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Stratifications on the nilpotent cone of the moduli space of Hitchin pairs

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Abstract

We consider the problem of finding the limit at infinity (corresponding to the downward Morse flow) of a Higgs bundle in the nilpotent cone under the natural \(\mathbb {C}^*\)-action on the moduli space. For general rank we provide an answer for Higgs bundles with regular nilpotent Higgs field, while in rank three we give the complete answer. Our results show that the limit can be described in terms of data defined by the Higgs field, via a filtration of the underlying vector bundle.

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Notes

  1. See Atiyah & Bott [1, Sect. 14] for general holomorphic bundles, Hitchin [10, Sect. 3] for Higgs bundles and also Hausel & Thaddeus [9, Sect. 8] for Hitchin pairs.

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Acknowledgements

The authors are members of the Vector Bundles and Algebraic Curves (vbac) research group.

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Authors and Affiliations

  1. Centro de Matemática da Universidade do Porto CMUP and Departamento de Matemática, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre, s/n, 4197-007, Porto, Portugal

    Peter B. Gothen

  2. Centro de Investigaciones Matemáticas y Metamatemáticas CIMM y, Centro de Investigación en Matemática Pura y Aplicada CIMPA, Escuela de Matemática EMat, Universidad de Costa Rica UCR, Calle Novillos, s/n, San José, 11501, Costa Rica

    Ronald A. Zúñiga-Rojas

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  1. Peter B. Gothen
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Correspondence to Ronald A. Zúñiga-Rojas.

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Peter B. Gothen: partially supported by Centro de Matemática da Universidade do Porto (CMUP), financed by national funds through FCT—Fundação para a Ciência e a Tecnologia, I.P., under the Project UIDB/00144/2020.

Ronald A. Zúñiga-Rojas: supported by Universidad de Costa Rica through Escuela de Matemática, specifically through CIMM and CIMPA, under Projects 820-B5-202, 820-B8-224 and 821-C1-010; and partially supported by FCT (Portugal) through grant SFRH/BD/51174/2010.

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Gothen, P.B., Zúñiga-Rojas, R.A. Stratifications on the nilpotent cone of the moduli space of Hitchin pairs. Rev Mat Complut 35, 311–321 (2022). https://doi.org/10.1007/s13163-021-00400-3

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