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]

References in corpus (3)

• Surface group representations in PU(p,q) and Higgs bundles
• Higgs bundles and surface group representations in the real symplectic group
• A Hitchin-Kobayashi correspondence for Kaehler fibrations

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