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Existence of an integrable complex structure on S6 (Hopf problem)

Determine whether the six-dimensional sphere S6 admits an integrable complex structure.

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Background

The classical Hopf problem asks which even-dimensional spheres admit integrable complex structures. Borel and Serre showed that only S2 and S6 admit almost complex structures; S2 is known to admit an integrable complex structure, while the integrability on S6 remains unresolved.

In this paper, the authors paper generalized complex structures on S6 and prove non-existence results for certain induced structures, but they explicitly note that the underlying classical question—whether S6 carries an integrable complex structure—remains unknown.

References

In respect of strong generalized complex structures, even though it is not known if there is an integrable complex structure on S6 we can study the problem on the bundle TS6 and determine if there are some particular integrable generalized complex structures.

— About generalized complex structures on $\mathbb S^6$ (2405.05681 - Etayo et al., 9 May 2024) in Section 1 (Introduction)