The monodromy of families of subvarieties on abelian varieties

TL;DR

This paper demonstrates that non-isotrivial families of subvarieties within abelian varieties exhibit large monodromy groups when twisted by generic rank one local systems, extending previous results to higher codimensions.

Contribution

It generalizes existing work by establishing big monodromy for subvarieties of codimension at least half the ambient dimension, using geometric and representation-theoretic methods.

Findings

Non-isotrivial families have big monodromy with generic local systems.

Results extend to subvarieties of higher codimension.

Tannaka groups of intersection complexes are shown to be big.

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.