Browse › Algebraic and complex geometry › 046 Shafarevich counterexamples in dimension two and with large fundamental group
A projective fourfold with large fundamental group and non-Stein universal cover
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
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.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Related prior literature mentioned by others
- Fedor Bogomolov and Ludmil Katzarkov, Complex projective surfaces and infinite groups (arXiv:alg-geom/9703002, 1997). The AI Proved What? exposition identifies this construction of potential Shafarevich counterexamples, conditional on group-theoretic questions, as a predecessor to the two manuscripts. [family-level]
- Philippe Eyssidieux, Ludmil Katzarkov, Tony Pantev and Mohan Ramachandran, Linear Shafarevich conjecture (Annals of Mathematics 176, 2012). The AI Proved What? exposition contrasts the manuscripts' unrestricted fundamental groups with this affirmative holomorphic-convexity theorem for projective manifolds with linear fundamental groups. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Ya Deng, Topology, Hyperbolicity, and the Shafarevich Conjecture for Complex Algebraic Varieties (arXiv:2512.24458, December 2025). This survey states as Conjecture 1.5 the form "large fundamental group implies Stein universal cover", which Theorem 1.1 of this paper addresses, and describes the conjecture as open in full generality.
- Philippe Eyssidieux and Louis Funar, Orbifold Kähler Groups related to Mapping Class groups (arXiv:2112.06726, December 2021). The abstract states that the paper settles most of the Bogomolov–Katzarkov candidate counterexamples; this paper's introduction compares its obstruction with those candidates and calls their group-infiniteness statements conjectural.
- Rodolfo Aguilar and Cristhian Garay, A Real Shafarevich Conjecture for Universal Covers (arXiv:2603.17939, March 2026). Proposes a real-variety refinement of the holomorphic-convexity conjecture and proves it for curves and nilpotent fundamental groups; this paper's Theorem 1.1 bears on the complex statement that the refinement starts from.
- Yuan Liu, On the holomorphic convexity of nilpotent coverings over compact Kähler surfaces (arXiv:2411.15744, 24 November 2024). Section 2.2 discusses failures of holomorphic convexity for two-ended infinite cyclic covers, providing a possible comparison for the manuscript's abelian-surface cover diffeomorphic to $\mathbb R\times(S^1)^3$.
Digests and expositions
- AI Proved What?, Shafarevich refuted? Counterexamples to holomorphic convexity (Episode 046, October 2026). This AI-assisted video and transcript explain both the surface's infinite chain of rational curves and the fourfold's lifted abelian-surface obstruction, with outlines of the two constructions. [AI-generated per source] [family-level]
Elsewhere
- Emergent Mind: family 046
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026,
author = {{OpenAI}},
title = {{A projective fourfold with large fundamental group and non-Stein universal cover}},
howpublished = {OpenAI Math Release preprint
\href{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}{OAI:A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026}},
year = {2026}
}Family 046: Shafarevich counterexamples in dimension two and with large fundamental group
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. A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.
Other papers in this family:
- A surface counterexample to Shafarevich holomorphic convexity · September 23, 2026