Ends of the moduli space of Higgs bundles

TL;DR

This paper constructs a family of solutions to Hitchin's equations that converge to singular limiting configurations, providing new insights into the boundary structure of the Higgs bundle moduli space on Riemann surfaces.

Contribution

It introduces a desingularization method for Higgs bundles using gluing techniques, revealing the structure of the moduli space's boundary.

Findings

Constructed solutions converge to singular limits as parameter tends to infinity.

Identified a dense open subset of the Higgs moduli space boundary.

Provided a new proof for the structure of the moduli space ends.

Abstract

We associate to each stable Higgs pair $(A_0,\Phi_0)$ on a compact Riemann surface $X$ a singular limiting configuration $(A_\infty,\Phi_\infty)$, assuming that $\det \Phi$ has only simple zeroes. We then prove a desingularization theorem by constructing a family of solutions $(A_t,t\Phi_t)$ to Hitchin's equations which converge to this limiting configuration as $t \to \infty$. This provides a new proof, via gluing methods, for elements in the ends of the Higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space.