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Quiver bundles and wall crossing for chains

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Abstract

Holomorphic chains on a Riemann surface arise naturally as fixed points of the natural \(\mathbb {C}^*\)-action on the moduli space of Higgs bundles. In this paper we associate a new quiver bundle to the \({{\mathrm{Hom}}}\)-complex of two chains, and prove that stability of the chains implies stability of this new quiver bundle. Our approach uses the Hitchin–Kobayashi correspondence for quiver bundles. Moreover, we use our result to give a new proof of a key lemma on chains (due to Álvarez-Cónsul–García-Prada–Schmitt), which has been important in the study of Higgs bundle moduli; this proof relies on stability and thus avoids the direct use of the chain vortex equations.

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Acknowledgements

We thank Steve Bradlow for useful discussions and we thank the referee for insightful comments which helped improve the exposition.

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Authors and Affiliations

  1. Centro de Matemática da Universidade do Porto, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre, s/n, 4169-007, Porto, Portugal

    P. B. Gothen

  2. Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Faculdade de Ciências da Universidade de Lisboa, Edf. C6, Campo Grande, 1749-016, Lisbon, Portugal

    A. Nozad

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  1. P. B. Gothen
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  2. A. Nozad
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Correspondence to P. B. Gothen.

Additional information

Partially supported by CMUP (UID/MAT/00144/2013) (first author), CMAF-CIO (UID/MAT/04561/2013) and Grant SFRH/BD/51166/2010 (second author), and the Project PTDC/MAT-GEO/2823/2014 (both authors) funded by FCT (Portugal) with national funds. The authors acknowledge support from U.S. National Science Foundation Grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric structures And Representation varieties” (the GEAR Network).

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Gothen, P.B., Nozad, A. Quiver bundles and wall crossing for chains. Geom Dedicata 199, 137–146 (2019). https://doi.org/10.1007/s10711-018-0341-6

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