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Implosion for hyperkähler manifolds

Part of: Global differential geometry Symplectic geometry, contact geometry Algebraic groups

Published online by Cambridge University Press:  28 June 2013

Andrew Dancer ,
Frances Kirwan  and
Andrew Swann
Andrew Dancer
Affiliation:
Jesus College, Oxford, OX1 3DW, UK email dancer@maths.ox.ac.uk
Frances Kirwan
Affiliation:
Balliol College, Oxford, OX1 3BJ, UK email kirwan@maths.ox.ac.uk
Andrew Swann
Affiliation:
Department of Mathematics, Aarhus University, Ny Munkegade 118, Bldg 1530, DK-8000 Aarhus C, Denmark email swann@imf.au.dk CP3-Origins, Centre of Excellence for Cosmology and Particle Physics Phenomenology, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark email swann@imf.au.dk
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Abstract

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We introduce an analogue in hyperkähler geometry of the symplectic implosion, in the case of $\mathrm{SU} (n)$ actions. Our space is a stratified hyperkähler space which can be defined in terms of quiver diagrams. It also has a description as a non-reductive geometric invariant theory quotient.

Keywords

symplectic implosionquiver diagramhyperkähler quotientnon-reductive geometric invariant theoryKostant variety

MSC classification

Secondary: 53C26: Hyper-Kähler and quaternionic Kähler geometry, “special” geometry 53D20: Momentum maps; symplectic reduction 14L24: Geometric invariant theory

Information

Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 9 , September 2013 , pp. 1592 - 1630
Copyright
© The Author(s) 2013 

References

Bielawski, R., Asymptotic metrics for $\mathit{SU}(N)$ monopoles with maximal symmetry breaking, Comm. Math. Phys. 199 (1998), 297–325.Google Scholar
Bielawski, R., Monopoles and the Gibbons-Manton metric, Comm. Math. Phys. 194 (1998), 297–321.Google Scholar
Biquard, O., Sur les équations de Nahm et la structure de Poisson des algèbres de Lie semi-simples complexes, Math. Ann. 304 (1996), 253–276.Google Scholar
Chriss, N. and Ginzburg, V., Representation theory and complex geometry (Birkhäuser, Boston, MA, 1997).Google Scholar
Dancer, A. and Swann, A., Hyperkähler metrics associated to compact Lie groups, Math. Proc. Cambridge Philos. Soc. 120 (1996), 61–69.CrossRefGoogle Scholar
Dancer, A. and Swann, A., The geometry of singular quaternionic Kähler quotients, Internat. J. Math. 8 (1997), 595–610.CrossRefGoogle Scholar
Dancer, A. and Swann, A., Modifying hyperkähler manifolds with circle symmetry, Asian J. Math. 10 (2006), 815–826.CrossRefGoogle Scholar
Doran, B. and Kirwan, F., Towards non-reductive geometric invariant theory, Pure Appl. Math. Q. 3 (2007), 61–105.Google Scholar
Feix, B., Hyperkähler metrics on cotangent bundles, J. Reine Angew. Math. 532 (2001), 33–46.Google Scholar
Guillemin, V., Jeffrey, L. and Sjamaar, R., Symplectic implosion, Transform. Groups 7 (2002), 155–184.CrossRefGoogle Scholar
Kempf, G. and Ness, L., The length of vectors in representation spaces, in Algebraic geometry (Proc. summer meeting, Copenhagen, 1978), Lecture Notes in Mathmatics, vol. 732 (Springer, Berlin, 1979), 233–243.CrossRefGoogle Scholar
Kirwan, F., Cohomology of quotients in symplectic and algebraic geometry (Princeton University Press, Princeton, NJ, 1985).Google Scholar
Kobak, P. and Swann, A., Classical nilpotent orbits as hyperkähler quotients, Internat. J. Math. 7 (1996), 193–210.CrossRefGoogle Scholar
Kostant, B., Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.Google Scholar
Kovalev, A., Nahm’s equations and complex adjoint orbits, Q. J. Math. 47 (1996), 41–58.CrossRefGoogle Scholar
Kraft, H. and Procesi, C., Minimal singularities in $G{L}_{n} $, Invent. Math. 62 (1981), 503–515.Google Scholar
Kronheimer, P. B., A hyperkähler structure on the cotangent bundle of a compact Lie group, MSRI preprint (1986), eprint version arXiv:math/0409253 [math.DG].Google Scholar
Kronheimer, P. B., Instantons and the geometry of the nilpotent variety, J. Differential Geom. 32 (1990), 473–490.Google Scholar
Kronheimer, P. B., A hyper-kählerian structure on coadjoint orbits of a semisimple complex group, J. Lond. Math. Soc. (2) 42 (1990), 193–208.Google Scholar
Mumford, D., Fogarty, J. and Kirwan, F., Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, vol. 34 (Springer, Berlin, 1994).Google Scholar
Nakajima, H., Instantons on ALE spaces, quiver varieties and Kac-Moody algebras, Duke Math. J. 76 (1994), 365–416.Google Scholar
Sjamaar, R. and Lerman, E., Stratified symplectic spaces and reduction, Ann. of Math. (2) 134 (1991), 375–422.CrossRefGoogle Scholar
Thomas, R. P. W., Notes on GIT and symplectic reduction for bundles and varieties, Surveys in Differential Geometry, vol. 10 (International Press, Somerville, MA, 2006), 221–273.Google Scholar