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Critical Loci for Higgs Bundles

  • Nigel HitchinEmail author
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Abstract

The paper studies the locus in the moduli space of rank 2 Higgs bundles over a curve of genus g corresponding to points which are critical for d of the Poisson commuting functions defining the integrable system. These correspond to the Higgs field vanishing on a divisor D of degree d. The degree d critical locus is shown to have an induced integrable system related to K(−D)-twisted Higgs bundles. It is embedded in the singular part of the fibration and a description of these singular fibres using Hecke curves is given. The methods used are also applied to give information about the cohomology classes of components of the nilpotent cone and of the first critical locus. It is shown that whereas in the extreme case d = 2g − 2 the locus is a hyperkähler submanifold this does not hold in general. The example of genus 2 is studied concretely and the d = 1 integrable system is seen to be described by a pencil of Kummer surfaces.

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Notes

Acknowledgements

The author wishes to thank A. Oliveira for useful conversations and the Engineering and Physical Sciences Research Council and the Instituto de Ciencias Matemáticas, Madrid for support during the preparation of this work.

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Authors and Affiliations

  1. 1.Mathematical InstituteOxfordUK

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