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Are there examples of non-Kahler complex manifolds with holomorphically trivial canonical bundle?

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Yes, you might look at the following paper by J. Fine and D. Panov: http://arxiv.org/abs/0905.3237

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This is covered in Andrei Halanay's answer, but it's worth mentioning the simplest examples, which are primary Kodaira surfaces. For the simplest of these:

Take C^2 and quotient by the group generated by these a_k:

a_1 : z -> z + 1

a_2 : z -> z + i

a_3 : w -> w + z + 1

a_4 : w -> w - iz + i

(I think this is it.)

The quotient group is nonabelian. Here z is the fiber and w the base.

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  • $\begingroup$ Let me add that it's obvious that dz dw is preserved in the quotienting (CY). Also, b_1 = 3, meaning it's non-Kahler. Here is a link to Kodaira's paper: jstor.org/pss/2373157 $\endgroup$ – Eric Zaslow Nov 19 '10 at 12:59
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There is a reasonably extensive literature on non-Kahler Calabi-Yau threefolds. They are of interest in string theory; see for example http://xxx.lanl.gov/abs/hep-th/0301161, as well as http://xxx.lanl.gov/abs/0809.4748 for an analogue of Calabi's conjecture in this context.

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Any non-trivial principal elliptic bundle $\pi:X \to B$ over a Calabi-Yau basis is non-Kaehler but has trivial canonical bundle (because $\mathcal{K}_X \simeq \pi^*\mathcal{K}_B$).

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  • $\begingroup$ Nice example! How do you see that it is non-Kahler? Do you have a reference? $\endgroup$ – JME Jul 30 '11 at 10:53
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    $\begingroup$ This is an old result of Andre Blanchard dating back in the '50s. A very thorough discussion may be found in a paper of Thomas Hofer: projecteuclid.org/DPubS/Repository/1.0/… $\endgroup$ – Andrei Halanay Aug 1 '11 at 19:32
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One can ask for non-Kähler compact manifolds with holomorphically trivial tangent bundle, and still get many examples. By a result of Wang ( Proc. AMS 5 ), these are quotients of a complex Lie group $G$ by a discrete subgroup $\Gamma$.

If the quotient is compact Kähler then $G$ must be abelian. Indeed, every vector subspace of the Lie algebra of $G$ gives rise to a holomorphic differential form on $G/\Gamma$. If $G$ is not abelian then one can choose a subspace not closed under the Lie bracket. The corresponding differential form is clearly not closed, what cannot happen in compact Kähler manifolds.

For a thorough study of examples of this kind see this book by Winkelmann.

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More examples are given by nilmanifolds.

Barberis, Dotti and Verbitsky proved in Theorem 2.7 in http://www.ams.org/mathscinet-getitem?mr=2496748 that nilmanifolds endowed with invariant complex structures have trivial canonical bundle. See also Cavalcanti and Gualtieri's Theorem 3.1 in http://www.ams.org/mathscinet-getitem?mr=2131642

On the other hand, non-tori nilmanifolds never admit a Kaehler structure, because of Benson and Gordon, http://www.ams.org/mathscinet-getitem?mr=976592 , or Hasegawa, http://www.ams.org/mathscinet-getitem?mr=946638 , or ...

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