Wild nonabelian Hodge theory on curves
Olivier Biquard, Philip Boalch
TL;DR
This work extends nonabelian Hodge theory to curves with irregular singularities by constructing a hyperKähler moduli framework for connections with fixed irregular polar parts and their meromorphic Higgs counterparts. The authors develop a weighted Sobolev space gauge-theoretic approach to control highly singular coefficients, prove a self-duality (Hitchin) system solvable in this setting, and establish a 1–1 correspondence between stable irregular integrable connections and stable parabolic Higgs bundles, including canonical extensions and stability correspondences. They formulate and relate the DeRham and Dolbeault moduli spaces via complex structures and demonstrate completeness of the hyperKähler metric under semisimple residues. The analysis yields precise asymptotics, local model behavior, and a global equivalence of analytic and algebraic moduli, effectively generalizing Simpson–Corlette-type correspondences to the wild (irregular) case. This work has implications for wild Riemann–Hilbert theory, hyperKähler geometry, and the geometric Langlands program in the irregular regime.
Abstract
On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. The moduli spaces of these objects are obtained by fixing at each singularity the polar part of the connection. We prove that they carry hyperKahler metrics, which are complete when the residue of the connection if semisimple.