we-solve (beta)

Math-2369 · mathematics · Algebraic geometry

Classification of Class-VII Complex Surfaces

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Classify compact complex surfaces of class $\mathrm{VII}_0^+$: minimal compact complex surfaces with first Betti number one, negative Kodaira dimension, and positive second Betti number. Determine their geometric structure.

Formulation references: [S1], [S2].

Short English statement

Classify compact complex surfaces of class $\mathrm{VII}_0^+$: minimal compact complex surfaces with first Betti number one, negative Kodaira dimension, and positive second Betti number. Determine their geometric structure.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2369-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2369

Manuscript 1 (family 060, 2026-09-24): The global-shell theorem proves a precise structural assertion for every minimal class VII surface with positive b2, and its corollary fixes deformation and topology. The problem is explicitly a classification fragment with no defined equivalence or requested invariants; a solved classification label would require silently supplying that missing scope.

Classification of Class-VII Complex Surfaces

OpenAI's claim

Every minimal class-VII compact complex surface with positive second Betti number is claimed to contain a global spherical shell. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $X$ be a complex surface that is compact and connected, satisfies $b_1(X)=1$ and $b_2(X)>0$, and has $\kappa(X)=-\infty$, and assume it has no rational curve that is smooth and has self-intersection $-1$. OpenAI claims a biholomorphism $\phi:U\to\Sigma\subset X$, where $U$ is an open neighborhood of $S^3=\{z\in\mathbb C^2:|z|=1\}$ in $\mathbb C^2\setminus\{0\}$, such that \[ X\setminus\Sigma\text{ is connected}. \] This is a global spherical shell for each class-VII surface assumed minimal and satisfying $b_2>0$.

Outline of the argument

Manuscript 1 (family 060, 2026-09-24): On an infinite cyclic cover, weighted Dolbeault and Fourier--Laplace estimates construct effective divisors in the necessary compact Chern classes. Residue currents, cylinder estimates, and cycle compactness make their components compact; the resulting rational-curve count invokes the established shell criteria.

Formal verification

Family 060 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

Go to contribution →