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Topology, Hyperbolicity, and the Shafarevich Conjecture for Complex Algebraic Varieties

Ya Deng

TL;DR

The work surveys deep connections between the Shafarevich conjecture, non‑abelian Hodge theory, hyperbolicity, and the topology of complex algebraic varieties, with a focus on linear notions of big/large fundamental groups. It develops both archimedean and non‑archimedean Hodge frameworks, constructs Shafarevich morphisms for reductive and positive‑characteristic settings, and uses canonical currents and harmonic maps to derive holomorphic convexity and hyperbolicity results. Central contributions include proving linear versions of Chern–Hopf–Thurston and Kollár conjectures, establishing deformation stability of big fundamental groups, and applying these tools to Campana’s and Kollár’s conjectures, with far‑reaching implications for the geometry of varieties and their universal covers. The methods integrate Nevanlinna theory, spectral coverings, period maps, and Bruhat–Tits buildings to connect topology, algebraic geometry, and analysis in both finite and infinite settings, highlighting a unified strategy to derive positivity, hyperbolicity, and structure theorems in diverse contexts.

Abstract

This survey presents recent developments concerning the Shafarevich conjecture, non-abelian Hodge theories, hyperbolicity, and the topology of complex algebraic varieties, as well as the interplay among these areas. More precisely, we present the main ideas and techniques involved in the linear versions of the following conjectures: the Shafarevich conjecture, the Chern-Hopf-Thurston conjecture, Kollár's conjecture on the holomorphic Euler characteristic, the de Oliveira-Katzarkov-Ramachandran conjecture, and Campana's nilpotency conjecture. In addition, we discuss characterizations of the hyperbolicity of complex quasi-projective varieties via representations of their fundamental groups, together with the generalized Green-Griffiths-Lang conjecture in the presence of a big local system.

Topology, Hyperbolicity, and the Shafarevich Conjecture for Complex Algebraic Varieties

TL;DR

The work surveys deep connections between the Shafarevich conjecture, non‑abelian Hodge theory, hyperbolicity, and the topology of complex algebraic varieties, with a focus on linear notions of big/large fundamental groups. It develops both archimedean and non‑archimedean Hodge frameworks, constructs Shafarevich morphisms for reductive and positive‑characteristic settings, and uses canonical currents and harmonic maps to derive holomorphic convexity and hyperbolicity results. Central contributions include proving linear versions of Chern–Hopf–Thurston and Kollár conjectures, establishing deformation stability of big fundamental groups, and applying these tools to Campana’s and Kollár’s conjectures, with far‑reaching implications for the geometry of varieties and their universal covers. The methods integrate Nevanlinna theory, spectral coverings, period maps, and Bruhat–Tits buildings to connect topology, algebraic geometry, and analysis in both finite and infinite settings, highlighting a unified strategy to derive positivity, hyperbolicity, and structure theorems in diverse contexts.

Abstract

This survey presents recent developments concerning the Shafarevich conjecture, non-abelian Hodge theories, hyperbolicity, and the topology of complex algebraic varieties, as well as the interplay among these areas. More precisely, we present the main ideas and techniques involved in the linear versions of the following conjectures: the Shafarevich conjecture, the Chern-Hopf-Thurston conjecture, Kollár's conjecture on the holomorphic Euler characteristic, the de Oliveira-Katzarkov-Ramachandran conjecture, and Campana's nilpotency conjecture. In addition, we discuss characterizations of the hyperbolicity of complex quasi-projective varieties via representations of their fundamental groups, together with the generalized Green-Griffiths-Lang conjecture in the presence of a big local system.
Paper Structure (50 sections, 60 theorems, 150 equations, 2 figures)

This paper contains 50 sections, 60 theorems, 150 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview of the Paper
  3. Algebraic varieties with big fundamental groups
  4. Large fundamental groups: motivation, conjectures and results
  5. Big fundamental groups
  6. Deformation of big fundamental groups
  7. Hyperbolicity and fundamental groups: conjectures and main results
  8. Notions of hyperbolicity
  9. Generalized Green-Griffiths-Lang conjecture
  10. Hyperbolicity of compactifications after taking finite étale coverings
  11. How Fundamental Groups Determine Hyperbolicity
  12. On GGL conjecture
  13. Non-Abelian Hodge Theories and the Shafarevich Conjecture
  14. Conjecture and some histories
  15. Non-Abelian Hodge theory: archimedean setting
  16. ...and 35 more sections

Key Result

Lemma 1.2

Let $X$ be a smooth complex projective variety. If $X$ is aspherical or if its universal covering $\widetilde{X}$ is a Stein manifold, then $X$ has a large fundamental group.

Figures (2)

  • Figure 1: Hyperbolicity from different viewpoints
  • Figure 2: Bruhat-Tits building for ${\rm SL}_2(\mathbb{Q}_p)$ with $p=2$.

Theorems & Definitions (149)

  • Conjecture 1.1: Chern--Hopf--Thurston
  • Lemma 1.2
  • Definition 1.3: Large fundamental group
  • Remark 1.4
  • proof : Proof of \ref{['lem:asp']}
  • Conjecture 1.5: Shafarevich
  • Theorem 1.6: EKPR12
  • Conjecture 1.7: Arapura-Wang
  • Theorem 1.8: DW24
  • Definition 1.9: Big fundamental group
  • ...and 139 more