It is known that a compact Calabi-Yau manifold can be defined as a compact Kahler manifold $M$ with trivial canonical bundle, or alternatively, a reduction of the structure group from $U(n)$ to $SU(n)$, where $n$ is the complex dimension of $M$. Suppose that I take $M$ to be a complex manifold which however is not Kahler, but it also has trivial canonical bundle. Then $M$ shouldn't be a Calabi-Yau manifold. Do you know an explicit example of this situation? Namely, a compact complex manifold with trivial canonical bundle which is not a Calabi-Yau manifold.