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Conditional Oka Complex Structures of the Six-Sphere
Abstract
Alpöge's construction produces a period family of compact complex threefolds fibred over ℙ1, 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 S6, these results yield uncountably many pairwise nonbiholomorphic Oka complex structures on the six-sphere.
Related
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BibTeX
@misc{chen2026conditional,
title = {{Conditional Oka Complex Structures of the Six-Sphere}},
author = {Zhangchi Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26706},
eprint = {2609.26706},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26706},
}
RIS
TY - UNPB
TI - Conditional Oka Complex Structures of the Six-Sphere
AU - Zhangchi Chen
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.26706
UR - https://arxiv.org/abs/2609.26706
AB - Alpöge's construction produces a period family of compact complex threefolds fibred over 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.
ER -