Convexity and toric classification for Hamiltonian torus actions

Several major theorems about Hamiltonian torus actions lie beyond the present page and are not asserted here as consequences of regular reduction:

• the Atiyah--Guillemin--Sternberg convexity theorem, that the image of the moment map of a Hamiltonian torus action on a compact connected symplectic manifold is a convex polytope, and that its fibres are connected;
• Delzant's classification of symplectic toric manifolds by their moment polytopes;
• localization formulas for Hamiltonian torus actions and the associated fixed-point and residue theory;
• equivariant cohomology, its relation to the cohomology of the reduced spaces and the Kirwan surjectivity programme.

Nothing on this page states, uses or presupposes these results. They are recorded here only to mark the boundary of the selected scope and to identify the directions in which the material of this pair is developed later: the finite-dimensional moment-map algebra, the cotangent and coadjoint models and regular symplectic reduction. In particular the examples on the companion page compute moment maps for specific circle and torus actions without classifying their images as polytopes or their reduced spaces as toric varieties.