Speculations on Hodge theory

Recall that a Mumford–Tate domain DM is an open set in its compact dual ĎM — the latter is a rational homogeneous variety defined over Q (think of the upper half plane H ⊂ P1). Thus a polarized Hodge structure (PHS) φ has an “upstairs” field of definition k(φ). If H•,• ⊂ T •,• is a subalgebra then the Noether-Lefschetz locus NLH = {φ : Hg•,• φ ⊇ H} is defined over Q. Its components are defined over a number field and there is a Galois action on them. Let Γ ⊂M be an arithmetic subgroup and MM := Γ\DM be the moduli space of Γ-equivalence classes of PHS whose Mumford–Tate group is contained in M . In the classical case (Shimura varieties), MM is projective and there is “arithmetic downstairs” related to arithmetic properties of automorphic forms. Let S be a quasi-projective variety defined over a number field k and Φ: S →MM a variation of Hodge structure whose monodromy group is Q-Zariski dense in M . Then it is known that Φ(Γ\NLH) is an algebraic subvariety of S.