---
title: "The (3, 4, infinity) modular family of 2-tori, completed at its three special points, is a complex structure on S^6"
authors:
  - "Levent Alpoge"
year: 2026
venue: "preprint (alpo.ge)"
arxiv: null
doi: null
source: "https://alpo.ge/s6.pdf"
topics:
  - "complex-geometry"
  - "arithmetic-geometry"
  - "modular-forms"
seed_rank: 1197
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 10
lineage: arithmetic-geometry
cites:
  - title: "Corrigendum: The algebraic dimension of compact complex threefolds with vanishing second Betti number"
    url: "https://doi.org/10.1112/S0010437X20007113"
    year: 2020
    arxiv: "1904.11179"
    doi: "10.1112/S0010437X20007113"
  - title: "The algebraic dimension of compact complex threefolds with vanishing second Betti number"
    url: null
    year: 1998
    arxiv: null
    doi: null
  - title: "Zur Topologie der komplexen Mannigfaltigkeiten"
    url: null
    year: 1948
    arxiv: null
    doi: null
  - title: "An analytic construction of degenerating abelian varieties over complete rings"
    url: null
    year: 1972
    arxiv: null
    doi: null
  - title: "On the history of the Hopf problem"
    url: "https://arxiv.org/abs/1708.01068"
    year: 2018
    arxiv: "1708.01068"
    doi: "10.1016/j.difgeo.2017.10.010"
see:
  []
---

# The (3, 4, infinity) modular family of 2-tori, completed at its three special points, is a complex structure on S^6

## One-sentence takeaway

Alpoge constructs a compact complex 3-fold diffeomorphic to S^6 by completing the (3,4,infinity) modular family of 2-tori at its three special points — a claimed positive solution of Hopf's problem, posted only at alpo.ge/s6.pdf.

## Why it's lovely

Almost-complex S^6 is classical (octonions); integrable complex structures have been claimed and withdrawn for decades (Etesi, Atiyah). This one is a modular family of non-algebraic 2-tori over an orbifold P^1, filled by a Mumford toric degeneration at the cusp (hexagonal dP6 glued to itself) and Kodaira logarithmic transforms at the order-3 and order-4 points, glued so that pi1 is trivial and H_*(X) = H_*(S^6).

## Problem

Hopf (1948): does S^6 admit a complex structure? Orthogonal / nearly-Kaehler candidates fail. Campana-Demailly-Peternell 1998/2020 claimed that a compact complex threefold with b2=0 cannot have algebraic dimension 1 in the way this construction needs; the paper locates the gap (non-normal toric fibre, R^2 f_*(T_X tensor L) nonzero for every L).

## Design

Triangle group Delta(3,4,infinity) acts on the upper half-plane; a rank-4 lattice representation with automorphisms of orders 3 and 4 (unipotent product) gives period functions (tau, mu, beta) and a family of complex 2-tori over the thrice-punctured sphere. Complete at the cusp by Mumford's toric degeneration (A2-triangulation to dP6 with opposite sides identified, e(W)=2) and at the elliptic points by logarithmic transforms of multiplicities 3 and 4. Twists (ell0,ell1,ell2)=(0,1,-1) make pi1(X) = Z/|12 ell0 - 4 ell1 - 3 ell2| = 1; remaining torus directions die by monodromy. Then X is diffeomorphic to S^6, a(X)=1, K_X is not torsion, Aut^0 isomorphic to C*.

## Evidence

108-page writeup at https://alpo.ge/s6.pdf (extracted text in this ingest). No arXiv id and no DOI as of 2026-08-24 — do not invent one. Theorem package: simply connected, integral homology of S^6, algebraic reduction = the fibration, Hodge numbers, c3=2. Last section is a line-by-line disagreement with CDP20 Cor. 2.3.

## Limitations

Not on arXiv; not refereed. Prior S^6 claims (Etesi, Atiyah) did not stick. The very general fibre is non-algebraic (NS = Z.eta indefinite). Independent check of the gluing / pi1 computation is the whole game. Course notes and tweets are not this paper.

## Implications for Broadside

First mint of the batch, rank 1197. Curiosity arithmetic-geometry: a CUDA-OG / Morgan-Prize number theorist writing a complex-geometry paper that tries to settle a 1948 problem. Cite bibliography-only; see later Alpoge cards for the number-theory spine. Source is the user PDF.

## Bottom line

Mint Alpoge 2026 S^6 first. No arXiv. Next mint 1198+.

## Links

- PDF: https://alpo.ge/s6.pdf
- Author: https://alpo.ge/