Are there any symplectic but not holomorphic Calabi-Yau manifolds in real dimensions 4 and 6? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T00:51:31Z http://mathoverflow.net/feeds/question/71661 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71661/are-there-any-symplectic-but-not-holomorphic-calabi-yau-manifolds-in-real-dimensi Are there any symplectic but not holomorphic Calabi-Yau manifolds in real dimensions 4 and 6? Thom 2011-07-30T16:23:19Z 2011-07-30T17:37:21Z <p>Are there any symplectic but not complex Calabi- Yau manifolds in real dimensions 4 and 6?</p> http://mathoverflow.net/questions/71661/are-there-any-symplectic-but-not-holomorphic-calabi-yau-manifolds-in-real-dimensi/71666#71666 Answer by Dmitri for Are there any symplectic but not holomorphic Calabi-Yau manifolds in real dimensions 4 and 6? Dmitri 2011-07-30T17:03:09Z 2011-07-30T17:37:21Z <p>First of all, the notion <em>Symplectic Calabi-Yau</em> is quite new. A few persons who use it (including myself) usually mean by this symplectic manifolds with \$c_1=0\$, (this is just to make sure that we speak about the same thing) </p> <p>In real dimension \$4\$ we know for the moment only two types of symplectic Calabi-Yau manifolds - \$K3\$ surfaces and \$T^2\$ bundles over \$T^2\$. These manifolds have as well the structure of a complex manifold with a non-vanishing holomorphic volume form. It is conjectured that there are no other symplectic Calabi Yau manifolds in dimension \$4\$. </p> <p>In real dimension six there are quite a lot of symplectic CY manifolds coming from the twistor construction (you can check here: <a href="http://arxiv.org/abs/0802.3648" rel="nofollow">http://arxiv.org/abs/0802.3648</a>), and some of them do have a complex structure, but this is not known for all of them.</p> <p>At the same time, probably you know that in dimension \$2n\ge 6\$ the following question is open:</p> <p><strong>Question</strong>. <em>Is it true that every manifold \$M^{2n}\$ that has an almost complex structure \$J\$ has as well a holomorphic structure homothopic to J?</em></p> <p>This is an old question and apparently no one has an idea of how to answer it. Now, the answer to your question in dimension \$6\$ depends on what you mean by a complex Calabi-Yau. This notation is not used in math literature. If by such a manifold you mean a complex manfiold with \$c_1=0\$, then you would not be able (for the moment) to get any example in dimension \$6\$ where the answer to your question is no (because the above <strong>Question</strong> is open). On the other hand, if by complex Calabi-Yau you mean a complex manifold with a non-vanishing holomorphic volume form, then the answer to your question is yes, an example is given in <a href="http://arxiv.org/abs/0905.3237" rel="nofollow">http://arxiv.org/abs/0905.3237</a>. There is a symplectic Calabi Yau 6-manifold in this paper, that has \$b_3=0\$, hence it can not have a holomorphic volume form of top degree for any complex structure. One can construct further such examples.</p>