p-curvature criterion for \pi_1(S)-invariant subvarieties

Establish that for a smooth proper morphism \mathscr{X}\to S over a finitely generated integral \mathbb{Z}-algebra R, an R-point s\in S, and a closed substack Z⊂\mathscr{M}_{dR}(\mathscr{X}/S,r)_s, the complex points Z(\mathbb{C}) are invariant under a finite-index subgroup of \pi_1(S,s) whenever the formal isomonodromic deformation of Z has an integral model.

Background

This is a higher-dimensional non-abelian extension of the p-curvature philosophy, predicting that integrality of formal isomonodromic deformations forces global \pi_1(S)-invariance of subvarieties in the de Rham moduli. It parallels Katz’s equivalence of p-curvature criteria and monodromy Zariski closures.