Hopf's problem: does the six-sphere admit a complex structure?

The six-sphere is, together with the two-sphere, one of only two spheres admitting almost complex structures, yet it is not known whether it admits a complex structure; this is Hopf's problem, open since Hopf's 1947-1948 essay. The standard almost complex structure induced by octonion multiplication on the imaginary unit octonions is not integrable, and every known non-existence result excludes integrability only under auxiliary hypotheses: orthogonality with respect to the round metric or metrics close to it, preservation of the standard 2-form, or first-order covariant-derivative conditions such as those introduced by Wilson (2018). Conversely, the geometry of a hypothetical complex structure on $S^6$ is strongly constrained (non-Kahler, algebraic dimension at most one, restricted Hodge numbers and automorphism group) but no contradiction is known, and claimed proofs of existence (Etesi) and of non-existence (Atiyah; Clemente) have not been accepted by the community. The unrestricted question - does there exist an integrable almost complex structure on the standard smooth six-sphere? - remains open, as do the derived obstruction sub-questions posed in Wilson (2018).

Background

An almost complex structure on a smooth $2n$-dimensional manifold $M$ is a smooth vector-bundle endomorphism $J\colon TM\to TM$ with $J^2=-\mathrm{Id}$, that is, a smoothly varying choice of linear complex structure on each tangent space. The structure is integrable, and $M$ a complex manifold, if these tangent-space complex structures arise from holomorphic local coordinates; by the theorem of Newlander and Nirenberg this holds exactly when the Nijenhuis tensor $N_J(X,Y)=[JX,JY]-[X,Y]-J[X,JY]-J[JX,Y]$ vanishes identically. Every complex manifold is almost complex, but not conversely. By a classical theorem of Borel and Serre, the only spheres carrying almost complex structures at all are $S^2$ and $S^6$; on $S^2$ every almost complex structure is integrable, while on $S^6$ the question of integrability is exactly what is at stake.

The six-sphere carries a canonical almost complex structure: identifying $S^6$ with the unit imaginary octonions (the octonions being the eight-dimensional non-associative real division algebra), multiplication by an imaginary unit octonion defines such a $J$, constructed by Kirchhoff in 1947. Together with the round metric this gives $S^6$ its standard nearly Kahler structure - an almost Hermitian structure (a metric preserved by $J$) whose fundamental 2-form $\omega(X,Y)=g(X,JY)$ has totally skew-symmetric covariant derivative, a first-order condition strictly weaker than the Kahler condition $d\omega=0$ - and classical results from the early 1950s show that this particular $J$ is not integrable. Whether some other almost complex structure on $S^6$ is integrable was asked by Hopf: in his 1947-1948 essay he asked whether every orientable even-dimensional real-analytic manifold admits local complex-analytic coordinates, answered this negatively for several spheres himself, and explicitly left the six-sphere unresolved. The question in its standard modern formulation - does there exist a complex structure on $S^6$? - has been open since; its early history is documented in the study of Agricola, Bazzoni, Goertsches, Konstantis, and Rollenske, the historical introduction to the 2017 Marburg conference devoted to the problem.

Two complementary research lines have grown around the problem. The first proves non-existence under auxiliary hypotheses. Blanchard in 1953 (rediscovered by LeBrun in 1987) showed that no almost complex structure on $S^6$ orthogonal for the round metric is integrable; Bor and Hernandez-Lamoneda exhibited an open set of almost complex structures, containing all orthogonal ones, in which integrability is impossible, and Kruglikov extended the orthogonal non-existence to metrics close to the round one; Tang excluded structures satisfying a curvature-positivity condition; an argument of S.-S. Chern, exposed by Bryant, excludes structures preserving the standard fundamental 2-form; and Wilson excluded structures whose covariant derivatives $\nabla_XJ$ and $\nabla_{JX}J$ commute for all $X$, as well as all structures that are strongly integrable (i.e. $\nabla_{JX}J=J\nabla_XJ$) with respect to the Levi-Civita connection of the round metric, in each case with refined intrinsic first-order differential inequalities forcing non-integrability. None of these results touches an arbitrary complex structure. The second line studies the geometry of a hypothetical complex manifold $X$ with underlying smooth manifold $S^6$: since $H^2(S^6,\mathbb{R})=0$, such an $X$ could not be Kahler, because a Kahler metric would make its fundamental 2-form a closed non-degenerate 2-form representing a nonzero cohomology class; Campana, Demailly, and Peternell showed that its algebraic dimension - the transcendence degree of its field of meromorphic functions - is at most one; and its Hodge numbers, Frolicher spectral sequence (the spectral sequence converging from Dolbeault cohomology to de Rham cohomology), and automorphism group are strongly constrained, as surveyed by Angella and by Lehn, Rollenske, and Schinko. These constraints are mutually consistent, and no contradiction has been derived. Several claimed resolutions in both directions - existence constructions by Etesi, and short non-existence arguments by Atiyah and, via curvature-obstruction integrability criteria, by Clemente - have not been accepted; in the Clemente case, Guan, Li, and Wang showed that the integrability criterion used is too strong, so the claimed non-existence is not established.

Problem Statement

Does the standard smooth six-dimensional sphere $S^6$ admit a complex structure? Equivalently: does there exist a smooth field of endomorphisms $J\colon TS^6\to TS^6$ with $J^2=-\mathrm{Id}$ whose Nijenhuis tensor

$N_J(X,Y)=[JX,JY]-[X,Y]-J[X,JY]-J[JX,Y]$

vanishes identically on $S^6$, so that by the Newlander-Nirenberg theorem $J$ is induced by holomorphic local coordinates? The question is unrestricted: compatibility with the round metric or any other Riemannian metric, preservation of any differential form, and first-order covariant-derivative conditions are hypotheses of the known partial results, not of the problem. The question is open and may be resolved in either direction. This is Hopf's problem in its standard formulation; the derived obstruction sub-questions raised in Wilson (2018) (relating the known non-integrability loci, localizing the $H^2(S^6)=0$-based corollaries via Stokes' theorem on $J$-holomorphic curves, and the sharpness of the first-order inequalities) are not part of this question.

References

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