Oka Geometry of Complex Six-Spheres

Abstract

Alp\"oge constructed a complex structure on $S^6$ through a torus fibration with exactly three special fibres. We prove that every member of the associated admissible period family is Oka and that the family projection over each relatively compact parameter disc is an Oka map. The Oka property survives the deletion or simultaneous blow-up of any finite set of distinct points in any member. Blowing up a single point therefore produces an Oka complex manifold diffeomorphic, but not biholomorphic, to $\mathbb{P}^3$. The proofs combine explicit cyclic coverings with analytic extension and the invariance of the Oka property under holomorphic coverings.