View source for Shafarevich's holomorphic-convexity conjecture

'''Shafarevich's holomorphic-convexity conjecture''' asks whether the universal cover of every smooth connected projective complex variety is holomorphically convex. For varieties with large fundamental group, whose universal cover contains no positive-dimensional compact analytic subvariety, it takes a direct form: is the universal cover Stein? Two manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 23 September and 5 October 2026, claim counterexamples: a surface whose universal cover is not holomorphically convex, and a fourfold with large fundamental group whose universal cover is not Stein. The family has no Lean formalization in the repository's catalogue. This page summarizes the published claims 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 introductions of the manuscripts:
** Shafarevich asked the question in his book on algebraic geometry. Gurjar and Shastri proved it for smooth projective elliptic surfaces, and Katzarkov for smooth projective varieties with virtually nilpotent fundamental group. Kollár's account explains both the conjecture and its form for large fundamental groups; the two are related by Remmert reduction, since a holomorphically convex manifold maps properly onto a Stein space with connected compact fibres.
** Linearity of the fundamental group supports strong affirmative results. Katzarkov and Ramachandran treated projective surfaces whose fundamental group has a faithful representation with reductive Zariski closure, and Eyssidieux the reductive case in every dimension. Eyssidieux, Katzarkov, Pantev and Ramachandran proved the linear Shafarevich conjecture, for smooth projective varieties whose fundamental group has a faithful finite-dimensional complex representation. Campana, Claudon and Eyssidieux extended it to compact Kähler manifolds; Deng and Yamanoi treated reductive covers of normal projective varieties, and Bakker, Brunebarbe and Tsimerman developed linear Shafarevich theory for normal algebraic spaces and quasiprojective varieties.
** The almost-holomorphic reduction maps of Campana and Kollár describe, through a very general point, the subvarieties whose fundamental groups have finite image; they do not by themselves give a proper holomorphic map from the whole universal cover to a Stein space.
** Bogomolov and Katzarkov proposed potential counterexamples from degenerating families of curves, in which an infinite chain of compact curves, an ''infinite Nori string'', would obstruct holomorphic convexity; the required infiniteness of a group quotient was left conjectural. Eyssidieux and Funar later proved holomorphic convexity for an associated uniform-ramification construction outside explicit exceptional ranges.

== The original results ==
* '''Claim (surface).''' Theorem 1.1 of ''A surface counterexample to Shafarevich holomorphic convexity'' states that there is a smooth connected projective complex surface containing a connected nodal curve with smooth rational components whose fundamental-group image is infinite. In the universal cover, a connected component above this curve is a closed, noncompact, locally finite union of compact rational curves; the holomorphic hull of any one of its points is noncompact, so the universal cover is not holomorphically convex.
** The manuscript notes that, by the linear Shafarevich theorem, the fundamental group of this surface has no faithful finite-dimensional complex representation. Using two other results of the collection, it also states that the surface is not special in Campana's sense (see [[Campana's abelianity conjecture]]) and that its universal cover is not biholomorphic to a semialgebraic open subset of a projective variety (see [[The Kollár–Pardon conjecture on semialgebraic universal covers]]).
* '''Claim (fourfold with large fundamental group).''' Theorem 1.1 of ''A projective fourfold with large fundamental group and non-Stein universal cover'' states that there exist a smooth connected projective complex fourfold with large fundamental group and a simple abelian surface in it whose fundamental group has infinite cyclic image in that of the fourfold. The universal cover contains no positive-dimensional compact analytic subvariety and is not Stein. The manuscript states that, through Remmert reduction, this also disproves the holomorphic-convexity conjecture.
* ''Method, as described:''
** For the surface, a family of spheres with 48 moving punctures comes from multiplication by 4 on the elliptic curve with an automorphism of order three; near one parameter value the punctures collide in 24 pairs, and a double base change and blowups give a main sphere with 24 attached spheres. A weighted Golod–Shafarevich inequality, for 48 generators and 24 relations that impose the sign identifications, leaves a margin that keeps the quotient infinite after deep monodromy relations and high powers of the boundary meridians. After compactification of the family, a finite quotient injective on each local boundary-group image gives a cover in which the infinite quotient survives normalization and resolution; the resulting smooth projective surface keeps a rational special fibre, and a connected component of that fibre carries the infinite image.
** For the fourfold, the obstruction is topological: the cover of the abelian surface inside the universal cover is diffeomorphic to ℝ × (''S''<sup>1</sup>)<sup>3</sup> and has nonzero third homology, while a Stein surface has the homotopy type of a CW complex of real dimension at most two. A genus-two curve with simple Jacobian, from Zarhin's theorem, is realized as an exceptional fibre of a pencil in a complex hyperbolic arrangement in an arithmetic ball quotient, where added hyperplanes make the image of its fundamental group cyclic. Local covers modelled on the infinite dihedral group, with Stover and Toledo's sign covers, and a local path criterion on uniformly Gromov-hyperbolic graphs show that this cyclic image is infinite. Hamm and Lê's Lefschetz theorem passes to a smooth projective variety containing the abelian surface, and an ample complete intersection in a product with an abelian variety gives large fundamental group.
* '''Sources''' (openai/math repository, commit <code>7ef5c0c</code>): [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026 A surface counterexample to Shafarevich holomorphic convexity] (23 September 2026); [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026 A projective fourfold with large fundamental group and non-Stein universal cover] (5 October 2026).

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 30 October 2026)
|-
| 2026-10-12 || ''AI Proved What?'', episode 046 [https://aiprovedwhat.com/episodes/046-shafarevich-counterexamples-dimension-two-large/ Shafarevich refuted?], remade version of 26 October 2026 || Reading (exposition) || Video explainer of both claims and their history. The episode states that the linear Shafarevich theorem is untouched, since the surface's fundamental group has no faithful finite-dimensional complex representation, and that the fourfold paper does not claim a counterexample with large fundamental group in dimension two or three. The page describes a short script that checks the puncture counts, the weighted Golod–Shafarevich count in exact fractions, the Betti numbers of the three-torus and the fourfold paper's incidence count. || 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 preprints: OpenAI, dated September 23 (surface) and October 5, 2026 (fourfold), released October 6; under expert review: not yet; peer reviewed: not yet; checked in Lean: no. The page states that neither paper has been withdrawn or revised and that it found no public assessment by specialists.
|-
| 2026-10-16
| J. Jardine, Lattice24, referee reviews [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_383.json rank_383] (surface) and [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_088.json rank_088] (fourfold) in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Reports no gap in either manuscript; both reviews state that they read every section. The surface review adds one minor note: a lemma on distinguished fibre meridians is stated tersely, and the review rebuilds the homotopy and finds that the argument holds. The fourfold review adds two minor notes, neither an error: the uniform hyperbolicity of the chart graphs comes almost for free, so the infinite-order conclusion rests on the local path criterion and a prefix-compatibility check, which it rederived and names as the natural first target for a formal check; and several range conditions are stated tersely, which it checked against the stated constants.
| Part of an audit that its README describes as complete for the collection's manuscripts without a Lean formalization. The reviews are published under CC BY 4.0.
|}

== Notes ==
* The background summarizes the manuscripts' 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.

[[Category:OpenAI results]]