Symplectic torus actions with non-contractible orbits

2026/07/23 by Rei Henigman, Yael Karshon · 1 voice
#math.SG #math.DG

paper · pdf

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 T² 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.

Discussions

• Rei Henigman, Yael Karshon: Symplectic torus actions with non-contractible orbits https://arxiv.org/abs/2607.21159 https://arxiv.org/pdf/2607.21159 https://arxiv.org/html/2607.21159 [bsky, 0 points, 0 comments]

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