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Moduli spaces of toric manifolds


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.

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    In the literature, these manifolds are usually called equivariantly symplectomorphic. However, the same name is sometimes also used for the notion in Definition 5, and so we use different names to distinguish the two.


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We would like to thank the anonymous referee who made many useful comments and clarifications which have significantly improved an earlier version of the paper. AP is grateful to Helmut Hofer for discussions and support. He also thanks Isabella Novik for discussions concerning general polytope theory, and Problem 4, during a visit to the University of Washington in 2010. The authors are also grateful to Victor Guillemin and Allen Knutson for helpful advice.

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Corresponding author

Correspondence to S. Sabatini.

Additional information

Á.P. was partly supported by NSF Grants DMS-0965738 and DMS-0635607, an NSF CAREER Award, a Leibniz Fellowship, Spanish Ministry of Science Grant MTM 2010-21186-C02-01, and by the Spanish National Research Council. A.R.P. was partly supported by an AMS-Simons Travel Grant. T.S.R. was partly supported by a MSRI membership, Swiss NSF grant 200021-140238, a visiting position at IHES, and by the government grant of the Russian Federation for support of research projects implemented by leading scientists, Lomonosov Moscow State University under the agreement No. 11.G34.31.0054.

Appendix: Polytopes

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.

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Pelayo, Á., Pires, A.R., Ratiu, T.S. et al. Moduli spaces of toric manifolds. Geom Dedicata 169, 323–341 (2014).

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  • Toric manifold
  • Delzant polytope
  • Moduli space
  • Metric space

Mathematics Subject Classification (2000)

  • MSC 53D20
  • MSC 53D05