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Geometric local systems on very general curves and isomonodromy

Aaron Landesman, Daniel Litt

TL;DR

The paper proves a sharp lower bound $\operatorname{rk}\mathbb{V}\ge 2\sqrt{g+1}$ for non-isotrivial local systems of geometric origin on analytically very general $n$-pointed curves of genus $g$, provided the local system has infinite monodromy and underlying an $\,\mathscr{O}_K$-local system for some number field $K$. The authors develop a stability framework for isomonodromic deformations using parabolic Atiyah bundles, Harder–Narasimhan theory, and Clifford-type arguments, and they reveal intricate behavior by constructing a Hodge-theoretic counterexample to a prior conjecture (via the Kodaira–Parshin trick). They show that low-rank VHS cannot arise on very general curves, and that, while isomonodromic deformations can fail to be semistable in general, they are forced to be (parabolically) semistable when the rank is small relative to the genus, yielding concrete corollaries for abelian schemes and for maps to Hilbert modular varieties. The results tightly constrain the topology of maps to curves and challenge previous claims about isomonodromic semistability, with implications for non-density results of geometric local systems in character varieties. The paper thus connects stability theory, VHS, and arithmetic geometry to provide new bounds and open directions on non-abelian Hodge loci and related moduli problems.

Abstract

We show that the minimum rank of a non-isotrivial local system of geometric origin, on a suitably general $n$-pointed curve of genus $g$, is at least $2\sqrt{g+1}$. We apply this result to resolve conjectures of Esnault-Kerz and Budur-Wang. The main input is an analysis of stability properties of flat vector bundles under isomonodromic deformation, which additionally answers questions of Biswas, Heu, and Hurtubise.

Geometric local systems on very general curves and isomonodromy

TL;DR

The paper proves a sharp lower bound for non-isotrivial local systems of geometric origin on analytically very general -pointed curves of genus , provided the local system has infinite monodromy and underlying an -local system for some number field . The authors develop a stability framework for isomonodromic deformations using parabolic Atiyah bundles, Harder–Narasimhan theory, and Clifford-type arguments, and they reveal intricate behavior by constructing a Hodge-theoretic counterexample to a prior conjecture (via the Kodaira–Parshin trick). They show that low-rank VHS cannot arise on very general curves, and that, while isomonodromic deformations can fail to be semistable in general, they are forced to be (parabolically) semistable when the rank is small relative to the genus, yielding concrete corollaries for abelian schemes and for maps to Hilbert modular varieties. The results tightly constrain the topology of maps to curves and challenge previous claims about isomonodromic semistability, with implications for non-density results of geometric local systems in character varieties. The paper thus connects stability theory, VHS, and arithmetic geometry to provide new bounds and open directions on non-abelian Hodge loci and related moduli problems.

Abstract

We show that the minimum rank of a non-isotrivial local system of geometric origin, on a suitably general -pointed curve of genus , is at least . We apply this result to resolve conjectures of Esnault-Kerz and Budur-Wang. The main input is an analysis of stability properties of flat vector bundles under isomonodromic deformation, which additionally answers questions of Biswas, Heu, and Hurtubise.
Paper Structure (39 sections, 46 theorems, 120 equations)

This paper contains 39 sections, 46 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Main Hodge-theoretic results
  4. Main results on isomonodromic deformations
  5. Motivation
  6. Idea of proof
  7. Organization of the paper
  8. Acknowledgments
  9. Background on parabolic bundles
  10. Definition of parabolic bundles
  11. Definition of parabolic sheaves
  12. Induced subbundles and quotient bundles
  13. Stability of parabolic and coparabolic bundles
  14. Harder-Narasimhan filtrations
  15. Serre duality
  16. ...and 24 more sections

Key Result

Theorem 1.2.5

Let $K$ be a number field with ring of integers $\mathscr{O}_K$. Suppose $(C, x_1, \cdots, x_n)$ is an analytically very general $n$-pointed hyperbolic curve of genus $g$, and $\mathbb{V}$ is a $\mathscr{O}_K$-local system on $C\setminus\{x_1, \cdots, x_n\}$ with infinite monodromy. Suppose addition

Theorems & Definitions (153)

  • Definition 1.2.1
  • Remark 1.2.2
  • Definition 1.2.3
  • Remark 1.2.4
  • Theorem 1.2.5
  • Remark 1.2.6
  • Corollary 1.2.7
  • Corollary 1.2.8
  • Remark 1.2.9
  • Remark 1.2.10
  • ...and 143 more