Limiting configurations for solutions of Hitchin's equation

TL;DR

This paper reviews the compactification of the moduli space of Hitchin's equations, analyzing degeneration behavior and desingularization of limiting configurations, with connections to Prym varieties and symmetric solutions.

Contribution

It introduces a detailed analysis of limiting configurations near the moduli space boundary and relates them to Prym varieties, extending Hitchin's ideas.

Findings

Desingularization of limiting configurations near the moduli space boundary.

Relation of boundary strata to Prym varieties.

Analysis of rotationally symmetric solutions on complex plane.

Abstract

We review recent work on the compactification of the moduli space of Hitchin's self-duality equation. We study the degeneration behavior near the ends of this moduli space in a set of generic directions by showing how limiting configurations can be desingularized. Following ideas of Hitchin, we can relate the top boundary stratum of this space of limiting configurations to a Prym variety. A key r\^ole is played by the family of rotationally symmetric solutions to the self-duality equation on $\mathbb C$, which we discuss in detail here.