The Zilber-Pink conjecture for varieties not defined over $\overline{\mathbb Q}$
Bruno Klingler, Salim Tayou
TL;DR
The paper extends the Zilber–Pink conjecture to subvarieties of mixed Shimura varieties and to $\mathbb{Z}$-variations of mixed Hodge structures (ZVMHS) of geometric origin. It combines geometric finiteness results for atypical Hodge loci with spreading arguments to reduce to zero-period-dimension cases, then uses field-of-definition and absoluteness assumptions to force non-density of atypical loci; a key innovation is the reduction to zero-period components and the inductive argument on the transcendence degree of the base field. Under strong non-$\overline{\mathbb{Q}}$-definability in the Shimura setting, ZP holds for all such subvarieties, while under absoluteness for geometric origin VMHS, ZP also holds whenever no nontrivial quotient of the period map factors over $\overline{\mathbb{Q}}$, with equivalence to the $\overline{\mathbb{Q}}$-defined case. The results bridge arithmetic definability and Hodge-theoretic atypicality, broadening the scope of ZP in the mixed and geometric-origin contexts with practical implications for understanding atypical subvarieties in period maps.
Abstract
In this note, we prove the Zilber--Pink conjecture for subvarieties of mixed Shimura varieties, which are not defined over~$\overline{\mathbb Q}$ in a strong sense. We prove similar results for general variations of mixed Hodge structure of geometric origin, assuming furthermore that they are absolute.