A Conditional Oka Complex Structure of the Six-Sphere

Alpöge constructs a compact complex threefold $Y$ fibred over ${\mathbb{P}}^1$ by complex two-tori away from three special points. We prove that $Y$ is an Oka manifold. The proof applies to every admissible period parameter with the same logarithmic twists and cusp gluing. If one also accepts the identification of the underlying smooth manifold with $S^6$ in Alpöge’s Theorem~8.1, then the six-sphere admits an Oka complex structure. This topological identification is not used in the Oka proof.