Differential geometry Symplectic geometry Theorem

Delzant's theorem on toric manifolds

Builds on Moment map Symplectic manifold Atiyah–Guillemin–Sternberg convexity theorem Marsden–Weinstein reduction

Statement

A symplectic toric manifold is a compact connected symplectic manifold $(M, \omega)$ of dimension $2n$ with two data. These are an effective action of the torus $T^n = (S^1)^n$ and a moment map $\mu \colon M \to \R^n$ for it. A Delzant polytope in $\R^n$ is a convex polytope $\Delta$ such that at each vertex $v$ :

exactly $n$ edges meet at $v$ ;
the edges at $v$ have the form $v + t u_i$ , $t \ge 0$ , with $u_1, \ldots, u_n \in \Z^n$ ;
these vectors $u_1, \ldots, u_n$ can be chosen to form a basis of the lattice $\Z^n$ .

Then the image $\mu(M)$ of a symplectic toric manifold is a Delzant polytope. The map $(M, \omega, \mu) \mapsto \mu(M)$ is a bijection from symplectic toric manifolds onto Delzant polytopes in $\R^n$ . Here toric manifolds are taken up to $T^n$ -equivariant symplectomorphisms that intertwine the moment maps.

Remarks

That $\mu(M)$ is a convex polytope is the Atiyah–Guillemin–Sternberg convexity theorem; the local normal form at fixed points gives the three conditions. For surjectivity, Delzant builds $M$ from $\Delta$ as a symplectic reduction of $\C^d$ by a subtorus of $T^d$ , where $d$ is the number of facets of $\Delta$ . Injectivity uses the local structure near each orbit and a gluing argument.

Example: the simplex with vertices $0, e_1, \ldots, e_n$ is a Delzant polytope, and it corresponds to $\mathbb{CP}^n$ ; the square $[0,1]^2$ corresponds to $S^2 \times S^2$ . The triangle with vertices $(0,0)$ , $(1,0)$ , $(0,2)$ is not Delzant. At $(1,0)$ the primitive edge vectors are $(-1,0)$ and $(-1,2)$ , with determinant $-2$ , so they do not form a basis of $\Z^2$ .

Going deeper

For a student

The theorem gives a dictionary: faces of the polytope correspond to orbit closures, and vertices to fixed points of the torus. Every symplectic toric manifold has a compatible complex structure that makes it a smooth toric variety. Rational simple polytopes with a positive integer label on each facet correspond in the same way to toric orbifolds, by work of Lerman and Tolman.

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Convex hull concept
Marsden–Weinstein reduction concept
Moment map concept
Symplectic manifold concept
Atiyah–Guillemin–Sternberg convexity theorem statement
Delzant's theorem on toric manifolds statement

Tag	`WBKCGFMT0P`
URI	`m2://WBKCGFMT0P`
Kind	statement, theorem
Writing	W0 stub
MSC	53D20, 14M25, 52B20
Status	This copy of the site has no build data. Run `m2 status WBKCGFMT0P`.
About	Moment map
Uses	Moment map, Symplectic manifold, Atiyah–Guillemin–Sternberg convexity theorem, Marsden–Weinstein reduction
Changed	2 November 2026

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