Wild non-abelian Hodge theory on curves. (English) Zbl 1051.53019

Biquard, Olivier; Boalch, Philip

Compos. Math. 140, No. 1, 179-204 (2004).

The paper studies non-abelian Hodge theory on complex curves with irregular singularities. A connection with irregular singularity has the local form (after a suitable gauge transformation) \[ d+A_n\frac{dz}{z^n}+\cdots +A_1\frac{dz}{z}+ \text{holomorphic terms} \]
where \(n>1\) and \(A_i\) are constant diagonal matrices by assumption. A Higgs field with irregular singularity can be similarly defined. The main result in the paper states that there is a one-to-one correspondence between stable integrable connections with irregular singularities and stable parabolic Higgs bundles with irregular singularities. The proof involves some interesting analysis on weighted Sobolev spaces, following the same line of argument of minimizing the Donaldson functional as done previously by the authors for the case of \(n=1\). The paper also shows that the moduli space of these integrable connections has the standard hyper-Kähler structure.

Reviewer: Shuguang Wang (Columbia)

MSC:

53C07	Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
34M55	Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
32G34	Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)

Keywords:

Hermitian-Einstein metric; integrable connection; irregular singularity; Higgs bundle