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Asymptotic Geometry of the Moduli Space of Rank Two Irregular Higgs Bundles over the Projective Line

Gao Chen, Nianzi Li

Abstract

We study the asymptotic behavior of Hitchin's hyperkähler metric on the moduli space of rank two irregular Higgs bundles over $\mathbb{C}P^1$. Along a generic curve, we prove that the Hitchin metric is asymptotic to the semiflat metric at an arbitrary polynomial order. When there are no weakly parabolic singularities, the rate is exponential. In the case of four-dimensional moduli spaces, we prove that the semiflat metric is asymptotic to an ALG/ALG$^\ast$ model metric.

Asymptotic Geometry of the Moduli Space of Rank Two Irregular Higgs Bundles over the Projective Line

Abstract

We study the asymptotic behavior of Hitchin's hyperkähler metric on the moduli space of rank two irregular Higgs bundles over . Along a generic curve, we prove that the Hitchin metric is asymptotic to the semiflat metric at an arbitrary polynomial order. When there are no weakly parabolic singularities, the rate is exponential. In the case of four-dimensional moduli spaces, we prove that the semiflat metric is asymptotic to an ALG/ALG model metric.
Paper Structure (22 sections, 31 theorems, 289 equations, 2 tables)

This paper contains 22 sections, 31 theorems, 289 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Irregular Higgs bundles and the moduli space
  5. Explicit descriptions
  6. Approximate Solutions
  7. Harmonic metric
  8. A curve in $\mathcal{M}$
  9. Decoupled harmonic metric
  10. Normal forms
  11. Approximate solutions
  12. Fiducial Solutions
  13. Model solution near $P_w$
  14. Linear Analysis
  15. Deforming to a Solution
  16. ...and 7 more sections

Key Result

Theorem 1.1

Fix a generic curve $[(\bar{\partial}_E,\varphi_t)]$ in $\mathcal{M}$, and an infinitesimal deformation $[(\dot{\eta},\dot{\varphi})]\in T_{[(\bar{\partial}_E,\varphi_t)]}\mathcal{M}$. As $t\to \infty$, for any $N>0$ we have If, moreover there are no weakly parabolic points, there exist positive constants $c,\sigma$ independent of $t$ and $[(\dot{\eta},\dot{\varphi})]$ such that the above differe

Theorems & Definitions (67)

  • Theorem 1.1
  • Definition 2.1: biquard_boalch_2004
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 57 more