Holomorphic Ellipticity vs. Subellipticity: Do the Classes Coincide?

Determine whether, in the holomorphic category, the classes of holomorphically elliptic manifolds and holomorphically subelliptic manifolds coincide.

Background

In the algebraic category, Kaliman and Zaidenberg proved that algebraic subellipticity and algebraic ellipticity coincide, unifying key algebraic Oka properties.

The analogous equivalence in the holomorphic category remains unsettled. Resolving this would clarify the precise landscape of holomorphic flexibility properties and unify the theory across algebraic and analytic settings.

References

However, it is not known whether the same is true in the holomorphic category.

— Oka-1 manifolds: New examples and properties  (2402.09798 - Forstneric et al., 2024) in Section 4 (Simply connected algebraically elliptic manifolds are aOka-1)

Our proof establishes only the Oka assertion. It leaves open whether Y_{c_0} admits such a spray.

— A Conditional Oka Complex Structure of the Six-Sphere  (2609.26706 - Chen, 22 Sep 2026) in Section \ref{sec:questions} (Further questions)