Shafarevich counterexamples in dimension two and with large fundamental group

What OpenAI 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. A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.

Papers (2)

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

   5 October 2026 · Preprint, not peer reviewed · PDF

   OpenAI's 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.

2. A surface counterexample to Shafarevich holomorphic convexity

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.