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 $$A_5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>5</mml:mn> </mml:msub> </mml:math> 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-Oort 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 $$A_5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>5</mml:mn> </mml:msub> </mml:math> and classify the positive-dimensional families of cubic threefolds satisfying our condition. In particular, we discover two examples of positive-dimensional special subvarieties in the intermediate Jacobian locus that contain the intermediate Jacobian of the Klein cubic threefold.