The asymptotics of the SL₂(ℂ)-Hitchin metric on the singular locus: subintegrable systems

2025/06/05 by Siqi He, He, Siqi, Johannes Horn +3
Mathematics · #53C07 #53C26 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.04957

openalex publication_date 2025/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the asymptotic hyperkähler geometry of the SL₂(ℂ)-Hitchin moduli space over the singular fibers of the Hitchin fibration. We extend the previously known exponential convergence results for solutions to the Hitchin equation to the class of locally fiducial Higgs bundles defined by a special local description at the singularities of the spectral curve. This condition is satisfied by the Higgs bundles contained in certain subintegrable systems introduced by Hitchin. We prove that the restriction of the hyperkähler metric to the subintegrable system converges exponentially fast to the corresponding semi-flat metric along a ray (E,tφ). This answers a question posed by Hitchin in \citeHitchin2021subintegrable_special_Kaehler. More generally, we prove that for each stratum of quadratic differentials there is a closed subset of the corresponding Hitchin fibers, such that the restricted hyperkähler metric converges to a generalized semi-flat metric.

Citations

• Semi-flat metrics of the moduli spaces of Higgs bundles in the non-zero degree case
• Comparison of the Hitchin metric and the semi-flat metric in the rank two case
• Asymptotic behaviour of the Hitchin metric on the moduli space of Higgs bundles
• Asymptotic behaviour of large-scale solutions of Hitchin's equations in higher rank
• Asymptotic Geometry of the Moduli Space of Rank Two Irregular Higgs Bundles over the Projective Line
• mathfraksl(2)-type singular fibres of the symplectic and odd\n orthogonal Hitchin system
• Integrable systems and Special Kähler metrics
• Critical loci for Higgs bundles
• PSL(2;C) connections on 3-manifolds with L2 bounds on curvature
• Wall-crossing, Hitchin Systems, and the WKB Approximation
• The Self-Duality Equations on a Riemann Surface

Related

• Asymptotic behaviour of the Hitchin metric on the moduli space of Higgs bundles
• Asymptotic Geometry of the Moduli Space of Rank Two Irregular Higgs Bundles over the Projective Line
• Asymptotic behaviour of large-scale solutions of Hitchin's equations in higher rank
• Métriques hyperkählériennes pliées
• Generic Ends of the Moduli Space of SL(n,ℂ)-Higgs Bundles