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A Metric on the Moduli Space of Bodies

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A Metric on the Moduli Space of Bodies A METRIC ON THE MODULI SPACE OF BODIES HAJIME FUJITA, KAHO OHASHI Abstract. We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of sym- plectic toric manifolds. We also discuss related problems. 1. Introduction An n-dimensional Delzant polytope is a convex polytope in Rn which is simple, rational and smooth1 at each vertex. There exists a natu- ral bijective correspondence between the set of n-dimensional Delzant polytopes and the equivariant isomorphism classes of 2n-dimensional symplectic toric manifolds, which is called the Delzant construction [4]. Motivated by this fact Pelayo-Pires-Ratiu-Sabatini studied in [14] 2 the set of n-dimensional Delzant polytopes Dn from the view point of metric geometry. They constructed a metric on Dn by using the n-dimensional volume of the symmetric difference. They also studied the moduli space of Delzant polytopes Den, which is constructed as a quotient space with respect to natural action of integral affine trans- formations. It is known that Den corresponds to the set of equivalence classes of symplectic toric manifolds with respect to weak isomorphisms [10], and they call the moduli space the moduli space of toric manifolds. In [14] they showed that, the metric space D2 is path connected, the moduli space Den is neither complete nor locally compact. Here we use the metric topology on Dn and the quotient topology on Den. They also determined the completion of D2. Though in [14] they did not consider a metric on Den it is natural arXiv:1804.05161v2 [math.MG] 29 Aug 2018 to ask whether Den has a natural metric or not. In this paper we give an answer for this question. Our main result is a construction of a metric on the moduli space Den. We also show that its metric topology is homeomorphic to the quotient topology. Strictly speaking our con- struction does not rely on any conditions of the Delzant polytope so 2010 Mathematics Subject Classification. Primary 54E35, Secondary 53D05, 52B11. 1 A vertex of a convex polytope in Rn is called smooth if directional vectors of edges at the vertex can be chosen as a basis of Zn. 2 In [14] they use the notation DT for the set of Delzant polytopes instead of Dn. 1 2 H. FUJITA, K. OHASHI that it is possible to apply our proof for all bodies, that is, subsets in Rn obtained as the closure of bounded open subsets. Such a generalization for non-Delzant case has been increasing in importance in recent years in several areas. For example there exists an integrable system on a flag manifold, called the Gelfand-Cetlin system [7], which is studied from several context such as representation theory, algebraic geometry including mirror symmetry and so on (See [2], [8] or [13] for example). The Gelfand-Cetlin system is equipped with a torus action but it is not a symplectic toric manifold, however, it associates a non-simple convex polytope (as a base space of the integrable system), called the Gelfand- Cetlin polytope. More generally the theory of Newton-Okounkov bodies associated to divisors on varieties is developed recently, and it has great success in representation theory and algebraic geometry, especially for toric degeneration of varieties (See [9] , [11] or [12] for example). This paper is organized as follows. In Section 2 we introduce the moduli space of bodies with respect to the group of integral affine transformations. In Section 3 we construct a metric de on the moduli space following [14]. We also show that the metric topology and the quotient topology are homeomorphic to each other. In Section 4 we give a proof of our main theorem (Theorem 3.2), that is, de is a metric on the moduli space. In Section 5 we propose several related problems. 2. The moduli space of bodies Let Bn be the set of all bodies (i.e., compact subsets obtained as the closure of open subsets) in Rn. Now we introduce the moduli space of bodies following [14]. Let Gn := AGL(n; Z) be the integral affine n transformation group. Namely Gn = GL(n; Z) × R as a set and the multiplication is defined by (A1; t1) · (A2; t2) = (A1A2;A1t2 + t1) for each (A1; t1); (A2; t2) 2 Gn. This group Gn acts on Bn in a natural way, and A 2 Bn and B 2 Bn are called Gn-congruent if A and B are contained in the same Gn-orbit. Definition 2.1. The moduli space of the bodies in Rn with respect to the Gn-congruence Ben is defined by the quotient Ben := Bn=Gn: Remark 2.2. Bn contains an important subset, the set of n-dimensional Delzant polytopes Dn. It is well known that there exists a bijective cor- respondence, the Delzant construction, between the set of all equivari- ant isomorphism classes of 2n-dimensional symplectic toric manifolds. On the other hand, the quotient space Den := Dn=Gn corresponds to the weak equivalence classes of 2n-dimensional symplectic toric mani- folds. It is called the moduli space of toric manifolds in [14]. 3 3. Construction of the metric Let d : Bn × Bn ! R be the function defined by Z Z d(A; B) := voln(A 4 B) = χA4Bdλ = jχA − χBjdλ Rn Rn for each A; B 2 Bn, where A4B := (ArB)[(B rA) is the symmetric difference of A and B, voln(X) denotes the n-dimensional volume of X with respect to the n-dimensional Lebesgue measure dλ and χX : Rn ! R denotes the characteristic function of a subset X of Rn. Then one can see that d is a metric on Bn. Note that the function Ben × Ben 3 ([P1]; [P2]) 7! d(P1;P2) is not well- defined. So we introduce the function by taking the infimum among the values of d. Definition 3.1. Define a function de: Ben × Ben ! R by de(α; β) := inffd(P1;P2) j [P1] = α; [P2] = βg for (α; β) 2 Ben × Ben. For g = (A; t) 2 Gn, since det A = ±1 we have d(gP1; gP2) = d(P1;P2) for any P1;P2 2 Bn. When we fix representatives P1 2 Bn of α 2 Ben and P2 2 Bn of β 2 Ben it follows that de(α; β) = inffd(g1P1; g2P2) j g1; g2 2 Gng = inffd(P1; gP2) j g 2 Gng: Clearly deis a symmetric function. The following is the main theorem of the present paper. Theorem 3.2. de is a metric on Ben. We show this theorem by proving the triangle inequality in Proposi- tion 4.1 and the positiveness3 in Proposition 4.6. Due to Theorem 3.2, we can consider the metric topology O of B . de en On the other hand by using the metric topology of Bn and the natural projection π : Bn ! Ben, we also have the quotient topology Oπ of Ben. We have the following. Theorem 3.3. Two topological spaces (B ; O ) and (B ; O ) are home- en de en π omorphic to each other. In particular (Ben; Oπ) is a Hausdorff space. This theorem follows from the following general proposition. 3It is known that the function defined as in Definition 3.1 is a pseudo-metric and it satisfies the positiveness if and only if each Gn-orbit is a closed subset in Bn. See [15, p.80] for example. In fact our proof of Proposition 4.6 is a proof to show the closedness of the orbit. 4 H. FUJITA, K. OHASHI Proposition 3.4. Suppose that (X; d) is a metric space and a group G acts on X in an isometric way. Let Xe := X=G be the quotient space and assume that the function de: Xe × Xe ! R defined by de([x1]; [x2]) = inffd(x1; gx2) j g 2 Gg (([x1]; [x2]) 2 Xe × Xe) is a metric on Xe. Then the quotient topology Oπ and the metric topol- ogy O on X are homeomorphic to each other. de e Proof. Suppose that U 2 Oπ and take α 2 U. Fix a representative x 2 π−1(α) ⊂ π−1(U). Since π−1(U) is an open subset in X there exists δ > 0 such that Bδ(x)(⊂ X), the open ball of radius δ centered −1 at x, is contained in π (U). For this δ let Beδ(α) ⊂ Xe be the open ball of radius δ centered at α. For each β 2 Beδ(α) and a representative x0 2 π−1(β) since de(α; β) = inffd(x; gx0)g < δ g there exists g 2 G such that d(x; gx0) < δ. It implies that gx0 2 −1 0 Bδ(x) ⊂ π (U), and hence, β = π(gx ) 2 U. So we have Beδ(α) ⊂ U, and it means U 2 O . de Conversely suppose that U 2 O and take x 2 π−1(U). Then there de exists δ > 0 such that Beδ(π(x)) ⊂ U. For this δ > 0 consider the 0 0 open ball Bδ(x) in X. Take x 2 Bδ(x) then we have de(π(x); π(x )) ≤ 0 0 0 d(x; x ) < δ, and hence, π(x ) 2 Beδ(π(x)) ⊂ U. It implies that x 2 −1 −1 π (U). So we have Bδ(x) ⊂ π (U), and it means U 2 Oπ.
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