A preprint by Andrei Marshakov and Antti Niemi appeared on hep-th this evening making a remarkable claim. According to this preprint, a few weeks before passing away recently at the age of 93, Shiing-Shen Chern completed a preprint entitled “On the Non-existence of a Complex Structure on the Six Sphere”.
Whether or not a given manifold defined using real coordinates can be given the structure of a complex manifold is often a difficult problem. For the case of a d-dimensional sphere, clearly you can’t do this in odd dimensions, but for even dimensions, you certainly can for the case d=2. For the cases d=4 and d=8 or more, there is a topological obstruction to even finding an “almost complex structure”. In other words, you can’t find a continous choice for each point on the sphere of what it means to multiply elements of the tangent space by the square root of minus one. The case d=6 is special: you can use the octonions to construct an almost complex structure, but this complex structure is not “integrable”, it doesn’t come from any local choice of complex coordinates. One of the most famous open problems in geometry has long been the following: is there another almost complex structure on the six-sphere that is actually integrable?
It has long been conjectured that there is no such integrable almost complex structure, but no one has ever been able to prove this. Chern’s preprint contains a purported proof, but Marshakov and Niemi devote only a paragraph to the non-trivial part of his argument. From their preprint you can’t tell whether Chern has a valid argument.
I’ve heard via e-mail from a knowledgeable authority on the subject who points out that there are serious flaws in the manuscript that was privately circulated. His opinion is that Chern’s argument actually does prove something interesting, but not the full result Chern claims, so the conjecture about the non-existence of a complex structure on the six-sphere remains open.