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 \(\mathbb{V} = R^{2i} f_{*} \mathbb{Z}(i)\) 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 \(\mathbb{V}_{s'}\) with \(s' \in S(\mathbb{C}) \setminus S(\overline{\mathbb{Q}})\)? When \(\mathbb{V}\) induces a quasi-finite period map \(\varphi : S \to \Gamma \backslash D\), 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.