Symplectic actions of non-Hamiltonian type

Hamiltonian symplectic actions of tori on compact symplectic manifolds have been extensively studied in the past thirty years, and a number of classifications have been achieved, for instance in the case that the acting torus is $n$-dimensional and the symplectic manifold is $2n$-dimensional. In this case the $n$-dimensional orbits are Lagrangian, so it is natural to wonder whether there are interesting classes of symplectic actions with Lagrangian orbits, and that are not Hamiltonian. It turns out that there are many such classes which contain for example the Kodaira variety, and which can be classified in terms of symplectic invariants. The paper reviews several classifications, which include symplectic actions having a Lagrangian orbit or a symplectic orbit of maximal dimension. We make an emphasis on the construction of the symplectic invariants, and their computation in examples.

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