∑math20

Algebraic and complex geometry · Result 046

Shafarevich counterexamples in dimension two and with large fundamental group

2 papers

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.

Watch

▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 2

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

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

OpenAI · October 5, 2026

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.

Read PDF Download Source

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}
}
First page of A surface counterexample to Shafarevich holomorphic convexity

A surface counterexample to Shafarevich holomorphic convexity

OpenAI · September 23, 2026

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.

Read PDF Download Source

Cite
@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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.