Murphy's law in non-abelian Hodge theory

Abstract

We construct explicit examples of non-integral variations of $\mathbb{Q}$-Hodge structures. Our approach leverages Fenchel--Nielsen-type parameterizations, due to Kabaya and Maskit, of the Teichm\"uller component of relative character varieties. Additionally, we discuss various Diophantine results concerning the $\mathcal{O}_{K,S}$-integral points of such character varieties, and give a new proof of Beauville's classical theorem on families of elliptic curves. We conclude by collecting \emph{pathological behaviors} of the Hodge locus of non-integral $\mathbb{Q}$VHS, in particular the failure of the Cattani--Deligne--Kaplan theorem and the Andr\'e--Oort conjecture; our results indicate that for $\mathbb{Q}$VHS which are not $\mathbb{Z}$VHS, what can go wrong must go wrong.