Delzant's theorem

In mathematics, Delzant's theorem, introduced by Thomas Delzant (1988), classifies effective Hamiltonian actions of a torus on a compact connected symplectic manifold of twice the dimension by their image under the momentum mapping (Delzant polytope). A Delzant polytope is a convex polytope in Rⁿ such that the slopes of the edges of each vertex are given by a basis of Zⁿ.

As a corollary, these symplectic manifolds have a complex structure and can be promoted as toric varieties, with invariant Kähler structures.