Conditional Oka Complex Structures of the Six-Sphere

Abstract

Alpöge'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 Alpöge's identification of the underlying smooth manifolds with $S^6$, these results yield uncountably many pairwise nonbiholomorphic Oka complex structures on the six-sphere.

arXiv comment21 pages. With 2 open problems at the end. Du-Guo-Kusakabe-Xie already solved some problems in the first version