Let be $X$ a Calabi-Yau manifold (i.e. Kahler, compact with trivial canonical bundle) with $\dim(X) = 3$. Let $\phi : A \mapsto B$ a proper holomoprhic submersion such that $X_{t_{0}} := \phi^{-1}(t_{0}) = X$ for some $t_{0} \in B$.
I consider the following map $t \mapsto H^{3,0}(X_{t}) \subset H^{3}(X_{t}) \simeq H^{3}(X)$ which is valued in $\mathbb{P}(H^{3}(X))$ (since $H^{3,0}(X_{t})$ is of dimension $1$ by Serre duality). I would like to show near $t_{0}$, the image is a manifold.
This comes from Voisin's book Hodge theory and complex algebraic geometry, chapter 10. I already know from this book that this map is a holomorphic immersion and that is differential at $t_{0}$ maps $T_{t_{0}}B$ in $\operatorname{Hom}(H^{3,0}(X), H^{2,1}(X))$.
I wish you a good day.