The Algebraic and Analytic Compactifications of the Hitchin Moduli Space

Following the work of Mazzeo-Swoboda-Weiss-Witt and Mochizuki, there is a map $\bar{\mathrm{\Xi }}$ between the algebraic compactification of the Dolbeault moduli space of $\mathsf{S}\mathsf{L}(2,\mathbb{C})$ 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 $\mathsf{S}\mathsf{U}(2)$ 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 $\bar{\mathrm{\Xi }}$ 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.