The monodromy of families of subvarieties on abelian varieties

Motivated by work of Lawrence, Venkatesh, and Sawin, we show that non-isotrivial families of subvarieties in abelian varieties have big monodromy when twisted by generic rank-1 local systems. While Lawrence and Sawin discuss the case of subvarieties of codimension 1, 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.