TITLE: CY fibration over $\mathbb P^1$ without any singular fibers
QUESTION [7 upvotes]: Let's call a smooth projective vartiety $M$ as Calabi-Yau (CY) manifold if it has trivial canonical class and
$h^i(M, \mathcal O_M ) = 0$ for $0 < i < \dim(M)$.
In this definition, a CY 1-fold is an elliptic curve,   a CY 2-fold is a projective $K3$ surface and etc.
For each $n$, I looking for a smooth projective variety $X$ of dimension  $n$ with a fibration $\pi: X \rightarrow \mathbb P^1$ such that

$\pi$ has no singular fibers,
any fiber of $\pi$ is a CY $(n-1)$-fold and
$X$ is not a product of $\mathbb P^1$ and a CY $(n-1)$-fold.

For $n=2$, it is known that such  $X$ (an ellitic surface) does not exist.
I put a question regarding the case of $n=3$ here but didn't get an answer.
For some $n$, does such a variety $X$ exist?

REPLY [6 votes]: To complement YangMills's answer:

Viehweg and Zuo ("On the
isotriviality of families of projective manifolds over curves") proved the following:


Theorem. Let $X$ be a complex projective manifold of non-negative Kodaira dimension. Then a surjective morphism $X\to\mathbf{P}^1$ has at least 3 singular fibres.

This predates Tosatti-Zhang and implies immediately what you want.

The Theorem of the Fixed Part implies that the variation of Hodge structure induced by $X\to \mathbf{P}^1$ is constant. Therefore, by infinitesimal Torelli the fibration is formally locally trivial, and hence etale locally trivial. Again we can use Lemma 17 from Kollar-Larsen to conclude.

The theorem is false in positive characteristic, see Schroer "Some Calabi-Yau threefolds with obstructed deformations over the Witt vectors" (the example, based on a construction of Moret-Bailly, is a K3 fibration over $\mathbf{P}^1$).