Family 046Algebraic and complex geometryNo LeanNot yet digested
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 in this family · 2
A projective fourfold with large fundamental group and non-Stein universal cover
Oct 5, 2026·large-fundamental-group-non-stein.pdf
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
Sep 23, 2026·paper.pdf
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.