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For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$.

If there are no mistakes in this argument, the base of an elliptically fibered Calabi-Yau threefold $\pi:X\to B$ has to be rational if $X$ has a trivial fundamental group, but if the fundamental group is not trivial, the base $B$ might be an Enriques surface.

Is it actually possible for the base to be an Enriques surface? Are there restrictions on the fundamental group of the Calabi-Yau variety in that case?

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    $\begingroup$ Let $\nu:\widetilde{B}\to B$ be an etale, degree-$2$ cover of an Enriques surface $B$ by a K3 surface $\widetilde{B}$. Denote by $i:\widetilde{B}\to \widetilde{B}$ the associated involution. Let $(E,0)$ be an elliptic curve with involution $j:(E,0)\to (E,0).$ Let $\widetilde{X}$ be the product $\widetilde{B}\times E$ with the involution $(i,j)$. Denote the quotient by this involution as $q:\widetilde{X}\to X$. The projection $\text{pr}_{\widetilde{B}}$ on $\widetilde{X}$ induces a morphism $\pi:X\to B$ such that $\pi\circ q$ equals $\nu\circ \text{pr}_{\widetilde{B}}$. $\endgroup$ – Jason Starr Aug 6 '18 at 0:28

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