Hypertoric Manifolds and HyperKähler Moment Maps

Andrew Francis Swann; Andrew Dancer

doi:10.1007/978-3-319-67519-0_5

Hypertoric Manifolds and HyperKähler Moment Maps

Andrew Francis Swann, Andrew Dancer

Institut for Matematik

Publikation: Bidrag til bog/antologi/rapport/proceeding › Bidrag til bog/antologi › Forskning › peer review

5 Citationer (Scopus)

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.

Originalsprog	Engelsk
Titel	Special Metrics and Group Actions in Geometry
Redaktører	S. Chiossi, A. Fino, E. Musso, F. Podestà, L. Vezzoni
Antal sider	21
Forlag	Springer
Publikationsdato	nov. 2017
Sider	107-127
ISBN (Trykt)	978-3-319-67518-3
DOI	https://doi.org/10.1007/978-3-319-67519-0_5
Status	Udgivet - nov. 2017

Navn	Springer INdAM Series
Vol/bind	23

Adgang til dokumentet

10.1007/978-3-319-67519-0_5

Andre filer og links

Link to publication in Scopus

Torus symmetry and Einstein metrics
Swann, A. F. (Deltager)
Danmarks Frie Forskningsfond
01/11/2016 → 31/12/2019
Projekter: Projekt › Forskning

Citationsformater

@inbook{79f3dac5d682449ca39f1132b7ac1869,

title = "Hypertoric Manifolds and HyperK{\"a}hler Moment Maps",

abstract = "We discuss various aspects of moment map geometry in symplectic and hyperK{\"a}hler geometry. In particular, we classify complete hyperK{\"a}hler 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 hyperK{\"a}hler 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 hyperK{\"a}hler manifolds such that the hyperK{\"a}hler moment map is not surjective and has some fibres that are not connected. We also discuss relationships to symplectic cuts, hyperK{\"a}hler modifications and implosion constructions.",

keywords = "Complete metric, Disconnected fibres, HyperK{\"a}hler manifold, Infinite topology, Moment map, Non-surjectivity, Toric manifold",

author = "Swann, {Andrew Francis} and Andrew Dancer",

year = "2017",

month = nov,

doi = "10.1007/978-3-319-67519-0_5",

language = "English",

isbn = "978-3-319-67518-3",

series = "Springer INdAM Series",

publisher = "Springer",

pages = "107--127",

editor = "S. Chiossi and A. Fino and E. Musso and F. Podest{\`a} and L. Vezzoni",

booktitle = "Special Metrics and Group Actions in Geometry",

address = "Netherlands",

}

Hypertoric Manifolds and HyperKähler Moment Maps. / Swann, Andrew Francis; Dancer, Andrew.
Special Metrics and Group Actions in Geometry. red. / S. Chiossi; A. Fino; E. Musso; F. Podestà; L. Vezzoni. Springer, 2017. s. 107-127 (Springer INdAM Series, Bind 23).

Publikation: Bidrag til bog/antologi/rapport/proceeding › Bidrag til bog/antologi › Forskning › peer review

TY - CHAP

T1 - Hypertoric Manifolds and HyperKähler Moment Maps

AU - Swann, Andrew Francis

AU - Dancer, Andrew

PY - 2017/11

Y1 - 2017/11

N2 - 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.

AB - 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.

KW - Complete metric

KW - Disconnected fibres

KW - HyperKähler manifold

KW - Infinite topology

KW - Moment map

KW - Non-surjectivity

KW - Toric manifold

UR - http://www.scopus.com/inward/record.url?scp=85037830007&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-67519-0_5

DO - 10.1007/978-3-319-67519-0_5

M3 - Book chapter

SN - 978-3-319-67518-3

T3 - Springer INdAM Series

SP - 107

EP - 127

BT - Special Metrics and Group Actions in Geometry

A2 - Chiossi, S.

A2 - Fino, A.

A2 - Musso, E.

A2 - Podestà, F.

A2 - Vezzoni, L.

PB - Springer

ER -

Hypertoric Manifolds and HyperKähler Moment Maps

Abstract

Adgang til dokumentet

Andre filer og links

Fingeraftryk

Projekter

Torus symmetry and Einstein metrics

Citationsformater