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Oka Geometry of Complex Six-Spheres
Abstract
Alpöge constructed a complex structure on S6 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 ℙ3. The proofs combine explicit cyclic coverings with analytic extension and the invariance of the Oka property under holomorphic coverings.
Related
- Photon spheres, elliptic surfaces, and Seiberg-Witten geometry
- The Morse Index, Nullity and Jacobi Fields of Constant-Curvature Minimal 2-Spheres in Complex Projective Spaces
- Complex curves in o-minimal geometry
- Invisible singularities in complex algebraic geometry
- Blaschke Conjecture and Complex Geometry
BibTeX
@misc{du2026geometry,
title = {{Oka Geometry of Complex Six-Spheres}},
author = {Yun-Heng Du and Bin Guo and Yuta Kusakabe and Song-Yan Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00250},
eprint = {2610.00250},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00250},
}
RIS
TY - UNPB
TI - Oka Geometry of Complex Six-Spheres
AU - Yun-Heng Du
AU - Bin Guo
AU - Yuta Kusakabe
AU - Song-Yan Xie
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.00250
UR - https://arxiv.org/abs/2610.00250
AB - Alpöge 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 ℙ^3. The proofs combine explicit cyclic coverings with analytic extension and the invariance of the Oka property under holomorphic coverings.
ER -