Symplectic actions of $2$-tori on $4$-manifolds

In this paper we classify symplectic actions of $2$-tori on compact connected symplectic $4$-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a collection of invariants of the topology of the manifold, of the torus action and of the symplectic form. We construct explicit models of such symplectic manifolds with torus actions, defined in terms of these invariants.

We also classify, up to equivariant symplectomorphisms, symplectic actions of $(2n-2)$-dimensional tori on compact connected $2n$-dimensional symplectic manifolds, when at least one orbit is a $(2n-2)$-dimensional symplectic submanifold. Then we show that a compact connected $2n$-dimensional symplectic manifold $(M, \sigma)$ equipped with a free symplectic action of a $(2n-2)$-dimensional torus with at least one symplectic orbit is equivariantly diffeomorphic to $M/T \times T$ equipped with the translational action of $T$. Thus two such symplectic manifolds are equivariantly diffeomorphic if and only if their orbit spaces are surfaces of the same genus.

The paper also contains a description of symplectic actions of a torus $T$ on compact connected symplectic manifolds with at least one $\operatorname{dim}T$-dimensional symplectic orbit, and where the torus is not necessarily $(2n-2)$-dimensional.