Meromorphic Parahoric Higgs Torsors and Filtered Stokes G-local Systems on Curves

TL;DR

This paper establishes a wild nonabelian Hodge correspondence for principal G-bundles on curves, confirming a conjecture by Boalch under a specific condition, linking Higgs torsors, connections, and Stokes local systems.

Contribution

It proves a new nonabelian Hodge correspondence for parahoric Higgs torsors, connections, and Stokes local systems on curves, extending previous results to a broader setting.

Findings

Confirmed Boalch's conjecture under the 'very good' condition.

Established a Kobayashi--Hitchin correspondence for degree zero objects.

Reduced to known results when G=GL(n,C).

Abstract

In this paper, we consider the wild nonabelian Hodge correspondence for principal $G$-bundles on curves, where $G$ is a connected complex reductive group. We establish the correspondence under a ``very good" condition introduced by Boalch, and thus confirm one of his conjectures. We first give a version of Kobayashi--Hitchin correspondence, which induces a one-to-one correspondence between stable meromorphic parahoric Higgs torsors of degree zero (Dolbeault side) and stable meromorphic parahoric connections of degree zero (de Rham side). Then, by introducing a notion of stability condition on filtered Stokes local systems, we prove a one-to-one correspondence between stable meromorphic parahoric connections of degree zero (de Rham side) and stable filtered Stokes $G$-local systems of degree zero (Betti side). When $G={\rm GL}_n(\mathbb{C})$, the main result in this paper reduces to Biquad--Boalch's result.