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Hitchin fibrations are Ngô fibrations

Mark Andrea de Cataldo, Roberto Fringuelli, Andres Fernandez Herrero, Mirko Mauri

TL;DR

The paper proves that the Hitchin fibration for $ obreak \\mathcal{L}$-valued $G$-Higgs bundles is a Ngô fibration in characteristic $0$, with a strong weak-abelian fibration structure supplied by a group scheme of symmetries $P^o$ acting fiberwise on the Hitchin base. It develops the geometry of cameral covers, semistable loci, and moduli spaces, and extends the framework to isotrivial reductive group schemes, establishing descent, theta-stratification, and delta-regularity results. A cohomological bound for the Hitchin morphism and a Decomposition Theorem–style description are proven via a comparison between the stacky and good-moduli-space pictures, including the construction of canonical morphisms $\

Abstract

We study the geometry of the Hitchin fibration for $\mathcal{L}$-valued $G$-Higgs bundles over a smooth projective curve of genus $g$, where $G$ is a reductive group and $\mathcal{L}$ is a suitably positive line bundle. We show that the Hitchin fibration admits the structure of a weak Abelian fibration. In the case when the line bundle $\mathcal{L}$ is a twist of the canonical bundle of the curve by a (possibly empty) reduced effective divisor, we prove a cohomological bound and $δ$-regularity of the weak Abelian fibration.

Hitchin fibrations are Ngô fibrations

TL;DR

The paper proves that the Hitchin fibration for -valued -Higgs bundles is a Ngô fibration in characteristic , with a strong weak-abelian fibration structure supplied by a group scheme of symmetries acting fiberwise on the Hitchin base. It develops the geometry of cameral covers, semistable loci, and moduli spaces, and extends the framework to isotrivial reductive group schemes, establishing descent, theta-stratification, and delta-regularity results. A cohomological bound for the Hitchin morphism and a Decomposition Theorem–style description are proven via a comparison between the stacky and good-moduli-space pictures, including the construction of canonical morphisms $\

Abstract

We study the geometry of the Hitchin fibration for -valued -Higgs bundles over a smooth projective curve of genus , where is a reductive group and is a suitably positive line bundle. We show that the Hitchin fibration admits the structure of a weak Abelian fibration. In the case when the line bundle is a twist of the canonical bundle of the curve by a (possibly empty) reduced effective divisor, we prove a cohomological bound and -regularity of the weak Abelian fibration.

Paper Structure

This paper contains 37 sections, 105 theorems, 149 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and setup
  3. Lie algebras
  4. The Chevalley Restriction Theorem
  5. Local regular centralizer group schemes
  6. The stack of G-Higgs bundles
  7. The stack of L-valued G-Higgs bundles
  8. The Hitchin fibration
  9. The semistable locus
  10. Group scheme of symmetries of the Hitchin fibration
  11. Cameral Covers
  12. The loci of reduced and smooth cameral covers A heart and A diamond.
  13. The group scheme of symmetries of the Hitchin fibration
  14. Action on the semistable locus
  15. Polarizability of the group of symmetries of the Hitchin fibration
  16. ...and 22 more sections

Key Result

Proposition 1.2

Let $(X,P,B)$ be a Ngô fibration as in defn: ngo fibration intro over an algebraically closed field $k$. There exists an isomorphism in the constructible bounded derived category $D^b_c(B, \overline{\mathbb{Q}}_ \ell)$ where $L^d_{\sigma}$ is a lisse sheaf on an open and dense subscheme of $Z_{\sigma}$, and $\mathrm{codim}(Z_{\sigma}) = \delta_{\sigma}$. Suppose in addition that the field $k$ is t

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (272)

  • Definition 1.1: Ngô fibration
  • Proposition 1.2: Decomposition Theorem for Ngô fibrations
  • proof
  • Theorem A: The Hitchin fibration is a Ngô fibration
  • proof
  • Remark 1.3
  • Remark 1.4
  • Theorem B: \ref{['P:Higgs-prop']} + \ref{['P:Higgs-integral']} + \ref{['P:Higgs-normal']}
  • Theorem C: \ref{['thm: adequate moduli spaces for higgs']} + \ref{['cor: higgs moduli space properties']} + \ref{['prop: fibers hitchin pure dimensional']} + \ref{['prop: smoothness of the semistable hitchin stack']}
  • Theorem D: Weak abelian fibration structure: \ref{['thm: weak Abelian fibration structure']}
  • ...and 262 more