Symplectic torus actions with coisotropic principal orbits

[Actions symplectiques toriques avec des orbites principales coïsotropes]

Duistermaat, Johannes Jisse ; Pelayo, Alvaro

Annales de l'Institut Fourier, Tome 57 (2007), p. 2239-2327 / Harvested from Numdam

Résumé

Dans cet article nous donnons une classification complète des actions symplectiques d’un tore $T$ sur une variété compacte connexe symplectique $(M,\sigma )$ pour laquelle une, et donc toute orbite principale est une variété coïsotrope de $(M,\sigma )$. 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 $M/T$. En utilisant une généralisation du théorème de Tietze-Nakayama à ce que nous appelons aussi espaces $V$-parallèles, nous obtenons que $M/T$ est isomorphe au produit cartésien d’un polytope de Delzant avec un tore.

Nous construisons alors les champs de vecteurs spéciaux dans $M$ qui se projettent sur les champs de vecteurs constants sur $M/T$, à l’aide desquels le modèle de la variété symplectique avec action torique est défini.

Abstract

In this paper we completely classify symplectic actions of a torus $T$ on a compact connected symplectic manifold $(M,\sigma )$ when some, hence every, principal orbit is a coisotropic submanifold of $(M,\sigma )$. 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 $M/T$. Using a generalization of the Tietze-Nakajima theorem to what we call $V$-parallel spaces, we obtain that $M/T$ is isomorphic to the Cartesian product of a Delzant polytope with a torus.

We then construct special lifts of the constant vector fields on $M/T$, in terms of which the model of the symplectic manifold with the torus action is defined.

Détail

Publié le : 2007-01-01

MR 2394542 | Zbl pre05249486 | Zbl 1197.53114

DOI : https://doi.org/10.5802/aif.2333
Classification:  53D35,  35J05,  35J10,  17B30,  22E25
Mots clés: symplectique, actions toriques, orbites coïsotropes, classification

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