Fibrations on the 6-sphere and Clemens threefolds

Let $Z$ be a compact, connected $3$-dimensional complex manifold with vanishing first and second Betti numbers and non-vanishing Euler characteristic. We prove that there is no holomorphic mapping from $Z$ onto any $2$-dimensional complex space. In other words, $Z$ can only possibly fiber over a curve. This result applies in particular to a class of threefolds, known as Clemens threefolds, which are diffeomorphic to a connected sum $k\mathrm{\#}(S^3\times S^3)$ for $k\ge 2$. This result also gives a new restriction on any hypothetical complex structure on the $6$-sphere $S^6$.