A vanishing theorem for co-Higgs bundles on the moduli space of bundles

Original Paper

Abstract

We consider smooth moduli spaces of semistable vector bundles of fixed rank and determinant on a compact Riemann surface X of genus at least 3. The choice of a Poincaré bundle for such a moduli space M induces an isomorphism between X and a component of the moduli space of semistable sheaves over M. We prove that \(\dim H^0(M,\, \text {End}({\mathcal {E}})\otimes TM)\,=\, 1\) for any vector bundle \(\mathcal {E}\) on M coming from this component. Furthermore, there are no nonzero integrable co-Higgs fields on \(\mathcal {E}\).

Keywords

Co-Higgs bundle Integrability Higgs bundle Moduli space Poincaré bundle 

Mathematics Subject Classification (2010)

14H60 14D20 14D21 

Notes

Acknowledgements

We thank the referee for helpful comments. The first author acknowledges support of a J. C. Bose Fellowship. The second author acknowledges the support of a New Faculty Recruitment Grant from the University of Saskatchewan.

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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.School of MathematicsTata Institute of Fundamental ResearchBombayIndia
  2. 2.Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoonCanada

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