The Alpöge–Claude manuscript claims a compact complex threefold, obtained by completing a modular family of complex two-tori, whose underlying smooth manifold is diffeomorphic to S⁶.
1 record · How we rate progress
The Alpöge–Claude manuscript claims a compact complex threefold, obtained by completing a modular family of complex two-tori, whose underlying smooth manifold is diffeomorphic to S⁶. Philip Engel supplies a separate geometric exposition and self-contained proof of the construction.
Read original source Original Alpöge–Claude constructionalpo.ge
Original report: 2026-08-23The Alpöge–Claude manuscript claims a compact complex threefold, obtained by completing a modular family of complex two-tori, whose underlying smooth manifold is diffeomorphic to S⁶. Philip Engel supplies a separate geometric exposition and self-contained proof of the construction.
The six-sphere admits an almost complex structure, but the longstanding Hopf problem asks whether it admits an integrable complex structure. An almost complex structure alone does not answer that question.
If correct, resolves the longstanding complex-structure existence problem for the six-sphere and provides a construction with further geometric research directions.
A sourced advance or substantial formalization with the exact target, hypotheses, baseline and AI role stated. Importance and verification are assessed separately.
Background review: 2026-10-11. Selected sources:
Publication dates and editorial changes.
Alpöge–Claude reports: a complex structure on the six-sphere
Allow optional Google Analytics to help improve AIProofWatch? Privacy details.
