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ISRN Geometry
VolumeΒ 2011Β (2011), Article IDΒ 642834, 24 pages
http://dx.doi.org/10.5402/2011/642834
Research Article

Characteristic Number Associated to Mass Linear Pairs

Departamento de FΓ­sica, Universidad de Oviedo, Avda Calvo Sotelo, 33007 Oviedo, Spain

Received 12 July 2011; Accepted 4 August 2011

Academic Editor: A.Β Fino

Copyright Β© 2011 AndrΓ©s ViΓ±a. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let Ξ” be a Delzant polytope in ℝ𝑛 and π›βˆˆβ„€π‘›. Let 𝐸 denote the symplectic fibration over 𝑆2 determined by the pair (Ξ”,𝐛). Under certain hypotheses, we prove the equivalence between the fact that (Ξ”,𝐛) is a mass linear pair (McDuff and Tolman, 2010) and the vanishing of a characteristic number of 𝐸. Denoting by Ham(𝑀Δ), the Hamiltonian group of the symplectic manifold defined by Ξ”, we determine loops in Ham(𝑀Δ) that define infinite cyclic subgroups in πœ‹1(Ham(𝑀Δ)) when Ξ” satisfies any of the following conditions: (i) it is the trapezium associated with a Hirzebruch sur-face, (ii) it is a Δ𝑝 bundle over Ξ”1, and (iii) Ξ” is the truncated simplex associated with the one point blowup of ℂ𝑃𝑛.

1. Introduction

Let (𝑁,Ξ©) be a closed connected symplectic 2𝑛-manifold. By Ham(𝑁,Ξ©), we denote the Hamiltonian group of (𝑁,Ξ©) [1, 2]. Associated with a loop πœ“ in Ham(𝑁,Ξ©), there exist characteristic numbers which are invariant under deformation of πœ“. These invariants are defined in terms of characteristic classes of fibre bundles, and their explicit values are not easy to calculate, in general. Here, we will consider a particular invariant 𝐼, whose definition will be recalled below. By proving the nonvanishing of 𝐼 for certain loops, we will deduce the existence of infinity cyclic subgroups of πœ‹1(Ham(𝑁,Ξ©)), when 𝑁 is a toric manifold. The vanishing of the invariant 𝐼 on particular loops in Ham(𝑁,Ξ©) is related with the concept of mass linear pair, which has been developed in [3]. In this introduction, we will state the main results of the paper and will give a schematic exposition of the concepts involved in these statements.

A loop πœ“ in Ham(𝑁,Ξ©) determines a Hamiltonian fibre bundle 𝐸→𝑆2 with standard fibre 𝑁, via the clutching construction. Various characteristic numbers for the fibre bundle 𝐸 have been defined in [4]. These numbers give rise to topological invariants of the loop πœ“. In this paper, we will consider only the following characteristic number: ξ€œπΌ(πœ“)∢=𝐸𝑐1(VTE)𝑐𝑛,(1.1) where VTE is the vertical tangent bundle of 𝐸, and π‘βˆˆπ»2(𝐸,ℝ) is the coupling class of the fibration 𝐸→𝑆2 [1, 5]. 𝐼(πœ“) depends only on the homotopy class of the loop πœ“. Moreover, the map πΌβˆΆπœ“βˆˆπœ‹1(Ham(𝑁,Ξ©))⟼𝐼(πœ“)βˆˆβ„(1.2) is an ℝ-valued group homomorphism [4].

Our purpose is to study this characteristic number when 𝑁 is a toric manifold and πœ“ is a 1-parameter subgroup of Ham(𝑁) defined by the toric action. The referred 1-parameter subgroup is determined by an element 𝐛 in the integer lattice of the Lie algebra of the corresponding torus. On the other hand, a toric symplectic manifold is determined by its moment polytope. For a general polytope, a mass linear function on it is a linear function β€œwhose value on the center of mass of the polytope depends linearly on the positions of the supporting hyperplanes” [3]. In this paper, we will relate the vanishing of the number 𝐼(πœ“) with the fact that 𝐛 defines a mass linear function on the polytope associated with the toric manifold. In the following paragraphs, we provide a more detailed exposition of this relation.

Let 𝑇 be the torus (U(1))𝑛 and Ξ”=Ξ”(𝐧,π‘˜) the polytope in π”±βˆ— with π‘š facets defined by Ξ”(𝐧,π‘˜)=π‘šξ™π‘—=1ξ€½π‘₯βˆˆπ”±βˆ—βˆΆξ«π‘₯,π§π‘—ξ¬β‰€π‘˜π‘—ξ€Ύ,(1.3) where π‘˜π‘—βˆˆβ„ and the π§π‘—βˆˆπ”± are the outward conormals to the facets. The facet defined by the equation ⟨π‘₯,π§π‘—βŸ©=π‘˜π‘— will be denoted 𝐹𝑗, and we put Cm(Ξ”) for the mass center of the polytope Ξ”.

In [3], the chamber π’žΞ” of Ξ” is defined as the set of π‘˜ξ…žβˆˆβ„π‘š such that the polytope Ξ”ξ…žβˆΆ=Ξ”(𝐧,π‘˜ξ…ž) is analogous to Ξ”; that is, the intersection βˆ©π‘—βˆˆπ½πΉπ‘— is nonempty if and only if βˆ©π‘—βˆˆπ½πΉξ…žπ‘—β‰ βˆ… for any π½βŠ‚{1,…,π‘š}. When we consider only polytopes which belong to the chamber of a fixed polytope, we delete the 𝐧 in the notation introduced in (1.3).

Further, McDuff and Tolman [3] introduced the concept of mass linear pair. Given the polytope Ξ” and π›βˆˆπ”±, the pair (Ξ”,𝐛) is mass linear if the map π‘˜βˆˆπ’žΞ”βŸΌβŸ¨Cm(Ξ”(π‘˜)),π›βŸ©βˆˆβ„(1.4) is linear. That is, ξ“βŸ¨Cm(Ξ”(π‘˜)),π›βŸ©=π‘—π‘…π‘—π‘˜π‘—+𝐢,(1.5) where 𝑅𝑗 and 𝐢 are constant.

Let us assume that Ξ” is a Delzant polytope [6]. We shall denote by (𝑀Δ,πœ”Ξ”,πœ‡Ξ”) the toric manifold determined by Ξ” (πœ‡Ξ”βˆΆπ‘€β†’π”±βˆ— being the corresponding moment map). Given 𝐛, an element in the integer lattice of 𝔱, we shall write πœ“π› for the loop of Hamiltonian diffeomorphisms of (𝑀Δ,πœ”Ξ”) defined by 𝐛 through the toric action. We will let 𝐼(Ξ”;𝐛) for the characteristic number 𝐼(πœ“π›). When we consider only polytopes in the chamber of a given polytope, we will write 𝐼(π‘˜;𝐛) instead of 𝐼(Ξ”(π‘˜);𝐛) for π‘˜ in this chamber.

The group 𝐺 of the translations defined by the elements of π”±βˆ— acts freely on π’žΞ”. We put π‘ŸβˆΆ=π‘šβˆ’π‘› for the dimension of the quotient π’žΞ”/𝐺. Thus, π‘Ÿ is the number of effective parameters which characterize the polytopes in π’žΞ” considered as β€œphysical bodies.”

We will prove the following theorem.

Theorem 1.1. Let (Ξ”,𝐛) be a pair consisting of a Delzant polytope in π”±βˆ— and an element in the integer lattice of 𝔱. If π‘Ÿβ‰€2, the following statements are equivalent: (a)𝐼(π‘˜;𝐛)=0, for all π‘˜βˆˆπ’žΞ”.(b)(Ξ”,𝐛) is a mass linear pair as in (1.5), with βˆ‘π‘—π‘…π‘—=0.

In [7], by direct computation, we proved the equivalence between the vanishing of 𝐼(π‘˜;𝐛) on π’žΞ” and the fact that (Ξ”,𝐛) is a mass linear pair, when Ξ” satisfies any of the following conditions:(i)it is the trapezium associated with a Hirzebruch surface,(ii)it is a Δ𝑝 bundle over Ξ”1 [3], and(iii)Ξ” is the truncated simplex associated with the one point blowup of ℂ𝑃𝑛.

On the other hand, when Ξ” is any of these polytopes (i)–(iii), the number π‘Ÿ is equal to 2; thus, from Theorem 1.1 and the result of [7], it follows that condition βˆ‘π‘—π‘…π‘—=0 is satisfied by all the mass linear pairs (Ξ”,𝐛). This fact can also be proved by direct calculation (Propositions 4.1, 4.6, and 4.9). So, Theorem 1.1, together with these propositions, generalize the result proved in [7].

Although the homotopy type of the Hamiltonian groups Ham(𝑁,Ξ©) is known only for some symplectic manifolds [8], the invariant 𝐼 allows us to identify nontrivial elements in πœ‹1(Ham(𝑁,Ξ©)). As 𝐼 is a group homomorphism, from Theorem 1.1, we deduce that a sufficient condition for πœ“π› to generate an infinite cyclic subgroup in πœ‹1(Ham(𝑀Δ,πœ”Ξ”)) is that the above condition (b) does not hold for (Ξ”,𝐛). More precisely, we have the following consequence of Theorem 1.1.

Theorem 1.2. Given the Delzant polytope Ξ” and 𝐛 an element in the integer lattice of 𝔱. If π‘Ÿβ‰€2 and (Ξ”,𝐛) is not mass linear, then πœ“π› generates an infinite cyclic subgroup in πœ‹1(Ham(𝑀Δ(π‘˜),πœ”Ξ”(π‘˜))), for all π‘˜βˆˆπ’žΞ”.

In the proof of Theorem 1.1, a formula for the characteristic number 𝐼(πœ“π›) obtained in [9] plays a crucial role. This formula gives 𝐼(πœ“π›) in terms of the integrals, on the facets of the polytope, of the normalized Hamiltonian function corresponding to the loop πœ“π› (see (2.13)). From this expression for 𝐼(πœ“π›), we will deduce a relation between the directional derivative of map (1.4) along the vector (1,…,1) of β„π‘š, the Euclidean volume of Ξ”(π‘˜) and 𝐼(π‘˜;𝐛) (see (3.5)). From this relation, it is easy to complete the proof of Theorem 1.1.

This paper is organized as follows. In Section 2, we study the characteristic number 𝐼(π‘˜;𝐛) when (Ξ”,𝐛) is a linear pair and π‘˜ varies in the chamber of Ξ”; we prove that 𝐼(π‘˜;𝐛) is a homogeneous polynomial of the π‘˜π‘— (Proposition 2.4).

In Section 3, we prove Theorem 1.1. In Proposition 3.5, a sufficient geometric condition for the Delzant polytope Ξ” to admit a mass linear pair (Ξ”,𝐛) is given. For a Delzant polytope Ξ”, Proposition 3.6 gives a necessary condition for the vanishing of 𝐼(π‘˜;𝐛) on π’žΞ”. We also express βˆ‘π‘—π‘…π‘— in terms of the displacement of the center of mass Cm(Ξ”(π‘˜)) produced by the change π‘˜π‘—β†’π‘˜π‘—+1 (Proposition 3.7).

Section 4 concerns the form which Theorem 1.2 adopts, when Ξ” is a Delzant polytope of the particular types (i)–(iii) mentioned above (see Corollary 4.2, Theorems 4.5, and 4.8). We also prove that, in these particular cases, if (Ξ”,𝐛) is a mass linear pair, then βˆ‘π‘—π‘…π‘—=0.

2. A Characteristic Number

Let us suppose that the polytope Ξ” defined in (1.3) is a Delzant polytope in π”±βˆ—. Following [10], we recall some points of the construction of (𝑀Δ,πœ”Ξ”,πœ‡Ξ”) from the polytope Ξ”. We put ξ‚π‘‡βˆΆ=(𝑆1)π‘šβˆ’π‘›. The 𝐧𝑖 determine weights π‘€π‘—βˆˆΜƒπ”±βˆ—, 𝑗=1,…,π‘š for a 𝑇-action on β„‚π‘š. Then moment map for this action is π½βˆΆπ‘§βˆˆβ„‚π‘šβŸΌπ½(𝑧)=πœ‹π‘šξ“π‘—=1||𝑧𝑗||2π‘€π‘—βˆˆΜƒπ”±βˆ—.(2.1) The π‘˜π‘– define a regular value 𝜎 for 𝐽, and the manifold 𝑀Δ is the following orbit space 𝑀Δ=ξ‚†π‘§βˆˆβ„‚π‘šβˆ‘βˆΆπœ‹π‘šπ‘—=1||𝑧𝑗||2𝑀𝑗=πœŽξ‚π‘‡,(2.2) where the relation defined by 𝑇 is ξ€·π‘§π‘—ξ€Έβ‰ƒξ€·π‘§ξ…žπ‘—ξ€ΈΜƒiffthereisπ²βˆˆπ”±suchthatπ‘§ξ…žπ‘—=𝑧𝑗𝑒2πœ‹π‘–βŸ¨π‘€π‘—,𝐲⟩for𝑗=1,…,π‘š.(2.3)

Identifying Μƒπ”±βˆ— with β„π‘Ÿ, 𝜎=(𝜎1,…,πœŽπ‘Ÿ) and each πœŽπ‘Ž is a linear combination of the π‘˜π‘—.

Given a facet 𝐹 of Ξ”, we choose a vertex 𝑝 of 𝐹. After a possible change in numeration of the facets, we can assume that 𝐹1,…,𝐹𝑛 intersect at 𝑝. In this numeration, 𝐹=𝐹𝑗, for some π‘—βˆˆ{1,…,𝑛}.

If we write π‘§π‘Ž=πœŒπ‘Žπ‘’π‘–πœƒπ‘Ž, then the symplectic form can be written on {[𝑧]βˆˆπ‘€βˆΆπ‘§π‘Žβ‰ 0,forallπ‘Ž}πœ”Ξ”=12𝑛𝑖=1π‘‘πœŒ2π‘–βˆ§π‘‘πœ‘π‘–,(2.4) with πœ‘π‘– an angular variable, linear combination of the πœƒπ‘Ž.

The action of 𝑇=(𝑆1)𝑛 on 𝑀Δ𝛼1,…,𝛼𝑛𝑧1,…,π‘§π‘šξ€»ξ€Ίπ›ΌβˆΆ=1𝑧1,…,𝛼𝑛𝑧𝑛,𝑧𝑛+1,…,π‘§π‘šξ€»(2.5) endows 𝑀Δ with a structure of toric manifold. Identifying π”±βˆ— with ℝ𝑛, the moment map πœ‡Ξ”βˆΆπ‘€Ξ”β†’π”±=ℝ𝑛 is defined by πœ‡Ξ”([𝑧]ξ€·πœŒ)=πœ‹21,…,𝜌2𝑛+𝑑1,…,𝑑𝑛,(2.6) where the constants 𝑑𝑖 are linear combinations of the π‘˜π‘— and imπœ‡Ξ”=Ξ”.(2.7) The facet 𝐹=𝐹𝑗 of Ξ” is the image by πœ‡Ξ” of the submanifold 𝐷𝑗=𝑧1,…,π‘§π‘šξ€»βˆˆπ‘€Ξ”βˆ£π‘§π‘—ξ€Ύ.=0(2.8)

We write π‘₯π‘–βˆΆ=πœ‹πœŒ2𝑖, then ξ€œπ‘€Ξ”ξ€·πœ”Ξ”ξ€Έπ‘›ξ€œ=𝑛!Δ𝑑π‘₯1⋯𝑑π‘₯𝑛.(2.9)

Let 𝐛 be an element in the integer lattice of 𝔱. The normalized Hamiltonian of the circle action generated by 𝐛 is the function 𝑓 determined by 𝑓=βŸ¨πœ‡Ξ”ξ€œ,π›βŸ©+constant,π‘€Ξ”π‘“ξ€·πœ”Ξ”ξ€Έπ‘›=0.(2.10) That is, 𝑓=βŸ¨πœ‡Ξ”,π›βŸ©βˆ’βŸ¨Cm(Ξ”),π›βŸ©, where ∫⟨Cm(Ξ”),π›βŸ©=π‘€βŸ¨πœ‡Ξ”ξ€·πœ”,π›βŸ©Ξ”ξ€Έπ‘›βˆ«π‘€ξ€·πœ”Ξ”ξ€Έπ‘›.(2.11) Moreover, ξ€œπ‘€Ξ”βŸ¨πœ‡Ξ”ξ€·πœ”,π›βŸ©Ξ”ξ€Έπ‘›ξ€œ=𝑛!Δ𝑛𝑖=1𝑏𝑖π‘₯𝑖𝑑π‘₯1⋯𝑑π‘₯𝑛.(2.12)

An expression for the value of the invariant 𝐼(πœ“π›) in terms of integrals of the Hamiltonian function has been obtained in Section 4 of [9] (see also [11, 12]) ξ€·πœ“πΌ(Ξ”;𝐛)∢=𝐼𝐛=βˆ’π‘›πΉfacet𝑁(𝐹),(2.13) where the contribution 𝑁(𝐹) of the above facet 𝐹=𝐹𝑗 (with 𝑗=1,…,𝑛) is π‘π‘—ξ€œβˆΆ=𝑁(𝐹)=(π‘›βˆ’1)!𝐹𝑗𝑓𝑑π‘₯1β‹―ξ‚Šπ‘‘π‘₯𝑗⋯𝑑π‘₯π‘›ξƒ©ξ€œ=(π‘›βˆ’1)!πΉπ‘—βŸ¨πœ‡Ξ”,π›βŸ©π‘‘π‘₯1β‹―ξ‚Šπ‘‘π‘₯𝑗⋯𝑑π‘₯π‘›ξ€œβˆ’βŸ¨Cm(Ξ”),π›βŸ©πΉπ‘—π‘‘π‘₯1β‹―ξ‚Šπ‘‘π‘₯𝑗⋯𝑑π‘₯𝑛ξƒͺ,(2.14) with 𝑑π‘₯1β‹―ξ‚Šπ‘‘π‘₯𝑗⋯𝑑π‘₯π‘›βˆΆ=𝑑π‘₯1⋯𝑑π‘₯π‘—βˆ’1𝑑π‘₯𝑗+1⋯𝑑π‘₯𝑛.

Given Ξ”=Ξ”(𝐧,π‘˜), we consider the polytope Ξ”ξ…ž=Ξ”(𝐧,π‘˜ξ…ž) obtained from Ξ” by the translation defined by a vector π‘Ž of π”±βˆ—. As we said, we write 𝐼(π‘˜;𝐛) and 𝐼(π‘˜ξ…ž;𝐛) for the corresponding characteristic numbers. According to the construction of the respective toric manifolds, 𝑀Δ′=𝑀Δ,πœ”Ξ”β€²=πœ”Ξ”,πœ‡Ξ”ξ…ž=πœ‡Ξ”+π‘Ž.(2.15) But the normalized Hamiltonians 𝑓 and π‘“ξ…ž corresponding to the action of 𝐛 on 𝑀Δ and 𝑀Δ′ are equal. Thus, it follows from (2.13) that 𝐼(π‘˜;𝐛)=𝐼(π‘˜ξ…ž;𝐛). More precisely, we have the evident proposition.

Proposition 2.1. If π‘Ž is an arbitrary vector of π”±βˆ—, then 𝐼(π‘˜;𝐛)=𝐼(π‘˜ξ…ž;𝐛), for π‘˜ξ…žπ‘—=π‘˜π‘—+βŸ¨π‘Ž,π§π‘—βŸ©, 𝑗=1,…,π‘š.

By Proposition 2.1, we can assume that all 𝑑𝑗 in (2.6) are zero for the determination of 𝐼(π‘˜;𝐛).

The following lemma is elementary.

Lemma 2.2. If 𝑆𝑛π‘₯(𝜏)∢=1,…,π‘₯π‘›ξ€Έβˆˆβ„π‘›βˆ£π‘›ξ“π‘–=1π‘₯π‘–β‰€πœ,0≀π‘₯𝑗,,βˆ€π‘—(2.16) then ξ€œπ‘†π‘›(𝜏)𝑓π‘₯1,…,π‘₯𝑛𝑑π‘₯1⋯𝑑π‘₯𝑛=⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©πœπ‘›π‘πœπ‘›!if𝑓=1𝑛+𝑐(𝑛+𝑐)!if𝑓=π‘₯π‘π‘–πœ,𝑐=1,2𝑛+2(𝑛+2)!if𝑓=π‘₯𝑖π‘₯𝑗,𝑖≠𝑗.(2.17)

More general, if 𝑐1,…,π‘π‘›βˆˆβ„>0, we put 𝑆𝑛π‘₯(𝑐,𝜏)∢=1,…,π‘₯π‘›ξ€Έβˆˆβ„π‘›βˆ£π‘›ξ“π‘–=1𝑐𝑖π‘₯π‘–β‰€πœ,0≀π‘₯𝑗,,βˆ€π‘—(2.18) then ξ€œπ‘†π‘›(𝑐,𝜏)𝑑π‘₯1⋯𝑑π‘₯𝑛=1𝑛!𝑛𝑖=1πœπ‘π‘–,ξ€œπ‘†π‘›(𝑐,𝜏)π‘₯𝑗𝑑π‘₯1⋯𝑑π‘₯𝑛=1𝜏(𝑛+1)!𝑐𝑗𝑛𝑖=1πœπ‘π‘–.(2.19) Thus, in the particular case that Ξ”=𝑆𝑛(𝑐,𝜏), the integral βˆ«π‘€Ξ”(πœ”Ξ”)𝑛 is a monomial of degree 𝑛 in 𝜏, and βˆ«π‘€Ξ”βŸ¨πœ‡Ξ”,π›βŸ©(πœ”Ξ”)𝑛 is a monomial of degree 𝑛+1.

We return to the general case in which Ξ” is the polytope defined in (1.3). Its vertices are the solutions to π‘₯,π§π‘—π‘Žξ¬=π‘˜π‘—π‘Ž,π‘Ž=1,…,𝑛;(2.20) hence, the coordinates of any vertex of Ξ” are linear combinations of the π‘˜π‘—.

A hyperplane in ℝ𝑛 through a vertex (π‘₯01,…,π‘₯0𝑛) of Ξ” is given by an equation of the form π‘₯⟨π‘₯,𝐧⟩=0,𝐧=βˆΆπœ….(2.21) Thus, the independent term πœ… is a linear combination (l. c.) of the π‘˜π‘—. Moreover, the coordinates of the common point of 𝑛 hyperplanes Μƒπ§βŸ¨π‘₯,π‘–βŸ©=πœ…π‘–,(2.22) with πœ…π‘– l. c. of, the π‘˜π‘— are also l. c. of the π‘˜π‘—.

By drawing hyperplanes through vertices of Ξ” (or more generally, through points which are the intersection of 𝑛 hyperplanes as (2.22)), we can obtain a family {𝛽𝑆} of subsets of Ξ” such that(a)each 𝛽𝑆 is the transformed of a simplex 𝑆𝑛(𝑏,𝜏) by an element of the group of Euclidean motions in ℝ𝑛.(b)For 𝛼≠𝛽, π›Όπ‘†βˆ©π›½π‘† is a subset of the border of 𝛼𝑆.(c)⋃𝛽𝛽𝑆=Ξ”.

Thus, by construction, each facet of 𝛽𝑆 is contained in a hyperplane πœ‹ of the form ⟨π‘₯,𝐧⟩=πœ…, with πœ… l. c. of the π‘˜π‘—.

On the other hand, the hyperplane πœ‹ is transformed by an element of SO(𝑛) in an hyperplane ⟨π‘₯,π§ξ…žβŸ©=πœ…. If 𝒯 is a translation in ℝ𝑛 which applies 𝑆𝑛(𝑏,𝜏) onto 𝛽𝑆, then this transformation maps (0,…,0) in a vertex π‘Ž=(π‘Ž1,…,π‘Žπ‘›) of 𝛽𝑆. So, the translation 𝒯 transforms πœ‹ in ⟨π‘₯,𝐧⟩=πœ…+βŸ¨π‘Ž,𝐧⟩=βˆΆπœ…ξ…ž. As each π‘Žπ‘— is a l. c. of the π‘˜π‘—, so is πœ…ξ…ž. Hence, any element of the group of Euclidean motions in ℝ𝑛 which maps 𝑆𝑛(𝑏,𝜏) onto 𝛽𝑆 transforms the hyperplane πœ‹ξ«π‘₯,π§ξ…žξ¬=πœ…ξ…ž,(2.23) with πœ…ξ…ž a l. c. of the π‘˜π‘—.

Let assume that (π‘…π’―π‘Ž)(𝑆(𝑏,𝜏))=𝛽𝑆, with π‘…βˆˆSO(𝑛) and π’―π‘Ž the translation defined by π‘Ž. Then the oblique facet of 𝑆(𝑏,𝜏), contained in the hyperplane βˆ‘π‘π‘–π‘₯𝑖=𝜏, is the image by π‘‡βˆ’π‘Žπ‘…βˆ’1 of a facet of 𝛽𝑆, which in turn is contained in a hyperplane of (2.23) (πœ…ξ…ž being a l. c. of the π‘˜π‘—). The argument of the preceding paragraph applied to π‘…βˆ’1 and π’―βˆ’π‘Ž proves that 𝜏 is a l. c. of the π‘˜π‘—. Hence, by (2.19), the integral ξ€œπ›½π‘†π‘‘π‘₯1⋯𝑑π‘₯𝑛=ξ€œπ‘†π‘›(𝑏,𝜏)𝑑π‘₯1⋯𝑑π‘₯𝑛(2.24) is a monomial of degree 𝑛 of a l. c. of the π‘˜π‘—. Thus, ξ€œπ‘€ξ€·πœ”Ξ”ξ€Έπ‘›=ξ“π›½ξ€œπ›½π‘†π‘‘π‘₯1⋯𝑑π‘₯𝑛,(2.25) is a homogeneous polynomial of degree 𝑛 of the π‘˜π‘—.

Similarly, ξ€œπ‘€Ξ”βŸ¨πœ‡Ξ”ξ€·πœ”,π›βŸ©Ξ”ξ€Έπ‘›(2.26) is a homogeneous polynomial of degree 𝑛+1 of the π‘˜π‘—. Analogous results hold for ξ€œπΉπ‘—π‘‘π‘₯1β‹―ξ‚Šπ‘‘π‘₯𝑗⋯𝑑π‘₯𝑛,ξ€œπΉπ‘—βŸ¨πœ‡Ξ”,π›βŸ©π‘‘π‘₯1β‹―ξ‚Šπ‘‘π‘₯𝑗⋯𝑑π‘₯𝑛.(2.27)

From formulas (2.9)–(2.14) together with the preceding argument, it follows the following proposition.

Proposition 2.3. Given a Delzant polytope Ξ”, if   𝐛 belongs to the integer lattice of   𝔱, then 𝐼(π‘˜;𝐛) is a rational function of the π‘˜π‘—, for π‘˜βˆˆπ’žΞ”.

Analogously, we have the following proposition.

Proposition 2.4. If (Ξ”,𝐛) is mass linear pair, then 𝐼(π‘˜;𝐛) is a homogeneous polynomial in the π‘˜π‘— of degree 𝑛, when π‘˜βˆˆπ’žΞ”.

We will use the following simple lemma in the proof of Theorem 1.1.

Lemma 2.5. If Μ‚π‘˜π‘—=π‘ π‘˜π‘— for 𝑗=1,…,π‘š, with π‘ βˆˆβ„, then Μ‚Cm(Ξ”(𝐧,π‘˜))=𝑠Cm(Ξ”(𝐧,π‘˜)).

Proof. The vertices of Ξ”(𝐧,π‘˜) are the solutions of (2.20), and the vertices of Μ‚Ξ”(𝐧,π‘˜) are the solutions of ⟨π‘₯,π§π‘—π‘ŽβŸ©=π‘ π‘˜π‘—π‘Ž, with π‘Ž=1,…,𝑛. Thus, the vertices of Μ‚Ξ”(𝐧,π‘˜) are those of Ξ”(𝐧,π‘˜) multiplied by 𝑠.

The lemma also follows from the fact that (2.25) and (2.26) are homogeneous polynomials of degree 𝑛 and 𝑛+1, respectively.

3. Proof of Theorem 1.1

Let us assume that the polytope Ξ” defined by (1.3) is Delzant and let π‘˜ be an element of π’žΞ”. We denote by 𝑀(π‘˜), πœ”(π‘˜), and πœ‡(π‘˜), the manifold, the symplectic structure and the moment map (resp.) determined by Ξ”(π‘˜). The facets of Ξ”(π‘˜) will be denoted by 𝐹(π‘˜)𝑗.

Let 𝐛 be an element in the integer lattice of 𝔱. We put 𝐴(π‘˜)ξ€œβˆΆ=𝑀(π‘˜)ξ«πœ‡(π‘˜)πœ”,𝐛(π‘˜)𝑛,𝐡(π‘˜)ξ€œβˆΆ=𝑀(π‘˜)ξ€·πœ”(π‘˜)𝑛.(3.1)

By (2.9), (1/𝑛!)𝐡(π‘˜) is the Euclidean volume of the polytope Ξ”(π‘˜). Given a facet 𝐹(π‘˜)𝑗, we can assume that π‘—βˆˆ{1,…,𝑛} (see third paragraph of Section 2). So, 𝐹(π‘˜)𝑗 is defined by the equation π‘₯𝑗=0. If we make an infinitesimal variation of the facet 𝐹(π‘˜)𝑗, by means of the translation defined by π‘˜π‘—β†’π‘˜π‘—+πœ– (keeping unchanged the other π‘˜π‘–), then the volume of Ξ”(π‘˜) changes according to 1𝐡𝑛!(π‘˜)⟢1𝐡𝑛!(π‘˜)ξ€œ+πœ–πΉ(π‘˜)𝑗𝑑π‘₯1β‹―ξ‚Šπ‘‘π‘₯𝑗⋯𝑑π‘₯π‘›ξ€·πœ–+𝑂2ξ€Έ.(3.2) We write 𝑑𝑋𝑗 for 𝑑π‘₯1β‹―ξ‚Šπ‘‘π‘₯𝑗⋯𝑑π‘₯𝑛. Thus, πœ•π΅(π‘˜)πœ•π‘˜π‘—ξ€œ=𝑛!𝐹(π‘˜)𝑗𝑑𝑋𝑗,πœ•π΄(π‘˜)πœ•π‘˜π‘—ξ€œ=𝑛!𝐹(π‘˜)π‘—ξ«πœ‡(π‘˜),𝐛𝑑𝑋𝑗.(3.3) So, by (2.11), πœ•πœ•π‘˜π‘—βŸ¨Cm(Ξ”(π‘˜)),π›βŸ©=𝑛!𝐡(π‘˜)ξ€Έ2𝐡(π‘˜)ξ€œπΉ(π‘˜)π‘—ξ«πœ‡(π‘˜),π›π‘‘π‘‹π‘—βˆ’π΄(π‘˜)ξ€œπΉ(π‘˜)𝑗𝑑𝑋𝑗ξƒͺ.(3.4) From (2.13) and (2.14), it follows π‘šξ“π‘—=1πœ•πœ•π‘˜π‘—βŸ¨Cm(Ξ”(π‘˜)),π›βŸ©=βˆ’1𝐡(π‘˜)𝐼(π‘˜;𝐛).(3.5) Thus, we have proved the following proposition.

Proposition 3.1. 𝐼(π‘˜;𝐛)=0 for all π‘˜βˆˆπ’žΞ” if and only if   βˆ‘π‘šπ‘—=1(πœ•/πœ•π‘˜π‘—)⟨Cm(Ξ”(π‘˜)),π›βŸ©=0, for all π‘˜βˆˆπ’žΞ”.

Next, we will parametrize the quotient π’žΞ”/𝐺 (of classes of polytopes in π’žΞ” module translation) defined in Section 1.

After a possible renumbering, we may assume that the intersection of facets 𝐹1,…,𝐹𝑛 is a vertex of Ξ”. Thus, the conormals 𝐧1,…,𝐧𝑛 are linearly independent in 𝔱. So, given π‘˜βˆˆπ’žΞ”, there is a unique π‘£βˆˆπ”±βˆ—, such that βŸ¨π‘£,π§π‘–βŸ©+π‘˜π‘–=0,𝑖=1,…,𝑛.(3.6) (Expressing the 𝐧𝑖 in terms of a basis of 𝔱 and 𝑣 in the dual basis, (3.6) is a compatible and determined system of linear equations for the coordinates of 𝑣.) Moreover 𝑣=𝑣(π‘˜) depends linearly on the π‘˜π‘–; that is, βŸ¨π‘£(π‘˜),𝐜⟩ is a linear function of π‘˜1,…,π‘˜π‘›, for all πœβˆˆπ”±.

If π‘šβˆ’π‘›=2, we write πœ†=π‘˜π‘›+1+𝑣(π‘˜),𝐧𝑛+1,𝜏=π‘˜π‘š+βŸ¨π‘£(π‘˜),π§π‘šβŸ©,(3.7) where 𝑣(π‘˜) the element in π”±βˆ— defined by (3.6). From the linearity of 𝑣 with respect to the π‘˜π‘–, it follows that πœ† and 𝜏 are linear combinations of π‘˜1,…,π‘˜π‘š.

The polytope in π’žΞ” defined by (π‘˜ξ…ž1=0,…,π‘˜ξ…žπ‘›=0,πœ†,𝜏) will be denoted by Ξ”0(πœ†,𝜏). It is the result of the translation of Ξ”(π‘˜) by the vector 𝑣(π‘˜); that is, Ξ”0(πœ†,𝜏)=Ξ”(π‘˜)+𝑣(π‘˜).(3.8)

Let 𝐛 an element in the integer lattice of 𝔱, we define the function 𝑔 by 𝑔Δ(πœ†,𝜏)∢=Cm0(πœ†,𝜏),𝐛.(3.9) The function 𝑔 is defined on the pairs (πœ†,𝜏) such that (0,…,0,πœ†,𝜏)βˆˆπ’žΞ”. By Lemma 2.5, it follows 𝑔(π‘ πœ†,π‘ πœ)=𝑠𝑔(πœ†,𝜏),(3.10) for any real number 𝑠 such that (π‘ πœ†,π‘ πœ) belongs to the domain of 𝑔. This property implies that 𝑔=πœ†πœ•π‘”πœ•πœ†+πœπœ•π‘”.πœ•πœ(3.11)

Theorem 3.2. If 𝐼(π‘˜;𝐛)=0, for all π‘˜βˆˆπ’žΞ” and π‘Ÿ=2, then βˆ‘βŸ¨Cm(Ξ”(π‘˜)),π›βŸ©=π‘—π‘…π‘—π‘˜π‘—, with 𝑅𝑗 constant (i.e., (Ξ”,𝐛) is a mass linear pair) and βˆ‘π‘—π‘…π‘—=0.

Proof. We set 𝑓(π‘˜1,…,π‘˜π‘š)∢=⟨Cm(Ξ”(π‘˜)),π›βŸ©. It follows from (3.8) that 𝑓(π‘˜)=𝑔(πœ†,𝜏)βˆ’βŸ¨π‘£(π‘˜),π›βŸ©.(3.12)
By the hypothesis and Proposition 3.1, π‘šξ“π‘—=1πœ•π‘“πœ•π‘˜π‘—=0.(3.13)
Since π‘šξ“π‘—=1πœ•π‘“πœ•π‘˜π‘—=πœ•π‘”πœ•πœ†π‘šξ“π‘—=1πœ•πœ†πœ•π‘˜π‘—+πœ•π‘”πœ•πœπ‘šξ“π‘—=1πœ•πœπœ•π‘˜π‘—βˆ’ξƒ‘πœ•π‘£πœ•π‘˜π‘—ξƒ’,,𝐛(3.14) from (3.13), we deduce π‘πœ•π‘”πœ•πœ†+π‘žπœ•π‘”πœ•πœβˆ’π‘‘=0,(3.15) where 𝑝, π‘ž, 𝑑 stand for the following constants 𝑝=π‘šξ“π‘—=1πœ•πœ†πœ•π‘˜π‘—,π‘ž=π‘šξ“π‘—=1πœ•πœπœ•π‘˜π‘—ξƒ‘,𝑑=πœ•π‘£πœ•π‘˜π‘—ξƒ’.,𝐛(3.16)
Since π‘žπœ†βˆ’π‘πœ and π‘‘πœβˆ’π‘žπ‘” are first integrals of (3.15), the general solution of this equation is 𝑑𝑔(πœ†,𝜏)=π‘žξ‚Άπœ+Ξ¦(π‘žπœ†βˆ’π‘πœ),(3.17) where Ξ¦ is a derivable function of one variable.
It follows from (3.11) and (3.17) that Ξ¦(𝑒)=π‘’Ξ¦ξ…ž(𝑒).(3.18) Thus, Ξ¦(𝑒)=𝛼𝑒, with 𝛼 constant. We have for 𝑓𝑏𝑓(π‘˜)=π‘žξ‚Άπœ+𝛼(π‘žπœ†βˆ’π‘πœ)βˆ’βŸ¨π‘£(π‘˜),π›βŸ©.(3.19) In other words, 𝑓 is a linear function of the π‘˜π‘—; that is, βˆ‘π‘“(π‘˜)=π‘—π‘…π‘—π‘˜π‘—, with 𝑅𝑗 constant. From (3.13), it follows βˆ‘π‘—π‘…π‘—=0.

Remark 3.3. The proof of Theorem 3.2 can be adapted to the simpler case when π‘Ÿ=1. In this case, the function 𝑔(πœ†)=⟨Cm(Ξ”0(πœ†)),π›βŸ© satisfies 𝑝(d𝑔/dπœ†)βˆ’π‘‘=0 and 𝑔(π‘ πœ†)=𝑠𝑔(πœ†). So, 𝑔(πœ†)=(𝑑/𝑝)πœ† and 𝑓(π‘˜)=(𝑑/𝑝)πœ†+βŸ¨π‘£(π‘˜),π›βŸ© is a linear map of the variables π‘˜π‘—.
On the other hand, the proof of this theorem does not admit an adaptation to the case π‘Ÿ>2. In fact, the corresponding function Ξ¦ would be a function of π‘Ÿβˆ’1 variables Ξ¦(𝑒1,…,π‘’π‘Ÿβˆ’1). The equation which corresponds to (3.18) in this case would be Ξ¦=π‘Ÿβˆ’1𝑖=1π‘’π‘–πœ•Ξ¦πœ•π‘’π‘–.(3.20) But this condition does not imply the linearity of Ξ¦.
When (Ξ”,𝐛) is a mass linear pair as in (1.5), by (3.5) 𝐼(π‘˜;𝐛)=βˆ’π΅(π‘˜)𝑗𝑅𝑗,(3.21) for all π‘˜βˆˆπ’žΞ”. From (3.21), we deduce the following proposition.

Proposition 3.4. Let (Ξ”,𝐛) be a mass linear pair. 𝐼(π‘˜;𝐛)=0 for all π‘˜βˆˆπ’žΞ” if and only if βˆ‘π‘—π‘…π‘—=0.

Proof of Theorem 1.1. It is a direct consequence of Proposition 3.4, Theorem 3.2, and the remark above.

We will deduce a sufficient condition for a Delzant polytope Ξ” to admit mass linear functions. We write Μ‡dCm(Ξ”(π‘˜))∢=|||dπœ–πœ–=0Cm(Ξ”(π‘˜+ΜŒπœ–)),(3.22) with ΜŒπœ–=(πœ–,…,πœ–).

Proposition 3.5. If all points Μ‡Cm(Ξ”(π‘˜)), for π‘˜βˆˆπ’žΞ”, belong to a hyperplane of (ℝ𝑛)βˆ— with a conormal vector in ℀𝑛 and π‘Ÿβ‰€2, then Ξ” admits a mass linear function.

Proof. Let π›βˆˆβ„€π‘› be a conormal vector to the hyperplane, then ̇=0=Cm(Ξ”(π‘˜)),π›π‘—πœ•πœ•π‘˜π‘—ξ„•.Cm(Ξ”(π‘˜)),𝐛(3.23) By (3.5), 𝐼(π‘˜,𝐛)=0; Theorem 3.2 applies and (Ξ”,𝐛) is a mass linear pair.

Proposition 3.6. Let Ξ” be a Delzant polytope, such that π‘˜=0 belongs to the closure of π’žΞ”. If π‘Ÿβ‰€2, a necessary condition for the vanishing of 𝐼(π‘˜;𝐛) on π’žΞ” is d|||dπœ–πœ–=0ξ‚­Cm(Ξ”(ΜŒπœ–)),𝐛=0.(3.24)

Proof. If 𝐼(π‘˜;𝐛) vanishes on π’žΞ”, then (Ξ”,𝐛) is a linear pair, by Theorem 1.1. Thus, βˆ‘βŸ¨Cm(Ξ”(π‘˜)),π›βŸ©=π‘—π‘…π‘—π‘˜π‘—+𝐢, on π’žΞ”. So, given π‘˜βˆˆπ’žΞ” and πœ– small enough ξ“βŸ¨Cm(Ξ”(π‘˜+ΜŒπœ–)),π›βŸ©=π‘—π‘…π‘—π‘˜π‘—ξ“+πœ–π‘—π‘…π‘—+𝐢.(3.25) By Theorem 1.1, βˆ‘π‘—π‘…π‘—=0. Thus, for any π‘˜βˆˆπ’žΞ”, d|||dπœ–πœ–=0⟨Cm(Ξ”(π‘˜+ΜŒπœ–)),π›βŸ©=0.(3.26) Taking the limit as π‘˜β†’0, 0=limπ‘˜β†’0d|||dπœ–πœ–=0d⟨Cm(Ξ”(π‘˜+ΜŒπœ–)),π›βŸ©=|||dπœ–πœ–=0ξ‚­.Cm(Ξ”(ΜŒπœ–)),𝐛(3.27)

Next, we will describe a geometric interpretation of the number βˆ‘π‘—π‘…π‘—. Given an arbitrary Delzant polytope Ξ”. If π‘Ž is a vector of π”±βˆ—, then ξ€·Ξ”ξ€·π‘˜Cmξ…žξ€Έξ€Έ=Cm(Ξ”(π‘˜))+π‘Ž,(3.28) if π‘˜ξ…žπ‘—=π‘˜π‘—+βŸ¨π‘Ž,π§π‘—βŸ©.

We will denote by 𝑑 the element of π”±βˆ— defined by the following relation ξ€·Ξ”ξ€·Μƒπ‘˜Cmξ€Έξ€Έ=Cm(Ξ”(π‘˜))+𝑑,(3.29) with Μƒπ‘˜π‘—=π‘˜π‘—+1 for all 𝑗.

From (3.28) and (3.29), we have ξ€·Ξ”ξ€·π‘˜Cm𝑗+𝑑,π§π‘—ξ€·Ξ”ξ€·π‘˜ξ¬ξ€Έξ€Έ=Cmπ‘—ξ€·Ξ”ξ€·Μƒπ‘˜ξ€Έξ€Έ+𝑑=Cm𝑗=π‘˜π‘—.+1ξ€Έξ€Έ(3.30)

Now, we assume that (Ξ”,𝐛) is a mass linear pair. From (1.5), it follows ξ«ξ€·Ξ”ξ€·π‘˜Cm𝑗+𝑑,𝐧𝑗=𝑅,π›π‘—π‘˜π‘—+𝑅𝑗𝑑,π§π‘—ξ¬ξ«ξ€·Ξ”ξ€·π‘˜+𝐢,Cm𝑗=𝑅+𝑑,π›π‘—π‘˜π‘—ξ«ξ€·Ξ”ξ€·Μƒπ‘˜ξ¬=𝑅+βŸ¨π‘‘,π›βŸ©+𝐢,Cmξ€Έξ€Έ,π›π‘—π‘˜π‘—+𝑅𝑗+𝐢.(3.31) These formulas allow us to state the following proposition that gives an interpretation of the sum βˆ‘π‘—π‘…π‘— in terms of the variation of Cm(Ξ”(π‘˜)) with the π‘˜π‘—.

Proposition 3.7. Let (Ξ”,𝐛) be a mass linear pair as in (1.5). Then, 𝑗𝑅𝑗𝑑,𝐧𝑗=βŸ¨π‘‘,π›βŸ©=𝑗𝑅𝑗,(3.32)𝑑 being the element of π”±βˆ— defined by (3.29).

4. Examples

In this section, we will deduce the particular form which adopts Theorem 1.2 when Ξ” is a polytope of the types (i)–(iii) mentioned in the introduction. For each case, we will determine the center of mass of the corresponding polytope Ξ”(π‘˜) and the condition for (Ξ”,𝐛) to be a mass linear pair. We will dedicate a subsection to each type.

4.1. Hirzebruch Surfaces

Given π‘Ÿβˆˆβ„€>0 and 𝜏,πœ†βˆˆβ„>0 with 𝜎∢=πœβˆ’π‘Ÿπœ†>0, in [7], we considered the Hirzebruch surface 𝑁 determined by these numbers. 𝑁 is the quotient ξ‚†π‘§βˆˆβ„‚4∢||𝑧1||2||𝑧+π‘Ÿ2||2+||𝑧4||2||𝑧=𝜏/πœ‹,2||2+||𝑧3||2=πœ†/πœ‹π•‹,(4.1) where the equivalence defined by 𝕋=(𝑆1)2 is given by 𝑧(π‘Ž,𝑏)β‹…1,𝑧2,𝑧3,𝑧4ξ€Έ=ξ€·π‘Žπ‘§1,π‘Žπ‘Ÿπ‘π‘§2,𝑏𝑧3,π‘Žπ‘§4ξ€Έ,(4.2) for (π‘Ž,𝑏)∈(𝑆1)2.

The manifold 𝑁 equipped with the following (U(1))2 action ξ€·πœ–1,πœ–2𝑧𝑗=ξ€Ίπœ–1𝑧1,πœ–2𝑧2,𝑧3,𝑧4ξ€»,(4.3) is a toric manifold. The corresponding moment polytope Ξ” is the trapezium in ℝ2 with vertices 𝑃1=(0,0),𝑃2=(0,πœ†),𝑃3=(𝜏,0),𝑃4=(𝜎,πœ†).(4.4) That is, 𝑁 is the toric manifold 𝑀Δ determined by the trapezium Ξ”.

As the conormals to the facets of Ξ” are the vectors 𝐧1=(βˆ’1,0), 𝐧2=(0,βˆ’1), 𝐧3=(0,1), and 𝐧4=(1,π‘Ÿ), the facets of a generic polytope Ξ”(π‘˜) in π’žΞ” are on the straights βˆ’π‘₯=π‘˜1,βˆ’π‘¦=π‘˜2,𝑦=π‘˜3,π‘₯+π‘Ÿπ‘¦=π‘˜4.(4.5) The vertices of Ξ”(π‘˜) are the points ξ€·βˆ’π‘˜1,βˆ’π‘˜2ξ€Έ,ξ€·βˆ’π‘˜1,π‘˜3ξ€Έ,ξ€·π‘˜4βˆ’π‘Ÿπ‘˜3,π‘˜3ξ€Έ,ξ€·π‘˜4+π‘Ÿπ‘˜2,βˆ’π‘˜2ξ€Έ.(4.6) Thus, the translation in the plane π‘₯, 𝑦 defined by (βˆ’π‘˜1,βˆ’π‘˜2) transforms the trapezium determined by the vertices (4.4) in Ξ”(π‘˜) if 𝜏=π‘˜4+π‘Ÿπ‘˜2+π‘˜1,πœ†=π‘˜3+π‘˜2.(4.7) So, ξ€·Cm(Ξ”(π‘˜))=Cm(Ξ”)+βˆ’π‘˜1,βˆ’π‘˜2ξ€Έ.(4.8) Moreover, the mass center of Ξ” is ξ‚΅Cm(Ξ”)=3𝜏2βˆ’3π‘Ÿπœπœ†+π‘Ÿ2πœ†2,3(2πœβˆ’π‘Ÿπœ†)3πœ†πœβˆ’2π‘Ÿπœ†2ξ‚Ά3(2πœβˆ’π‘Ÿπœ†).(4.9)

The chamber π’žΞ” consists of the points (π‘˜1,…,π‘˜4) such that πœβˆ’π‘Ÿπœ†>0, with 𝜏 and πœ† given by (4.7). So, the point π‘˜=0 belongs to the closure of π’žΞ”. From (4.8), together with (4.7) and (4.9), it follows ξ‚΅π‘ŸCm(Ξ”(ΜŒπœ–))=2πœ–,12βˆ’π‘Ÿπœ–6ξ‚Ά,(4.10) where ΜŒπœ–=(πœ–,πœ–,πœ–,πœ–). By Proposition 3.6, if 𝐼(π‘˜;𝐛) with 𝐛=(𝑏1,𝑏2)βˆˆβ„€2 vanishes on the chamber π’žΞ”, then π‘Ÿπ‘1βˆ’2𝑏2=0.

On the other hand, from (4.9) and (4.8), it follows ξ€·βŸ¨Cm(Ξ”(π‘˜)),π›βŸ©=3𝜏2βˆ’3π‘Ÿπœπœ†+π‘Ÿ2πœ†2𝑏1+ξ€·3πœ†πœβˆ’2π‘Ÿπœ†2𝑏23(2πœβˆ’π‘Ÿπœ†)βˆ’π‘˜1𝑏1βˆ’π‘˜2𝑏2.(4.11) By (4.7), expression (4.11) is linear in the π‘˜π‘– if and only if ξ€·3𝜏2βˆ’3π‘Ÿπœπœ†+π‘Ÿ2πœ†2𝑏1+ξ€·3πœ†πœβˆ’2π‘Ÿπœ†2𝑏23(2πœβˆ’π‘Ÿπœ†)(4.12) is linear in 𝜏, πœ†. That is, if and only if there exist constants 𝐴, 𝐡 such that for al 𝜏, πœ†ξ€·3𝜏2βˆ’3π‘Ÿπœπœ†+π‘Ÿ2πœ†2𝑏1+ξ€·3πœ†πœβˆ’2π‘Ÿπœ†2𝑏2=3(2πœβˆ’π‘Ÿπœ†)(𝐴𝜏+π΅πœ†).(4.13) From this relation, it follows the above condition π‘Ÿπ‘1=2𝑏2. In this case (4.11) reduces to ⟨Cm(Ξ”(π‘˜),π›βŸ©=βˆ’π‘12π‘˜1+𝑏12π‘˜4.(4.14)

Comparing (1.5) with (4.14), we obtain 𝑅1=βˆ’π‘…4=βˆ’π‘1/2, 𝑅2=𝑅3=0; so, βˆ‘π‘—π‘…π‘—=0. That is, the condition βˆ‘π‘—π‘…π‘—=0 holds for all the mass pairs (Ξ”,𝐛) when Ξ” is the polytope associated to a Hirzebruch surface. Hence, we have following proposition.

Proposition 4.1. (Ξ”,𝐛) is a mass linear pair if and only if π‘Ÿπ‘1=2𝑏2. Moreover, in this case βˆ‘π‘—π‘…π‘—=0.

By Theorem 1.2, we have the following corollary.

Corollary 4.2. If π‘Ÿπ‘1β‰ 2𝑏2, then πœ“π› generates an infinite cyclic subgroup in πœ‹1(Ham(𝑀Δ,πœ”Ξ”)).

Remark 4.3. We denote by πœ™π‘‘ the following isotopy of π‘€Ξ”πœ™π‘‘[𝑧]=𝑒2πœ‹π‘–π‘‘π‘§1,𝑧2,𝑧3,𝑧4ξ€»,(4.15) where πœ™ is a loop in the Hamiltonian group of 𝑀Δ. By πœ™ξ…ž, we denote the Hamiltonian loop πœ™ξ…žπ‘‘[𝑧]=𝑧1,𝑒2πœ‹π‘–π‘‘π‘§2,𝑧3,𝑧4ξ€».(4.16) In Theorem 8 of [11], we proved that 𝐼(πœ™ξ…ž)=(βˆ’2/π‘Ÿ)𝐼(πœ™). If 𝐛=(𝑏1,𝑏2)βˆˆβ„€2, then πΌξ€·πœ“π›ξ€Έ=𝑏1𝐼(πœ™)+𝑏2πΌξ€·πœ™ξ…žξ€Έ=𝑏1βˆ’ξ‚€2π‘Ÿξ‚π‘2𝐼(πœ™).(4.17) That is, 𝐼(πœ“π›)=0 if and only if π‘Ÿπ‘1=2𝑏2, which is in agreement with Proposition 4.1 and Theorem 1.1.

4.2. Δ𝑝 Bundle over Ξ”1

Given the integer 𝑝>1, as McDuff and Tolman in [3], we consider the following vectors in ℝ𝑝+1: 𝐧𝑖=βˆ’π‘’π‘–,𝑖=1,…,𝑝,𝐧𝑝+1=𝑝𝑖=1𝑒𝑖,𝐧𝑝+2=βˆ’π‘’π‘+1,𝐧𝑝+3=𝑒𝑝+1βˆ’π‘ξ“π‘–=1π‘Žπ‘–π‘’π‘–,(4.18) where 𝑒1,…,𝑒𝑝+1 is the standard basis of ℝ𝑝+1 and π‘Žπ‘–βˆˆβ„€. We write ξ€·π‘ŽπšβˆΆ=1,…,π‘Žπ‘ξ€Έβˆˆβ„€π‘,𝐴∢=𝑝𝑖=1π‘Žπ‘–,πšβ‹…πš=𝑝𝑖=1π‘Ž2𝑖.(4.19)

Let πœ†, 𝜏 be real positive numbers with πœ†+π‘Žπ‘–>0, for 𝑖=1,…,𝑝. In this subsection, we will consider the polytope Ξ” in (ℝ𝑝+1)βˆ— defined by the above conormals 𝐧𝑗 and the following π‘˜π‘—: π‘˜1=β‹―=π‘˜π‘=π‘˜π‘+2=0,π‘˜π‘+1=𝜏,π‘˜π‘+3=πœ†.(4.20)

This polytope will be also denoted by Ξ”0(πœ†,𝜏). It is a Δ𝑝 bundle on Ξ”1 (see [3]). When 𝑝=2, Ξ”=Ξ”0(πœ†,𝜏) is the prism whose base is the triangle of vertices (0,0,0), (𝜏,0,0), and (0,𝜏,0) and whose ceiling is the triangle determined by (0,0,πœ†), (𝜏,0,πœ†+π‘Ž1𝜏), and (0,𝜏,πœ†+π‘Ž2𝜏) (see Figure 1).

642834.fig.001
Figure 1: Ξ”2 bundle over Ξ”1.

We assume that the above polytope Ξ” is a Delzant polytope. The manifold (2.2) is in this case 𝑀Δ=ξ‚†π‘§βˆˆβ„‚π‘+3βˆΆβˆ‘π‘+1𝑖=1||𝑧𝑖||2βˆ‘=𝜏/πœ‹,βˆ’π‘π‘—=1π‘Žπ‘—||𝑧𝑗||2+||𝑧𝑝+2||2+||𝑧𝑝+3||2=πœ†/πœ‹β‰ƒ,(4.21) where (𝑧𝑗)≃(π‘§ξ…žπ‘—) if and only if there are 𝛼,π›½βˆˆπ‘ˆ(1) such that π‘§ξ…žπ‘—=π›Όπ›½βˆ’π‘Žπ‘—π‘§π‘—,𝑗=1,…,𝑝,π‘§ξ…žπ‘+1=𝛼𝑧𝑝+1,π‘§ξ…žπ‘˜=π›½π‘§π‘˜,π‘˜=𝑝+2,𝑝+3.(4.22) Thus, 𝑀Δ is the total space of the fibre bundle β„™(𝐿1βŠ•β‹―βŠ•πΏπ‘βŠ•β„‚)→ℂ𝑃1, where 𝐿𝑗 is the holomorphic line bundle over ℂ𝑃1 with Chern number π‘Žπ‘—.

The symplectic form (2.4) is πœ”Ξ”=12ξ€·πœŽ1+β‹―+πœŽπ‘+πœŽπ‘+2ξ€Έ,(4.23) where πœŽπ‘˜=π‘‘πœŒ2π‘˜βˆ§π‘‘πœ‘π‘˜.

And the moment map πœ‡Ξ”([𝑧]ξ€·π‘₯)=1,…,π‘₯𝑝,π‘₯𝑝+2ξ€Έ,(4.24) where π‘₯π‘–βˆΆ=πœ‹πœŒ2𝑖.

Proposition 4.4. The coordinates π‘₯𝑗 of Cm(Ξ”0(πœ†,𝜏)) are given by π‘₯π‘˜=πœπœ†ξ€·π‘+2(𝑝+2)+𝜏𝐴+π‘Žπ‘˜ξ€Έπœ†(𝑝+1)+𝜏𝐴,forπ‘˜=1,…,𝑝,(4.25)π‘₯𝑝+2=12(𝑝+1)(𝑝+2)πœ†2ξ€·+2(𝑝+2)π΄πœ†πœ+πšβ‹…πš+𝐴2ξ€Έπœ2.(𝑝+2)((𝑝+1)πœ†+𝐴𝜏)(4.26)

Proof. Since the points [𝑧]βˆˆπ‘€Ξ” satisfy |𝑧𝑝+2|2βˆ‘β‰€πœ†/πœ‹+𝑝𝑗=1π‘Žπ‘—|𝑧𝑗|2, by (2.9) and Lemma 2.2, we have ξ€œπ‘€Ξ”ξ€·πœ”Ξ”ξ€Έπ‘+1ξ€œ=(𝑝+1)!𝑆𝑝(𝜏)ξƒ©πœ†+𝑝𝑗=1π‘Žπ‘—π‘₯𝑗ξƒͺξ‚΅=(𝑝+1)!πœ†πœπ‘+πœπ‘!𝑝+1𝐴.(𝑝+1)!(4.27) Similarly, for π‘˜=1,…,π‘ξ€œπ‘€Ξ”π‘₯π‘˜ξ€·πœ”Ξ”ξ€Έπ‘+1=(𝑝+1)!πœ†πœπ‘+1+𝜏(𝑝+1)!𝑝+2(𝑝+2)!π‘—β‰ π‘˜π‘Žπ‘—+2πœπ‘+2π‘Žπ‘˜ξƒͺ.(𝑝+2)!(4.28) The π‘˜th coordinate of Cm(Ξ”), π‘₯π‘˜, is the quotient of (4.28) by (4.27); that is, π‘₯π‘˜=πœπœ†ξ€·π‘+2(𝑝+2)+𝜏𝐴+π‘Žπ‘˜ξ€Έ.πœ†(𝑝+1)+𝜏𝐴(4.29)
For the 𝑝+2-coordinate of Cm(Ξ”), we need to calculate βˆ«π‘€π‘₯𝑝+2(πœ”Ξ”)𝑝+1. By Lemma 2.2, 1ξ€œ(𝑝+1)!𝑀π‘₯𝑝+2ξ€·πœ”Ξ”ξ€Έπ‘+1=12ξ€œπ‘†π‘(𝜏)ξƒ©πœ†+𝑝𝑗=1π‘Žπ‘—π‘₯𝑗ξƒͺ2=12ξƒ©πœ†2πœπ‘+𝑝!2π΄πœ†πœπ‘+1+ξ€·(𝑝+1)!πšβ‹…πš+𝐴2ξ€Έπœπ‘+2ξƒͺ.(𝑝+2)!(4.30) Formula (4.26) is a consequence of (4.27) together with (4.30).

The translation in (ℝ𝑝+1)βˆ— defined by the vector (βˆ’π‘˜1,…,βˆ’π‘˜π‘,βˆ’π‘˜π‘+2) transforms the hyperplanes ⟨π‘₯,𝐧𝑝+3⟩=πœ† and ⟨π‘₯,𝐧𝑝+1⟩=𝜏 in π‘₯,𝐧𝑝+3=πœ†βˆ’π‘˜π‘+2+𝑝𝑗=1π‘Žπ‘—π‘˜π‘—,π‘₯,𝐧𝑝+1=πœβˆ’π‘ξ“π‘—=1π‘˜π‘—,(4.31) respectively.

Let Ξ”(π‘˜) be a polytope with π‘˜=(π‘˜1,…,π‘˜π‘+3) generic in the chamber π’žΞ”. From (4.31), it follows that Ξ”(π‘˜) is the image of the polytope Ξ”0(πœ†,𝜏) by the translation determined by (βˆ’π‘˜1,…,βˆ’π‘˜π‘,βˆ’π‘˜π‘+2), whenever π‘˜π‘+2βˆ’π‘ξ“π‘—=1π‘Žπ‘—π‘˜π‘—+π‘˜π‘+3=πœ†,𝑝𝑗=1π‘˜π‘—+π‘˜π‘+1=𝜏.(4.32) In this case, ξ€·Ξ”Cm(Ξ”(π‘˜))=Cm0ξ€Έβˆ’ξ€·π‘˜(πœ†,𝜏)1,…,π‘˜π‘,π‘˜π‘+2ξ€Έ.(4.33)

According to (4.32), the coordinates of the mass center Cm(Ξ”(ΜŒπœ–)), with ΜŒπœ–=(πœ–,…,πœ–), can be obtained substituting in (4.25) and in (4.26) πœ† by πœ–βˆ’π‘ξ“π‘—=1π‘Žπ‘—πœ–+πœ–=(2βˆ’π΄)πœ–(4.34) and 𝜏 by (𝑝+1)πœ–, and finally take into account (4.33). These operations give π‘₯π‘—πœ–(Ξ”(ΜŒπœ–))=ξ€·2(𝑝+2)(𝑝+1)π‘Žπ‘—ξ€Έβˆ’π΄,𝑗=1,…,𝑝,π‘₯𝑝+2πœ–(Ξ”(ΜŒπœ–))=ξ€·4(𝑝+2)βˆ’π΄2ξ€Έ.+(𝑝+1)(πšβ‹…πš)(4.35)

Given 𝐛=(𝑏1,…,𝑏𝑝̂̇,𝑏)≑(𝐛,𝐛), with ̂𝐛=(𝑏1,…,𝑏𝑝,0) and ̇𝐛=(0,…,0,𝑏),d|||dπœ–πœ–=0ξ‚­=1Cm(Ξ”(ΜŒπœ–)),𝐛̂,4(𝑝+2)(𝑝+1)2πšβ‹…π›βˆ’π‘πšβ‹…πšβˆ’π΄(2𝐡+𝑏𝐴)(4.36) where Μ‚βˆ‘πšβ‹…π›=𝑝𝑗=1π‘Žπ‘—π‘π‘— and βˆ‘π΅=𝑝𝑗=1𝑏𝑗.

By Proposition 3.6, we have the following theorem.

Theorem 4.5. Let Ξ” be the Δ𝑝 bundle over Ξ”1 defined by (4.18) and (4.20). Given ̂̇𝐛=(𝐛,𝐛)βˆˆβ„€π‘+1, if ξ€·Μ‚ξ€Έ(𝑝+1)2πšβ‹…π›βˆ’π‘πšβ‹…πšβˆ’π΄(2𝐡+𝑏𝐴)β‰ 0,(4.37) then πœ“π› defines an infinite cyclic subgroup in the fundamental group πœ‹1(Ham(𝑀Δ,πœ”Ξ”)).

It is straightforward to check that ξ€·Μ‚ξ€Έ(𝑝+1)2πšβ‹…π›βˆ’π‘πšβ‹…πšβˆ’π΄(2𝐡+𝑏𝐴)=0(4.38) is also a sufficient condition for (Ξ”,𝐛) to be a mass linear pair.

Since ΔCm0=Δ(πœ†,𝜏),𝐛Cm0ξ€Έ,̂𝐛+Δ(πœ†,𝜏)Cm0ξ€Έ,̇𝐛,(πœ†,𝜏)(4.39) if (4.38) holds, using (4.25) and (4.26), one obtains ΔCm0=(πœ†,𝜏),π›π‘πœ†2+𝑏2ξ€·πšβ‹…πš+𝐴2ξ€Έ+ξ€·Μ‚ξ€Έ(𝑝+2)π΄πšβ‹…π›+𝐴𝐡ξƒͺ(𝑝+2)𝐴𝜏.(4.40) By (4.33), for π‘˜βˆˆπ’žΞ”, ξ«ξ€·Ξ”βŸ¨Cm(Ξ”(π‘˜)),π›βŸ©=Cm0ξ€Έξ¬βˆ’(πœ†,𝜏),𝐛𝑝𝑗=1π‘π‘—π‘˜π‘—βˆ’π‘π‘˜π‘+2,(4.41) with πœ† and 𝜏 given by (4.32).

If ̂𝐛𝐛=, the condition (4.38) reduces to Μ‚(𝑝+1)πšβ‹…π›=𝐴𝐡 and ξ€·Μ‚ξ€ΈβŸ¨Cm(Ξ”(π‘˜)),π›βŸ©=πšβ‹…π›+𝐴𝐡(𝑝+2)𝐴𝑝𝑗=1π‘˜π‘—+π‘˜π‘+1ξƒͺβˆ’π‘ξ“π‘—=1π‘π‘—π‘˜π‘—.(4.42) Hence, βˆ‘π‘…βŸ¨Cm(Ξ”(π‘˜)),π›βŸ©=π‘—π‘˜π‘—, where 𝑅𝑗=ξ€·Μ‚ξ€Έπšβ‹…π›+𝐴𝐡(𝑝+2)π΄βˆ’π‘π‘—,𝑗=1,…,𝑝,𝑅𝑝+1=ξ€·Μ‚ξ€Έπšβ‹…π›+𝐴𝐡(𝑝+2)𝐴,𝑅𝑝+2=𝑅𝑝+3=0.(4.43) So, 𝑝+3𝑗=1𝑅𝑗=(̂𝑝+1)πšβ‹…π›βˆ’π΄π΅(𝑝+2)𝐴=0.(4.44)

A similar calculation for the case ̇𝐛𝐛= shows that the corresponding βˆ‘π‘—π‘…π‘— vanishes. That is, we have the following proposition.

Proposition 4.6. Let Ξ” be a Δ𝑝 bundle over Ξ”1. If (Ξ”,𝐛) is a mass linear pair, then βˆ‘π‘—π‘…π‘—=0.

For 𝑝=2, let 𝐛 be the following linear combination of the conormal vectors 𝐛=𝛾1𝐧1+𝛾2𝐧2+𝛾3𝐧3 with 𝛾1+𝛾2+𝛾3=0. By (4.18), 𝐛=(𝑏1,𝑏2,0) with 𝑏1=𝛾3βˆ’π›Ύ1, 𝑏2=𝛾3βˆ’π›Ύ2. In this case, condition (4.38) reduces to 3ξ€·π‘Ž1𝑏1+π‘Ž2𝑏2ξ€Έ=ξ€·π‘Ž1+π‘Ž2𝑏1+𝑏2ξ€Έ.(4.45) Or in terms of the π›Ύπ‘–π‘Ž1𝛾1+π‘Ž2𝛾2=0.(4.46) This is a necessary and sufficient condition for (Ξ”,𝐛) to be mass linear. This result is the statement of Lemma 4.8 in [3].

4.3. One Point Blowup of ℂ𝑃𝑛

In this subsection, Δ≑Δ0(πœ†,𝜏) will be ξƒ―ξ€·π‘₯Ξ”=1,…,π‘₯π‘›ξ€Έβˆˆβ„π‘›βˆ£π‘›ξ“π‘–=1π‘₯π‘–β‰€πœ,0≀π‘₯𝑖,π‘₯𝑛,β‰€πœ†(4.47) where 𝜏,πœ†βˆˆβ„>0 and 𝜎∢=πœβˆ’πœ†>0. That is, Ξ” is the polytope obtained truncating the simplex 𝑆𝑛(𝜏), defined in Lemma 2.2, by a β€œhorizontal” hyperplane through the point (0,…,0,πœ†). The manifold 𝑀Δ associated with Ξ” is the one point blowup of ℂ𝑃𝑛.

The mass center of the simplex 𝑆𝑛(𝜏) is the point 𝑆Cm𝑛=𝜏(𝜏)𝑛+1𝑀,(4.48) with 𝑀=(1,…,1).

As the volume of 𝑆𝑛(𝜏) is πœπ‘›/𝑛!, it follows from (4.48) (πœπ‘›βˆ’πœŽπ‘›)Cm(Ξ”)=πœπ‘›πœπ‘›+1π‘€βˆ’πœŽπ‘›ξ‚€πœŽπ‘›+1𝑀+πœ†π‘’π‘›ξ‚.(4.49) That is, 1Cm(Ξ”)=πœπ‘›βˆ’πœŽπ‘›πœξ‚΅ξ‚΅π‘›+1βˆ’πœŽπ‘›+1𝑛+1π‘€βˆ’πœ†πœŽπ‘›π‘’π‘›ξ‚Ά.(4.50)

Given π‘˜=(π‘˜1,…,π‘˜π‘›+2)βˆˆπ’žΞ”, the facets of Ξ”(π‘˜) are in the following hyperplanes: βˆ’π‘₯𝑗=π‘˜π‘—,𝑗=1,…,𝑛;𝑝𝑖=1π‘₯π‘˜=π‘˜π‘›+1;π‘₯𝑛+1=π‘˜π‘›+2.(4.51) As in the preceding subsections, Ξ”(π‘˜)=Ξ”0ξ€·π‘˜(πœ†,𝜏)βˆ’1,…,π‘˜π‘›ξ€Έ,(4.52) provided that πœ†=π‘˜π‘›+π‘˜π‘›+2 and βˆ‘πœ=𝑛+1𝑖=1π‘˜π‘–.

The pair (Ξ”,𝐛=(𝑏1,…,𝑏𝑛)) is mass linear if and only if there exist 𝐴,𝐡,πΆβˆˆβ„ such that π‘›βˆ’1𝑗=1π‘π‘—πœπ‘›+1βˆ’πœŽπ‘›+1𝑛+1+π‘π‘›ξ‚΅πœπ‘›+1βˆ’πœŽπ‘›+1𝑛+1βˆ’(πœβˆ’πœŽ)πœŽπ‘›ξ‚Ά=(𝐴𝜏+𝐡𝜎+𝐢)(πœπ‘›βˆ’πœŽπ‘›),(4.53) for all 𝜏, 𝜎 β€œadmissible”. A simple calculation proves the following proposition.

Proposition 4.7. The pair (Ξ”,𝐛) is mass linear if and only if 𝑏𝑛=1π‘›π‘›βˆ’1𝑗=1𝑏𝑗.(4.54)

From Theorem 1.2 together with Proposition 4.7, it follows the following theorem.

Theorem 4.8. If 𝐛=(𝑏1,…,𝑏𝑛)βˆˆβ„€π‘› and βˆ‘π‘›βˆ’1𝑗=1𝑏𝑗≠𝑛𝑏𝑛, then πœ“π› generates an infinite cyclic subgroup in πœ‹1(Ham(𝑀Δ,πœ”Ξ”)).

For π‘˜βˆˆπ’žΞ”, by (4.52) ξ«ξ€·Ξ”βŸ¨Cm(Ξ”(π‘˜)),π›βŸ©=Cm0ξ€Έξ¬βˆ’(πœ†,𝜏),𝐛𝑛𝑗=1π‘π‘—π‘˜π‘—.(4.55) If (Ξ”,𝐛) is a mass linear pair, by (4.50) and Proposition 4.7, we have ⟨Cm(Ξ”0(πœ†,𝜏)),π›βŸ©=π‘π‘›πœ. Thus, ⟨Cm(Ξ”(π‘˜)),π›βŸ©=𝑛+1𝑗=1π‘…π‘—π‘˜π‘—,(4.56) where 𝑅𝑗=π‘π‘›βˆ’π‘π‘—, for 𝑗=1,…,𝑛 and 𝑅𝑛+1=𝑏𝑛. Hence, we have the following proposition.

Proposition 4.9. Let Ξ” be the polytope obtained by truncating the standard 𝑛-simplex 𝑆𝑛(𝜏) by a horizontal hyperplane. If (Ξ”,𝐛) is a mass linear pair, then βˆ‘π‘—π‘…π‘—=0.

Remark 4.10. When 𝑛=3, the toric manifold 𝑀Δ is 𝑀Δ=ξ‚†π‘§βˆˆβ„‚5∢||𝑧1||2+||𝑧2||2+||𝑧3||2+||𝑧5||2||𝑧=𝜏/πœ‹,3||2+||𝑧4||2=πœ†/πœ‹π•‹,(4.57) where the action of 𝕋=(U(1))2 is defined by 𝑧(π‘Ž,𝑏)1,𝑧2,𝑧3,𝑧4,𝑧5ξ€Έ=ξ€·π‘Žπ‘§1,π‘Žπ‘§2,π‘Žπ‘π‘§3,𝑏𝑧4,π‘Žπ‘§5ξ€Έ,(4.58) for π‘Ž,π‘βˆˆU(1).
We consider the following loops in the Hamiltonian group of (𝑀Δ,πœ”Ξ”): πœ“π‘‘[𝑧]=𝑧1𝑒2πœ‹π‘–π‘‘,𝑧2,𝑧3,𝑧4,𝑧5ξ€»,πœ“ξ…žπ‘‘[𝑧]=𝑧1,𝑧2𝑒2πœ‹π‘–π‘‘,𝑧3,𝑧4,𝑧5ξ€»,ξ‚πœ“π‘‘[𝑧]=𝑧1,𝑧2,𝑧3𝑒2πœ‹π‘–π‘‘,𝑧4,𝑧5ξ€».(4.59) In [9] (Remark 4.10), we gave formulas that relate the characteristic numbers associated with these loops ξ€·πœ“πΌ(πœ“)=πΌξ…žξ€Έ=ξ‚€βˆ’13𝐼.ξ‚πœ“(4.60) So, for 𝐛=(𝑏1,𝑏2,𝑏3)βˆˆβ„€3, πΌξ€·πœ“π›ξ€Έ=𝑏1+𝑏2βˆ’3𝑏3𝐼(πœ“).(4.61) By Proposition 4.7, the vanishing of 𝐼(πœ“π›) in (4.61) is equivalent to the fact that (Ξ”,𝐛) is a mass linear pair. This equivalence is a new checking of Theorem 1.1.

Acknowledgment

This work has been partially supported by Ministerio de Ciencia y TecnologΓ­a, Grant FPA2009-11061.

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