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In mathematics, Delzant's theorem, introduced by Thomas Delzant (1988), classifies 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.
References

Delzant, Thomas (1988), "Hamiltoniens périodiques et images convexes de l'application moment", Bulletin de la Société Mathématique de France 116 (3): 315–339, ISSN 0037-9484, MR 984900

Mathematics Encyclopedia