Complex structure on the isoparametric product S^4 times the focal submanifold

Establish whether the isoparametric hypersurface diffeomorphic to S^4 times the focal submanifold M_+ in the definite OT–FKM case with first multiplicity m=4 admits a complex structure.

Background

The paper discusses the isoparametric approach of Tang–Yan and Qian–Tang–Yan. In the definite OT–FKM case with first multiplicity m=4, a complex structure is known on the focal submanifold M_+, while the associated isoparametric hypersurface is diffeomorphic to S4 times M_+. The existence of a complex structure on this product is explicitly identified as unresolved.

References

In the definite OT--FKM case with first multiplicity m=4, a complex structure on the focal submanifold M_+ was constructed in . The associated isoparametric hypersurface is diffeomorphic to S4\times M_+, but the existence of a complex structure on this product was left open.

— A Complex Structure on $T^2\times S^4$: The Genus-One Case of Calabi's Problem  (2610.06067 - Gao et al., 5 Oct 2026) in Section 1, Introduction