In this paper we completely classify symplectic actions of a torus on a compact connected symplectic manifold when some, hence every, principal orbit is a coisotropic submanifold of . That is, we construct an explicit model, defined in terms of certain invariants, of the manifold, the torus action and the symplectic form. The invariants are invariants of the topology of the manifold, of the torus action, or of the symplectic form.
In order to deal with symplectic actions which are not Hamiltonian, we develop new techniques, extending the theory of Atiyah, Guillemin-Sternberg, Delzant, and Benoist. More specifically, we prove that there is a well-defined notion of constant vector fields on the orbit space . Using a generalization of the Tietze-Nakajima theorem to what we call -parallel spaces, we obtain that is isomorphic to the Cartesian product of a Delzant polytope with a torus.
We then construct special lifts of the constant vector fields on , in terms of which the model of the symplectic manifold with the torus action is defined.
Dans cet article nous donnons une classification complète des actions symplectiques d’un tore sur une variété compacte connexe symplectique pour laquelle une, et donc toute orbite principale est une variété coïsotrope de . Cela veut dire que nous construisons un modèle explicite, défini en termes de certains invariants de la variété, l’action torique et de la forme symplectique.
Pour traiter des actions symplectiques qui ne sont pas hamiltoniennes, nous développons des techniques nouvelles, étendant la théorie d’Atiyah, Guillemin-Sternberg, Delzant et Benoist. En particulier, nous démontrons qu’il y a une notion bien définie de champs de vecteurs constants sur l’espace des orbites . En utilisant une généralisation du théorème de Tietze-Nakayama à ce que nous appelons aussi espaces -parallèles, nous obtenons que est isomorphe au produit cartésien d’un polytope de Delzant avec un tore.
Nous construisons alors les champs de vecteurs spéciaux dans qui se projettent sur les champs de vecteurs constants sur , à l’aide desquels le modèle de la variété symplectique avec action torique est défini.
Keywords: Symplectic, torus actions, coisotropic orbits, classification
Mot clés : symplectique, actions toriques, orbites coïsotropes, classification
Duistermaat, Johannes Jisse 1; Pelayo, Alvaro 2
@article{AIF_2007__57_7_2239_0, author = {Duistermaat, Johannes Jisse and Pelayo, Alvaro}, title = {Symplectic torus actions with coisotropic principal orbits}, journal = {Annales de l'Institut Fourier}, pages = {2239--2327}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {7}, year = {2007}, doi = {10.5802/aif.2333}, zbl = {1197.53114}, mrnumber = {2394542}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2333/} }
TY - JOUR AU - Duistermaat, Johannes Jisse AU - Pelayo, Alvaro TI - Symplectic torus actions with coisotropic principal orbits JO - Annales de l'Institut Fourier PY - 2007 SP - 2239 EP - 2327 VL - 57 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2333/ DO - 10.5802/aif.2333 LA - en ID - AIF_2007__57_7_2239_0 ER -
%0 Journal Article %A Duistermaat, Johannes Jisse %A Pelayo, Alvaro %T Symplectic torus actions with coisotropic principal orbits %J Annales de l'Institut Fourier %D 2007 %P 2239-2327 %V 57 %N 7 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2333/ %R 10.5802/aif.2333 %G en %F AIF_2007__57_7_2239_0
Duistermaat, Johannes Jisse; Pelayo, Alvaro. Symplectic torus actions with coisotropic principal orbits. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2239-2327. doi : 10.5802/aif.2333. https://aif.centre-mersenne.org/articles/10.5802/aif.2333/
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