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

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.

49 pages; v3: various corrections and improvements; this paper supersedes Part 1 of arXiv:0809.0576v2 [math.AG]

Cited by in corpus (32)

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• Involutions of the moduli spaces of $G$-Higgs bundles over elliptic curves
• Higgs bundles and higher Teichmüller spaces
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• Higgs bundles over elliptic curves for complex reductive Lie groups
• Mirror symmetry with branes by equivariant Verlinde formulae
• Commuting elements in reductive groups and Higgs bundles on abelian varieties
• Non-Abelian Hodge Theory and Related Topics
• A general Cayley correspondence and higher Teichmüller spaces
• Birationality of moduli spaces of twisted $\mathrm{U}(p,q)$-Higgs bundles
• Higgs bundles and the real symplectic group
• Higgs bundles over elliptic curves
• Studying Deformations of Fuchsian Representations with Higgs Bundles
• Higgs bundles over elliptic curves for real groups
• Stability of Quadric bundles
• Finite group actions on Higgs bundle moduli spaces
• Maximal Higgs bundles for adjoint forms via Cayley correspondence
• Model Higgs bundles in exceptional components of the $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-character variety
• Morse Theory, Higgs fields and Yang-Mills-Higgs functionals
• Quadric bundles and their moduli spaces
• $\mathfrak{sl}(2)$-type singular fibres of the symplectic and odd orthogonal Hitchin system
• The SO(3) Vortex Equations over Orbifold Riemann Surfaces
• Quadric bundles applied to non-maximal Higgs bundles
• Hitchin Pairs for non-compact real Lie groups
• A properness result for degenerate Quadratic and Symplectic Bundles on a smooth projective curve
• The moduli space of generalized quivers
• Harmonic maps from Kähler manifolds