Delzant theorem (symplectic toric classification)

Intuition [Beginner]

A symplectic toric manifold is a closed symplectic manifold $M$ of dimension $2n$ together with a rotational action of an $n$-torus $T^n=(S^1{)}^n$ that preserves the symplectic form and is generated by Hamiltonian functions. The torus is exactly half the dimension of the manifold, which is the smallest dimension a torus can have while still leaving room for the symplectic structure. The image of the moment map is a flat polytope sitting inside ${\mathbb{R}}^n$, and Atiyah-Guillemin-Sternberg 05.04.03 tells you it is convex.

Delzant's 1988 theorem says the polytope alone determines the manifold. Two symplectic toric manifolds are equivariantly the same precisely when their polytopes match up to a translation. The polytopes that arise are exactly those satisfying three conditions: at every corner, $n$ edges meet (simplicity); the edges point in rational directions (rationality); and the edge vectors at each corner form a basis of the integer lattice (smoothness).

This is one of the few complete classifications in symplectic geometry. The picture: pure combinatorics on the right, a $2n$-dimensional curved manifold on the left, and a perfect dictionary between them.

Visual [Beginner]

The dictionary direction is left-to-right (a polytope determines the manifold) and right-to-left (a manifold determines its polytope). The standard simplex in ${\mathbb{R}}^n$ corresponds to ${\mathbb{CP}}^n$; the unit cube corresponds to a product of spheres; a trapezoid in ${\mathbb{R}}^2$ corresponds to a Hirzebruch surface.

The picture you should keep is a polytope drawn beside a manifold, with arrows pointing both ways: the moment map sends manifold to polytope, and the Delzant construction sends polytope back to manifold.

Worked example [Beginner]

The standard $2$-simplex in ${\mathbb{R}}^2$ has three vertices: $(0,0)$, $(1,0)$, and $(0,1)$. Its three edges are simple (two edges meet at each vertex, matching $n=2$), rational (the edge directions are integer vectors), and smooth (at each vertex, the two edge vectors form a basis of ${\mathbb{Z}}^2$). For instance at $(0,0)$ the edges run along $(1,0)$ and $(0,1)$; these are the standard basis. So the simplex is a Delzant polytope.

Delzant's theorem says there is exactly one symplectic toric four-manifold whose moment polytope is this simplex, and that manifold is ${\mathbb{CP}}^2$ with its Fubini-Study form. The three vertices correspond to the three coordinate fixed points $[1:0:0]$, $[0:1:0]$, $[0:0:1]$. Counting the dimensions: the polytope lives in ${\mathbb{R}}^2$, the manifold has real dimension $4=2\cdot 2$, the torus has dimension $2$.

What this tells us: a flat triangle, three numbers, and three conditions on lattice geometry are enough to reconstruct a curved four-real-dimensional Kähler manifold. Every other Delzant polytope plays the same game with a different recipe.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $T^n={\mathbb{R}}^n/{\mathbb{Z}}^n$ be the standard $n$-torus, with Lie algebra $\mathfrak{t}\cong {\mathbb{R}}^n$ and integer lattice $\Lambda ={\mathbb{Z}}^n\subset \mathfrak{t}$.

Definition (symplectic toric manifold). A symplectic toric manifold is a quadruple $(M,\omega ,T^n,\mu )$, where $(M,\omega )$ is a closed connected symplectic manifold of dimension $2n$, the torus $T^n$ acts effectively on $M$ by symplectomorphisms, and $\mu :M\to {\mathfrak{t}}^{\ast }\cong {\mathbb{R}}^n$ is a moment map for the action. Two symplectic toric manifolds $(M_1,{\omega }_1,T^n,{\mu }_1)$ and $(M_2,{\omega }_2,T^n,{\mu }_2)$ are equivariantly symplectomorphic when there exists a $T^n$-equivariant diffeomorphism $\phi :M_1\to M_2$ with ${\phi }^{\ast }{\omega }_2={\omega }_1$ and ${\mu }_2\circ \phi ={\mu }_1+c$ for some constant $c\in {\mathfrak{t}}^{\ast }$.

Definition (Delzant polytope). A convex polytope $\Delta \subset {\mathfrak{t}}^{\ast }\cong {\mathbb{R}}^n$ is a Delzant polytope when:

• (Simple) at each vertex of $\Delta$, exactly $n$ edges meet;
• (Rational) each edge of $\Delta$ has a primitive lattice direction vector in $\Lambda$;
• (Smooth) at each vertex $v$, the $n$ primitive inward-pointing edge vectors $u_1(v),\dots ,u_n(v)\in \Lambda$ form a $\mathbb{Z}$-basis of $\Lambda$.

The smoothness condition is equivalent to requiring that the inward-pointing primitive lattice normals $v_1,\dots ,v_n$ to the $n$ facets meeting at $v$ form a $\mathbb{Z}$-basis of $\Lambda$. Equivalently, the determinant $\det (v_1,\dots ,v_n)=\pm 1$ at every vertex.

Sign convention. We use $d{\mu }^{\xi }={\iota }_{X_{\xi }}\omega$ for the moment map, matching the Hamiltonian-vector-field convention of 05.02.01 and 05.04.01. The standard $T^d$-action on ${\mathbb{C}}^d$ by $(t_1,\dots ,t_d)\cdot (z_1,\dots ,z_d)=(e^{2\pi i{t}_1}{z}_1,\dots ,e^{2\pi i{t}_d}{z}_d)$ then has moment map

$${\mu }_d(z)=\pi (|z_1{|}^2,\dots ,|z_d{|}^2)$$

up to additive constants, with respect to the symplectic form ${\omega }_0=\frac{i}{2}\sum d{z}_j\wedge d{\bar{z}}_j$.

Counterexamples to common slips.

• A square pyramid in ${\mathbb{R}}^3$ is not simple at its apex (four edges meet, not three), so it fails the simplicity condition and is not Delzant.
• The regular hexagon centred at the origin in ${\mathbb{R}}^2$ has edge directions $(\cos (k\pi /3),\sin (k\pi /3))$, which are not lattice vectors after any rescaling that preserves regularity.
• The polytope $\{(x,y):0\le x\le 1,0\le y\le 1,x+y\le 1.5\}$ has vertex $(1,0.5)$ where the two edge vectors are $(0,1)$ and $(-1,1)$; their determinant is $1$, but the vertex $(0.5,1)$ has edges $(1,0)$ and $(1,-1)$ with determinant $-1$, fine — yet this polytope still fails because the vertex coordinates $(1,0.5)$ involve a half-integer that breaks rationality of the cut.

Key theorem with proof [Intermediate+]

Theorem (Delzant 1988). The map $(M,\omega ,T^n,\mu )\mapsto \mu (M)$ descends to a bijection

$$\{\text{symplectic toric manifolds}\}/\sim ⟷\{\text{Delzant polytopes}\}/\text{translation},$$

where the left side identifies equivariantly symplectomorphic manifolds and the right side identifies polytopes that differ by a translation in $\mathfrak{t}^$.*

The forward direction is the Atiyah-Guillemin-Sternberg theorem 05.04.03 together with the local-normal-form check that the moment polytope of a symplectic toric manifold is Delzant. The construction of the inverse — sending a Delzant polytope back to a manifold — is the technical core of the proof.

Proof. The argument proceeds in two parts: existence (every Delzant polytope arises) and uniqueness (the manifold is determined up to equivariant symplectomorphism).

Existence: the Delzant construction. Let $\Delta \subset {\mathbb{R}}^n$ be a Delzant polytope with $d$ facets $F_1,\dots ,F_d$. Each facet has a primitive inward-pointing lattice normal $v_i\in {\mathbb{Z}}^n$, and $\Delta$ admits the description

$$\Delta =\{x\in {\mathbb{R}}^n:⟨x,v_i⟩\ge {\lambda }_i\text{ for }i=1,\dots ,d\}$$

for some constants ${\lambda }_1,\dots ,{\lambda }_d\in \mathbb{R}$. The construction has five steps.

Step 1 (the surjection $\beta$). Define a linear map $\beta :{\mathbb{R}}^d\to {\mathbb{R}}^n$ by $\beta (e_i)=v_i$, where $e_1,\dots ,e_d$ is the standard basis of ${\mathbb{R}}^d$. Since $\Delta$ has full $n$-dimensional interior, the normals $v_1,\dots ,v_d$ span ${\mathbb{R}}^n$, so $\beta$ is surjective. The map $\beta$ takes ${\mathbb{Z}}^d$ into ${\mathbb{Z}}^n$, hence descends to a surjective Lie-group morphism $\bar{\beta }:T^d\to T^n$ between the corresponding tori. The smoothness condition at every vertex of $\Delta$ ensures that $\bar{\beta }$ has connected kernel.

Step 2 (the kernel torus $K$). Set $\mathfrak{k}:=\ker \beta \subset {\mathbb{R}}^d$, a vector subspace of dimension $d-n$. The smoothness condition at each vertex implies $\mathfrak{k}\cap {\mathbb{Z}}^d$ is a sublattice of full rank in $\mathfrak{k}$, so $K:=\mathfrak{k}/(\mathfrak{k}\cap {\mathbb{Z}}^d)\subset T^d$ is a closed connected $(d-n)$-dimensional subtorus. The exact sequence

$$1⟶K⟶T^d\stackrel{\bar{\beta }}{\to }{T}^n⟶1$$

makes $T^n=T^d/K$ as Lie groups.

Step 3 (the ambient ${\mathbb{C}}^d$). Equip ${\mathbb{C}}^d$ with the standard symplectic form ${\omega }_0=\frac{i}{2}{\sum }_{j=1}^{d}d{z}_j\wedge d{\bar{z}}_j$, equivalently ${\sum }_{j=1}^{d}d{x}_j\wedge d{y}_j$ in real coordinates $z_j=x_j+i{y}_j$. The torus $T^d$ acts by $(t_1,\dots ,t_d)\cdot (z_1,\dots ,z_d)=(e^{2\pi i{t}_1}{z}_1,\dots ,e^{2\pi i{t}_d}{z}_d)$. This action is Hamiltonian with moment map

$${\mu }_d:{\mathbb{C}}^d\to {\mathbb{R}}^d,{\mu }_d(z)=\pi (|z_1{|}^2,\dots ,|z_d{|}^2)-({\lambda }_1,\dots ,{\lambda }_d).$$

The constants ${\lambda }_i$ are exactly the polytope offsets from Step 1, included so the eventual residual moment map will land on $\Delta$ rather than on a translate.

Step 4 (restriction to $K$). The subtorus $K\subset T^d$ acts on ${\mathbb{C}}^d$ via the same formula, and the moment map for the $K$-action is the composition

$${\mu }_K={\iota }^{\ast }\circ {\mu }_d:{\mathbb{C}}^d\to {\mathfrak{k}}^{\ast },$$

where $\iota :\mathfrak{k}\hookrightarrow {\mathbb{R}}^d$ is the inclusion and ${\iota }^{\ast }$ its dual restriction. Equivalently, for $\xi \in \mathfrak{k}$ written as $\xi =({\xi }_1,\dots ,{\xi }_d)\in {\mathbb{R}}^d$ with $\sum {\xi }_i{v}_i=0$,

$${\mu }_K^{\xi }(z)=\pi \sum _{i=1}^d{\xi }_i|z_i{|}^2-\sum _{i=1}^d{\xi }_i{\lambda }_i.$$

The level set $Z:={\mu }_K^{-1}(0)\subset {\mathbb{C}}^d$ is precisely the set of $z$ for which $\pi (|z_1{|}^2-{\lambda }_1/\pi \cdot \pi ,\dots ,|z_d{|}^2-{\lambda }_d/\pi \cdot \pi )$ projects to zero in ${\mathfrak{k}}^{\ast }$, that is, ${\mu }_d(z)\in {\beta }^{\ast }({\mathbb{R}}^n)\subset {\mathbb{R}}^d$. The defining inequalities for $\Delta$ translate exactly to $|z_i{|}^2\ge 0$ being automatic and ${\mu }_d(z)\in \mathrm{image}({\beta }^{\ast })$ being the moment-level constraint. The Delzant smoothness condition makes $0$ a regular value of ${\mu }_K$ and the $K$-action on $Z$ free.

Step 5 (the symplectic quotient). Apply Marsden-Weinstein-Meyer reduction 05.04.02 at the regular value $0$ of ${\mu }_K$ with $K$ acting freely on $Z$. The quotient

$$M_{\Delta }:=Z/K={\mu }_K^{-1}(0)/K$$

is a closed connected symplectic manifold of real dimension $2d-2(d-n)=2n$, with reduced symplectic form ${\omega }_{\Delta }$ uniquely characterised by ${\pi }^{\ast }{\omega }_{\Delta }={\iota }_Z^{\ast }{\omega }_0$. The residual $T^n=T^d/K$ acts on $M_{\Delta }$ by symplectomorphisms; this action is Hamiltonian with moment map ${\mu }_{\Delta }:M_{\Delta }\to {\mathbb{R}}^n$ characterised by ${\mu }_{\Delta }\circ \pi =({\beta }^{\ast }{)}^{-1}\circ {\mu }_d{|}_Z$, where $({\beta }^{\ast }{)}^{-1}$ is the inverse of ${\beta }^{\ast }:{\mathbb{R}}^n\hookrightarrow {\mathbb{R}}^d$ on its image. A direct computation in coordinates shows ${\mu }_{\Delta }(M_{\Delta })=\Delta$. The compactness of $\Delta$ implies $Z/K$ is compact. Effectiveness of the residual $T^n$-action follows from $K$ being the full preimage of the identity in $T^d\to T^n$.

This completes the existence half: the polytope $\Delta$ produces the symplectic toric manifold $(M_{\Delta },{\omega }_{\Delta },T^n,{\mu }_{\Delta })$ with ${\mu }_{\Delta }(M_{\Delta })=\Delta$.

Uniqueness. Suppose $(M,\omega ,T^n,\mu )$ is a symplectic toric manifold with $\mu (M)=\Delta$. The goal is to produce a $T^n$-equivariant symplectomorphism $M\to M_{\Delta }$. The argument has three steps.

Step U1 (matching on the open dense orbit). Over the interior ${\Delta }^{\circ }$, both $M$ and $M_{\Delta }$ restrict to principal $T^n$-bundles via the moment map. Pick a section of each (this uses contractibility of ${\Delta }^{\circ }$); pulling back action coordinates produces a $T^n$-equivariant symplectomorphism ${\Phi }^{\circ }:{\mu }^{-1}({\Delta }^{\circ })\to {\mu }_{\Delta }^{-1}({\Delta }^{\circ })$ over the identity of ${\Delta }^{\circ }$.

Step U2 (extension across the boundary). Near a vertex $v\in \Delta$, the local-normal-form for Hamiltonian torus actions (a consequence of the equivariant Darboux theorem) gives an equivariant symplectomorphism between a neighbourhood of ${\mu }^{-1}(v)$ in $M$ and a neighbourhood of $0$ in ${\mathbb{C}}^n$ with the standard $T^n$-action. The smoothness condition at $v$ guarantees the linear identifications of the two normal models agree. The same statement at edge and facet strata, glued by the orbit-type stratification, extends ${\Phi }^{\circ }$ to a homeomorphism $\Phi :M\to M_{\Delta }$ that is smooth and equivariant on each stratum.

Step U3 (Moser argument). The map $\Phi$ obtained from U2 is a $T^n$-equivariant diffeomorphism with ${\mu }_{\Delta }\circ \Phi =\mu$ but ${\Phi }^{\ast }{\omega }_{\Delta }$ may differ from $\omega$. Since both forms have the same moment map for the same action and are cohomologous (their classes are determined by $\mu$ and $\Delta$ alone via the Duistermaat-Heckman formula), the family ${\omega }_t=(1-t)\omega +t{\Phi }^{\ast }{\omega }_{\Delta }$ is a smooth path of symplectic forms with constant moment map. The standard Moser argument 05.01.04 integrates a $T^n$-equivariant time-dependent vector field $V_t$ to produce an equivariant isotopy ${\rho }_t$ with ${\rho }_t^{\ast }{\omega }_t={\omega }_0$. Then $\Phi \circ {\rho }_1$ is the desired equivariant symplectomorphism.

Hence $\mu (M)={\mu }_{\Delta }(M_{\Delta })$ implies $(M,\omega ,T^n,\mu )\cong (M_{\Delta },{\omega }_{\Delta },T^n,{\mu }_{\Delta })$. $\square$

Bridge. Delzant's theorem builds toward 05.03.01 (coadjoint orbit), where the moment polytope of the maximal-torus action on a regular coadjoint orbit of $U(n)$ appears again in the form of a permutohedron — the Schur-Horn polytope is a Delzant polytope precisely when the orbit type is regular, and the Delzant manifold reconstruction recovers the orbit with its Kirillov-Kostant-Souriau form. The construction generalises the Atiyah-Guillemin-Sternberg image statement: where AGS 05.04.03 only says $\mu (M)$ is convex, Delzant says the polytope is a complete invariant when $\dim T=\frac{1}{2}\dim M$. The five-step kernel-torus reduction is dual to the polytope inclusion $\Delta \hookrightarrow {\mathbb{R}}^n$, with the surjection $T^d\to T^n$ realising ${\mathfrak{t}}^{\ast }$ as the affine span of $\Delta$. Putting these together, the foundational reason classification works is exactly that the Delzant smoothness condition on edge vectors is the same condition that makes $0$ a regular value of ${\mu }_K$ and $K$ act freely on the level set — the combinatorial smoothness of the polytope is the analytic smoothness of the symplectic reduction. The bridge is the kernel torus $K$: the polytope's facet normals determine $K$, and $K$ determines the manifold via reduction. The theorem identifies the category of symplectic toric $2n$-manifolds with the category of $n$-dimensional Delzant polytopes.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib does not currently have the symplectic-reduction or Hamiltonian-torus-action infrastructure necessary to state Delzant's theorem. The skeleton below sketches the statement at the type level; the proof would require the full apparatus described in lean_mathlib_gap.

[object Promise]

Advanced results [Master]

Delzant's theorem belongs to a family of classification results in equivariant symplectic geometry. Several refinements and generalisations exist, each respecting the polytope-to-manifold dictionary at the heart of the original.

Lerman-Tolman extension (1997). When the torus-action condition is relaxed from effective to quasi-effective (allowing finite stabilisers), the polytope condition relaxes to a labeled polytope in which each facet carries a positive-integer label encoding the stabiliser order. The classification becomes a bijection with labeled rational simple polytopes. This produces the symplectic toric orbifolds, with the unlabeled Delzant case as the smooth subcategory.

Karshon classification of Hamiltonian $S^1$-spaces (1999). Yael Karshon classified compact connected symplectic four-manifolds with Hamiltonian $S^1$-actions (one rank lower than toric) by labeled multigraphs. The Delzant theorem appears as the special case where the multigraph has the structure of a polygon edge list. The classification cleanly extends to mass-and-genealogy Hamiltonian $S^1$-spaces of higher dimension by Karshon-Tolman.

Symplectic cuts and birational toric geometry. Lerman's symplectic cut construction (1995) shows that polytope operations like edge cuts and vertex truncations correspond to symplectic operations on the manifold (blow-ups along $T^k$-invariant submanifolds). The Hirzebruch surfaces ${\mathbb{F}}_n$ for $n=0,1,2,\dots$ are obtained from ${\mathbb{CP}}^1\times {\mathbb{CP}}^1$ and ${\mathbb{CP}}^2$ by sequences of toric blow-ups; the polytope side displays this as edge insertions.

Algebraic-geometric counterpart. Every Delzant polytope determines a smooth projective toric variety in the sense of algebraic geometry, by associating to it a fan: take the inward normals at the polytope vertices to generate the maximal cones of the fan. The smooth toric varieties of complex dimension $n$ correspond bijectively to complete simplicial smooth fans in ${\mathbb{R}}^n$. The Delzant manifold and the toric variety have the same underlying complex structure when one chooses a compatible Kähler form; the polytope encodes the Kähler class while the fan encodes the complex structure. This gives a bridge from symplectic toric geometry to combinatorial commutative algebra (Stanley-Reisner ring) and to mirror symmetry (the Strominger-Yau-Zaslow picture for toric Calabi-Yau).

Equivariant cohomology. The equivariant cohomology ring $H_{T^n}^{\ast }(M_{\Delta };\mathbb{Q})$ of a symplectic toric manifold has a presentation purely in terms of the polytope: it is the Stanley-Reisner ring of the polytope's face fan modulo the linear relations coming from the facet normals. Atiyah-Bott / Berline-Vergne localisation specialises here to the Brion-Vergne polytope-vertex-sum formulas.

Duistermaat-Heckman. For a symplectic toric manifold, the Duistermaat-Heckman measure on $\Delta$ (the pushforward of Liouville measure under $\mu$) is exactly Lebesgue measure on $\Delta$, scaled to give total mass equal to the symplectic volume of $M$. The total Liouville volume is therefore $(2\pi {)}^n\cdot \mathrm{vol}(\Delta )$, an entirely combinatorial number.

Kostant-Kirillov compatibility. When $\Delta$ is the Newton polytope of a regular dominant weight $\lambda \in {\mathfrak{t}}_{+}^{\ast }$ for $U(n)$, the Delzant manifold $M_{\Delta }$ is equivariantly symplectomorphic to the coadjoint orbit ${\mathcal{O}}_{\lambda }\subset \mathfrak{u}(n{)}^{\ast }$ with its Kirillov-Kostant-Souriau form 05.03.01, the $T^n$-action by maximal-torus conjugation, and the moment map by diagonal projection. This is the Schur-Horn theorem 05.04.03 read as a Delzant identification.

Synthesis. Delzant's theorem identifies a piece of polyhedral combinatorics with a piece of symplectic geometry, and the bridge is a symplectic-reduction construction: every Delzant polytope determines a kernel torus $K\subset T^d$ via the linear surjection $\beta :{\mathbb{R}}^d\to {\mathbb{R}}^n$ on facet normals, and the symplectic quotient ${\mathbb{C}}^d//K$ at the level prescribed by the polytope offsets is exactly the toric manifold. The five-step construction is dual to the polytope's facet description: facets correspond to coordinate hyperplanes in ${\mathbb{C}}^d$, vertices correspond to dimension-$0$ orbits (fixed points), and the smoothness condition on edge vectors is exactly the regularity-and-freeness condition on the $K$-action. Read in the opposite direction, the moment polytope of a symplectic toric manifold is the image of a piece of algebraic-geometric data: the Kähler class of an associated toric variety, with the polytope encoding the fan together with a polarisation.

The foundational reason classification works is exactly the local-normal-form for Hamiltonian torus actions at fixed points: every fixed point of an effective Hamiltonian $T^n$-action on a $2n$-manifold has an equivariantly Darboux-standard neighbourhood ${\mathbb{C}}^n$ with the diagonal $T^n$-action, and the smoothness condition at the corresponding polytope vertex is exactly the condition that the linearised local model is the standard one. This is the bridge between the global statement (polytope determines manifold) and the local one (vertex determines neighbourhood). Putting these together, one sees that the entire classification reduces to checking that the global glueing of standard local models is controlled by the polytope's combinatorial structure — which is exactly the content of Delzant's argument. The theorem identifies symplectic toric $2n$-manifolds with $n$-dimensional Delzant polytopes in the precise categorical sense that the moment-image-and-reconstruction functors form an equivalence of groupoids. The central insight is that for half-dimensional torus actions, all the relevant geometry localises onto the moment polytope, because the torus action is just barely transitive enough to reduce all of symplectic geometry to combinatorics.

Full proof set [Master]

Lemma (moment map for the standard $T^d$-action on ${\mathbb{C}}^d$). The standard $T^d$-action on ${\mathbb{C}}^d$ by $(t_1,\dots ,t_d)\cdot (z_1,\dots ,z_d)=(e^{2\pi i{t}_1}{z}_1,\dots ,e^{2\pi i{t}_d}{z}_d)$ with respect to ${\omega }_0={\sum }_{j}d{x}_j\wedge d{y}_j$ is Hamiltonian with moment map ${\mu }_d(z)=\pi (|z_1{|}^2,\dots ,|z_d{|}^2)+c$ for any constant $c\in {\mathbb{R}}^d$.

Proof. The fundamental vector field of $\xi =({\xi }_1,\dots ,{\xi }_d)\in {\mathbb{R}}^d$ is $X_{\xi }={\sum }_{j}2\pi {\xi }_j(x_j{\partial }_{y_j}-y_j{\partial }_{x_j})$. Compute ${\iota }_{X_{\xi }}{\omega }_0={\sum }_{j}2\pi {\xi }_j(x_{j}d{x}_j+y_{j}d{y}_j)=\pi {\sum }_j{\xi }_{j}d(x_j^2+y_j^2)=d(\pi {\sum }_j{\xi }_j|z_j{|}^2)$, identifying ${\mu }_d^{\xi }=\pi {\sum }_j{\xi }_j|z_j{|}^2$ as a moment-map component. Equivariance is automatic since $T^d$ is abelian. Choice of constant $c$ is the moment-map ambiguity. $\square$

Lemma (kernel torus is connected). Let $\Delta$ be a Delzant polytope with facet normals $v_1,\dots ,v_d$, and let $\beta :{\mathbb{R}}^d\to {\mathbb{R}}^n$ be defined by $\beta (e_i)=v_i$. Then $K:=\ker (\bar{\beta }:T^d\to T^n)$ is a connected closed subtorus of dimension $d-n$.

Proof. The kernel of the linear map $\beta$ has dimension $d-n$ since $\beta$ is surjective. To see $K$ is connected, note $K=(\ker \beta )/(\ker \beta \cap {\mathbb{Z}}^d)$, so connectedness is equivalent to $\ker \beta \cap {\mathbb{Z}}^d$ being a lattice of full rank $d-n$ in $\ker \beta$. The smoothness condition at any single vertex of $\Delta$ implies the $n$ normals $v_{i_1},\dots ,v_{i_n}$ at that vertex form a $\mathbb{Z}$-basis of ${\mathbb{Z}}^n$, so the matrix expressing the remaining $d-n$ normals in this basis has integer entries, and the kernel of $\beta$ on ${\mathbb{Z}}^d$ has full rank. Hence $K$ is connected. $\square$

Lemma (regularity and freeness of $K$ on ${\mu }_K^{-1}(0)$). In the setup of the Delzant construction (Steps 1-5 above), $0$ is a regular value of $\mu_K : \mathbb{C}^d \to \mathfrak{k}^$and$K$actsfreelyon$Z := \mu_K^{-1}(0)$.*

Proof. Fix $z\in Z$ and let $S(z)=\{i:z_i=0\}$. The image of ${\mu }_K^{-1}(0)$ in $\Delta$ via the residual moment map lands on the face $F(S(z))={\bigcap }_{i\in S(z)}{F}_i$ of $\Delta$. The differential $d{\mu }_K{|}_z:T_z{\mathbb{C}}^d\to {\mathfrak{k}}^{\ast }$ has kernel containing the directions parallel to the coordinate planes for $i\in S(z)$ (these are tangent to the orbit-stratum) and surjects onto ${\mathfrak{k}}^{\ast }$ iff the projections $\{v_i{\}}_{i\notin S(z)}$ span a complement to the span of $\{v_i{\}}_{i\in S(z)}$ in ${\mathbb{R}}^n$. The Delzant simplicity condition propagated to all faces — every face $F$ of $\Delta$ is itself a simple lattice polytope, with facet normals at any vertex $v\in F$ forming a $\mathbb{Z}$-basis — gives exactly this spanning condition. Hence ${\mu }_K$ is submersive at every $z\in Z$.

The $K$-stabiliser of $z$ is generated, modulo ${\mathbb{Z}}^d$, by the elements $t=(t_1,\dots ,t_d)\in \mathfrak{k}$ with $t_i\in \mathbb{Z}$ for $i\notin S(z)$. Identifying $\mathfrak{k}=\ker \beta$, this stabiliser is $\ker \beta \cap ({\mathbb{Z}}^{S(z{)}^c}\times {\mathbb{R}}^{S(z)})$, which intersects $\ker \beta$ only at the origin modulo ${\mathbb{Z}}^d$ iff the $\{v_i{\}}_{i\in S(z)}$ are part of a $\mathbb{Z}$-basis of ${\mathbb{Z}}^n$ — exactly the Delzant smoothness condition. Hence $K$ acts freely on $Z$. $\square$

Lemma (image of the residual moment map is $\Delta$). The reduced symplectic toric manifold $M_{\Delta }=Z/K$ has residual moment map $\mu_\Delta : M_\Delta \to \mathfrak{t}^ \cong \mathbb{R}^n$with$\mu_\Delta(M_\Delta) = \Delta$.*

Proof. The defining inequalities of $\Delta$ are $⟨x,v_i⟩\ge {\lambda }_i$, equivalently ${\sum }_i(x\cdot v_i-{\lambda }_i)e_i^{\ast }\in ({\mathbb{R}}_{\ge 0}{)}^d$ as a vector in ${\mathbb{R}}^d$. The map ${\mu }_d(z)=\pi (|z_1{|}^2,\dots ,|z_d{|}^2)-({\lambda }_1,\dots ,{\lambda }_d)$ takes values in the closed orthant $({\mathbb{R}}_{\ge -\lambda }{)}^d$. The condition $z\in Z$ is ${\mu }_d(z)\in {\beta }^{\ast }({\mathbb{R}}^n)-({\lambda }_1,\dots ,{\lambda }_d)$, equivalently ${\mu }_d(z)+({\lambda }_1,\dots ,{\lambda }_d)\in {\beta }^{\ast }({\mathbb{R}}^n)$. Pre-composing with $({\beta }^{\ast }{)}^{-1}$ on its image gives $({\beta }^{\ast }{)}^{-1}({\mu }_d(z))=(⟨{\mu }_{\Delta }(\bar{z}),v_i⟩-{\lambda }_i)$ in coordinates dual to $\{v_i\}$, where $\bar{z}$ is the $K$-orbit. Non-negativity in each coordinate is exactly the inequality $⟨{\mu }_{\Delta }(\bar{z}),v_i⟩\ge {\lambda }_i$, i.e., ${\mu }_{\Delta }(\bar{z})\in \Delta$. Surjectivity onto $\Delta$ follows because the orthant $({\mathbb{R}}_{\ge 0}{)}^d$ is filled by $|z_i{|}^2\ge 0$ choices and the polytope inequalities can each be saturated separately. $\square$

Theorem (Delzant 1988, full statement and proof). The map $\Delta \mapsto M_{\Delta }$ from Delzant polytopes (mod translation) to symplectic toric manifolds (mod equivariant symplectomorphism) is a bijection, with inverse the moment map.

Proof. Existence (the Delzant construction) is the five-step procedure of the Intermediate proof, with the four lemmas above supplying rigour for the moment-map formula, the kernel torus structure, regularity-and-freeness of the $K$-action, and the residual moment map's image being $\Delta$.

Uniqueness has three components. Local model: near any fixed point $p\in M$ of an effective Hamiltonian $T^n$-action, the equivariant Darboux theorem (the local-normal-form for symplectic torus actions, due to Marle and Guillemin-Sternberg) gives an equivariant symplectomorphism between a neighbourhood of $p$ in $M$ and a neighbourhood of $0$ in ${\mathbb{C}}^n$ with the standard $T^n$-action. The smoothness condition at $\mu (p)\in \Delta$ ensures this local model is the same standard ${\mathbb{C}}^n$ for $M$ and for $M_{\Delta }$.

Stratification glueing: the orbit-type stratification of $M$ by faces of $\Delta$ matches the corresponding stratification of $M_{\Delta }$ exactly; equivariant tubular neighbourhood arguments (Koszul / Marle / Guillemin-Sternberg local models extended to all face strata) glue the local symplectomorphisms across face strata, producing an equivariant diffeomorphism $\Phi :M\to M_{\Delta }$ with ${\mu }_{\Delta }\circ \Phi =\mu$.

Moser deformation: the two symplectic forms $\omega$ and ${\Phi }^{\ast }{\omega }_{\Delta }$ on $M$ have the same moment map for the same $T^n$-action, hence the same Duistermaat-Heckman cohomology class, hence the path ${\omega }_t=(1-t)\omega +t{\Phi }^{\ast }{\omega }_{\Delta }$ is symplectic for all $t\in [0,1]$ (non-degeneracy is preserved on a closed manifold by openness, and ${\omega }_t-{\omega }_0=t\cdot \mathrm{exact}$ form has constant moment map). The standard Moser argument 05.01.04 applied equivariantly yields a $T^n$-equivariant isotopy ${\rho }_t$ with ${\rho }_t^{\ast }{\omega }_t={\omega }_0$. Then $\Phi \circ {\rho }_1:M\to M_{\Delta }$ is a $T^n$-equivariant symplectomorphism with ${\mu }_{\Delta }\circ (\Phi \circ {\rho }_1)=\mu$. $\square$

Connections [Master]

• Atiyah-Guillemin-Sternberg 05.04.03. Delzant's theorem is the toric specialisation of AGS: where AGS asserts only that $\mu (M)$ is a convex polytope, Delzant asserts it is a complete invariant in the half-dimensional torus case. The proof uses AGS as input — the moment polytope is convex by AGS, and the Delzant condition is read off from the local-normal-form at vertices.

• Symplectic reduction 05.04.02. The Delzant construction is a symplectic reduction of ${\mathbb{C}}^d$ by the kernel torus $K\subset T^d$ at a regular value of ${\mu }_K$ with $K$ acting freely. The unit's regular Marsden-Weinstein-Meyer reduction theorem is what makes Step 5 of the construction produce a smooth symplectic manifold.

• Moment map 05.04.01. The forward direction of Delzant's bijection — sending $(M,\omega ,T^n,\mu )$ to $\mu (M)$ — uses the moment map's three defining conditions to extract the polytope structure. The inverse uses the moment map of the standard $T^d$-action on ${\mathbb{C}}^d$ as the construction's starting point.

• Coadjoint orbit 05.03.01. Regular coadjoint orbits of $U(n)$ are symplectic toric manifolds for the maximal-torus action; their moment polytopes are the permutohedra. Delzant's theorem applied to a permutohedron recovers the coadjoint orbit, providing a constructive bridge between the Kirillov-Kostant-Souriau form and symplectic reduction.

• Symplectic manifold 05.01.02. The Delzant construction relies on the standard symplectic structure on ${\mathbb{C}}^d$, on the regular reduction theorem, and on the Moser argument — all symplectic-manifold infrastructure. The classification statement is purely symplectic: it does not invoke complex structure, even though the resulting manifolds happen to be Kähler.

• Hamiltonian vector field 05.02.01. Each component ${\mu }^{\xi }$ of the moment map generates the one-parameter subgroup $\exp (t\xi )\subset T^n$; the unit's defining identity ${\iota }_{X_H}\omega =dH$ is exactly what makes the $T^n$-action Hamiltonian, which is the input condition for both AGS and Delzant.

Historical & philosophical context [Master]

Thomas Delzant's 1988 paper Hamiltoniens périodiques et images convexes de l'application moment (Bull. Soc. Math. France 116, 315-339) ^{[Delzant 1988]} established the classification of compact symplectic toric manifolds by their moment polytopes. Delzant's argument built directly on Atiyah's 1982 Convexity and commuting Hamiltonians (Bull. London Math. Soc. 14) ^{[Atiyah 1982]} and Guillemin-Sternberg's 1982 Convexity properties of the moment mapping (Invent. Math. 67), which had established the convexity of moment-map images. Delzant's contribution was twofold: identifying the precise lattice-combinatorial conditions on a polytope that characterise the toric case, and producing the explicit reconstruction of a manifold from its polytope via the kernel-torus reduction of ${\mathbb{C}}^d$.

The construction has algebraic-geometric antecedents in the toric variety theory developed by Demazure (1970), Mumford (1973, in Toroidal Embeddings), and Danilov (1978, The geometry of toric varieties, Russian Math. Surveys 33) ^{[Danilov 1978]}. The smooth projective toric varieties are classified by complete simplicial smooth fans, and a Delzant polytope is exactly the convex hull of generators of such a fan together with a polarisation. Delzant's theorem reveals the symplectic side of this picture: where the algebraic geometers had classified the underlying complex variety, Delzant classified the additional Kähler structure encoded by the polytope.

Michèle Audin's 1991 monograph Topology of Torus Actions on Symplectic Manifolds ^{[Audin 1991]} gave the canonical textbook presentation of Delzant's theorem. Victor Guillemin's 1994 Moment Maps and Combinatorial Invariants of Hamiltonian $T^n$-spaces ^{[Guillemin 1994]} developed the equivariant-cohomological consequences and the Stanley-Reisner / face-ring presentations. Eugene Lerman's 1995 Symplectic cuts (Math. Res. Lett. 2) ^{[Lerman 1995]} showed that polytope operations (cutting an edge, truncating a vertex) correspond to symplectic operations (toric blow-up, cut), giving the polytope-side dictionary for birational toric symplectic geometry. Lerman-Tolman 1997 Hamiltonian torus actions on symplectic orbifolds and toric varieties (Trans. AMS 349) ^{[LermanTolman1997]} extended the classification to the orbifold setting via labeled polytopes.

Bibliography [Master]

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