While reading [this paper of Kollár](http://arxiv.org/abs/1206.5721), the following question came up. If $f:X\to Y$ is a fiber space (i.e. surjective holomorphic map with connected fibers) with $X,Y$ smooth projective manifolds, with $K\_X\sim\_{\mathbb{Q}} 0$, and with $f$ a submersion everywhere, does there exist a finite étale cover $Y'\to Y$ such that $X\times\_{Y} Y'$ is isomorphic to a product family (and so $K\_Y\sim\_{\mathbb{Q}}0$ too)?


Note that since $f$ is a submersion, every fiber $F$ satisfies $K\_F\sim\_{\mathbb{Q}}0$.


The case when $\mathrm{dim}X=\mathrm{dim}Y+1$ (which includes the case when $\mathrm{dim}X=2$) is easy, since then the fibers are elliptic curves, and the $j$-invariant of each fiber gives a holomorphic function on $Y$, hence constant. It then follows from Fisher-Grauert that $f$ is a holomorphic fiber bundle, and you can apply for example Lemma 17 in [this paper of Kollár-Larsen](http://arxiv.org/abs/math/0701466) to conclude.


This same argument shows that in arbitrary dimension it is enough to show that all the fibers of $f$ are biholomorphic.


Thanks to [work of Kawamata](http://dx.doi.org/10.1515/crll.1985.363.1) we know that $\kappa(Y)\leq 0$. We can also apply Theorem 4.4 in [this paper of Fujino-Gongyo](http://arxiv.org/abs/1003.1483) and get that $-K\_Y$ is semiample.
So if $\kappa(Y)=0$ then we obtain that $K\_Y\sim\_{\mathbb{Q}}0$. At this point one can apply the arguments in Theorem 4.8 of [this paper of Ambro](http://arxiv.org/abs/math/0308143) and conclude.


So the question is reduced to the following problem: is it possible to have a holomorphic submersion $f:X\to Y$ with connected fibers, where $X,Y$ are smooth projective manifolds with $K\_X\sim\_{\mathbb{Q}} 0$ and $\kappa(Y)<0$ ?


If we assume that $Y$ is simply connected and that $X$ has $K\_X\cong\mathcal{O}\_X$ and $h^{p,0}(X)=0$ for $p=1,...,\mathrm{dim} X-1$ (so $X$ is Calabi-Yau in the stronger sense), then Corollary 2.5 in [this paper of Zhang-Zuo](http://arxiv.org/abs/math/0504141) implies that this is impossible.


In general I don't know the answer to this, but when $f$ is not assumed to be a submersion (so there might be singular fibers) then $\kappa(Y)<0$ can happen, for example when $X$ is an elliptically fibered $K3$ surface and $Y=\mathbb{P}^1$.



Calabi-Yau fiber space without singular fibers implies finite quotient of product?
I am not familiar with this topic, but I remember seeing [this recent paper](http://arxiv.org/abs/1309.3773). It claims that if the map has no singular fiber, it is a fiber bundle (all the fibers isomorphic). Moreover, if the total space is algebraic, it can be a product family after a suitable base change, as you claimed.


**Edit**
I made my answer more informative, as the comments below suggested. I thought the abstract of the paper would answer to the question.


