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The Global Spherical Shell conjecture for minimal class VII surfaces with positive second Betti number

A compact complex surface is of class VII if b1=1b_1=1 and its Kodaira dimension is −∞-\infty; these are the non-Kahler surfaces left open by the Enriques-Kodaira classification. With b2=0b_2=0 the minimal ones are Hopf and Inoue surfaces (Bogomolov, Li-Yau-Zheng, Teleman). A global spherical shell is a holomorphic embedding of a neighborhood of S3⊂C2∖{0}S^3\subset\mathbb C^2\setminus\{0\} with connected complement; Kato showed surfaces with one are degenerations of blown-up Hopf surfaces. Nakamura's classification program conjectures (Working Hypothesis 5.5) that shells always exist when b2>0b_2>0; Teleman proved it for b2=1b_2=1, and Dloussky-Oeljeklaus-Toma reduced it to finding b2b_2 rational curves. Does every minimal class VII surface with b2>0b_2>0 contain a global spherical shell?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Complex geometry; non-Kahler compact complex surfaces of class VII
Posed by
Iku Nakamura (Working Hypothesis 5.5, Sugaku 36, 1984)
Year posed
1984
Years open
42y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
50 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: every minimal class VII surface with b2>0b_2>0 contains a global spherical shell. Corollaries: by Kato, each such surface deforms to blown-up primary Hopf surfaces, has π1≅Z\pi_1\cong\mathbb Z and is diffeomorphic to (S1×S3)#b2CP‾2(S^1\times S^3)\#b_2\overline{\mathbb{CP}}^2; and aspherical compact complex surfaces satisfy χ≥95∣σ∣\chi\ge\tfrac95|\sigma|. The proof builds divisors on the infinite cyclic cover and shows all their components are compact rational curves, then applies Dloussky-Oeljeklaus-Toma. It does not give a complete biholomorphic classification of the resulting Kato surfaces beyond what Kato's theory provides.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Nakamura's Working Hypothesis 5.5; it is stated for every connected compact complex surface with b1=1b_1=1, b2>0b_2>0, κ=−∞\kappa=-\infty and no (−1)(-1)-curve, and gives a global spherical shell. The proof was not refereed. No Lean formalization of this manuscript is in the release (there is no lean/docs/060.md at the pinned commit). The argument relies on Enoki's theorem and the Dloussky-Oeljeklaus-Toma criterion, both published, and constructs the needed rational curves on the infinite cyclic cover by weighted Fourier-Laplace analysis.

Source

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