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

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

2009-09-24

arxiv.org/abs/0909.4487v2

Mathematics

Differential Geometry

47 pages; this paper supersedes Part 1 of arXiv:0809.0576v2 [math.AG]

Scientific paper

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.

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