Very stable bundles and properness of the Hitchin map

Abstract

Let X be a smooth complex projective curve of genus \(g\ge 2\) and let K be its canonical bundle. In this note we show that a stable vector bundle E on X is very stable, i.e. E has no non-zero nilpotent Higgs field, if and only if the restriction of the Hitchin map to the vector space of Higgs fields \(H^0(X, \mathrm {End}(E) \otimes K)\) is a proper map.

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Correspondence to Ana Peón-Nieto.

Additional information

The second author was supported by a post-doctoral Grant associated to the Marie Curie Project GEOMODULI of the Programme FP7/PEOPLE/2013/CIG, Project Number 618471.

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Pauly, C., Peón-Nieto, A. Very stable bundles and properness of the Hitchin map. Geom Dedicata 198, 143–148 (2019). https://doi.org/10.1007/s10711-018-0333-6

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Keywords

  • Higgs bundles
  • Hitchin map
  • Moduli space
  • Vector bundle

Mathematics Subject Classification (2000)

  • Primary 14H60
  • 14H70