  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.
