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Semi-flat metrics of the moduli spaces of Higgs bundles in the non-zero degree case

Takuro Mochizuki

TL;DR

This paper extends the asymptotic comparison between the Hitchin metric and the semi-flat metric from degree zero to non-zero degree in the moduli space of Higgs bundles. It introduces horizontal deformations for the non-zero degree case, establishes an integrable connection on the Hitchin fibration over the smooth spectral-curve locus, and constructs the corresponding semi-flat metric $g_{sf}^{X,n,d}$. The main result proves exponential convergence of the Hitchin metric to the semi-flat metric along rays $(E,t\theta)$ as $t\to\infty$, thereby generalizing prior d=0 results and providing a unified picture of the hyperkähler asymptotics. The framework relies on translating horizontality from line bundles on spectral curves to Higgs bundles, analyzing deformation theory, and exploiting covering maps to relate non-zero degree cases to the zero-degree setting.

Abstract

We study horizontal deformations of a Higgs bundle whose spectral curve is smooth. It allows us to define a natural integrable connection of the Hitchin fibration on the locus where the spectral curves are smooth. Then, in the non-zero degree case, we introduce the semi-flat metric, and compare the asymptotic behaviour of the semi-flat metric and the Hitchin metric along the ray $(E,tθ)$ $(t\to\infty)$.

Semi-flat metrics of the moduli spaces of Higgs bundles in the non-zero degree case

TL;DR

This paper extends the asymptotic comparison between the Hitchin metric and the semi-flat metric from degree zero to non-zero degree in the moduli space of Higgs bundles. It introduces horizontal deformations for the non-zero degree case, establishes an integrable connection on the Hitchin fibration over the smooth spectral-curve locus, and constructs the corresponding semi-flat metric . The main result proves exponential convergence of the Hitchin metric to the semi-flat metric along rays as , thereby generalizing prior d=0 results and providing a unified picture of the hyperkähler asymptotics. The framework relies on translating horizontality from line bundles on spectral curves to Higgs bundles, analyzing deformation theory, and exploiting covering maps to relate non-zero degree cases to the zero-degree setting.

Abstract

We study horizontal deformations of a Higgs bundle whose spectral curve is smooth. It allows us to define a natural integrable connection of the Hitchin fibration on the locus where the spectral curves are smooth. Then, in the non-zero degree case, we introduce the semi-flat metric, and compare the asymptotic behaviour of the semi-flat metric and the Hitchin metric along the ray .
Paper Structure (36 sections, 36 theorems, 52 equations)

This paper contains 36 sections, 36 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Moduli space of stable Higgs bundles
  3. Semi-flat metric in the case $d=0$
  4. Comparison of the Hitchin metric and the semi-flat metric in the case $d=0$
  5. The main result of this paper
  6. Acknowledgement
  7. Horizontal deformations of line bundles in a general setting
  8. Preliminaries
  9. Roots of line bundles
  10. Horizontal line bundles with respect to a prescribed holomorphic line bundle
  11. Pull back by a covering map
  12. Horizontal deformations of Higgs bundles
  13. Horizontal deformations of line bundles on smooth spectral curves
  14. Basic properties of smooth spectral curves
  15. Horizontal line bundles
  16. ...and 21 more sections

Key Result

Theorem 1.1

For any $(E,\theta)\in \mathcal{M}'_{H}(X,n,d)$, there exists $\epsilon>0$ such that the following estimate holds as $t\to\infty$ with respect to $(g_{\mathop{\rm sf}\nolimits}^{X,n,d})_{|(E,t\theta)}$:

Theorems & Definitions (42)

  • Theorem 1.1: Theorem \ref{['thm;25.1.22.40']}
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • Proposition 2.8
  • Lemma 2.9
  • ...and 32 more