Moduli spaces of toric manifolds

Abstract

We construct a distance on the moduli space of symplectic toric manifolds of dimension four. Then we study some basic topological properties of this space, in particular, path-connectedness, compactness, and completeness. The construction of the distance is related to the Duistermaat–Heckman measure and the Hausdorff metric. While the moduli space, its topology and metric, may be constructed in any dimension, the tools we use in the proofs are four-dimensional, and hence so is our main result.

Appendix: Polytopes

Let $V$ be a finite dimensional real vector space. A convex polytope $S$ in $V$ is the closed convex hull of a finite set $\{v_1, \ldots , v_n\}$, i.e., the smallest convex set containing $S$ or, equivalently,

$$\begin{aligned} \text{ Conv }\{v_1, \ldots , v_n\} : = \left\{ \sum _{i=1}^n a_i v_i\;\left| \; a_i \in [0,1], \;\sum _{i=1}^n a_i = 1 \right\} \right. . \end{aligned}$$

The dimension of $\text{ Conv }\{v_1, \ldots , v_n\}$ is the dimension of the vector space $\text{ span }_ \mathbb R \{v_1, \ldots , v_n\}$. A polytope is full dimensional if its dimension equals the dimension of $V$.

Note that the definition implies that a convex polytope is a compact subset of $V$. An extreme point of a convex subset $C \subseteq V$ is a point of $C$ which does not lie in any open line segment joining two points of $C$. Thus, a convex polytope is the closed convex hull of its extreme points (by the Krein–Milman  [19] Theorem) called vertices. In particular, the set of vertices is contained in $\{v_1, \ldots , v_n\}$. Clearly, there are infinitely many descriptions of the same polytope as a closed convex hull of a finite set of points. However, the description of a polytope as the convex hull of its vertices is minimal and unique.

There is another description of convex polytopes in terms of intersections of half-spaces. Let $V^*$ be the dual of $V$ and denote by $\left\langle \,, \right\rangle :V ^*\times V \rightarrow \mathbb R$ the natural non-degenerate duality pairing. The positive (negative) half-space defined by $\alpha \in V^*$ and $a \in \mathbb R$ is defined by

$$\begin{aligned} V_{\alpha , a}^{\pm }: =\left\{ v\in V \,\left| \,\left\langle \alpha ,v\right\rangle \gtreqless a\right\} \right. . \end{aligned}$$

Traditionally, in the theory of convex polytopes, the half spaces are chosen to be of the form $V_{\alpha , a}^{-}$. With these definitions, a convex polytope is given as a finite intersection of half-spaces. As for the convex hull representation, there are infinitely many representations of the same convex polytope as a finite intersection of half-spaces, but unlike it, a distinguished one that is minimal exists only for full dimensional polytopes, we will describe it in the next paragraph.

A face of a convex polytope is an intersection with a half-space satisfying the following condition: the boundary of the half-space does not contain any interior point of the polytope. Thus the faces of a convex polytope are themselves polytopes (and hence compact sets). Let $m$ be the dimension of a convex polytope. Then the whole polytope is the unique $m$-dimensional face, or body, the $(m-1)$-dimensional faces are called facets, the $1$-dimensional faces are the edges, and the $0$-dimensional faces are the vertices of the polytope. If the convex polytope is full-dimensional, its minimal and unique description as an intersection of half-spaces is given when the boundary of those half-spaces contain the facets.