# Complex manifolds with trivial canonical bundle

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.

I think the answers you are looking for are in this 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!

• Thanks. But from that proposition I fail to see that every compact complex manifold with trivial canonical bundle is Calabi-Yau. Plus, wouldn't the complex nilmanifold be a counterexample? – Bilateral Feb 16 '15 at 12:44

I think I might be able to answer my own question. I would appreciate if someone can confirm that I am interpreting this correctly. In reference

http://intlpress.com/site/pub/files/_fulltext/journals/mrl/2009/0016/0002/MRL-2009-0016-0002-a010.pdf

they explicitly say:

"We prove that a complex nilmanifold has trivial canonical bundle"

"This condition is quite strong. For instance, any compact complex surface with trivial canonical bundle is isomorphic to a K3 surface, a torus, or a Kodaira surface; the first two are Kahler, and the latter is a nilmanifold"

Therefore it looks like there are actually many examples, namely complex nilmanifolds, of compact, complex manifolds with trivial canonical bundle which however are not Calabi-Yau manifolds.

• Yes, the nilmanifolds are counterexamples. You could read Barth, Peters and van den Ven for details proving that the Kodaira surfaces are not Kaehler, because they had odd first Betti number. – Ben McKay Feb 16 '15 at 17:03
• See also the long list of references on page 1 of arxiv.org/pdf/1401.4797.pdf, which will provide you with many more examples of compact complex non-Kähler manifolds with trivial canonical bundle. – YangMills Feb 16 '15 at 21:37