Symplectic Toric Manifolds

• Symplectic toric manifolds are compact 2n-dimensional spaces equipped with effective Hamiltonian Tⁿ actions and a moment map linking geometry to combinatorics.
• Delzant's theorem provides a one-to-one correspondence between these manifolds and smooth, rational convex polytopes, enabling a clear classification framework.
• They are constructed via symplectic reduction and generalized to orbifolds and stacks, offering deep insights into topology, combinatorial geometry, and deformation theory.

A symplectic toric manifold is a compact, connected symplectic $2n$-manifold $(M, \omega)$ equipped with an effective Hamiltonian action of the $n$-torus $T^n$ and a moment map $\mu: M \to (\mathbb{R}^n)^*$. These manifolds form a central class in symplectic geometry, where their classification, construction, deformation theory, and generalizations (notably to orbifolds and stacks) integrate symplectic, combinatorial, and topological aspects. At the foundation of their study lies the Delzant correspondence between symplectic toric manifolds and simple, rational, smooth convex polytopes—so-called Delzant polytopes—which encodes the geometric data in purely combinatorial terms (Hochenegger et al., 2011, Fujita et al., 2020, Pabiniak et al., 2020, McDuff, 2010).

1. Hamiltonian Torus Actions and Moment Maps

A Hamiltonian action of $T^n$ on $(M, \omega)$ is defined by the existence of a moment map $\mu: M \to \mathfrak{t}^* \cong \mathbb{R}^n$ satisfying

$\iota_{X_\xi}\omega = -d\langle \mu, \xi \rangle$

for all $\xi \in \mathfrak{t}$. The moment map is $T^n$-equivariant and its fibers are connected for compact $M$ (Atiyah–Guillemin–Sternberg convexity theorem). For a symplectic toric manifold, $\dim M = 2n$, the torus acts effectively, and $\mu(M)$ is a convex polytope (Hochenegger et al., 2011, Azam et al., 2021). The polytope $\Delta = \mu(M)$ can be written

$$\Delta = \{ x \in \mathbb{R}^n \mid \langle x, v_i \rangle \geq \lambda_i,\ i=1, \dots, d \}$$

where $v_i \in \mathbb{Z}^n$ are primitive inward normals.

2. Delzant Polytopes and Classification

Delzant polytopes are convex $n$-dimensional polytopes satisfying:

1. Simplicity: Exactly $n$ facets meet at each vertex.
2. Rationality: All facet normals $v_i$ are primitive in $\mathbb{Z}^n$.
3. Smoothness: At each vertex, the primitive normals of meeting facets form a $\mathbb{Z}$-basis of $\mathbb{Z}^n$.

Delzant's theorem establishes a bijection (up to translation and $GL(n, \mathbb{Z})$-action) between isomorphism classes of symplectic toric manifolds and Delzant polytopes (Hochenegger et al., 2011, Fujita et al., 2020, McDuff, 2010): $(M, \omega, T^n, \mu) \longleftrightarrow \Delta.$ The classification is rigid: the moment polytope completely determines the toric symplectic structure up to $T^n$-equivariant symplectomorphism (modulo translation) (Hochenegger et al., 2011, Azam et al., 2021).

3. Construction via Symplectic Reduction

Given a Delzant polytope $\Delta \subset \mathbb{R}^n$ with $d$ facets and normals $v_i \in \mathbb{Z}^n$:

• Form the standard model $(\mathbb{C}^d, \omega_0)$, $\omega_0 = \frac{i}{2}\sum dz_j \wedge d\bar{z}_j$.
• $T^d$ acts coordinatewise with moment map $\mu_0(z) = \frac{1}{2}(|z_1|^2, \ldots, |z_d|^2)$.
• The linear map $\beta: \mathbb{R}^d \to \mathbb{R}^n$, $\beta(e_i) = v_i$, defines a kernel torus $N$.
• Level set $\nu = ι^*(\lambda)$ in $Lie(N)^*$ is a regular value for the composed moment map $\mu_0^N$, inducing the symplectic quotient:

$M_\Delta = (\mu_0^N)^{-1}(\nu)/N.$

• The residual $T^n = T^d/N$ acts effectively on $M_\Delta$, and the reduced form $\omega_\Delta$ satisfies $π^*(\omega_\Delta) = i^*(\omega_0)$.

This “Delzant construction” produces the unique symplectic toric manifold with moment image $\Delta$ (Hochenegger et al., 2011, Azam et al., 2021).

4. Generalizations: Orbifolds and Stacks

If $\Delta$ is only simple and rational (not smooth), $N$ fails to act freely and $M_\Delta$ is a symplectic orbifold. The Lerman–Tolman classification then describes symplectic toric orbifolds by "labelled" polytopes, where each facet carries an isotropy order reflecting the local stabilizer (Hochenegger et al., 2011). Going further, Lerman–Malkin introduced symplectic toric Deligne–Mumford stacks (DM stacks), classified by rational simple polytopes, with stacky structure encoding higher isotropy (Hoffman, 2019, Hochenegger et al., 2011). The stack perspective generalizes the Delzant correspondence: compact toric symplectic stacks correspond to simple convex polytopes with additional combinatorial label data (decorated stacky polytopes); in the smooth case, the stack structure collapses to a manifold (Hoffman, 2019).

5. Topological and Cohomological Aspects

The topology of symplectic toric manifolds is deeply controlled by the combinatorics of $\Delta$: their cohomology rings admit a description via the Stanley–Reisner presentation of the polytope (McDuff, 2010). There are finiteness results: for any fixed compact symplectic manifold $(M,\omega)$, there are finitely many inequivalent toric structures (McDuff, 2010). Cohomological rigidity questions—whether the integral cohomology ring and symplectic class determine the manifold up to symplectomorphism—have been settled for several families (notably Bott manifolds and products of $\mathbb{CP}^1$) using toric degeneration techniques (Pabiniak et al., 2020).

6. Deformation Theory and Metric Convergence

The stability of symplectic toric manifolds under deformations of the polytope is controlled by convex-geometry distances: Hausdorff, volume-symmetric-difference, and Wasserstein distances all induce the same topology on the space of Delzant polytopes (Fujita et al., 2020). If a sequence of Delzant polytopes $P_i \to P$ (with constant facet number), the corresponding symplectic toric manifolds converge in the $T^n$-equivariant Gromov–Hausdorff sense in the Riemannian metric induced by the Guillemin potential. Topological invariants such as the Euler number are semi-continuous in this process, and “collapsing facets” correspond to controlled degenerations of the toric structure (Fujita et al., 2020).

7. Extensions and Future Directions

The cutting construction extends the symplectic reduction approach to possibly non-compact symplectic toric manifolds by sequentially enforcing facet inequalities via symplectic cuts. This construction describes non-compact (and conic) toric geometries unaddressable by Delzant's original model (Okitsu, 2013). Further, toric log symplectic geometry generalizes the momentum map to tropical (log-affine) structures, classifying log symplectic toric manifolds by decorated log affine polytopes, extending the Delzant paradigm to the Poisson setting (Gualtieri et al., 2014).

Symplectic toric manifolds provide a laboratory for higher categorical, topological, and dynamical phenomena in symplectic geometry. Generalizations to stacks, orbifolds, and log structures robustly extend the reach of the polytope-based classification, integrating symplectic, algebraic, and topological methods into a unified theory (Hochenegger et al., 2011, Hoffman, 2019, Gualtieri et al., 2014).