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On the non-abelian Hodge locus I

Philip Engel, Salim Tayou

TL;DR

The paper addresses the problem of refining Deligne’s finiteness results and Simpson’s analyticity by proving an algebraicity statement for the non-abelian Hodge locus in the anisotropic (${\mathbb Q}$-anisotropic monodromy) setting. The authors combine a hyperbolic-geometry-based finiteness argument, which reduces to curves and uses Collar Lemmas to bound monodromy traces, with an algebraization framework built from Douady/Barlet spaces of polarized distribution manifolds and period-image spaces tangent to Griffiths distribution. The main accomplishment is the algebraicity of NHL_a in the relative de Rham moduli space $M_{\rm dR}({\mathcal Y}/{\mathcal S},\mathrm{GL}_n)$, as well as a relative finiteness result for monodromy representations, thereby partially resolving conjectures of Deligne and Simpson. The approach yields a robust geometry-to-algebra bridge via Moishezon parameter spaces and GAGA-type arguments, with implications for the structure of period maps and the mapping-class-group actions on families of varieties.

Abstract

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

On the non-abelian Hodge locus I

TL;DR

The paper addresses the problem of refining Deligne’s finiteness results and Simpson’s analyticity by proving an algebraicity statement for the non-abelian Hodge locus in the anisotropic (-anisotropic monodromy) setting. The authors combine a hyperbolic-geometry-based finiteness argument, which reduces to curves and uses Collar Lemmas to bound monodromy traces, with an algebraization framework built from Douady/Barlet spaces of polarized distribution manifolds and period-image spaces tangent to Griffiths distribution. The main accomplishment is the algebraicity of NHL_a in the relative de Rham moduli space , as well as a relative finiteness result for monodromy representations, thereby partially resolving conjectures of Deligne and Simpson. The approach yields a robust geometry-to-algebra bridge via Moishezon parameter spaces and GAGA-type arguments, with implications for the structure of period maps and the mapping-class-group actions on families of varieties.

Abstract

We partially resolve conjectures of Deligne and Simpson concerning -local systems on quasi-projective varieties that underlie a polarized variation of Hodge structure. For local systems with -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.
Paper Structure (13 sections, 32 theorems, 55 equations, 4 figures)

This paper contains 13 sections, 32 theorems, 55 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The non-abelian Hodge locus
  3. Strategy of the proof
  4. Finiteness of monodromy representations
  5. Algebraicity of ${\rm NHL}_a({\mathcal{Y}}/{\mathcal{S}},\mathop{\mathrm{GL}}\nolimits_n)$
  6. Organization of the paper
  7. Acknowledgements
  8. Variations of Hodge structures
  9. Monodromy and Mumford-Tate group
  10. Automorphic vector bundles
  11. Boundedness of monodromy representations
  12. Douady spaces of polarized distribution manifolds
  13. Algebraicity of the non-abelian Hodge locus

Key Result

Theorem 1.2

Let ${\mathcal{Y}}\to {\mathcal{S}}$ be a topologically trivial family of smooth quasi-projective varieties. Then the flat connections in $M_{\rm dR}({\mathcal{Y}}/{\mathcal{S}},\mathop{\mathrm{GL}}\nolimits_n)$ underlying a ${\mathbb Z}$-PVHS with ${\mathbb Q}$-anisotropic monodromy form an algebra

Figures (4)

  • Figure 1: A hyperbolic pair of pants $P(\ell_1,\ell_2,\ell_3)$, and its truncation $P^o(\ell_1,\ell_2,\ell_3)$. Distinguished boundary points on $P$, $P^o$ are shown in red.
  • Figure 2: The four possible configurations of the distinguished points $A,B$ and $A',B'$ which result from gluing two pairs of pants $P$ and $P'$ along a cuff.
  • Figure 3: Seam geodesics on three pairs of pants in red, green, orange, with distinguished points on the cuff in the same color. The decomposition of a loop $\delta$ into paths $\mu$ in pants and $\nu$ in cuffs, depicted in blue.
  • Figure 4: Homotopy of the representatives of $\gamma_j$.

Theorems & Definitions (81)

  • Definition 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Definition 3.1
  • Proposition 3.2
  • ...and 71 more