Moduli spaces of Delzant polytopes and symplectic toric manifolds

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).