Semi-abelian spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin system

Abstract

We describe spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin fibration with irreducible and reduced spectral curve. Using Hecke transformations we give a stratification of these singular spaces by fibre bundles over Prym varieties. By analysing the parameter spaces of Hecke transformations this describes the singular Hitchin fibres as compactifications of abelian group bundles over abelian torsors. We prove that a large class of singular fibres are themselves fibre bundles over Prym varieties. As applications we study irreducible components of singular Hitchin fibres and give a description of $\mathsf{SL}(2,\mathbb{R})$-Higgs bundles in terms of these semi-abelian spectral data.

arXiv comment43 pages, 1 figure; v3: Correction of special case in Section 6 and 8. To appear in International Mathematics Research Notices

References in corpus (15)

• Mirror symmetry, Langlands duality, and the Hitchin system
• Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces
• Spectral data for G-Higgs bundles
• Higgs bundles for real groups and the Hitchin-Kostant-Rallis section
• The geometry of maximal components of the PSp(4,R) character variety
• Prym varieties of spectral covers
• Unramified covers and branes on the Hitchin system
• Monodromy of the SL_2 Hitchin fibration
• The singular fibre of the Hitchin map
• SO(p,q)-Higgs bundles and higher Teichmüller components
• Langlands duality and G2 spectral curves
• Higgs bundles, Lagrangians and mirror symmetry
• Hecke modifications of Higgs bundles and the extended Bogomolny equation
• Orthogonal Higgs bundles with singular spectral curves
• Non-abelian vortices, Hecke modifications and singular monopoles

Cited by in corpus (2)

• The Algebraic and Analytic Compactifications of the Hitchin Moduli Space
• Mirror of Orbifold Singularities in the Hitchin Fibration: the case $({\text{SL}}_n,{\text{PGL}}_n)$