All Questions

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) ...

Are there any symplectic but not complex Calabi- Yau manifolds in real dimensions 4 and 6?