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Stable Higgs bundles on ruled surfaces

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Abstract

Let π : X = ℙC(E) → C be a ruled surface over an algebraically closed field k of characteristic 0, with a fixed polarization L on X. In this paper, we show that pullback of a (semi)stable Higgs bundle on C under π is a L-(semi)stable Higgs bundle. Conversely, if (V, θ) is a L-(semi)stable Higgs bundle on X with c1(V) = π* (d) for some divisor d of degree d on C and c2(V) = 0, then there exists a (semi)stable Higgs bundle (W, ψ) of degree d on C whose pullback under π is isomorphic to (V, θ). As a consequence, we get an isomorphism between the corresponding moduli spaces of (semi)stable Higgs bundles. We also show the existence of non-trivial stable Higgs bundle on X whenever g(C) ≥ 2 and the base field is ℂ.

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Acknowledgement

I would like to thank my advisor Prof. D. S. Nagaraj, IMSc Chennai for his constant guidance at every stage of this work. I would also like to thank Dr. Rohith Varma, IMSc Chennai for many useful discussions. This work is supported financially by a fellowship from IMSc, Chennai (HBNI), DAE, Government of India.

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  1. The Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai, 600 113, India

    Snehajit Misra

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  1. Snehajit Misra
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Misra, S. Stable Higgs bundles on ruled surfaces. Indian J Pure Appl Math 51, 735–747 (2020). https://doi.org/10.1007/s13226-020-0427-3

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  • DOI: https://doi.org/10.1007/s13226-020-0427-3

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