Linear Shafarevich conjecture in positive characteristic, hyperbolicity and applications

Ya Deng; Katsutoshi Yamanoi

doi:10.1515/crelle-2025-0078

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Linear Shafarevich conjecture in positive characteristic, hyperbolicity and applications

Ya Deng and Katsutoshi Yamanoi

Published/Copyright: November 18, 2025

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

Abstract

Given a complex quasi-projective normal variety 𝑋 and a linear representation ϱ : π 1 ⁢ ( X ) → GL N ⁡ ( K ) with 𝐾 any field of positive characteristic, we mainly establish the following results.

The construction of the Shafarevich morphism sh ϱ : X → Sh ϱ ⁢ ( X ) associated with 𝜚.
In cases where 𝑋 is projective, 𝜚 is faithful and the Γ-dimension of 𝑋 is at most two (e.g. dim X = 2 ), we prove that the Shafarevich conjecture holds for 𝑋: the universal covering of 𝑋 is holomorphically convex.
In cases where 𝜚 is big, we prove that the Green–Griffiths–Lang conjecture holds for 𝑋: 𝑋 is of log general type if and only it is pseudo-Picard or pseudo-Brody hyperbolic.
When 𝜚 is big and the Zariski closure of ϱ ⁢ ( π 1 ⁢ ( X ) ) is a semisimple algebraic group, we prove that 𝑋 is pseudo-Picard hyperbolic, and strongly of log general type.
If 𝑋 is special or ℎ-special, then ϱ ⁢ ( π 1 ⁢ ( X ) ) is virtually abelian.

We also prove Claudon–Höring–Kollár’s conjecture for complex projective manifolds with linear fundamental groups of any characteristic.

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: ANR-21-CE40-0010

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: 22K03286

Funding statement: Ya Deng acknowledges support from the ANR grant Karmapolis (ANR-21-CE40-0010). Katsutoshi Yamanoi acknowledges support from JSPS Grant-in-Aid for Scientific Research (C) 22K03286.

Acknowledgements

We would like to thank Michel Brion, Benoît Claudon, Philippe Eyssidieux and Andreas Höring for very helpful discussions. We also thank Benoît Cadorel and Yuan Liu for reading the paper and their helpful remarks. Finally, we sincerely thank the referees for very careful readings and helpful remarks, in particular for pointing out a simpler proof of Corollary G (see Remark 7.3).

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Received: 2024-09-23

Revised: 2025-07-25

Published Online: 2025-11-18

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https://doi.org/10.1515/crelle-2025-0078