The Hitchin fibration under degenerations to noded Riemann surfaces

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).

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Acknowledgments

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

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Correspondence to Jan Swoboda.

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Swoboda, J. The Hitchin fibration under degenerations to noded Riemann surfaces. Abh. Math. Semin. Univ. Hambg. 86, 189–201 (2016). https://doi.org/10.1007/s12188-016-0132-7

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Keywords

  • Hitchin fibration
  • Self-duality equations
  • Noded Riemann surface

Mathematics Subject Classification

  • 53C07
  • 32G13
  • 30F30