$\DeclareMathOperator{\pr}{pr}$Let $\phi: \mathcal X \to B$ be a proper submersion of complex manifolds, with central fiber $X = X_0$. Then by the Ehresmann theorem we may shrink $B$ such that there exists a $\mathcal C^\infty$-trivialisation $$T: \mathcal X \cong X \times B$$ such that $\phi = \pr_B \circ T$. In general, $T$ will not be holomorphic. However, according to Voisin[1], one can choose $T$ in such a way, that the fibers of the projection $\pi = \pr_X \circ T: \mathcal X \to X$ are complex submanifolds of $\mathcal X$. Assume such a $T$ is given. Then in the proof of Proposition 9.7 the following situation appears:
Suppose $t_1, \dotsc, t_r$ are local coordinates of $B$. As $\phi$ is a submersion, one can locally (in a neighbourhood of a point lying over $0 \in B$) choose additional coordinates $z_1, \dotsc, z_n$ such that $(z_1, \dotsc, z_n, \phi^* t_1, \dotsc, \phi^* t_r)$ is a local system of coordinates on $\mathcal X$, and $\phi$ is given by $(z, t) \mapsto t$ (dropping the $\phi^*$ by abuse of notation). Then $z_1|_X, \dotsc, z_n|_X$ also give local coordinates on $X$. Hence we can write $$\pi(z, t) = (\pi_1(z,t), \dotsc, \pi_n(z,t))$$ for $\pi: \mathcal X \to X$, where $\pi_j$ is the component corresponding to $z_j$.
Question: Voisin claims that the $\pi_j$ depend holomorphically on $t_1, \dotsc, t_r$. How can I see this? I guess it comes from the fact that the fibers of $\pi$ are complex manifolds, but I'm not sure.
[1] Claire Voisin, Hodge Theory and Complex Algebraic Geometry, I
I don't think this follows immediatly from the condition that the fibers of $\pi$ are holomorphic. Consider for example the case $r = n = 1$ with $$\pi(z,t) = \overline{t} - \overline{z}.$$ Then $\pi$ is neither holomorphic in $z$ nor in $t$. However, the fibers of $\pi$ are of the form $$\pi^{-1}(c) = \{(z,t) \,|\, t - z = \overline{c}\},$$ which are complex submanifolds of $\mathbb C^2$.
Maybe we have to make a certain kind of choice for the coordinates $z, t$?