The $(3,4,\infty)$ Modular Family of 2-Tori as a Complex Structure on $S^6$

Hopf's problem, posed in 1948: does the six-sphere $S^6$ admit an integrable complex structure? $S^6$ is one of only two spheres carrying an almost complex structure at all (the other is $S^2$), from the octonions' multiplication, but almost complex structures need not be integrable, and whether that one - or any other - integrates has stood open for 78 years through a history of disputed attempts, including a widely discussed 2016 argument by Atiyah that did not hold up. This paper claims yes: it builds an explicit compact complex threefold $X$, fibred over $\mathbb{P}^1$ by complex 2-tori degenerating at three points, and argues $X$ is simply connected with the integral homology of $S^6$, hence diffeomorphic to it.

Claims an explicit compact complex threefold $X$, fibred over $\mathbb{P}^1$ by complex 2-tori via period functions on the $(3,4,\infty)$ orbifold, degenerating to a del Pezzo-of-degree-six fibre (identified opposite sides of its hexagon) at one point and to bielliptic multiple fibres of multiplicities 3 and 4 at the other two. Argues $X$ is simply connected with $H_*(X;\mathbb{Z})=H_*(S^6;\mathbb{Z})$, hence diffeomorphic to $S^6$, with algebraic dimension exactly 1. This directly contradicts [CDP20, Cor. 2.3], a published (and once-corrected) theorem; the paper states this and argues where the two accounts diverge, rather than overlooking it. Posted hours before this entry, with no independent check, no formalisation, and no refutation yet in any venue found. Filed as a candidate specifically because none of that has happened, not because a problem with the argument has been found.