Geometriae Dedicata

, Volume 193, Issue 1, pp 145–154 | Cite as

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

  • Indranil BiswasEmail author
  • Steven Rayan
Original Paper


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 \(\mathcal {E}\).


Co-Higgs bundle Integrability Higgs bundle Moduli space Poincaré bundle 

Mathematics Subject Classification (2010)

14H60 14D20 14D21 



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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