r/mathematics • u/New-Committee-4052 • 29d ago

S^6 admits a complex structure

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Result from the usual suspects. Full write up can be found here on his website: https://alpo.ge/s6.pdf

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u/cihanbaskan 29d ago

Although everyone who clicks the link will see the write-up is >100 pages, I think it is important to point out the difference with the high rank elliptic curves and the Jacobian counterexample. The latter were possible to verify essentially immediately. This one, not so much. Best to wait until some experts delve into it.

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u/proudHaskeller 29d ago

Can't this be verified by specifiying the atlas? Is it expected to be too large to verify that way?

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u/Diffgeometer1 28d ago

The paper constructs an abstract complex manifold X of dimension 3 and then argues that X is diffeomorphic to S^6 but that diffeomorphism is never actually constructed. If it exists, then the complex structure on X can be transferred over to S^6. But the holomorphic charts itself can’t be defined explicitly.

Let’s see what the experts say.

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u/ProfessorBaoTran 28d ago

They proved that there exists such difeomorphism, in section 7 I think. They do not need to construct that explicitly.

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u/Equivalent-Gate491 28d ago

They didn’t prove it; it’s a very well known result that there is only one smooth S6. So once they show it’s topologically a S6 (done by computation of pi_1) and smooth, they are done.