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Global Spherical Shells on Minimal Surfaces of Class VII
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Abstract
We prove the Global Spherical Shell conjecture: every connected minimal compact complex surface of class VII with positive second Betti number contains a global spherical shell. This is a holomorphically embedded neighborhood of the standard three-sphere in $\mathbb C^2\setminus\{0\}$ whose complement is connected.
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Related prior literature mentioned by others
- Franc Forstnerič and Finnur Lárusson, Holomorphic flexibility properties of compact complex surfaces (arXiv:1207.4838; IMRN, 2014). The AI Proved What? exposition below identifies this classification of Oka minimal class VII surfaces, conditional on the global spherical shell conjecture, as an application of the manuscript’s claimed theorem.
Possibly relevant literature (found by the agent, not checked)
- A. Teleman, Gauge theoretical methods in the classification of non-Kaehlerian surfaces (arXiv:0804.0557, April 2008; announced for Banach Center Publications). Survey of the Donaldson-theory programme for producing curves on minimal class VII surfaces, including the conjecture that such surfaces contain exactly $b_2$ curves, related to the paper's gauge-theoretic discussion and its count of exactly $b_2(X)$ rational curves; not in the paper's bibliography, which cites Teleman 1994, 2005 and 2010.
Digests and expositions
- AI Proved What?, Every K3 flexible? Oka classification for minimal compact complex surfaces (Episode 042, October 2026). This exposition distinguishes the release’s K3 claim from its class VII Oka classification, explaining that the latter combines Forstnerič–Lárusson’s work with result 060’s claimed global spherical shell theorem. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 060
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Global-Spherical-Shells-on-Minimal-Surfaces-of-Class-VII-September-24-2026,
author = {{OpenAI}},
title = {{Global Spherical Shells on Minimal Surfaces of Class VII}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Global-Spherical-Shells-on-Minimal-Surfaces-of-Class-VII-September-24-2026/paper.pdf}{OAI:Global-Spherical-Shells-on-Minimal-Surfaces-of-Class-VII-September-24-2026}},
year = {2026}
}Family 060: The Global Spherical Shell conjecture
Every connected minimal compact complex surface of class VII with $b_2\gt 0$ contains a global spherical shell, proving the positive-b2 Global Spherical Shell conjecture. Such a shell is a holomorphically embedded neighborhood of the standard three-sphere in $\mathbb C^2\setminus\{0\}$ whose complement is connected.
This is the only paper in this family.