Complex-Structure Problem for the Six-Sphere

The six-sphere admits an almost complex structure, obtained for example from the octonionic cross product, but the classical problem asks whether any almost complex structure on $S^6$ is integrable and hence comes from a genuine complex manifold. Borel and Serre proved that among spheres only $S^2$ and $S^6$ can even carry an almost complex structure [BorelSerre1953Steenrod], which makes dimension six the unique unresolved sphere case.

An integrable structure on $S^6$ would necessarily be non-Kähler and is subject to unusually strong restrictions. Campana, Demailly, and Peternell derive constraints on its algebraic dimension and meromorphic geometry, illustrating how far a hypothetical complex $S^6$ would lie from familiar compact complex manifolds [CampanaDemaillyPeternell2020S6].

An unsigned, undated manuscript hosted by Alpoge and circulated in August 2026 claims to construct such an integrable structure [AlpogeHosted2026S6]. No independently verified proof or broadly accepted publication is yet available, so the manuscript is recorded as a pending claim and the existence question remains operationally open.