Delzant's theorem

In mathematics, a Delzant polytope is a convex polytope in $\mathbb {R} ^{n}$ such that for each vertex $v$, exactly $n$ edges meet at $v$ (that is, it is a simple polytope), and there are integer vectors parallel to these edges forming a $\mathbb {Z}$-basis of $\mathbb {Z} ^{n}$.

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 every symplectic toric manifold is a Delzant polytope, every Delzant polytope is the moment polytope of such a manifold, and any two such manifolds with equivalent moment polytopes (up to translations and $GL(n,\mathbb {Z} )$ transformations) 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, MR 0984900