$S^2\times S^4$ 上的一个复结构
A Complex Structure on $S^2\times S^4$
浏览论文内容
中文总结 AI 辅助
本文通过构造模族的紧化并验证微分同胚，证明了 $S^2\times S^4$ 上存在复结构，该证明由 AI 代理 Odin 发现。
中文摘要 AI 辅助
我们证明 $S^2\times S^4$ 允许一个复结构。从与 $(3,4,\infty)$ 三角形群相关联的复二维环面的模族以及 [Alpöge 2026] 中构造的紧化出发，我们将周期格替换为其唯一的单值性不变的指数二超格，并对所得族进行紧化。在积分 Mayer–Vietoris 计算中，一个原始局部类（其二倍是尖点分量的类）和由四重纤维贡献的唯一非零挠类在公共边界上限制为同一个二阶类。然后我们证明所得的紧复三维流形与 $S^2\times S^4$ 微分同胚。该证明由 Odin 自动人工智能研究代理发现。
英文摘要
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 [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 $S^2\times S^4$. The proof is discovered by the Odin Automatic AI Research Agent.
