Symplectic torus actions with coisotropic principal orbits

Johannes Jisse Duistermaat[1]; Alvaro Pelayo[2]

  • [1] Universiteit Utrecht Mathematisch Instituut P.O. Box 80010 3508 TA Utrecht (The Netherlands)
  • [2] Massachusetts Institute of Technology Department of Mathematics 77 Massachusetts Avenue Cambridge, MA 02139–4307 (USA)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 7, page 2239-2327
  • ISSN: 0373-0956

Abstract

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In this paper we completely classify symplectic actions of a torus T on a compact connected symplectic manifold ( M , σ ) when some, hence every, principal orbit is a coisotropic submanifold of ( M , σ ) . 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.

How to cite

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Duistermaat, Johannes Jisse, and Pelayo, Alvaro. "Symplectic torus actions with coisotropic principal orbits." Annales de l’institut Fourier 57.7 (2007): 2239-2327. <http://eudml.org/doc/10297>.

@article{Duistermaat2007,
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.},
affiliation = {Universiteit Utrecht Mathematisch Instituut P.O. Box 80010 3508 TA Utrecht (The Netherlands); Massachusetts Institute of Technology Department of Mathematics 77 Massachusetts Avenue Cambridge, MA 02139–4307 (USA)},
author = {Duistermaat, Johannes Jisse, Pelayo, Alvaro},
journal = {Annales de l’institut Fourier},
keywords = {Symplectic; torus actions; coisotropic orbits; classification; symplectic},
language = {eng},
number = {7},
pages = {2239-2327},
publisher = {Association des Annales de l’institut Fourier},
title = {Symplectic torus actions with coisotropic principal orbits},
url = {http://eudml.org/doc/10297},
volume = {57},
year = {2007},
}

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
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 7
SP - 2239
EP - 2327
AB - 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.
LA - eng
KW - Symplectic; torus actions; coisotropic orbits; classification; symplectic
UR - http://eudml.org/doc/10297
ER -

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