Tame parahoric nonabelian Hodge correspondence on curves

TL;DR

This paper extends the nonabelian Hodge correspondence to noncompact curves for general complex reductive groups by introducing parahoric torsors, bridging Higgs bundles and fundamental group representations beyond the classical GL_n case.

Contribution

It introduces parahoric group scheme torsors to establish a full nonabelian Hodge correspondence for general reductive groups on noncompact curves.

Findings

Established a full correspondence between Higgs bundles and fundamental group representations.

Extended nonabelian Hodge theory beyond GL_n to arbitrary complex reductive groups.

Connected parahoric torsors with existing Riemann-Hilbert results.

Abstract

The nonabelian Hodge correspondence for vector bundles over noncompact curves is adequately described by implementing a weighted filtration on the objects involved. In order to establish a full correspondence between a Dolbeault and a de Rham space for a general complex reductive group $G$, we introduce torsors given by parahoric group schemes in the sense of Bruhat--Tits. Combined with existing results on the Riemann--Hilbert correspondence for logarithmic parahoric connections, this gives a full nonabelian Hodge correspondence from Higgs bundles to fundamental group representations over a noncompact curve beyond the $\text{GL}_n(\mathbb{C})$-case.