Trivializations of moment maps
Annales de l'Institut Fourier, Volume 74 (2024) no. 1, pp. 307-347.

We study various trivializations of moment maps. First in the general framework of a complex reductive group G acting on a smooth affine variety. We prove that the moment map is a locally trivial fibration over a regular locus of the center of the Lie algebra of H a maximal compact subgroup of G. The construction relies on Kempf–Ness theory and Morse theory of the square norm of the moment map studied by Kirwan, Ness–Mumford and Sjamaar. Then we apply it together with ideas from Nakajima and Kronheimer to trivialize the hyperkähler moment map for Nakajima quiver varieties. Notice this trivialization result about quiver varieties was known and used by experts such as Nakajima and Maffei but we could not locate a proof in the literature.

Plusieurs trivialisations d’applications moment sont étudiées. Tout d’abord dans le cadre général de l’action d’un groupe réductif sur une variété affine lisse. Nous prouvons que l’application moment est une fibration localement triviale au dessus d’un lieu régulier du centre de l’algèbre de Lie d’un sous-groupe compact maximal. Les constructions reposent sur la théorie de Kempf–Ness et sur la théorie de Morse du carré de la norme de l’application moment étudiée par Kirwan, Ness–Mumford et Sjamaar. Ces constructions sont ensuite appliquées avec des idées de Nakajima et de Kronheimer pour trivialiser l’application moment hyperkähler pour les variétés de carquois de Nakajima. Ce résultat concernant les variétés de carquois était connu et utilisé par des experts comme Nakajima et Maffei mais nous n’avons pu trouver de preuve dans la littérature.

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Published online:
DOI: 10.5802/aif.3587
Classification: 53C26, 53D20, 16G20
Keywords: moment map, hyperkähler moment map, Nakajima quiver varieties
Mot clés : application moment, application moment hyperkähler, variétés de carquois de Nakajima
Ballandras, Mathieu 1

1 C. Nicolás Cabrera, 13-15 Campus de Cantoblanco, UAM 28049 Madrid (Spain)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Ballandras, Mathieu. Trivializations of moment maps. Annales de l'Institut Fourier, Volume 74 (2024) no. 1, pp. 307-347. doi : 10.5802/aif.3587.

[1] Atiyah, Michael F.; Hitchin, Nigel J.; Singer, Isadore M. Self-Duality in Four-Dimensional Riemannian Geometry, Proc. R. Soc. Lond., Ser. A, Volume 362 (1978) no. 1711, pp. 425-461 | MR | Zbl

[2] Crawley-Boevey, William Quiver algebras, weighted projective lines, and the Deligne–Simpson problem, Proceedings of the international congress of mathematicians (Madrid, 2006), Volume 2, European Mathematical Society, 2006

[3] Crawley-Boevey, William; Van den Bergh, Michel Absolutely indecomposable representations and Kac-Moody Lie algebras, Invent. Math., Volume 155 (2004) no. 3, pp. 537-559 | DOI | MR | Zbl

[4] Georgoulas, Valentina; Robbin, Joel; Salamon, Dietmar The moment-weight inequality and the Hilbert–Mumford criterion, 2013 (

[5] Ginzburg, Victor; Kaledin, Dmitry Poisson deformations of symplectic quotient singularities, Adv. Math., Volume 186 (2004) no. 1, pp. 1-57 | DOI | MR | Zbl

[6] Guillemin, Victor; Sternberg, Shlomo Convexity Properties of the Moment Mapping, Invent. Math., Volume 67 (1982), pp. 491-514 | DOI | MR

[7] Harada, Megumi; Wilkin, Graeme Morse theory of the moment map for representations of quivers, Geom. Dedicata, Volume 150 (2008), pp. 307-353 | DOI | MR | Zbl

[8] Hitchin, Nigel J. Hyperkähler manifolds, Séminaire Bourbaki, volume 1991/92 (Astérisque), Volume 34, Société Mathématique de France, 1991–1992, pp. 137-166 | Zbl

[9] Hitchin, Nigel J.; Karlhede, Anders; Lindström, Ulf; Roček, Martin HyperKähler Metrics and Supersymmetry, Commun. Math. Phys., Volume 108 (1987), pp. 535-589 | DOI | Zbl

[10] Hoskins, Victoria Stratifications associated to reductive group actions on affine spaces, Q. J. Math., Volume 65 (2013) no. 3, pp. 1011-1047 | DOI | MR | Zbl

[11] Kac, Victor G.; Peterson, Dale H. Unitary structure in representations of infinite-dimensional groups and a convexity theorem, Invent. Math., Volume 76 (1984), pp. 1-14 | MR

[12] Kempf, George; Ness, Linda The length of vectors in representation spaces, Algebraic Geometry (Lønsted, Knud, ed.), Springer (1979), pp. 233-243 | DOI | Zbl

[13] King, Alastair D. Moduli of representations of finite dimensional algebras, Q. J. Math., Volume 45 (1994) no. 4, pp. 515-530 | DOI | MR | Zbl

[14] Kirwan, Frances Clare Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes, 31, Princeton University Press, 1984 | MR

[15] Kronheimer, Peter B. The construction of ALE spaces as hyper-Kähler quotients, J. Differ. Geom., Volume 29 (1989) no. 3, pp. 665-683 | DOI | Zbl

[16] Maffei, Andrea A remark on quiver varieties and Weyl groups, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 1 (2002) no. 3, pp. 649-686 | Numdam | MR | Zbl

[17] Marsden, Jerrold; Weinstein, Alan Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., Volume 5 (1974) no. 1, pp. 121-130 | DOI | MR | Zbl

[18] Meyer, Kenneth R. Symmetries and integrals in mechanics, Dynamical systems (Bahia, 1971), Academic Press New York, 1973, pp. 259-273 | Zbl

[19] Migliorini, Luca Stability of homogeneous vector bundles, Boll. Unione Mat. Ital., VII. Ser., B, Volume 10 (1996) no. 4, pp. 963-990 | MR | Zbl

[20] Mumford, David; Fogarty, John Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 34, Springer, 1982 | DOI

[21] Nakajima, Hiraku Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J., Volume 76 (1994) no. 2, pp. 365-416 | DOI | MR | Zbl

[22] Ness, Linda; Mumford, David A Stratification of the Null Cone Via the Moment Map, Am. J. Math., Volume 106 (1984) no. 6, pp. 1281-1329 | DOI | MR | Zbl

[23] Onishchik, Arkadiĭ L.; Vinberg, Èrnest B.; Gorbatsevich, Vladimir V Lie groups and Lie algebras III. Structure of Lie groups and Lie algebras. Transl. from the Russian by V. Minachin, Encyclopaedia of Mathematical Sciences, 41, Springer, 1994

[24] Penrose, Roger Nonlinear Gravitons and Curved Twistor Theory, Gen. Relativ. Gravitation, Volume 7 (1976), pp. 31-52 | DOI | MR | Zbl

[25] Salamon, Simon Quaternionic Kähler Manifolds., Invent. Math., Volume 67 (1982), pp. 143-172 | DOI | Zbl

[26] Salamon, Simon Differential geometry of quaternionic manifolds, Ann. Sci. Éc. Norm. Supér., Volume 19 (1986) no. 1, pp. 31-55 | DOI | Numdam | MR | Zbl

[27] Sjamaar, Reyer Convexity Properties of the Moment Mapping Re-examined, Adv. Math., Volume 138 (1998) no. 1, pp. 46-91 | DOI | MR | Zbl

[28] Slodowy, Peter Four lectures on simple groups and singularities, Communications of the Mathematical Institute, Rijksuniversiteit Utrecht, 11, 1980, 64 pages | DOI | MR

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