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The behavior of sequences of solutions to the Hitchin-Simpson equations

Siqi He

Abstract

The Hitchin-Simpson equations are first-order non-linear equations for a pair consisting of a connection and a Higgs field. In this paper, we study the behavior of sequences of solutions to the Hitchin-Simpson equations on closed Kähler manifolds with unbounded $L^2$ norms of the Higgs fields. We prove a compactness result for the connections and renormalized Higgs fields, which generalizes the work of Taubes and Mochizuki. As applications, we prove that every $\mathbb{Z}/2$ harmonic 1-form on a Kähler manifold can be deformed into a sequence of solutions to the Hitchin-Simpson equations. Additionally, we solve the generalized Hitchin's WKB problem on any Kähler manifold.

The behavior of sequences of solutions to the Hitchin-Simpson equations

Abstract

The Hitchin-Simpson equations are first-order non-linear equations for a pair consisting of a connection and a Higgs field. In this paper, we study the behavior of sequences of solutions to the Hitchin-Simpson equations on closed Kähler manifolds with unbounded norms of the Higgs fields. We prove a compactness result for the connections and renormalized Higgs fields, which generalizes the work of Taubes and Mochizuki. As applications, we prove that every harmonic 1-form on a Kähler manifold can be deformed into a sequence of solutions to the Hitchin-Simpson equations. Additionally, we solve the generalized Hitchin's WKB problem on any Kähler manifold.

Paper Structure

This paper contains 42 sections, 59 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Applications
  4. Deformation problem of $\mathbb{Z}/2$ harmonic 1-forms
  5. Generalized Hitchin's WKB problem
  6. Higgs Bundles and the Hitchin-Simpson Equations on Kähler Manifolds
  7. Higgs Bundles on Kähler Manifolds
  8. The Hitchin-Simpson Equations and the Kobayashi-Hitchin Correspondence
  9. Hermitian Geometry on Kähler Manifolds
  10. The Hitchin-Simpson Equations on Kähler Manifolds
  11. Topologically Trivial Higgs Bundles
  12. Hitchin Morphism and the Spectral Variety
  13. Hitchin Morphism
  14. The Spectral Variety
  15. Rank-Two Spectral Base
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $X$ be a closed Kähler manifold, and let $E$ be a rank $r$ vector bundle with a Hermitian metric $H$. Let $(A_i, \phi_i)$ be a sequence of solutions to the Hitchin-Simpson equations Hitchin-Simpson, and let $r_i := \|\phi_i\|_{L^2(X)}$.

Theorems & Definitions (114)

  • Theorem 1.1
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • Proposition 2.5
  • ...and 104 more