Mathematics Lecture Notes

Theorem5.2Delzant's Theorem

The map sending a compact toric symplectic manifold $(M^{2n}, \omega, T^n, \mu)$ to its moment polytope $\Delta = \mu(M) \subseteq \mathbb{R}^n$ establishes a bijection between equivariant symplectomorphism classes of compact toric symplectic manifolds and Delzant polytopes.

Proof

Uniqueness: If two toric manifolds have the same Delzant polytope, they are equivariantly symplectomorphic. This follows from the equivariant version of Moser's argument: on the dense open orbit, the symplectic forms agree (both equal $\sum dI_i \wedge d\theta_i$ in action-angle coordinates), and an equivariant isotopy extends this to the boundary.

Existence: Given a Delzant polytope $\Delta = \{x \in \mathbb{R}^n : \langle x, v_i \rangle \geq \lambda_i, \, i = 1, \ldots, d\}$ , construct $M_\Delta$ via symplectic reduction of $\mathbb{C}^d$ : the standard $T^d$ -action on $(\mathbb{C}^d, \omega_{\text{std}})$ has moment map $\mu(z) = (\frac{1}{2}|z_1|^2, \ldots, \frac{1}{2}|z_d|^2)$ . The subtorus $K = \ker(\beta: T^d \to T^n)$ (where $\beta$ maps the $i$ -th circle to $v_i \in \mathbb{Z}^n$ ) acts on $\mathbb{C}^d$ , and $M_\Delta = \mu_K^{-1}(\lambda)/K$ is the desired toric manifold. $\square$

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ExampleBasic Delzant Polytopes

The interval $[0, a]$ gives $\mathbb{CP}^1$ with area $a$ .
The triangle with vertices $(0,0), (a,0), (0,a)$ gives $\mathbb{CP}^2$ with the Fubini-Study form scaled by $a$ .
The square $[0,a] \times [0,b]$ gives $\mathbb{CP}^1 \times \mathbb{CP}^1$ with symplectic form $a\omega_1 \oplus b\omega_2$ .

RemarkClassification Power

Delzant's theorem reduces the classification of toric symplectic manifolds to combinatorics. Questions about symplectic topology (e.g., which manifolds admit toric structures, computation of symplectic invariants) translate to questions about convex polytopes. This has been enormously fruitful in both symplectic geometry and algebraic geometry.