CY fibration over $\mathbb P^1$ without any singular fibers

Question

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

1. $\pi$ has no singular fibers,
2. any fiber of $\pi$ is a CY $(n-1)$-fold and
3. $X$ 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?

Answer 1

No such variety exists: first, you can use Remark 3.2 here to see that your map $\pi$ must be a holomorphic fiber bundle, and then Lemma 17 here gives you that this bundle becomes a trivial product after pulling back your family via a finite étale map of the base. Since the base is $\mathbb{P}^1$ this finite map is an isomorphism, so your original family is a product.

It is also interesting to observe that this fails when $X$ is a general compact complex manifold, for example the standard diagonal Hopf surface is an elliptic bundle over $\mathbb{P}^1$ which is not a product.

Answer 2

To complement YangMills's answer:

1. 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.

2. 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.

3. 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 $\mathbf{P}^1$).