Maximal symplectic torus actions
Rei Henigman
TL;DR
The paper classifies almost isotropy-maximal symplectic torus actions up to equivariant symplectomorphism, proving they are equivariantly diffeomorphic to a product of a symplectic toric manifold and a torus, thereby resolving Ishida's question in this broader setting. The key tool is the Duistermaat–Pelayo framework for symplectic torus actions with coisotropic principal orbits, with almost isotropy-maximal actions shown to have coisotropic orbits and governed by five invariants (the fifth invariant vanishes). The isotropy-maximal case is a special instance where the invariants reduce to the bilinear form $\sigma$ and the Delzant polytope $\Delta$, enabling a clean product construction and a simple equivalence criterion. The results illuminate the landscape of maximal symplectic torus actions, describe when actions can extend to higher-dimensional tori, and provide concrete structural decompositions for a broad class of symplectic manifolds.
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.