Maximal symplectic torus actions

TL;DR

This paper classifies maximal symplectic torus actions on smooth manifolds, providing new symplectic analogues of existing classifications and answering a question about their structure.

Contribution

It offers classifications of isotropy-maximal and almost isotropy-maximal symplectic torus actions up to equivariant symplectomorphism, extending previous work to new contexts.

Findings

Classified isotropy-maximal symplectic torus actions.

Proved almost isotropy-maximal actions are products of symplectic toric manifolds and tori.

Extended classification results to symplectic context using Duistermaat-Pelayo's work.

Abstract

There are several different notions of maximal torus actions on smooth manifolds, in various contexts: symplectic, Riemannian, complex. In the symplectic context, for the so-called isotropy-maximal actions, as well as for the weaker notion of almost isotropy-maximal actions, we give classifications up to equivariant symplectomorphism. These classification results give symplectic analogues of recent classifications in the complex and Riemannian contexts. Moreover, we deduce that every almost isotropy-maximal symplectic torus action is equivariantly diffeomorphic to a product of a symplectic toric manifold and a torus, answering a question of Ishida. The classification theorems are consequences of Duistermaat and Pelayo's classification of symplectic torus actions with coisotropic orbits.