Complex Foliations of Odd-Dimensional Spheres

Definitions and mathematical statement

Classification question. For which integers $n$ does the odd-dimensional sphere $S^{2n-1}$ admit a foliation by complex manifolds? Here $S^{2n-1}=\{x\in{\mathbb{R}}^{2n}:\|x\|=1\}$. The source does not state the leaf dimension, codimension, regularity, or whether singular foliations are permitted. Allowing zero-dimensional leaves gives the foliation by points on every sphere, so a nontrivial leaf-dimension requirement is essential.

Under the usual interpretation suggested by the cited Reeb-foliation context, take $n\ge2$ and ask for a regular real-codimension-one foliation $\mathcal F$ on $S^{2n-1}$, together with a smoothly varying complex structure $J$ on its rank-$2n-2$ tangent bundle $T\mathcal F$, such that $J^2=-\operatorname{id}$ and each leaf with the induced structure is a complex manifold of complex dimension $n-1$. Thus leafwise integrability is required; a leafwise almost complex structure alone is insufficient. Equivalently, one seeks a smooth foliation whose leafwise coordinate changes are holomorphic in the complex leaf variables, with the intended transverse regularity specified. This is a conditional codimension-one formulation, not a claim that the missing source explicitly imposes it. If other positive leaf dimensions or regularities are intended, they define different existence questions. No complex-manifold structure on the entire odd-dimensional sphere is being requested.

The supplied literature triage checked 2026-08-21 labels the classification open. It cites low-dimensional examples such as the Reeb foliation of $S^3$, and reports that the previously claimed $S^5$ construction was corrected and does not establish such a foliation of $S^5$. Failure of that construction is not an impossibility theorem for $S^5$. These dated claims and the exact foliation hypotheses of the cited correction require independent verification; no current classification is asserted here.

Short English statement

Determine which odd-dimensional spheres carry a nontrivial foliation with complex leaves, under the intended dimension and regularity conventions. The likely codimension-one interpretation is explicit. The supplied correction invalidates a claimed $S^5$ construction; it does not prove nonexistence.

Sources

• Local UnsolvedMath registry, permanent ID 30000122, code catalogue entry; Complex Foliations of Odd-Dimensional Spheres, Oberwolfach Reports Open Problems.
• Komplexe Analysis (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/42, pp. 2171--2216. https://doi.org/10.4171/owr/2004/42. Proposed year and year tag: 2004; no proposer named.
• Supplied literature label Meersseman--Verjovsky correction: https://annals.math.princeton.edu/2011/174-3/p11.
• Supplied literature label OWR report: https://doi.org/10.4171/owr/2004/42.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field concerns balanced notation and removal of a repeated question label. Verify the dated literature claims independently.