The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations

TL;DR

This paper establishes a comprehensive correspondence between solutions to Hermite-Einstein equations and polystable Higgs pairs on Riemann surfaces, linking geometric structures to surface group representations.

Contribution

It develops a complete Hitchin-Kobayashi correspondence for twisted pairs, clarifying polystability and establishing a homeomorphism between Higgs bundle moduli and character varieties.

Findings

Proves the bijective correspondence between polystable G-Higgs bundles and surface group representations.

Provides detailed examples for specific groups, simplifying stability conditions.

Establishes a rigorous isomorphism between moduli spaces and character varieties.

Abstract

We develop a complete Hitchin-Kobayashi correspondence for twisted pairs on a compact Riemann surface X. The main novelty lies in a careful study of the the notion of polystability for pairs, required for having a bijective correspondence between solutions to the Hermite-Einstein equations, on one hand, and polystable pairs, on the other. Our results allow us to establish rigorously the homemomorphism between the moduli space of polystable G-Higgs bundles on X and the character variety for representations of the fundamental group of X in G. We also study in detail several interesting examples of the correspondence for particular groups and show how to significantly simplify the general stability condition in these cases.