Is there a complex structure on the 6-sphere? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T01:32:05Z http://mathoverflow.net/feeds/question/1973 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere Is there a complex structure on the 6-sphere? Deane Yang 2009-10-22T23:16:31Z 2011-06-27T17:53:56Z <p>I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a number of published proofs that are not taken seriously, even though nobody seems to know exactly why they are wrong.</p> http://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere/1974#1974 Answer by Chris Schommer-Pries for Is there a complex structure on the 6-sphere? Chris Schommer-Pries 2009-10-22T23:21:24Z 2009-10-22T23:21:24Z <p>This is a famous open-problem. It is still unknown. </p> http://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere/1984#1984 Answer by Joel Fine for Is there a complex structure on the 6-sphere? Joel Fine 2009-10-23T00:04:52Z 2009-10-23T00:04:52Z <p>Of course, I'm not about to answer this question one way or the other, but there are at least a couple of interesting things one might point out. Firstly, it has been shown (although I forget by whom) that there is no complex structure on S<sup>6</sup> which is also orthogonal with respect to the round metric. The proof uses twistor theory. The twistor space of S<sup>6</sup> is the bundle whose fibre at a point p is the space of orthogonal almost complex structures on the tangent space at p. It turns out that the total space is a smooth quadric hypersurface Q in CP<sup>7</sup>. If I remember rightly, an orthogonal complex structure would correspond to a section of this bundle which is also complex submanifold of Q. Studying the complex geometry of Q allows you to show this can't happen.</p> <p>Secondly, there is a related question: does there exist a non-standard complex structure on CP<sup>3</sup>? To see the link, suppose there is a complex structure on S<sup>6</sup> and blow up a point. This gives a complex manifold diffeomorphic to CP<sup>3</sup>, but with a non-standard complex structure, which would seem quite a weird phenomenon. On the other hand, so little is known about complex threefolds (in particular those which are not Kahler) that it's hard to decide what's weird and what isn't.</p> <p>Finally, I once heard a talk by Yau which suggested the following ambitious strategy for finding complex structures on 6-manifolds. Assume we are working with a 6-manifold which has an almost complex structure (e.g. S<sup>6</sup>). Since the tangent bundle is a complex vector bundle it is pulled back from some complex Grassmanian via a classifying map. Requiring the structure to be integrable corresponds to a certain PDE for this map. One could then attempt to deform the map (via a cunning flow, continuity method etc.) to try and solve the PDE. I have no idea if anyone has actually tried to carry out part of this program. </p> http://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere/2536#2536 Answer by Fran Burstall for Is there a complex structure on the 6-sphere? Fran Burstall 2009-10-25T23:54:45Z 2009-10-25T23:54:45Z <p>A little more detail to Joel's first paragraph (I can't see how to add a comment to it, sorry!).</p> <p>The argument that there is no orthogonal complex structure on the 6-sphere is due to Claude Lebrun and the point is that such a thing, viewed as a section of twistor space, has as image a complex submanifold. Now, on the one hand, this submanifold is Kaehler, and so has non-trivial second cohomology, since the twistor space is Kaehler. On the other hand, the section itself provides a diffeomorphism of our submanifold with the 6-sphere which has trivial second cohomology. Neat, huh?</p> http://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere/12583#12583 Answer by algori for Is there a complex structure on the 6-sphere? algori 2010-01-21T23:02:23Z 2010-01-21T23:11:58Z <p>If such a complex structure exists, it would weird indeed! For example, as shown by Campana, Demailly and Peternell (Compositio 112, 77-91), if such a thing exists, then $S^6$ would have no non-constant meromorphic functions. In particular, $S^6$ can't be Moishezon, let alone algebraic.</p> http://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere/68949#68949 Answer by Al-burcas for Is there a complex structure on the 6-sphere? Al-burcas 2011-06-27T17:53:56Z 2011-06-27T17:53:56Z <p>Continuing Joel Fine and Fran Burstall's answer about, indeed "neat", Lebrun's result. Just want to recall that the "orthogonal" twistor space of any $2n$-dimensional pseudo-sphere $SO(2p+1,2q)/SO(2p,2q)$ can be written as $SO(2p+2,2q)/U(p+1,q)$. So the Kähler manifold in question, in case of the 6-sphere, is $SO(8)/U(4)$. One should think of each $j:T_xS^6\rightarrow T_xS^6$ as a linear map on $R^8$ with $j(x)=-1$ and $j(1)=x$. Well, proofs have been rewritten of LeBrun's result. I wish I had more opinion on this: <a href="http://arxiv.org/abs/math/0509442" rel="nofollow">http://arxiv.org/abs/math/0509442</a></p>