Contents1. Introduction 2. Arithmetic hyperbolicity and geometric hyperbolicity 3. Weakly bounded varieties and persistence of non-density 4. Hodge theory 5. Proof of Deligne-Schmid's theorem and Theorem 1.4 6. Locally symmetric varieties and Shimura varieties 7. The moduli of smooth hypersurfaces 8. Non-density of integral points on the Hilbert scheme References Abstract. We show that for a variety which admits a quasi-finite period map, finiteness (resp. non-Zariski-density) of S-integral points implies finiteness (resp. non-Zariski-density) of points over all Z-finitely-generated integral domains of characteristic zero. Our proofs rely on foundational results in Hodge theory due to Deligne, Griffiths, and Schmid, as well as the recent resolution of Griffiths's conjecture by Bakker-Brunebarbe-Tsimerman. We give straightforward applications to Shimura varieties, locally symmetric varieties, and the moduli space of smooth hypersurfaces in projective space. Using similar arguments and results of Viehweg-Zuo, we obtain similar arithmetic finiteness (resp. non-Zariski-density) statements for complete subvarieties of the moduli of canonically polarized varieties.
We show that the minimum rank of a non-isotrivial local system of geometric origin on a suitably general n-pointed curve of genus g is at least 2 g + 1. We apply this result to resolve conjectures of Esnault-Kerz and Budur-Wang. The main input is an analysis of stability properties of flat vector bundles under isomonodromic deformations, which additionally answers questions of Biswas, Heu, and Hurtubise.
Let Σ g,n be an orientable surface of genus g with n punctures. We study actions of the mapping class group Mod g,n of Σ g,n via Hodge-theoretic and arithmetic techniques. We show that if ρ : π 1 (Σ g,n ) → GL r (C) is a representation whose conjugacy class has finite orbit under Mod g,n , and r < g + 1, then ρ has finite image. This answers questions of Junho Peter Whang and Mark Kisin. We give applications of our methods to the Putman-Wieland conjecture, the Fontaine-Mazur conjecture, and a question of Esnault-Kerz.The proofs rely on non-abelian Hodge theory, our earlier work on semistability of isomonodromic deformations, and recent work of Esnault-Groechenig and Klevdal-Patrikis on Simpson's integrality conjecture for cohomologically rigid local systems. CONTENTS 1. Introduction 1 2. Representation-theoretic preliminaries 10 3. Hodge-theoretic preliminaries 20 4. The period map associated to a unitary representation 23 5. The main cohomological results 27 6. The asymptotic Putman-Wieland conjecture 30 7. Proof of the main theorem on MCG-finite representations 33 8. Consequences for arithmetic representations 39 9. Questions and examples 43
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