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<p>We partially resolve conjectures of Deligne and Simpson concerning <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {Z}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation>-local systems on quasi-projective varieties that underlie a polarized variation of Hodge structure. For local systems with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathbb {Q}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation>-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.</p>

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On the non-abelian Hodge locus I

  • 期刊论文

  • 发表日期：

    2026年8月5日

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  • Philip Engel,
  • Salim Tayou

摘要

We partially resolve conjectures of Deligne and Simpson concerning \({\mathbb {Z}}\) Z -local systems on quasi-projective varieties that underlie a polarized variation of Hodge structure. For local systems with \({\mathbb {Q}}\) 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.