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

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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 \(\mathcal {E}\).

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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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Authors and Affiliations

  1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay, 400005, India

    Indranil Biswas

  2. Department of Mathematics and Statistics, University of Saskatchewan, McLean Hall, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada

    Steven Rayan

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  1. Indranil Biswas
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  2. Steven Rayan
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Correspondence to Indranil Biswas.

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Biswas, I., Rayan, S. A vanishing theorem for co-Higgs bundles on the moduli space of bundles. Geom Dedicata 193, 145–154 (2018). https://doi.org/10.1007/s10711-017-0259-4

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  • DOI: https://doi.org/10.1007/s10711-017-0259-4

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