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](https://mathoverflow.net/questions/392831/an-example-of-threefold-with-k3-fibration) but didn't get an answer.


For some $n$, does such a variety $X$ exist?



CY fibration over $\mathbb P^1$ without any singular fibers
No such variety exists: first, you can use Remark 3.2 [here](https://arxiv.org/pdf/1309.3773.pdf) to see that your map $\pi$ must be a holomorphic fiber bundle, and then Lemma 17 [here](https://arxiv.org/pdf/math/0701466.pdf) 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.


