Shafarevich counterexamples in dimension two and with large fundamental group

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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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What the papers state

Theorem 1.1

In A projective fourfold with large fundamental group and non-Stein universal cover, 5 October 2026

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There exist a smooth connected projective complex fourfold X and a simple abelian surface A\subset X such that

1.
X has large fundamental group;
2.
\im\bigl(\pi_1(A)\to\pi_1(X)\bigr)\simeq\Z.

The universal cover \widetilde X contains no positive-dimensional compact complex analytic subvariety and is not Stein.

Theorem 1.1

In A surface counterexample to Shafarevich holomorphic convexity, 23 September 2026

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There is a smooth connected projective complex surface X containing a connected nodal curve Z_0 with smooth rational irreducible components such that

\im\bigl(\pi_1(Z_0)\longrightarrow\pi_1(X)\bigr) \quad\text{is infinite}.

In the simply connected universal cover of X, a connected component above Z_0 is a closed, connected, noncompact, locally finite union of compact rational curves. The holomorphic hull of any one of its points is noncompact. In particular, this universal cover is not holomorphically convex.

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Papers and explanations: Two current papers, each readable here as a page beside its PDF.

Papers

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

5 October 2026, current edition.

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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

23 September 2026, current edition.

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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.

Lean statements

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Independent comparisons and proof-verification reports

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Papers elsewhere in the release that this family’s papers cite

The abelianity conjecture for special compact Kähler manifoldsCurrent edition; no accompanying Lean statement Family Fundamental groups of special complex varieties and root orbifoldsCited by A surface counterexample to Shafarevich holomorphic convexity.
Semialgebraic universal covers of normal projective varietiesCurrent edition; no accompanying Lean statement Family Semialgebraic universal covers and bounded domainsCited by A surface counterexample to Shafarevich holomorphic convexity.

Papers elsewhere in the release that cite this family’s papers

The abelianity conjecture for special compact Kähler manifoldsCurrent edition; no accompanying Lean statement Family Fundamental groups of special complex varieties and root orbifoldsCites A surface counterexample to Shafarevich holomorphic convexity.

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An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
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