On the Transcendence of Period Images [article]

David Urbanik
2021 arXiv   pre-print
Let f : X → 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_*ℤ(i) the associated integral variation of Hodge structure on the degree 2i cohomology. We consider the following question: when can a fibre 𝕍_s above an algebraic point s ∈ S(ℚ) be isomorphic to a transcendental fibre 𝕍_s' with s' ∈ S(ℂ) ∖ S(ℚ)? When 𝕍 induces a quasi-finite period map φ : S →Γ\ D, conjectures in Hodge theory predict that such
more » ... somorphisms cannot exist. We introduce new differential-algebraic techniques to show this is true for all points s ∈ 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.
arXiv:2106.09342v1 fatcat:6a65zpaapnbo7kj64yviyjodnq