Special subvarieties in the locus of intermediate Jacobians of cubic threefolds
Abstract
We study special subvarieties, i.e., subvarieties containing a dense subset of CM points, of the moduli space A5 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 A5. 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.