Timeline for Which almost complex manifolds admit a complex structure?
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Feb 11, 2011 at 2:31 | comment | added | Joel Fine | I should add I think that this is a fascinating problem, but have not the slightest idea how to start trying to attack it. It seems to me like a question waiting for a "big idea". I would guess the total failure to understand the six-sphere is what puts most people off, but perhaps that is overly negative. Sometimes the less topology you have the harder things are. E.g, we know a lot about differential topology of various 4-manifolds, but still nothing about simply connected ones with b_2=0... | |
| Feb 11, 2011 at 2:25 | comment | added | Joel Fine | Added as a comment (because you've already got two great answers). In dim 4 you can get every finitely presented group as the fundamental group of an almost complex 4-mfd (symplectic even, by a thm of Gompf, but this is harder). On the other hand, the classification of complex surfaces tells you that the possible topology of complex surfaces is much more constrained. What changes drastically in dim 6 is that any finitely presented group arises as pi_1 of a closed complex threefold. This was first proved by Taubes as a corollary of an existence thm for self-dual metrics. | |
| Feb 2, 2011 at 19:13 | vote | accept | Gunnar Þór Magnússon | ||
| Feb 2, 2011 at 13:24 | answer | added | Misha Verbitsky | timeline score: 20 | |
| Jan 23, 2011 at 14:52 | answer | added | Francesco Polizzi | timeline score: 26 | |
| Jan 23, 2011 at 14:37 | history | asked | Gunnar Þór Magnússon | CC BY-SA 2.5 |