Delzant's theorem
In mathematics, a Delzant polytope is a convex polytope in [math]\displaystyle{ \mathbb{R}^n }[/math] such for each vertex [math]\displaystyle{ v }[/math], exactly [math]\displaystyle{ n }[/math] edges meet at [math]\displaystyle{ v }[/math], and these edges form a collection of vectors that form a [math]\displaystyle{ \mathbb{Z} }[/math]-basis of [math]\displaystyle{ \mathbb{Z}^n }[/math]. Delzant's theorem, introduced by Thomas Delzant (1988), classifies effective Hamiltonian torus actions on compact connected symplectic manifolds by the image of the associated moment map, which is a Delzant polytope.
The theorem states that there is a bijective correspondence between symplectic toric manifolds (up to torus-equivariant symplectomorphism) and Delzant polytopes -- more precisely, the moment polytope of a symplectic toric manifold is a Delzant polytope, every Delzant polytope is the moment polytope of such a manifold, and any two such manifolds with the equivalent moment polytopes (up to translations) admit a torus-equivariant symplectomorphism between them.
References
- Delzant, Thomas (1988), "Hamiltoniens périodiques et images convexes de l'application moment", Bulletin de la Société Mathématique de France 116 (3): 315–339, doi:10.24033/bsmf.2100, ISSN 0037-9484, http://www.numdam.org/item?id=BSMF_1988__116_3_315_0
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