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Fixed Points of Automorphisms of the Vector Bundle Moduli Space Over a Compact Riemann Surface

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Abstract

Let X be a compact Riemann surface of genus \(g\ge 2\) and let \(n\ge 3\) be an integer number. The group of automorphisms of the moduli space of vector bundles over X with rank n and trivial determinant is isomorphic to \(H^1(X,{\mathbb {Z}}/(n))\rtimes ({{\,\textrm{Out}\,}}({{\,\textrm{SL}\,}}(n,{\mathbb {C}}))\times {{\,\textrm{Aut}\,}}(X))\). Several papers have studied the subvarieties of fixed points for the action of the unique outer involution of \({{\,\textrm{SL}\,}}(n,{\mathbb {C}})\) on this moduli space. In this paper, explicit descriptions of the fixed points for the actions of the elements of \(H^1(X,{\mathbb {Z}}/(n))\), \(H^1(X,{\mathbb {Z}}/(n))\rtimes {{\,\textrm{Out}\,}}({{\,\textrm{SL}\,}}(n,{\mathbb {C}}))\), and \({{\,\textrm{Out}\,}}({{\,\textrm{SL}\,}}(n,{\mathbb {C}}))\times {{\,\textrm{Aut}\,}}(X)\) on the moduli space of rank n and trivial determinant vector bundles over X are provided. For the description of the fixed points for the action of the elements of \({{\,\textrm{Out}\,}}({{\,\textrm{SL}\,}}(n,{\mathbb {C}}))\times {{\,\textrm{Aut}\,}}(X)\), the notion of Galois bundle is introduced. Specifically, Galois bundles over X admitting a nontrivial automorphism which commutes with the Galois structure are constructed associated with an involution \(\sigma _X\) of X. Finally, it is discussed how the description of fixed points for the action of the elements of is covered by the descriptions above.

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  1. Department of Mathematics and Experimental Science, Fray Luis de León University College of Education, Catholic University of Ávila, C/Tirso de Molina, 44, 47010, Valladolid, Spain

    Álvaro Antón-Sancho

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Antón-Sancho, Á. Fixed Points of Automorphisms of the Vector Bundle Moduli Space Over a Compact Riemann Surface. Mediterr. J. Math. 21, 20 (2024). https://doi.org/10.1007/s00009-023-02559-z

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