delta \end{equation} such that all the fibers are smooth projective manifolds. The for every $\varphi \in \Delta$, we have a Hodge decomposition \begin{equation} H^n(X_{\varphi}, \mathbb{C})=\oplus_{p+q=n}H^{p,q}(X_{\varphi}) \end{equation} which determines a filtration \begin{equation} F^p_{\varphi}:=\oplus_{k \geq p}H^{k,n-k}(X_{\varphi}) \end{equation}
Question: lots of references claim that $H^{p,q}(X_{\varphi})$ varies smoothly with respect to $\varphi$, i.e. $\cup_{\varphi}H^{p,q}(X_{\varphi})$ only forms a smooth bundle over $\Delta$. But $F^p_{\varphi}$ varies holomorphically with respect to $\varphi$, i.e. $\cup_{\varphi} F^p_{\varphi}$ forms a holomorphic bundle over $\Delta$. Could anyone explain why? Any examples?