Result 046, Algebraic and complex geometry

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.

Disproof or counterexample

The bigger picture

Why it matters

Two unreviewed manuscripts claim counterexamples to the idea that unwrapping all loops in a smooth projective complex space produces an analytically well-behaved space. One example is already a surface; another defeats an additional hypothesis about the fundamental group.

What changes?

A universal cover is a simply connected space lying over the original one. The first manuscript constructs a smooth projective complex fourfold, of complex dimension four, with large fundamental group. Its universal cover contains no positive-dimensional compact analytic subvariety, yet is neither Stein nor holomorphically convex. The second constructs a separate smooth connected projective complex surface, of complex dimension two, whose universal cover is not holomorphically convex. It does not assert the large-fundamental-group condition for that surface.

What does that help mathematicians do?

Holomorphic convexity requires that enlarging a compact set by including points that obey all its bounds on holomorphic functions still gives a compact set. The fourfold shows that failure can occur even without compact analytic curves or higher-dimensional compact analytic pieces in the cover. Such pieces therefore cannot explain every obstruction. The separate surface example places failure of unrestricted Shafarevich holomorphic convexity already in complex dimension two.

Are there practical applications?

The immediate value is foundational: these claimed examples constrain which links between projective geometry, topology and complex analysis can hold universally. Researchers seeking holomorphic convexity would need hypotheses beyond those defeated here. In particular, the large-fundamental-group assumption alone would not suffice, and the absence of compact analytic subvarieties would not guarantee a Stein cover.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

A projective fourfold with large fundamental group and non-Stein universal cover

October 5, 2026 41 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

A surface counterexample to Shafarevich holomorphic convexity

September 23, 2026 26 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026,
  author = {{OpenAI}},
  title = {{A surface counterexample to Shafarevich holomorphic convexity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/paper.pdf}{OAI:A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 10 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.