Ends of the moduli space of Higgs bundles. (English) Zbl 1352.53018

Mazzeo, Rafe; Swoboda, Jan; Weiss, Hartmut; Witt, Frederik

Duke Math. J. 165, No. 12, 2227-2271 (2016).

Authors’ 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 zeros. 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 \(t\rightarrow \infty\). 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.

Reviewer: Ioan Pop (Iaşi)

MSC:

53C07	Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

moduli space; (stable) Higgs pair; Higgs bundle; Hitchin’s equation; limiting configuration; complex gauge transformation; desingularization