Hypertoric Manifolds and HyperKähler Moment Maps

Andrew Francis Swann, Andrew Dancer

Research output: Contribution to book/anthology/report/proceedingBook chapterResearchpeer-review

Abstract

We discuss various aspects of moment map geometry in symplectic and hyperKähler geometry. In particular, we classify complete hyperKähler manifolds of dimension 4 n 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.

Original languageEnglish
Title of host publicationSpecial Metrics and Group Actions in Geometry
EditorsS. Chiossi, A. Fino, E. Musso, F. Podestà, L. Vezzoni
Number of pages21
PublisherSpringer
Publication dateNov 2017
Pages107-127
ISBN (Print)978-3-319-67518-3
DOIs
Publication statusPublished - Nov 2017
SeriesSpringer INdAM Series
Volume23

Keywords

  • Complete metric
  • Disconnected fibres
  • HyperKähler manifold
  • Infinite topology
  • Moment map
  • Non-surjectivity
  • Toric manifold

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