On the Transcendence of Period Images

Abstract

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

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