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Logahoric Higgs torsors for a complex reductive group

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Abstract

In this article, a logahoric Higgs torsor is defined as a parahoric torsor with a logarithmic Higgs field. For a connected complex reductive group G, we introduce a notion of stability for logahoric \(\mathcal {G}_{\varvec{\theta }}\)-Higgs torsors on a smooth algebraic curve X, where \(\mathcal {G}_{\varvec{\theta }}\) is a parahoric group scheme on X. In the case when the group G is the general linear group \(\textrm{GL}_n\), we show that the stability condition of a parahoric torsor is equivalent to the stability of a parabolic bundle. A correspondence between semistable logahoric \(\mathcal {G}_{\varvec{\theta }}\)-Higgs torsors and semistable equivariant logarithmic G-Higgs bundles allows us to construct the moduli space explicitly. This moduli space is shown to be equipped with an algebraic Poisson structure.

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Acknowledgements

We are the most grateful to Philip Boalch for a series of constructive queries on a first draft of this article which have improved its content. We would also like to thank Pengfei Huang for his interest in this project and for useful discussions, as well as an anonymous referee for a careful reading of the manuscript and important remarks. G. Kydonakis is much obliged to the Max-Planck-Institut für Mathematik in Bonn for its hospitality and support during the production of this article. H. Sun is partially supported by National Key R &D Program of China (No. 2022YFA1006600) and NSFC (No. 12101243).

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Authors and Affiliations

  1. Alexander von Humboldt-Stiftung et Universität Heidelberg, Mathematisches Institut Universität Heidelberg, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany

    Georgios Kydonakis

  2. Department of Mathematics, South China University of Technology, 381 Wushan Rd, Tianhe Qu, Guangzhou, Guangdong, China

    Hao Sun

  3. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green St, Urbana, IL, 61801, USA

    Lutian Zhao

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  1. Georgios Kydonakis
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Correspondence to Hao Sun.

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To the memory of Professor M. S. Narasimhan

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Kydonakis, G., Sun, H. & Zhao, L. Logahoric Higgs torsors for a complex reductive group. Math. Ann. 388, 3183–3228 (2024). https://doi.org/10.1007/s00208-023-02605-x

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