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.


Thanks.



Complex manifolds with trivial canonical bundle
I think the answers you are looking for are in [this](http://arxiv.org/pdf/1401.4797.pdf) paper by V. Tosatti, see in particular Proposition 1.1, point (4) and Proposition 1.3. 


Warning (in view of the comment below by S.S.): the holonomy is computed with respect to the Chern connection of the hermitian metric, which is, since $(X,\omega)$ is not necessarily Kähler, not equal in general to the Levi-Civita connection of the underlying riemannian metric!


Have a nice reading!


