The Global Spherical Shell conjecture

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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.

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The paper states

Theorem 1.1 (Global Spherical Shell conjecture)

In Global Spherical Shells on Minimal Surfaces of Class VII, 24 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

The other possible matches are Corollary 1.3 (A sharpened Euler–signature inequality) in the same paper () and Corollary 1.2 (Deformation and smooth topology) in the same paper ().

Let X be a connected compact complex surface with

b_1(X)=1,\qquad b_2(X)>0,\qquad \kappa(X)=-\infty.

Assume that X is minimal: it contains no smooth rational curve of self-intersection -1. Then there are an open neighborhood U of

S^3=\{z\in\CC^2:|z|=1\} \quad\text{in }\CC^2\setminus\{0\},

an open subset \Sigma\subset X, and a biholomorphism \phi:U\longrightarrow\Sigma such that X\setminus\Sigma is connected.

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Global Spherical Shells on Minimal Surfaces of Class VII

24 September 2026, current edition.

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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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Papers elsewhere in the release that cite this family’s papers

Every complex K3 surface is OkaCurrent edition; no accompanying Lean statement Family Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VIICites Global Spherical Shells on Minimal Surfaces of Class VII.
The Singer conjecture in dimension fourCurrent edition; no accompanying Lean statement Family The four-dimensional Singer conjectureCites Global Spherical Shells on Minimal Surfaces of Class VII.

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31 references in Global Spherical Shells on Minimal Surfaces of Class VII

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