TITLE: Moishezon manifolds with vanishing first Chern class
QUESTION [17 upvotes]: Suppose $M$ is a Moishezon manifold with $c_1(M)=0$ in $H^2(M,\mathbb{R})$. Does it follow that $K_M$ is torsion in $\mathrm{Pic}(M)$?
This is true whenever $M$ is Kähler (and therefore projective) and was proved independently by Bogomolov, Fujiki and Lieberman. It is also a well-known consequence of Yau's solution of the Calabi Conjecture. Also when $\mathrm{dim}M=2$ then $M$ is automatically projective, so the question is really about dimensions $3$ or more.
The only examples that I know of non-Kähler Moishezon manifolds with $c_1(M)=0$ are obtained by applying a Mukai flop to a projective hyperkähler manifold, and so they have holomorphically trivial canonical bundle. They are described here.
Are there other simple examples of such manifolds?
The same question can also be asked for compact complex manifolds bimeromorphic to Kähler (i.e. in Fujiki's class $\mathcal{C}$).

REPLY [2 votes]: This problem is solved affirmatively in Theorem 1.5 in this paper. 
The idea is that after some blowups we obtain a compact Kähler manifold whose canonical bundle is effective after twisting by a numerically trivial line bundle. A seminal result of Simpson, recently extended to the Kähler case by Wang, then shows that the manifold has nonnegative Kodaira dimension (this is related to the proof of the abundance conjecture when the numerical dimension of the canonical bundle is zero).
Once we know that there is a nontrivial pluricanonical section, it is easy to conclude that this section must be never vanishing.