Non-Kahler "Calabi-Yau"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T15:27:33Z http://mathoverflow.net/feeds/question/46524 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46524/non-kahler-calabi-yau Non-Kahler "Calabi-Yau"? 680 2010-11-18T18:48:36Z 2011-08-01T06:40:00Z <p>Are there examples of non-Kahler complex manifolds with holomorphically trivial canonical bundle?</p> http://mathoverflow.net/questions/46524/non-kahler-calabi-yau/46528#46528 Answer by Andrei Moroianu for Non-Kahler "Calabi-Yau"? Andrei Moroianu 2010-11-18T19:12:29Z 2010-11-18T19:12:29Z <p>Yes, you might look at the following paper by J. Fine and D. Panov: <a href="http://arxiv.org/abs/0905.3237" rel="nofollow">http://arxiv.org/abs/0905.3237</a></p> http://mathoverflow.net/questions/46524/non-kahler-calabi-yau/46530#46530 Answer by Jim Bryan for Non-Kahler "Calabi-Yau"? Jim Bryan 2010-11-18T19:26:55Z 2010-11-18T19:26:55Z <p>There are Hopf surfaces which give an example. Topologically they are given by $S^3\times S^1$ and can be realized as a complex manifold by the quotient $\mathbb{C}^2-(0,0)/\mathbb{Z}$ where $n\in \mathbb{Z}$ acts by $(x,y)\mapsto (\lambda^n x,\beta^n y)$ for some fixed non-zero complex numbers $\lambda$ and $\beta$. The Picard group of these guys is a $\mathbb{C}^*$ (a line bundle is determined by its monodromy around the $S^1$ factor). The monodromy of the canonical line bundle on the above Hopf surface is $\lambda\cdot \beta$ and so if we take $\beta=\lambda^{-1}$, it will be trivial. Indeed, the section $dx\wedge dy$ of the canonical line bundle on $\mathbb{C}^2-(0,0)$ will then descend to the quotient.</p> http://mathoverflow.net/questions/46524/non-kahler-calabi-yau/46533#46533 Answer by Andrei Halanay for Non-Kahler "Calabi-Yau"? Andrei Halanay 2010-11-18T19:56:41Z 2011-08-01T06:40:00Z <p>Any non-trivial principal elliptic bundle $\pi:X \to B$ over a Calabi-Yau basis is non-Kaehler but has trivial canonical bundle (because $\mathcal{K}_X \simeq \pi^*\mathcal{K}_B$).</p> http://mathoverflow.net/questions/46524/non-kahler-calabi-yau/46543#46543 Answer by Eric Zaslow for Non-Kahler "Calabi-Yau"? Eric Zaslow 2010-11-18T21:54:54Z 2010-11-18T21:54:54Z <p>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:</p> <p>Take C^2 and quotient by the group generated by these a_k:</p> <p>a_1 : z -> z + 1</p> <p>a_2 : z -> z + i</p> <p>a_3 : w -> w + z + 1</p> <p>a_4 : w -> w - iz + i</p> <p>(I think this is it.)</p> <p>The quotient group is nonabelian. Here z is the fiber and w the base.</p> http://mathoverflow.net/questions/46524/non-kahler-calabi-yau/46548#46548 Answer by Balazs for Non-Kahler "Calabi-Yau"? Balazs 2010-11-18T22:22:53Z 2010-11-18T22:22:53Z <p>There is a reasonably extensive literature on non-Kahler Calabi-Yau threefolds. They are of interest in string theory; see for example <a href="http://xxx.lanl.gov/abs/hep-th/0301161" rel="nofollow">http://xxx.lanl.gov/abs/hep-th/0301161</a>, as well as <a href="http://xxx.lanl.gov/abs/0809.4748" rel="nofollow">http://xxx.lanl.gov/abs/0809.4748</a> for an analogue of Calabi's conjecture in this context. </p> http://mathoverflow.net/questions/46524/non-kahler-calabi-yau/46571#46571 Answer by jvp for Non-Kahler "Calabi-Yau"? jvp 2010-11-19T02:15:36Z 2010-11-19T02:49:11Z <p>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 $\Gamma$.</p> <p>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.</p> <p>For a thorough study of examples of this kind see this <a href="http://homepage.ruhr-uni-bochum.de/Joerg.Winkelmann/publ/papers/smf.html" rel="nofollow">book</a> by Winkelmann.</p>