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PreprintCC BY 4.0arXiv

A Complex Structure on S2× S4

  • Shengtao Guo
  • Ethan X. Fang
  • Junwei Lu
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Abstract

We show that S2× S4 admits a complex structure. Starting from a modular family of complex two-tori associated with the (3,4,∞) triangle group and the compactification constructed in [Alpöge 2026], we replace the period lattice by its unique monodromy-invariant index-two superlattice and compactify the resulting family. In the integral Mayer--Vietoris calculation, a primitive local class whose double is the class of a cusp component and the unique nonzero torsion class contributed by the multiplicity-four fibre restrict to the same class of order two on the common boundary. We then prove that the resulting compact complex threefold is diffeomorphic to S2× S4. The proof is discovered by the Odin Automatic AI Research Agent.

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Cite

BibTeXRIS
BibTeX
@misc{guo2026complex,
  title = {{A Complex Structure on $S^2\times S^4$}},
  author = {Shengtao Guo and Ethan X. Fang and Junwei Lu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05504},
  eprint = {2609.05504},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05504},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Complex Structure on S^2× S^4
AU  - Shengtao Guo
AU  - Ethan X. Fang
AU  - Junwei Lu
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.05504
UR  - https://arxiv.org/abs/2609.05504
AB  - We show that S^2× S^4 admits a complex structure. Starting from a modular family of complex two-tori associated with the (3,4,∞) triangle group and the compactification constructed in [Alpöge 2026], we replace the period lattice by its unique monodromy-invariant index-two superlattice and compactify the resulting family. In the integral Mayer--Vietoris calculation, a primitive local class whose double is the class of a cusp component and the unique nonzero torsion class contributed by the multiplicity-four fibre restrict to the same class of order two on the common boundary. We then prove that the resulting compact complex threefold is diffeomorphic to S^2× S^4. The proof is discovered by the Odin Automatic AI Research Agent.
ER  -