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Birationality of moduli spaces of twisted \(\mathrm {U}(p,q)\)-Higgs bundles

Abstract

A \(\mathrm {U}(p,q)\)-Higgs bundle on a Riemann surface (twisted by a line bundle) consists of a pair of holomorphic vector bundles, together with a pair of (twisted) maps between them. Their moduli spaces depend on a real parameter \(\alpha \). In this paper we study wall crossing for the moduli spaces of \(\alpha \)-polystable twisted \(\mathrm {U}(p,q)\)-Higgs bundles. Our main result is that the moduli spaces are birational for a certain range of the parameter and we deduce irreducibility results using known results on Higgs bundles. Quiver bundles and the Hitchin–Kobayashi correspondence play an essential role.

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Notes

  1. 1.

    Note that the stability parameter for the corresponding untwisted triple as considered in [9] is \(\alpha +\deg (L)\).

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Correspondence to Peter B. Gothen.

Additional information

Partially supported by CMUP (UID/MAT/00144/2013), the projects PTDC/MAT-GEO/0675/2012 and PTDC/MAT-GEO/2823/2014 (first author) and grant SFRH/BD/51166/2010 (second author), funded by FCT (Portugal) with national and where applicable European structural funds through the programme FEDER, under the partnership agreement PT2020. 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. Birationality of moduli spaces of twisted \(\mathrm {U}(p,q)\)-Higgs bundles. Rev Mat Complut 30, 91–128 (2017). https://doi.org/10.1007/s13163-016-0207-0

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Keywords

  • Higgs bundles
  • Quiver bundles
  • Indefinite unitary group
  • Birationality of moduli

Mathematics Subject Classification

  • Primary 14D20
  • Secondary 14H60
  • 53C07