Wild nonabelian Hodge theory on curves

TL;DR

This paper develops a nonabelian Hodge theory on complex curves, establishing a correspondence between irregular connections and meromorphic Higgs bundles, and studies their moduli spaces with hyperKahler metrics.

Contribution

It extends nonabelian Hodge theory to irregular singularities on curves, including the construction of associated moduli spaces with hyperKahler metrics.

Findings

Moduli spaces carry hyperKahler metrics

Metrics are complete when residues are semisimple

Established correspondence between irregular connections and Higgs bundles

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.