Conditional Oka Complex Structures of the Six-Sphere

Abstract

Alp\"oge's construction produces a period family of compact complex threefolds fibred over $\mathbb{P}^1$, with complex two-tori as general fibres. We give a direct proof that every admissible member is Oka, using only the complex-analytic construction. Du--Guo--Kusakabe--Xie independently established the Oka property and determined the exact biholomorphism classes in this family. Assuming Alp\"oge's identification of the underlying smooth manifolds with $S^6$, these results yield uncountably many pairwise nonbiholomorphic Oka complex structures on the six-sphere.