On the distribution of non-rigid families in the moduli spaces

This paper investigates the distribution of non-rigid families in a moduli space $\mathcal{M}$ of polarized projective manifolds for which the infinitesimal Torelli theorem holds. Guided by the analogy with unlikely intersection in Shimura varieties, we show that the image of any non-rigid classifying morphisms into $\mathcal{M}$ is contained in the Hodge locus as long as the derived Mumford-Tate group is $\mathbb{Q}$-simple and the period map is generically finite. If moreover the period domain is not Hermitian of rank at least 2, then the Hodge locus can be replaced by a closed subscheme, which yields a finiteness theorem of geometric Bombieri-Lang type. Inspired by the Zilber-Pink conjecture, we also characterize the geometry of non-rigid locus by the specialness of bi-Hom schemes and the finiteness of “structurally-atypical” intersections. Finally, we specialize to the moduli spaces of polarized Calabi-Yau manifolds, formulate an unobstructedness conjecture for non-rigid maps which implies the specialness of bi-Hom schemes, prove a geometric André-Oort theorem describing the Zariski closure of non-rigid locus, and test the theory and the conjecture for the explicit Viehweg-Zuo family of Calabi–Yau quintics in ${\mathbb{P}}^4$.