The moment map and line bundles over presymplectic toric manifolds

We apply symplectic methods in studying smooth toric varieties with a closed, invariant 2-form ω that may have degeneracies. Consider the push-forward of Liouville measure by the moment map. We show that it is a “twisted polytope” in t* which is determined by the winding numbers of a map Sn−1 → t* around points in t*. The index of an equivariant, holomorphic line-bundle with curvature ω is a virtual T-representation which can easily be read from this “twisted polytope”.