Period mappings are definable in the o-minimal structure $\mathbb{R}_{an,exp}$

(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 $\mathbb{R}_{an,exp}$ 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 $\phi:\bigtriangleup^* \rightarrow D/ \Gamma$, the lift of $\phi$ to $D$ lands in a union of finitely many Siegel sets. We rely heavily on classical work of Cattani-Kaplan-Schmid and Kashiwara.