On the non-abelian Hodge locus I

2023/05/01 by Philip Engel, Engel, Philip, Salim Tayou +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2305.00943

openalex publication_date 2023/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We partially resolve conjectures of Deligne and Simpson concerning ℤ-local systems on quasi-projective varieties that underlie a polarized variation of Hodge structure. For local systems with ℚ-anisotropic monodromy, we prove (1) a relative form of Deligne's finiteness theorem, for any family of quasi-projective varieties, and (2) algebraicity of the corresponding non-abelian Hodge locus.

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• Local systems which do not come from abelian varieties
• Hodge loci and atypical intersections: conjectures
• Local systems on proper algebraic V-manifolds
• A non-abelian version of Deligne's Fixed Part Theorem
• Non vanishing loci of Hodge numbers of local systems