Symplectic actions of 2-tori on 4-manifolds

We classify symplectic actions of 2-tori on compact, connected symplectic 4-manifolds, up to equivariant symplectomorphisms, hence extending the theory of Atiyah, Guillemin–Sternberg, Delzant and Benoist to actions of tori which are not necessarily Hamiltonian. The classification is in terms of a collection of up to nine invariants. We construct an explicit model of such symplectic manifolds with torus actions, defined in terms of these invariants, which are invariants of the topology of the manifold, of the torus action or of the symplectic form. To obtain this classification we combine item i) below with the classification of symplectic torus actions with coisotropic principal orbits in previous work by J. J. Duistermaat and the author. We also provide additional classifications in arbitrary dimensions: i) we classify, up to equivariant symplectomorphisms, symplectic actions of (2n − 2)-dimensional tori on 2ndimensional symplectic manifolds, when at least one orbit is a (2n − 2)-dimensional symplectic submanifold. ii) we classify, up to equivariant diffeomorphisms, free symplectic actions of (2n − 2)-dimensional tori on 2n-dimensional symplectic manifolds, when at least one orbit is a (2n − 2)-dimensional symplectic submanifold, generalizing a theorem of Khan.