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.

Manuscripts (2)

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

October 5, 2026 · source · cite as \cite{OAI:A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-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.

A surface counterexample to Shafarevich holomorphic convexity

September 23, 2026 · source · cite as \cite{OAI:A-surface-counterexample-to-Shafarevich-holomorphic-convexity-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.

Result 046 · Algebraic and complex geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.