The Algebraic and Analytic Compactifications of the Hitchin Moduli Space

Abstract

Following the work of Mazzeo-Swoboda-Weiss-Witt and Mochizuki, there is a map $\overline{\Xi }$ between the algebraic compactification of the Dolbeault moduli space of $\mathsf{SL}(2,\mathbb{C})$ Higgs bundles on a smooth projective curve coming from the ${\mathbb{C}}^{\ast }$ action, and the analytic compactification of Hitchin's moduli space of solutions to the $\mathsf{SU}(2)$ self-duality equations on a Riemann surface obtained by adding solutions to the decoupled equations, known as ``limiting configurations''. This map extends the classical Kobayashi-Hitchin correspondence. The main result of this paper is that $\overline{\Xi }$ fails to be continuous at the boundary over a certain subset of the discriminant locus of the Hitchin fibration. This suggests the possibility of a third, refined compactification which dominates both.

38 pages. Minor edits

References in corpus (4)

• Asymptotic Geometry of the Hitchin Metric
• Asymptotics of Hitchin's metric on the Hitchin section
• Exponential Decay for the Asymptotic Geometry of the Hitchin Metric
• Semi-abelian spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin system