Moduli spaces of Delzant polytopes and symplectic toric manifolds

TL;DR

The work establishes a deep bridge between modern geometric combinatorics and toric symplectic geometry, showing that the moduli spaces of Delzant polytopes $\mathcal{D}(n)$ and toric manifolds $\mathcal{M}(2n)$ are path-connected and intimately stratified by normal fans, while also proving that no finite Oda-type classification can extend to dimension $n\ge 3$. By leveraging triangulation theory, stellar subdivisions, and Minkowski paths, the authors construct explicit minimal 3-polytopes that resist blow-downs, and they translate these results via the Delzant correspondence to statements about symplectic toric manifolds, including higher-dimensional obstructions and implications for simple connectedness and CW-topology. The paper also develops a robust topological framework for these spaces, introducing two compatible metrics, a secondary-fan stratification, and a CW structure that yields weak contractibility, thereby clarifying the global geometry, completions, and moduli behavior. Collectively, this work significantly advances the understanding of the global topology of toric moduli spaces and settles longstanding questions about minimal models in higher dimensions, with substantial consequences for toric symplectic geometry and its classification problems.

Abstract

This paper introduces modern geometric combinatorial technology from the theory of triangulations in order to derive results in toric symplectic geometry. In the main part of the paper we prove a number of properties of the space $\mathcal{D}(n)$ of $n$-dimensional Delzant polytopes. Two highlights are the construction of examples showing that, in contrast with the classical work of Oda in dimension $2$, no classification of combinatorially minimal Delzant polytopes can be expected in dimension $3$ or higher, and a proof that the space of $n$-dimensional Delzant polytopes is path-connected. Our proof of the latter is based on the fact that every rational fan can be refined to a unimodular fan, which is a standard technique used for resolution of singularities of toric varieties. In the last part of the paper, using the Delzant correspondence, these results allow us to answer several open questions concerning the moduli space $\mathcal{M}(n)$ of symplectic toric manifolds of dimension $2n$, since this space is isometric to the space of Delzant polytopes. Our results imply that no classification of minimal models of symplectic toric manifolds is plausible in dimension $6$ or higher, which answers in the negative a long-standing folklore question originating in Oda's work (1978).

Figures (6)

• Figure 1: Two Delzant polytopes in dimension three. Under the Delzant correspondence, the first one (tetrahedron) corresponds to the toric varitety $\mathbb{C} P^3$.
• Figure 2: The Hirzebruch trapezoid ${\rm H}_{3/2,1,2}$ with vertices $(0, -\tfrac{1}{2})$, $(0, \tfrac{1}{2})$, $(\tfrac{5}{2},-\tfrac{1}{2})$, $(\tfrac{1}{2},\tfrac{1}{2})$ and its corresponding normal fan, with generators $(-1,0)$, $(0,1)$, $(0,-1)$, and $(1,2)$. See Corollary \ref{['coro:2dim']} for the meaning of its parameters.
• Figure 3: The only $2$-dimensional symplectic toric manifold, up to scaling, $(\mathbb{C} P^1 \cong \mathbb{S}^2, \omega = {\rm d}\theta \wedge {\rm d}h, {\mathbb T}^1=\mathbb{S}^1, \mu(\theta, h) =h)$. The momentum polytope $P=[-1,1] \subset \mathfrak{t}^* \cong \mathbb{R}$ classifies its $\mathbb{S}^1$-equivariant symplectomorphism class. The Delzant correspondence sends the class of $(\mathbb{S}^2, \omega, \mathbb{S}^1, \mu)$ to $P$.
• Figure 4: Left: a polytope $P$ (a quadrilateral) and a face $F$ of it (a vertex). Right: the tangent and normal cones of $F$ in $P$.
• Figure 5: The refinement constructed in the proof of Theorem \ref{['thm:dim3']}. Left: A $3$-dimensional cone of the original fan $\Sigma$, represented as a triangle, via an intersection with an affine plane. Center: its blow-up in $\Sigma"$, consisting of three $3$-dimensional cones separated via three new two-dimensional cones (dashed lines) with one new ray (center dot). Right: In $\Sigma'$ each $3$-cone from $\Sigma"$ is subdivided into six $3$-cones, introducing several new $2$-cones (pointed lines) and new rays of two types: one along each $2$-cone of $\Sigma"$, denoted $\alpha_{ij}$ in the proof, and one in each $3$-cone of $\Sigma"$, denoted $\alpha_{ijk}$ in the proof.

Theorems & Definitions (122)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5: A non-polytopal icosahedral fan
• Example 2.6: The smallest non-polytopal fan
• Lemma 2.7
• proof
• Definition 2.8
• Proposition 2.9