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

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 [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.