Mathematics / Milestone

A Complex Structure on the Six-Sphere

Does the smooth six-sphere admit an integrable complex structure?

The source trail 2

Findings, with context

DiscussionAug 24, 2026

A source discussing Complex six-sphere

aixiv.online contains the excerpt below in a discussion related to Complex six-sphere. Read the source for the surrounding context.

“The (3, 4, ∞) modular family of 2-tori, completed at its three special points, is a complex structure on S6.”

This excerpt is evidence of what the linked page says, not verification of its claims. Source reports an update on 2026-09-04; this is not necessarily the original posting date.

Source pageIndirect account · Found Sep 20

aixiv.online

PaperSep 13, 2026

Engel gives a geometric account of the Alpöge–Claude six-sphere claim

Philip Engel's manuscript presents a self-contained account of the claimed complex structure on S⁶, explaining the geometry and possible follow-up questions. He says he checked the mathematical content himself.

“give a self-contained proof, and remark on possible future research directions”

The manuscript is dated September 13 and discusses work attributed to Alpöge and Claude. We have not independently verified its proof or established a journal-review status.

Philip EngelOriginal source · Found Sep 19

Philip Engel · manuscript