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The monodromy of families of subvarieties on abelian varieties

Ariyan Javanpeykar, Thomas Krämer, Christian Lehn, Marco Maculan

TL;DR

The paper addresses monodromy questions for families of subvarieties in abelian varieties, extending big-monodromy results to higher codimension with a framework built on perverse sheaves and Tannaka groups. The authors develop a Tannakian approach, relate monodromy to the derived part of Tannaka groups, and deploy conormal-geometry via characteristic cycles to translate representation-theoretic information into geometric constraints. They prove a big monodromy theorem for subvarieties of codimension at least half the ambient dimension, subject to Euler-characteristic exclusions, and provide a complete wedge-power classification showing when Tannaka data arises from sums of curves. The results connect geometric positivity, symmetric/power maps, and Liouville-type constraints to monodromy phenomena, yielding a robust bridge between geometry, topology, and arithmetic via a Tannaka-geometry toolkit with potential arithmetic applications akin to those in LV and LS.

Abstract

Motivated by recent work of Lawrence-Venkatesh and Lawrence-Sawin, we show that non-isotrivial families of subvarieties in abelian varieties have big monodromy when twisted by generic rank one local systems. While Lawrence-Sawin discuss the case of subvarieties of codimension one, our results hold for subvarieties of codimension at least half the dimension of the ambient abelian variety. For the proof, we use a combination of geometric arguments and representation theory to show that the Tannaka groups of intersection complexes on such subvarieties are big.

The monodromy of families of subvarieties on abelian varieties

TL;DR

The paper addresses monodromy questions for families of subvarieties in abelian varieties, extending big-monodromy results to higher codimension with a framework built on perverse sheaves and Tannaka groups. The authors develop a Tannakian approach, relate monodromy to the derived part of Tannaka groups, and deploy conormal-geometry via characteristic cycles to translate representation-theoretic information into geometric constraints. They prove a big monodromy theorem for subvarieties of codimension at least half the ambient dimension, subject to Euler-characteristic exclusions, and provide a complete wedge-power classification showing when Tannaka data arises from sums of curves. The results connect geometric positivity, symmetric/power maps, and Liouville-type constraints to monodromy phenomena, yielding a robust bridge between geometry, topology, and arithmetic via a Tannaka-geometry toolkit with potential arithmetic applications akin to those in LV and LS.

Abstract

Motivated by recent work of Lawrence-Venkatesh and Lawrence-Sawin, we show that non-isotrivial families of subvarieties in abelian varieties have big monodromy when twisted by generic rank one local systems. While Lawrence-Sawin discuss the case of subvarieties of codimension one, our results hold for subvarieties of codimension at least half the dimension of the ambient abelian variety. For the proof, we use a combination of geometric arguments and representation theory to show that the Tannaka groups of intersection complexes on such subvarieties are big.
Paper Structure (54 sections, 73 theorems, 340 equations)

This paper contains 54 sections, 73 theorems, 340 equations.

Table of Contents

  1. Introduction
  2. Big monodromy
  3. Big Tannaka groups
  4. Sum morphisms
  5. Simplicity of Tannaka groups
  6. Wedge powers
  7. Spin representations
  8. Conventions and notation
  9. Gauss maps, positivity and nondegeneracy
  10. The stabilizer and the abelian variety generated
  11. Conormal varieties and Gauss maps
  12. Positivity and nondegeneracy of subvarieties
  13. Symmetric powers of curves in abelian varieties
  14. Bounds for the topological Euler characteristic
  15. Perverse sheaves on abelian varieties
  16. ...and 39 more sections

Key Result

Corollary 1

Suppose $\mathscr{X}_{\bar{\eta}}\subset A_{S, \bar{\eta}}$ is nondivisible, not constant up to translation, and one of the following holds: Then $V_{\underline{\chi}}$ has big monodromy for most $n$-tuples $\underline{\chi} \in \Pi(A, \mathbb{F})^n$ of torsion characters.

Theorems & Definitions (177)

  • Definition
  • Corollary
  • Definition
  • Corollary
  • Example
  • Theorem
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • ...and 167 more