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The Spectral base and quotients of bounded symmetric domains

Siqi He, Jie Liu, Ngaiming Mok

Abstract

In this article, we explore Higgs bundles on a projective manifold $X$, focusing on their spectral bases, a concept introduced by T.Chen and B.Ngô. The spectral base is a specific closed subscheme within the space of symmetric differentials. We observe that if the spectral base vanishes, then any reductive representation $ρ: π_1(X) \to \text{GL}_r(\mathbb{C})$ is both rigid and integral. Additionally, we prove that for $X=Ω/Γ$, a quotient of a bounded symmetric domain $Ω$ of rank at least $2$ by a torsion-free cocompact irreducible lattice $Γ$, the spectral base indeed vanishes, which generalizes a result of B.Klingler.

The Spectral base and quotients of bounded symmetric domains

Abstract

In this article, we explore Higgs bundles on a projective manifold , focusing on their spectral bases, a concept introduced by T.Chen and B.Ngô. The spectral base is a specific closed subscheme within the space of symmetric differentials. We observe that if the spectral base vanishes, then any reductive representation is both rigid and integral. Additionally, we prove that for , a quotient of a bounded symmetric domain of rank at least by a torsion-free cocompact irreducible lattice , the spectral base indeed vanishes, which generalizes a result of B.Klingler.
Paper Structure (15 sections, 19 theorems, 46 equations)

This paper contains 15 sections, 19 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Higgs bundles and Hitchin morphism
  3. Higgs bundle and non-abelian Hodge correspondence
  4. Hitchin morphism
  5. Spectral base
  6. Spectral variety and its decomposition
  7. The spectral base for a quotient of a bounded symmetric domain with $\mathrm{rank} \geq 2$.
  8. Finsler (pseudo-)metric
  9. Bounded symmetric domain and Finsler metric rigidity
  10. Rigidity and Integrality
  11. Rigidity and integrality from spectral base perspective
  12. Rigidity of the character variety
  13. Harmonic Maps into Bruhat-Tits Buildings
  14. Applications
  15. Examples

Key Result

Theorem 1.2

Let $X$ be a projective manifold such that $\mathcal{S}^r_X=0$ for some $r\geq 1$. Then the following statements hold:

Theorems & Definitions (53)

  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: hitchin1987selfdonaldson1987twistedSimpson1988Constructioncorlette1988flat
  • Theorem 2.4: hitchin1987selfsimpson1994moduli
  • Definition 2.5
  • Lemma 2.6
  • ...and 43 more