On the transcendence of period images

Abstract

L$\def\Q{\overline{\mathbb{Q}}}\def\V{\mathbb{V}}$Let $f : X \to S$ be a family of smooth projective algebraic varieties over a smooth connected base $S$, with everything defined over $\Q$. Denote by $\V =R^{2i} f_\ast \mathbb{Z}(i)$ the associated integral variation of Hodge structure on the degree $2i$ cohomology. We consider the following question: when can a fiber $\mathbb{V}_s$ above an algebraic point $s \in S(\Q)$ be isomorphic to a transcendental fiber $\mathbb{V}_{s^\prime}$ with $s^\prime \in S(\mathbb{C}) \setminus S(\Q)$? When $V$ induces a quasi-finite period map $\varphi : S \to \mathbb{D}$, conjectures in Hodge theory predict that such isomorphisms cannot exist. We introduce new differential-algebraic techniques to show this is true for all points $s \in S(\Q)$ outside of an explicit proper closed algebraic subset of $S$. As a corollary, we establish the existence of a canonical $\Q$-algebraic model for normalizations of period images.