Another way to get toric symplectic manifolds

The above construction relies on the short exact sequence of tori

which via the splitting defines an action of on the symplectic quotient where is the moment map.

Even without a polytope, we can construct symplectic toric varieties in a similar fashion by starting with a torus homomorphism as above.

Similarly to before, consider a torus with a Hamiltonian action on . Now let , let be a torus of lower dimension and consider an injective group morphism

Since we have a canonical identification
we conclude that the above injection can be represented by an matrix with integer entries, corresponding to the maps . The injection requirement says that has full rank .

Via , this defines a Hamiltonian action of on . The moment map of this action decomposes as

where and are the corresponding Lie-algebras, is the moment map of the action of , and denotes the transpose of , which represents the projection that is dual to .

If now is a regular value of the moment map, we can form the symplectic quotient of real dimension . This is in fact a toric symplectic manifold: Analoguous to the construction in section 2, we have an exact sequence of tori

Since the -action on is trivial on the image of , we get a torus action of .