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The Hitchin fibration under degenerations to noded Riemann surfaces

  • Jan SwobodaEmail author
Article
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Abstract

In this note we study some analytic properties of the linearized self-duality equations on a family of smooth Riemann surfaces \(\Sigma _R\) converging for \(R\searrow 0\) to a surface \(\Sigma _0\) with a finite number of nodes. It is shown that the linearization along the fibres of the Hitchin fibration \(\mathcal M_d\rightarrow \Sigma _R\) gives rise to a graph-continuous Fredholm family, the index of it being stable when passing to the limit. We also report on similarities and differences between properties of the Hitchin fibration in this degeneration and in the limit of large Higgs fields as studied in Mazzeo et al. (Duke Math. J. 165(12):2227–2271, 2016).

Keywords

Hitchin fibration Self-duality equations Noded Riemann surface 

Mathematics Subject Classification

53C07 32G13 30F30 

Notes

Acknowledgments

The author would like to thank Hartmut Weiß for a number of valuable comments and useful discussions.

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Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematisches Institut der Ruprecht-Karls-Universität HeidelbergHeidelbergGermany

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