The global spherical shell conjecture

The global spherical shell conjecture concerns compact complex surfaces of class VII, the non-Kähler surfaces with first Betti number one and Kodaira dimension −∞. A global spherical shell is a holomorphically embedded neighbourhood of the unit three-sphere in C² ∖ {0} whose complement is connected; Kato introduced the notion, and Nakamura's classification programme includes the working hypothesis that every minimal class VII surface with positive second Betti number contains one. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 24 September 2026, claims this conjecture for every positive second Betti number. The family has no Lean formalization in the repository's catalogue. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.

Background

• As recounted in the introduction of the manuscript:
  • For minimal class VII surfaces with second Betti number zero, work of Bogomolov, Li–Yau–Zheng and Teleman leads to the classification by Hopf and Inoue surfaces.
  • Two curve criteria reduce the conjecture to producing curves: Enoki's theorem treats surfaces with a nonzero effective divisor of square zero, and Dloussky, Oeljeklaus and Toma proved that b₂ rational curves imply a global spherical shell.
  • Teleman proved the conjecture for b₂ = 1 and showed that a minimal class VII surface with b₂ = 2 contains a cycle of curves; Dloussky gave further criteria from additional geometric structure, and Kurnosov and Spicer obtained the conclusion for b₂ = 3 when two distinct singular holomorphic foliations are present.

The original result

• Claim. Theorem 1.1 of Global Spherical Shells on Minimal Surfaces of Class VII states that every connected minimal compact complex surface with b₁ = 1, b₂ > 0 and Kodaira dimension −∞ contains a global spherical shell. With Kato's deformation theorem the manuscript derives that each such surface admits a small deformation whose nonzero fibres are primary Hopf surfaces blown up b₂ times, has fundamental group Z, and is diffeomorphic to (S¹ × S³) # b₂ copies of the complex projective plane with reversed orientation. It also states that the theorem removes the possible class VII exception in the geography of aspherical complex surfaces studied by Albanese, Di Cerbo and Lombardi, giving χ ≥ (9/5)|σ| for every closed complex surface with contractible universal cover.
• Method, as described: after Enoki's reduction the manuscript works on the infinite cyclic cover of the surface with a height function that increases by one under the deck transformation. From finite cyclic quotients it builds nearly n b₂ compactly supported line-bundle classes, and Fourier–Laplace analysis on the periodic ends, following Taubes, gives each class a holomorphic section whose zero divisor is bounded above in height. Residue currents built from holomorphic differentials show that every divisor component has genus zero, and a uniform estimate on long cylinders together with a periodic gauge argument excludes open components, so every component is a compact rational curve. A count of translation orbits over Laurent polynomials then gives exactly b₂ rational curves, and the criterion of Dloussky, Oeljeklaus and Toma supplies the shell.
• Source (openai/math repository, commit 7ef5c0c): Global Spherical Shells on Minimal Surfaces of Class VII, dated 24 September 2026.
• Use by another manuscript of the collection. The introduction of Every complex K3 surface is Oka states that a separate class VII classification in that manuscript combines earlier Oka results with this theorem, and that the theorem is not an input to its K3 argument (see The Oka property of complex K3 surfaces).

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 30 October 2026)
2026-10-12	AI Proved What?, episode 060 A Hidden Shell?, remade version of 26 October 2026	Reading (exposition)	Video explainer of the claim and its history. The episode states that the theorem concerns minimal surfaces with positive second Betti number, the case of second Betti number zero having been settled before, and that it gives no new construction of surfaces, its conclusion being that every such surface is one of Kato's. It notes that the Oka classification of result 042 relies on the theorem and that its inputs include Donaldson's diagonalization, Enoki's theorem and the criterion of Dloussky, Oeljeklaus and Toma.	Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, dated September 24, 2026, released October 6; under expert review: not yet; peer reviewed: not yet; checked in Lean: no Lean proof published. The page states that it found no public assessment by specialists.
2026-10-16	J. Jardine, Lattice24, referee review rank_081 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	Reports no gap. Its one minor note concerns the continuation through the trivial character in the weighted analysis: the functional used there is said to come from pairing with the harmonic representative without naming the Sobolev scale, and a residue is asserted to be constant without proof; the review calls both standard and fixable and notes that a following remark checks the second independently. The reviewer states that it read the whole source and checked seventeen listed steps in detail, but not the cited external theorems beyond their quoted statements or every Sobolev constant.	Part of an audit that its README describes as complete for the collection's manuscripts without a Lean formalization. The review is published under CC BY 4.0.

Notes

• The background summarizes the manuscript's own account of earlier work.
• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.