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Hyperbolicity in presence of a large local system

Yohan Brunebarbe

TL;DR

<3-5 sentence high-level summary>The paper proves that projective varieties carrying a large complex local system satisfy a strengthened Green–Griffiths–Lang conjecture: the algebraic, abelian, and holomorphic special subsets coincide and form a Zariski-closed locus Sp(X) that detects general-type behavior. The authors develop a comprehensive framework combining non-abelian Hodge theory, Shafarevich morphisms, and a non-Archimedean detour via Katzarkov–Zuo reductions and Klingler’s local system, to reduce the general problem to base cases with semisimple monodromy and solvable fiber monodromy. They establish a canonical decomposition of X into a fibration over a general-type base with fibers of abelian-type, and prove Lang-type statements first for big local systems and then for large ones, including structural results and precise handling of semi-simple vs. general cases. The results yield not only theoretical insight into hyperbolicity phenomena in the presence of local systems but also a path toward a broader, birationally robust understanding of when varieties exhibit Green–Griffiths–Lang hyperbolicity in the presence of rich monodromy data.

Abstract

We prove that the projective complex algebraic varieties admitting a large complex local system satisfy a strong version of the Green-Griffiths-Lang conjecture.

Hyperbolicity in presence of a large local system

TL;DR

<3-5 sentence high-level summary>The paper proves that projective varieties carrying a large complex local system satisfy a strengthened Green–Griffiths–Lang conjecture: the algebraic, abelian, and holomorphic special subsets coincide and form a Zariski-closed locus Sp(X) that detects general-type behavior. The authors develop a comprehensive framework combining non-abelian Hodge theory, Shafarevich morphisms, and a non-Archimedean detour via Katzarkov–Zuo reductions and Klingler’s local system, to reduce the general problem to base cases with semisimple monodromy and solvable fiber monodromy. They establish a canonical decomposition of X into a fibration over a general-type base with fibers of abelian-type, and prove Lang-type statements first for big local systems and then for large ones, including structural results and precise handling of semi-simple vs. general cases. The results yield not only theoretical insight into hyperbolicity phenomena in the presence of local systems but also a path toward a broader, birationally robust understanding of when varieties exhibit Green–Griffiths–Lang hyperbolicity in the presence of rich monodromy data.

Abstract

We prove that the projective complex algebraic varieties admitting a large complex local system satisfy a strong version of the Green-Griffiths-Lang conjecture.
Paper Structure (23 sections, 33 theorems, 12 equations)

This paper contains 23 sections, 33 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Generalities on special sets
  3. Generalities on large local systems
  4. Monodromy groups
  5. Equivalent definitions of largeness
  6. Shafarevich morphisms
  7. Variations of Hodge structure with discrete monodromy
  8. Local systems with solvable monodromy
  9. A canonical decomposition
  10. Algebraic varieties with a large complex local system.
  11. The special subsets coincide
  12. A non-Archimedean detour
  13. Katzarkov-Zuo reductions.
  14. Klingler's local system
  15. Recollections on buildings
  16. ...and 8 more sections

Key Result

Theorem A

Let $X$ be a projective complex algebraic variety. Assume that there exists a large complex local system on $X$. Then,

Theorems & Definitions (55)

  • Theorem A
  • Theorem B
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • ...and 45 more