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

Period mappings are definable in the o-minimal structure ℝan,exp - Jacob Tsimerman

Jacob Tsimerman
University of Toronto
March 13, 2018

(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.