Let $f:X\rightarrow Y$ be a surjective map between smooth varieties with connected fibers. Assume that everything is over $\mathbb{C}$. $f$ is said to be isotrivial if all its smooth fibers are isomorphic.

When $\mathrm{dim}X=2$ and $\mathrm{dim}Y=1$, if $\mathrm{deg}(f_*\omega_{X/Y})=0$, then $f$ is isotrivial, in BPV,Chapter III. Theorem 17.3.

My question is that does this true for $\mathrm{dim}X\geq 3$ and $\mathrm{dim}Y=1$?

The tools to prove this are using the period map and Torelli's theorem for curves.

In case $\mathrm{dim}X=2$ and $\mathrm{dim}Y=1$: If $\mathrm{deg}(f_*\omega_{X/Y})=0$, then the period map $$\mathcal{P}:Y\rightarrow \overline{\Gamma/D}$$ is constant, where $D$ is the period domain and $\Gamma$ is the monodromy group. Thus $\mathcal{P}(Y^0)$ is a point in $\Gamma/D$, where $Y^0\subseteq Y$ such that $f^0:X^0=f^{-1}(Y^0)\rightarrow Y^0$ is smooth. By Torelli's theorem, all nonsingular fibers will be isomorphic.

In higher dimensional cases, it seems that Torelli's theorem is not true in general. Does this true that $\mathrm{deg}(f_*\omega_{X/Y})=0$ implies the associated period map is constant?