---
title: "Shafarevich counterexamples in dimension two and with large fundamental group"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 046; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebraic-and-complex-geometry"
  - "unformalized"
seed_rank: 1905
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "A projective fourfold with large fundamental group and non-Stein universal cover"
    url: "https://github.com/openai/math/blob/main/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "A surface counterexample to Shafarevich holomorphic convexity"
    url: "https://github.com/openai/math/blob/main/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Shafarevich counterexamples in dimension two and with large fundamental group

## One-sentence takeaway

OpenAI's result family 046 (Algebraic and complex geometry) claims: Constructs a smooth projective complex fourfold with large fundamental group whose universal cover contains no positive-dimensional compact analytic subvariety but is neither Stein nor holomorphically convex.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.
- *A projective fourfold with large fundamental group and non-Stein universal cover*: We construct a smooth projective complex fourfold with large fundamental group whose universal cover is not Stein. The universal cover contains no positive-dimensional compact complex-analytic subvariety, so this disproves Shafarevich's holomorphic-convexity conjecture even under the large-fundamental-group hypothesis.
- *A surface counterexample to Shafarevich holomorphic convexity*: We construct a smooth connected projective complex surface whose universal cover is not holomorphically convex. This disproves the Shafarevich conjecture on holomorphic convexity in complex dimension two.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [A projective fourfold with large fundamental group and non-Stein universal cover](https://github.com/openai/math/blob/main/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf)
- Manuscript: [A surface counterexample to Shafarevich holomorphic convexity](https://github.com/openai/math/blob/main/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math