(joint w. Ben Bakker) Period spaces are quotients of period domains by arithmetic groups that parametrize hodge structures. These are typically complex-analytic orbifolds, but in most cases cannot be equipped with an algebraic structure. As a substitute, we use Siegel sets to put a definable Ran,exp structure on period spaces, and show that period mappings from algebraic varieties are definable with respect to this structure. As a corollary, we obtain another proof of the result of Cattani-Deligne-Kaplan that Hodge loci are algebraic, as well as other finiteness results.

The proof depends primarily on a generalization of a result of Schmid, showing that lifts of one-dimensional period mappings land in Siegel sets. We show that for a period mapping ϕ:△∗→D/Γ, the lift of ϕ to D lands in a union of finitely many Siegel sets. We rely heavily on classical work of Cattani-Kaplan-Schmid and Kashiwara.