A Complex Structure on $S^2\times S^4$

Shengtao Guo; Ethan X. Fang; Junwei Lu

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arXiv · 2609.05504

A Complex Structure on $S^2\times S^4$

Shengtao Guo·Ethan X. Fang·Junwei Lu

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Abstract

We show that $S^2\times S^4$ admits a complex structure. Starting from a modular family of complex two-tori associated with the $(3,4,\infty)$ triangle group and the compactification constructed in [Alp\"oge 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 $S^2\times S^4$. The proof is discovered by the Odin Automatic AI Research Agent.

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BibTeX RIS

Shengtao Guo, Ethan X. Fang, Junwei Lu. 2026-08-28. A Complex Structure on $S^2\times S^4$. https://arxiv.org/abs/2609.05504

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