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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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. https://aif.centre-mersenne.org/articles/10.5802/aif.3587/

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