Symplectic torus actions with non-contractible orbits

Abstract

We prove that a symplectic $T^{n-1}$ action on a closed connected $2n$-dimensional symplectic manifold is Hamiltonian if and only if its orbits are contractible. This generalizes a result of Lalonde--McDuff--Polterovich on four-manifolds and theorems of McDuff and Kim on existence of fixed points. When the orbits are isotropic, we prove a stronger variant of this result, which implies non-extendability of certain symplectic circle actions on 6-manifolds to symplectic $T^2$ actions. Moreover, we prove that a symplectic $T^{n-1}$ action with isotropic orbits always splits into a maximal Hamiltonian action and a locally-free action. We end by posing several open questions on the topology of symplectic torus actions.

39 pages, comments are welcome!