Special subvarieties in the locus of intermediate Jacobians of cubic threefolds

We study special subvarieties, i.e., subvarieties containing a dense subset of CM points, of the moduli space $A_5$ of principally polarized abelian varieties of dimension five, generically contained in the locus of intermediate Jacobians of cubic threefolds. The analogous question for Jacobians of curves is related to a conjecture of Coleman-Ort and has been studied by Shimura, Mostow, De Jong-Noot, Rohde, Moonen, Oort, Frediani, Ghigi and others. Adapting methods of Frediani, Ghigi and Penegini, we give a sufficient condition ensuring that the closure of the image of a family of smooth cubic threefolds with prescribed automorphisms via the period map is a special subvariety of $A_5$. By the work of Allcock, Carlson and Toledo, it is known that the family of cyclic cubic threefolds gives rise to a special subvariety. Analyzing the action of subgroups of the automorphism group of the Klein cubic threefold, we discover new examples of positive-dimensional special subvarieties arising from families of smooth cubic threefolds that are not contained in the locus of cyclic cubic threefolds.