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?