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Hypertoric Manifolds and HyperKähler Moment Maps

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Special Metrics and Group Actions in Geometry

Part of the book series: Springer INdAM Series ((SINDAMS,volume 23))

Abstract

We discuss various aspects of moment map geometry in symplectic and hyperKähler geometry. In particular, we classify complete hyperKähler manifolds of dimension \(4n\) with a tri-Hamiltonian action of a torus of dimension \(n\), without any assumption on the finiteness of the Betti numbers. As a result we find that the hyperKähler moment in these cases has connected fibres, a property that is true for symplectic moment maps, and is surjective. New examples of hypertoric manifolds of infinite topological type are produced. We provide examples of non-Abelian tri-Hamiltonian group actions of connected groups on complete hyperKähler manifolds such that the hyperKähler moment map is not surjective and has some fibres that are not connected. We also discuss relationships to symplectic cuts, hyperKähler modifications and implosion constructions.

To Simon Salamon on the occasion of his 60th birthday

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Acknowledgements

Andrew Swann partially supported by the Danish Council for Independent Research, Natural Sciences. We thank Sue Tolman for discussions about the disconnected fibres of hyperKähler moment maps.

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Authors and Affiliations

  1. Jesus College, Oxford, OX1 3DW, UK

    Andrew Dancer

  2. Department of Mathematics, Aarhus University, Ny Munkegade 118, Bldg 1530, DK-8000, Aarhus C, Denmark

    Andrew Swann

Authors
  1. Andrew Dancer
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  2. Andrew Swann
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Correspondence to Andrew Swann .

Editor information

Editors and Affiliations

  1. GMA - IME, Universidade Federal Fluminense GMA - IME, Niteroi, Rio de Janeiro, Brazil

    Simon G. Chiossi

  2. Dipartimento di Mathematica "G. Peano", Università di Torino, Torino, Italy

    Anna Fino

  3. Dipartimento di Matematica e Informatica, Università di Firenze, Firenze, Firenze, Italy

    Emilio Musso

  4. Dipartimento di Scienze Matematiche, Politecnico di Torino, Torino, Italy

    Fabio Podestà

  5. Dipart. di Matematica, Giuseppe Peano, Università di Torino, Torino, Italy

    Luigi Vezzoni

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Dancer, A., Swann, A. (2017). Hypertoric Manifolds and HyperKähler Moment Maps. In: Chiossi, S., Fino, A., Musso, E., Podestà, F., Vezzoni, L. (eds) Special Metrics and Group Actions in Geometry. Springer INdAM Series, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-319-67519-0_5

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