On the transcendence of period images

LQ̄V Let f : X → S be a family of smooth projective algebraic varieties over a smooth connected base S, with everything defined over . Denote by =R²ⁱ f_ Z(i) the associated integral variation of Hodge structure on the degree $2i$ cohomology. We consider the following question: when can a fiber Vₛ above an algebraic point s ∈ S() be isomorphic to a transcendental fiber Vs^ with s^ ∈ S(C) S()? When V induces a quasi-finite period map φ : S → 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 ∈ S() outside of an explicit proper closed algebraic subset of S. As a corollary, we establish the existence of a canonical -algebraic model for normalizations of period images.