Shafarevich counterexamples in dimension two and with large fundamental group
OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
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.
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MANUSCRIPT 01
A projective fourfold with large fundamental group and non-Stein universal cover
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.
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A surface counterexample to Shafarevich holomorphic convexity
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.
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