Delzant's theorem

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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.


  • 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 
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