Murphy's law in non-abelian Hodge theory

2026/08/03 by Gregorio Baldi, Yeuk Hay Joshua Lam
Mathematics · #math.AG #math.NT

paper · pdf

24 pages, comments very welcome!

arxiv created 2026/08/03 · arxiv updated 2026/08/05

Abstract

We construct explicit examples of non-integral variations of ℚ-Hodge structures. Our approach leverages Fenchel--Nielsen-type parameterizations, due to Kabaya and Maskit, of the Teichmüller component of relative character varieties. Additionally, we discuss various Diophantine results concerning the 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 pathological behaviors of the Hodge locus of non-integral ℚVHS, in particular the failure of the Cattani--Deligne--Kaplan theorem and the André--Oort conjecture; our results indicate that for ℚVHS which are not ℤVHS, what can go wrong must go wrong.

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