Timeline for Are there any symplectic but not holomorphic Calabi-Yau manifolds in real dimensions 4 and 6?
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| Jul 30, 2011 at 18:25 | comment | added | Dmitri Panov | Sure, this is a nice article. There is a difference between non-Kahler and non-holomorphic cases (and your question was of course about non-holomorphic). The twistor construction, that I mentioned produces a huge amount on non-Kahler symplectic Calabi Yau six-manifolds -- for example because Kahler manifolds have quite restricted fundamental groups. But we don't know any non-trivial restrictions on complex manifolds... | |
| Jul 30, 2011 at 18:18 | comment | added | Thom | Maybe they admit complex structure. not sure about that. Probably this seems very difficult question. | |
| Jul 30, 2011 at 18:14 | comment | added | Thom | I was just checking the following papers on arxiv. They do contain some symplectic, but non Kahler Calabi-Yau 6 manifolds. Here are the links arxiv.org/pdf/1107.2623.pdf arxiv.org/abs/1105.3519 | |
| Jul 30, 2011 at 17:37 | history | edited | Dmitri Panov | CC BY-SA 3.0 |
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| Jul 30, 2011 at 17:37 | vote | accept | Thom | ||
| Jul 30, 2011 at 17:31 | comment | added | Dmitri Panov | No, there is no upper bound on the Betty number of symplectic Calabi-Yaus in dimension higher than $4$. On the other hand there is such a bound in dimension $4$, you can check it here: T. J. Li. Quaternionic vector bundles and Betti numbers of symplectic 4-manifolds with Kodaira dimension zero. Internat. Math. Res. Notices, (2006), 1–28. | |
| Jul 30, 2011 at 17:27 | history | edited | Dmitri Panov | CC BY-SA 3.0 |
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| Jul 30, 2011 at 17:16 | comment | added | Thom | Thanks! Do you know if there is an upper bound on the Betti numbers for symplectic Calabi-Yau? | |
| Jul 30, 2011 at 17:14 | history | edited | Dmitri Panov | CC BY-SA 3.0 |
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| Jul 30, 2011 at 17:03 | history | answered | Dmitri Panov | CC BY-SA 3.0 |