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,](http://iopscience.iop.org/0025-5726/8/1/A02) [Fujiki](https://doi.org/10.1007/BF01403162) and [Lieberman.](https://doi.org/10.1007/BFb0064399) It is also a [well-known consequence](http://projecteuclid.org/euclid.jdg/1214438181) 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.](http://books.google.com/books?id=frKJmxAr2xUC&lpg=PP1&ots=j2y7RTsg24&pg=PA170)


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 $\mathcal{C}$).



Moishezon manifolds with vanishing first Chern class
This problem is solved affirmatively in Theorem 1.5 in [this paper.](http://arxiv.org/abs/1401.4797) 


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](http://www.numdam.org/item?id=ASENS_1993_4_26_3_361_0), recently extended to the Kähler case by [Wang](http://arxiv.org/abs/1312.6619), 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](http://arxiv.org/abs/math/0405093) [bundle](http://projecteuclid.org/euclid.msjm/1389986103) [is](http://arxiv.org/abs/0912.0576) [zero](http://arxiv.org/abs/1002.2682)).


Once we know that there is a nontrivial pluricanonical section, it is easy to conclude that this section must be never vanishing.


