CATALOGUE 046
Counterexamples to Shafarevich holomorphic convexity
The question is whether the simply connected ordinary universal cover of every smooth connected projective complex variety is holomorphically convex. The first manuscript in official order constructs a fourfold with large fundamental group; the second constructs a surface with an infinite chain of rational curves upstairs. Both claim counterexamples in the projective setting, rather than examples only among general compact complex manifolds.
Papers in official catalogue order
These summaries give the principal scope; consult the cited original results for the complete hypotheses and statements.
Page counts include references. On narrow screens, scroll the table sideways.
| # | Manuscript and overview | Principal claim | Role and available overview |
|---|---|---|---|
| 01 | A projective fourfold with large fundamental group and non-Stein universal cover →Short name: A projective fourfold with large fundamental group and non-Stein universal coverOctober 5, 2026 · 41 pagesOriginal paper (GitHub) ↗ | A smooth connected projective complex fourfold X contains a simple abelian surface A, with large fundamental group and image π₁(A) ≅ ℤ. Its universal cover has no positive-dimensional compact analytic subvariety and is not Stein. Theorem 1.1 · p. 1 ↗ | Local covers and hyperbolic path tests preserve a cyclic image through projective realization; a final complete intersection gives largeness. The lifted closed surface has nonzero H₃, obstructing Steinness. Read the proof overview → |
| 02 | A surface counterexample to Shafarevich holomorphic convexity →Short name: A surface counterexample to Shafarevich holomorphic convexitySeptember 23, 2026 · 26 pagesOriginal paper (GitHub) ↗ | A smooth connected projective complex surface X contains a connected nodal curve Z₀ with rational components and infinite fundamental-group image. A singleton in the universal cover has noncompact holomorphic hull. Theorem 1.1 · p. 2 ↗ | A weighted group presentation ensures infinitude, while finite detection of local boundary groups preserves it through projective compactification. An infinite chain of rational curves obstructs holomorphic convexity. Read the proof overview → |
References on the arrows
- [OpenAI26a] OpenAI, A projective fourfold with large fundamental group and non-Stein universal cover, October 5, 2026, 41 pages. The topology of a closed lifted surface, and the use of largeness to deduce Steinness from hypothetical holomorphic convexity.
[OpenAI26a, Theorem 1.1, p. 1]
- [OpenAI26b] OpenAI, A surface counterexample to Shafarevich holomorphic convexity, September 23, 2026, 26 pages. The fundamental-group image of a whole rational nodal curve and the holomorphic-hull obstruction in its lift.
Two routes to the same negative conclusion
The two sides of the diagram concern separately constructed spaces; they do not represent proof dependence between the manuscripts.
On the fourfold side, a lifted simple abelian surface is a closed complex surface with nonzero third homology, and hence is not Stein. The universal cover additionally has no positive-dimensional compact analytic subvariety. If it were holomorphically convex, its Remmert reduction would be an isomorphism, making it Stein. This additional condition supplies the passage to failure of holomorphic convexity. See the fourfold proof overview.
[OpenAI26a, Proposition 2.1, proof, p. 6] [OpenAI26a, Section 1.1, p. 2]
On the surface side, the presence of compact rational curves alone is not the claimed obstruction. The entire connected lift is closed and noncompact, and every holomorphic function is constant on it, so it lies in the hull of any of its points. The surface proof overview explains how infinitude is retained through compactification before applying this analytic obstruction.
Sources and checking scope
The official inventory is OpenAI Research Catalog, October 6, 2026, entry 046 on PDF page 6, at fixed commit 1581d4cbdd69a0faf34b2089614fe2a814316a6a. The official subject is Algebraic and complex geometry. CONTENTS uses the alternative heading Shafarevich counterexamples in dimension two and with large fundamental group; the two manuscripts and their order agree.
OpenAI Research Catalog · Catalogue 046 · p. 6
We checked the authors' main statements against the passages connecting construction, preservation of the group image, and the analytic obstruction. Each article specifies checked and unresolved direct inputs. This is not independent verification of every external proof or of the complete manuscripts. No mutual citation or direct use was found within the checked passages. Inputs from other catalogues into consequences are distinguished in the surface article; uninvestigated relations are not recorded as absent.