Characteristic Number Associated to Mass Linear Pairs

Abstract

Let Δ be a Delzant polytope in ℝ^𝑛 and 𝐛∈ℤ^𝑛. Let 𝐸 denote the symplectic fibration over 𝑆² determined by the pair (Δ,𝐛). Under certain hypotheses, we prove the equivalence between the fact that (Δ,𝐛) is a mass linear pair (McDuff and Tolman, 2010) and the vanishing of a characteristic number of 𝐸. Denoting by Ham(𝑀_Δ), the Hamiltonian group of the symplectic manifold defined by Δ, we determine loops in Ham(𝑀_Δ) that define infinite cyclic subgroups in 𝜋₁(Ham(𝑀_Δ)) when Δ satisfies any of the following conditions: (i) it is the trapezium associated with a Hirzebruch sur-face, (ii) it is a Δ_𝑝 bundle over Δ₁, and (iii) Δ is the truncated simplex associated with the one point blowup of ℂ𝑃^𝑛.

1. Introduction

Let (𝑁,Ω) be a closed connected symplectic 2𝑛-manifold. By Ham(𝑁,Ω), we denote the Hamiltonian group of (𝑁,Ω) [1, 2]. Associated with a loop 𝜓 in Ham(𝑁,Ω), there exist characteristic numbers which are invariant under deformation of 𝜓. 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 𝜋₁(Ham(𝑁,Ω)), when 𝑁 is a toric manifold. The vanishing of the invariant 𝐼 on particular loops in Ham(𝑁,Ω) 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 𝜓 in Ham(𝑁,Ω) 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 𝜓. In this paper, we will consider only the following characteristic number: 𝐼(𝜓)∶=_𝐸𝑐₁(VTE)𝑐^𝑛,(1.1) where VTE is the vertical tangent bundle of 𝐸, and 𝑐∈𝐻²(𝐸,ℝ) is the coupling class of the fibration 𝐸→𝑆² [1, 5]. 𝐼(𝜓) depends only on the homotopy class of the loop 𝜓. Moreover, the map 𝐼∶𝜓∈𝜋₁(Ham(𝑁,Ω))⟼𝐼(𝜓)∈ℝ(1.2) is an ℝ-valued group homomorphism [4].

Our purpose is to study this characteristic number when 𝑁 is a toric manifold and 𝜓 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 “whose 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(𝜓) 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 Δ=Δ(𝐧,𝑘) the polytope in 𝔱^∗ with 𝑚 facets defined by Δ(𝐧,𝑘)=^𝑚_𝑗=1𝑥∈𝔱^∗∶𝑥,𝐧_𝑗≤𝑘_𝑗,(1.3) where 𝑘_𝑗∈ℝ and the 𝐧_𝑗∈𝔱 are the outward conormals to the facets. The facet defined by the equation ⟨𝑥,𝐧_𝑗⟩=𝑘_𝑗 will be denoted 𝐹_𝑗, and we put Cm(Δ) for the mass center of the polytope Δ.

In [3], the chamber 𝒞_Δ of Δ is defined as the set of 𝑘^∈ℝ^𝑚 such that the polytope Δ^∶=Δ(𝐧,𝑘^) is analogous to Δ; that is, the intersection ∩_𝑗∈𝐽𝐹_𝑗 is nonempty if and only if ∩_𝑗∈𝐽𝐹^𝑗≠∅ for any 𝐽⊂{1,…,𝑚}. 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 Δ and 𝐛∈𝔱, the pair (Δ,𝐛) is mass linear if the map 𝑘∈𝒞_Δ⟼⟨Cm(Δ(𝑘)),𝐛⟩∈ℝ(1.4) is linear. That is, ⟨Cm(Δ(𝑘)),𝐛⟩=_𝑗𝑅_𝑗𝑘_𝑗+𝐶,(1.5) where 𝑅_𝑗 and 𝐶 are constant.

Let us assume that Δ is a Delzant polytope [6]. We shall denote by (𝑀_Δ,𝜔_Δ,𝜇_Δ) the toric manifold determined by Δ (𝜇_Δ∶𝑀→𝔱^∗ being the corresponding moment map). Given 𝐛, an element in the integer lattice of 𝔱, we shall write 𝜓_𝐛 for the loop of Hamiltonian diffeomorphisms of (𝑀_Δ,𝜔_Δ) defined by 𝐛 through the toric action. We will let 𝐼(Δ;𝐛) for the characteristic number 𝐼(𝜓_𝐛). When we consider only polytopes in the chamber of a given polytope, we will write 𝐼(𝑘;𝐛) instead of 𝐼(Δ(𝑘);𝐛) for 𝑘 in this chamber.

The group 𝐺 of the translations defined by the elements of 𝔱^∗ acts freely on 𝒞_Δ. We put 𝑟∶=𝑚−𝑛 for the dimension of the quotient 𝒞_Δ/𝐺. Thus, 𝑟 is the number of effective parameters which characterize the polytopes in 𝒞_Δ considered as “physical bodies.”

We will prove the following theorem.

Theorem 1.1. Let (Δ,𝐛) be a pair consisting of a Delzant polytope in 𝔱^∗ and an element in the integer lattice of 𝔱. If 𝑟≤2, the following statements are equivalent: (a)𝐼(𝑘;𝐛)=0, for all 𝑘∈𝒞_Δ.(b)(Δ,𝐛) is a mass linear pair as in (1.5), with ∑_𝑗𝑅_𝑗=0.

In [7], by direct computation, we proved the equivalence between the vanishing of 𝐼(𝑘;𝐛) on 𝒞_Δ and the fact that (Δ,𝐛) is a mass linear pair, when Δ satisfies any of the following conditions:(i)it is the trapezium associated with a Hirzebruch surface,(ii)it is a Δ_𝑝 bundle over Δ₁ [3], and(iii)Δ is the truncated simplex associated with the one point blowup of ℂ𝑃^𝑛.

On the other hand, when Δ is any of these polytopes (i)–(iii), the number 𝑟 is equal to 2; thus, from Theorem 1.1 and the result of [7], it follows that condition ∑_𝑗𝑅_𝑗=0 is satisfied by all the mass linear pairs (Δ,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,Ω) is known only for some symplectic manifolds [8], the invariant 𝐼 allows us to identify nontrivial elements in 𝜋₁(Ham(𝑁,Ω)). As 𝐼 is a group homomorphism, from Theorem 1.1, we deduce that a sufficient condition for 𝜓_𝐛 to generate an infinite cyclic subgroup in 𝜋₁(Ham(𝑀_Δ,𝜔_Δ)) is that the above condition (b) does not hold for (Δ,𝐛). More precisely, we have the following consequence of Theorem 1.1.

Theorem 1.2. Given the Delzant polytope Δ and 𝐛 an element in the integer lattice of 𝔱. If 𝑟≤2 and (Δ,𝐛) is not mass linear, then 𝜓_𝐛 generates an infinite cyclic subgroup in 𝜋₁(Ham(𝑀_Δ(𝑘),𝜔_Δ(𝑘))), for all 𝑘∈𝒞_Δ.

In the proof of Theorem 1.1, a formula for the characteristic number 𝐼(𝜓_𝐛) obtained in [9] plays a crucial role. This formula gives 𝐼(𝜓_𝐛) in terms of the integrals, on the facets of the polytope, of the normalized Hamiltonian function corresponding to the loop 𝜓_𝐛 (see (2.13)). From this expression for 𝐼(𝜓_𝐛), we will deduce a relation between the directional derivative of map (1.4) along the vector (1,…,1) of ℝ^𝑚, the Euclidean volume of Δ(𝑘) and 𝐼(𝑘;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(𝑘;𝐛) when (Δ,𝐛) is a linear pair and 𝑘 varies in the chamber of Δ; we prove that 𝐼(𝑘;𝐛) is a homogeneous polynomial of the 𝑘_𝑗 (Proposition 2.4).

In Section 3, we prove Theorem 1.1. In Proposition 3.5, a sufficient geometric condition for the Delzant polytope Δ to admit a mass linear pair (Δ,𝐛) is given. For a Delzant polytope Δ, Proposition 3.6 gives a necessary condition for the vanishing of 𝐼(𝑘;𝐛) on 𝒞_Δ. We also express ∑_𝑗𝑅_𝑗 in terms of the displacement of the center of mass Cm(Δ(𝑘)) produced by the change 𝑘_𝑗→𝑘_𝑗+1 (Proposition 3.7).

Section 4 concerns the form which Theorem 1.2 adopts, when Δ 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 (Δ,𝐛) is a mass linear pair, then ∑_𝑗𝑅_𝑗=0.

2. A Characteristic Number

Let us suppose that the polytope Δ defined in (1.3) is a Delzant polytope in 𝔱^∗. Following [10], we recall some points of the construction of (𝑀_Δ,𝜔_Δ,𝜇_Δ) from the polytope Δ. We put 𝑇∶=(𝑆¹)^𝑚−𝑛. The 𝐧_𝑖 determine weights 𝑤_𝑗∈𝔱^∗̃, 𝑗=1,…,𝑚 for a 𝑇-action on ℂ^𝑚. Then moment map for this action is 𝐽∶𝑧∈ℂ^𝑚⟼𝐽(𝑧)=𝜋^𝑚_𝑗=1||𝑧_𝑗||²𝑤_𝑗∈𝔱^∗̃.(2.1) The 𝑘_𝑖 define a regular value 𝜎 for 𝐽, and the manifold 𝑀_Δ is the following orbit space 𝑀_Δ=𝑧∈ℂ^𝑚∶𝜋∑^𝑚𝑗=1||𝑧_𝑗||²𝑤_𝑗=𝜎𝑇,(2.2) where the relation defined by 𝑇 is 𝑧_𝑗≃𝑧^𝑗iffthereis𝐲∈𝔱̃suchthat𝑧^𝑗=𝑧_𝑗𝑒^{2𝜋𝑖⟨𝑤𝑗,𝐲⟩}for𝑗=1,…,𝑚.(2.3)

Identifying 𝔱^∗̃ with ℝ^𝑟, 𝜎=(𝜎₁,…,𝜎_𝑟) and each 𝜎_𝑎 is a linear combination of the 𝑘_𝑗.

Given a facet 𝐹 of Δ, we choose a vertex 𝑝 of 𝐹. After a possible change in numeration of the facets, we can assume that 𝐹₁,…,𝐹_𝑛 intersect at 𝑝. In this numeration, 𝐹=𝐹_𝑗, for some 𝑗∈{1,…,𝑛}.

If we write 𝑧_𝑎=𝜌_𝑎𝑒^𝑖𝜃𝑎, then the symplectic form can be written on {[𝑧]∈𝑀∶𝑧_𝑎≠0,forall𝑎}𝜔_Δ=12^𝑛_𝑖=1𝑑𝜌^2𝑖∧𝑑𝜑_𝑖,(2.4) with 𝜑_𝑖 an angular variable, linear combination of the 𝜃_𝑎.

The action of 𝑇=(𝑆¹)^𝑛 on 𝑀_Δ𝛼₁,…,𝛼_𝑛𝑧₁,…,𝑧_𝑚∶=𝛼₁𝑧₁,…,𝛼_𝑛𝑧_𝑛,𝑧_𝑛+1,…,𝑧_𝑚(2.5) endows 𝑀_Δ with a structure of toric manifold. Identifying 𝔱^∗ with ℝ^𝑛, the moment map 𝜇_Δ∶𝑀_Δ→𝔱=ℝ^𝑛 is defined by 𝜇_Δ([𝑧])=𝜋𝜌²¹,…,𝜌^2𝑛+𝑑₁,…,𝑑_𝑛,(2.6) where the constants 𝑑_𝑖 are linear combinations of the 𝑘_𝑗 and im𝜇_Δ=Δ.(2.7) The facet 𝐹=𝐹_𝑗 of Δ is the image by 𝜇_Δ of the submanifold 𝐷_𝑗=𝑧₁,…,𝑧_𝑚∈𝑀_Δ∣𝑧_𝑗=0.(2.8)

We write 𝑥_𝑖∶=𝜋𝜌^2𝑖, then _𝑀Δ𝜔_Δ^𝑛=𝑛!_Δ𝑑𝑥₁⋯𝑑𝑥_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=⟨𝜇_Δ,𝐛⟩+constant,_𝑀Δ𝑓𝜔_Δ^𝑛=0.(2.10) That is, 𝑓=⟨𝜇_Δ,𝐛⟩−⟨Cm(Δ),𝐛⟩, where ⟨Cm(Δ),𝐛⟩=∫^𝑀⟨𝜇_Δ,𝐛⟩𝜔_Δ^𝑛∫_𝑀𝜔_Δ^𝑛.(2.11) Moreover, _𝑀Δ⟨𝜇_Δ,𝐛⟩𝜔_Δ^𝑛=𝑛!_Δ𝑛_𝑖=1𝑏_𝑖𝑥_𝑖𝑑𝑥₁⋯𝑑𝑥_𝑛.(2.12)

An expression for the value of the invariant 𝐼(𝜓_𝐛) in terms of integrals of the Hamiltonian function has been obtained in Section 4 of [9] (see also [11, 12]) 𝐼(Δ;𝐛)∶=𝐼𝜓_𝐛=−𝑛_𝐹facet𝑁(𝐹),(2.13) where the contribution 𝑁(𝐹) of the above facet 𝐹=𝐹_𝑗 (with 𝑗=1,…,𝑛) is 𝑁_𝑗∶=𝑁(𝐹)=(𝑛−1)!_𝐹𝑗𝑓𝑑𝑥₁⋯𝑑𝑥_𝑗⋯𝑑𝑥_𝑛=(𝑛−1)!_𝐹𝑗⟨𝜇_Δ,𝐛⟩𝑑𝑥₁⋯𝑑𝑥_𝑗⋯𝑑𝑥_𝑛−⟨Cm(Δ),𝐛⟩_𝐹𝑗𝑑𝑥₁⋯𝑑𝑥_𝑗⋯𝑑𝑥_𝑛,(2.14) with 𝑑𝑥₁⋯𝑑𝑥_𝑗⋯𝑑𝑥_𝑛∶=𝑑𝑥₁⋯𝑑𝑥_𝑗−1𝑑𝑥_𝑗+1⋯𝑑𝑥_𝑛.

Given Δ=Δ(𝐧,𝑘), we consider the polytope Δ^=Δ(𝐧,𝑘^) obtained from Δ by the translation defined by a vector 𝑎 of 𝔱^∗. As we said, we write 𝐼(𝑘;𝐛) and 𝐼(𝑘^;𝐛) for the corresponding characteristic numbers. According to the construction of the respective toric manifolds, 𝑀_Δ^′=𝑀_Δ,𝜔_Δ^′=𝜔_Δ,𝜇_Δ=𝜇_Δ+𝑎.(2.15) But the normalized Hamiltonians 𝑓 and 𝑓^ corresponding to the action of 𝐛 on 𝑀_Δ and 𝑀_Δ^′ are equal. Thus, it follows from (2.13) that 𝐼(𝑘;𝐛)=𝐼(𝑘^;𝐛). More precisely, we have the evident proposition.

Proposition 2.1. If 𝑎 is an arbitrary vector of 𝔱^∗, then 𝐼(𝑘;𝐛)=𝐼(𝑘^;𝐛), for 𝑘^𝑗=𝑘_𝑗+⟨𝑎,𝐧_𝑗⟩, 𝑗=1,…,𝑚.

By Proposition 2.1, we can assume that all 𝑑_𝑗 in (2.6) are zero for the determination of 𝐼(𝑘;𝐛).

The following lemma is elementary.

Lemma 2.2. If 𝑆_𝑛(𝜏)∶=𝑥₁,…,𝑥_𝑛∈ℝ^𝑛∣^𝑛_𝑖=1𝑥_𝑖≤𝜏,0≤𝑥_𝑗,∀𝑗,(2.16) then _{𝑆𝑛(𝜏)}𝑓𝑥₁,…,𝑥_𝑛𝑑𝑥₁⋯𝑑𝑥_𝑛=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝜏^𝑛𝑐𝑛!𝜏if𝑓=1_𝑛+𝑐(𝑛+𝑐)!if𝑓=𝑥^𝑐𝑖𝜏,𝑐=1,2_𝑛+2(𝑛+2)!if𝑓=𝑥^𝑖𝑥_𝑗,𝑖≠𝑗.(2.17)

More general, if 𝑐₁,…,𝑐_𝑛∈ℝ_>0, we put 𝑆_𝑛(𝑐,𝜏)∶=𝑥₁,…,𝑥_𝑛∈ℝ^𝑛∣^𝑛_𝑖=1𝑐_𝑖𝑥_𝑖≤𝜏,0≤𝑥_𝑗,∀𝑗,(2.18) then _{𝑆𝑛(𝑐,𝜏)}𝑑𝑥₁⋯𝑑𝑥_𝑛=1𝑛!^𝑛_𝑖=1𝜏𝑐_𝑖,_{𝑆𝑛(𝑐,𝜏)}𝑥_𝑗𝑑𝑥₁⋯𝑑𝑥_𝑛=1(𝑛+1)!𝑐𝜏_{𝑗𝑛𝑖=1}𝜏𝑐_𝑖.(2.19) Thus, in the particular case that Δ=𝑆_𝑛(𝑐,𝜏), the integral ∫_𝑀Δ(𝜔_Δ)^𝑛 is a monomial of degree 𝑛 in 𝜏, and ∫_𝑀Δ⟨𝜇_Δ,𝐛⟩(𝜔_Δ)^𝑛 is a monomial of degree 𝑛+1.

We return to the general case in which Δ is the polytope defined in (1.3). Its vertices are the solutions to 𝑥,𝐧_𝑗𝑎=𝑘_𝑗𝑎,𝑎=1,…,𝑛;(2.20) hence, the coordinates of any vertex of Δ are linear combinations of the 𝑘_𝑗.

A hyperplane in ℝ^𝑛 through a vertex (𝑥⁰¹,…,𝑥^0𝑛) of Δ is given by an equation of the form ⟨𝑥,𝐧⟩=𝑥⁰,𝐧=∶𝜅.(2.21) Thus, the independent term 𝜅 is a linear combination (l. c.) of the 𝑘_𝑗. Moreover, the coordinates of the common point of 𝑛 hyperplanes ⟨𝑥,𝐧̃_𝑖⟩=𝜅_𝑖,(2.22) with 𝜅_𝑖 l. c. of, the 𝑘_𝑗 are also l. c. of the 𝑘_𝑗.

By drawing hyperplanes through vertices of Δ (or more generally, through points which are the intersection of 𝑛 hyperplanes as (2.22)), we can obtain a family {_𝛽𝑆} of subsets of Δ such that(a)each _𝛽𝑆 is the transformed of a simplex 𝑆_𝑛(𝑏,𝜏) by an element of the group of Euclidean motions in ℝ^𝑛.(b)For 𝛼≠𝛽, _𝛼𝑆∩_𝛽𝑆 is a subset of the border of _𝛼𝑆.(c)⋃_𝛽𝛽𝑆=Δ.

Thus, by construction, each facet of _𝛽𝑆 is contained in a hyperplane 𝜋 of the form ⟨𝑥,𝐧⟩=𝜅, with 𝜅 l. c. of the 𝑘_𝑗.

On the other hand, the hyperplane 𝜋 is transformed by an element of SO(𝑛) in an hyperplane ⟨𝑥,𝐧^⟩=𝜅. If 𝒯 is a translation in ℝ^𝑛 which applies 𝑆_𝑛(𝑏,𝜏) onto _𝛽𝑆, then this transformation maps (0,…,0) in a vertex 𝑎=(𝑎₁,…,𝑎_𝑛) of _𝛽𝑆. So, the translation 𝒯 transforms 𝜋 in ⟨𝑥,𝐧⟩=𝜅+⟨𝑎,𝐧⟩=∶𝜅^. As each 𝑎_𝑗 is a l. c. of the 𝑘_𝑗, so is 𝜅^. Hence, any element of the group of Euclidean motions in ℝ^𝑛 which maps 𝑆_𝑛(𝑏,𝜏) onto _𝛽𝑆 transforms the hyperplane 𝜋𝑥,𝐧^=𝜅^,(2.23) with 𝜅^ a l. c. of the 𝑘_𝑗.

Let assume that (𝑅𝒯_𝑎)(𝑆(𝑏,𝜏))=_𝛽𝑆, with 𝑅∈SO(𝑛) and 𝒯_𝑎 the translation defined by 𝑎. Then the oblique facet of 𝑆(𝑏,𝜏), contained in the hyperplane ∑𝑏_𝑖𝑥_𝑖=𝜏, is the image by 𝑇_−𝑎𝑅⁻¹ of a facet of _𝛽𝑆, which in turn is contained in a hyperplane of (2.23) (𝜅^ being a l. c. of the 𝑘_𝑗). The argument of the preceding paragraph applied to 𝑅⁻¹ and 𝒯_−𝑎 proves that 𝜏 is a l. c. of the 𝑘_𝑗. Hence, by (2.19), the integral _𝛽𝑆𝑑𝑥₁⋯𝑑𝑥_𝑛=_{𝑆𝑛(𝑏,𝜏)}𝑑𝑥₁⋯𝑑𝑥_𝑛(2.24) is a monomial of degree 𝑛 of a l. c. of the 𝑘_𝑗. Thus, _𝑀𝜔_Δ^𝑛=_𝛽_𝛽𝑆𝑑𝑥₁⋯𝑑𝑥_𝑛,(2.25) is a homogeneous polynomial of degree 𝑛 of the 𝑘_𝑗.

Similarly, _𝑀Δ⟨𝜇_Δ,𝐛⟩𝜔_Δ^𝑛(2.26) is a homogeneous polynomial of degree 𝑛+1 of the 𝑘_𝑗. Analogous results hold for _𝐹𝑗𝑑𝑥₁⋯𝑑𝑥_𝑗⋯𝑑𝑥_𝑛,_𝐹𝑗⟨𝜇_Δ,𝐛⟩𝑑𝑥₁⋯𝑑𝑥_𝑗⋯𝑑𝑥_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 Δ, if   𝐛 belongs to the integer lattice of   𝔱, then 𝐼(𝑘;𝐛) is a rational function of the 𝑘_𝑗, for 𝑘∈𝒞_Δ.

Analogously, we have the following proposition.

Proposition 2.4. If (Δ,𝐛) is mass linear pair, then 𝐼(𝑘;𝐛) is a homogeneous polynomial in the 𝑘_𝑗 of degree 𝑛, when 𝑘∈𝒞_Δ.

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

Lemma 2.5. If 𝑘̂_𝑗=𝑠𝑘_𝑗 for 𝑗=1,…,𝑚, with 𝑠∈ℝ, then Cm(Δ(𝐧,𝑘̂))=𝑠Cm(Δ(𝐧,𝑘)).

Proof. The vertices of Δ(𝐧,𝑘) are the solutions of (2.20), and the vertices of Δ(𝐧,𝑘̂) are the solutions of ⟨𝑥,𝐧_𝑗𝑎⟩=𝑠𝑘_𝑗𝑎, with 𝑎=1,…,𝑛. Thus, the vertices of Δ(𝐧,𝑘̂) are those of Δ(𝐧,𝑘) 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 Δ defined by (1.3) is Delzant and let 𝑘 be an element of 𝒞_Δ. We denote by 𝑀_(𝑘), 𝜔_(𝑘), and 𝜇_(𝑘), the manifold, the symplectic structure and the moment map (resp.) determined by Δ(𝑘). The facets of Δ(𝑘) will be denoted by 𝐹_(𝑘)𝑗.

Let 𝐛 be an element in the integer lattice of 𝔱. We put 𝐴_(𝑘)∶=_𝑀(𝑘)𝜇_(𝑘),𝐛𝜔_(𝑘)^𝑛,𝐵_(𝑘)∶=_𝑀(𝑘)𝜔_(𝑘)^𝑛.(3.1)

By (2.9), (1/𝑛!)𝐵_(𝑘) is the Euclidean volume of the polytope Δ(𝑘). Given a facet 𝐹_(𝑘)𝑗, we can assume that 𝑗∈{1,…,𝑛} (see third paragraph of Section 2). So, 𝐹_(𝑘)𝑗 is defined by the equation 𝑥_𝑗=0. If we make an infinitesimal variation of the facet 𝐹_(𝑘)𝑗, by means of the translation defined by 𝑘_𝑗→𝑘_𝑗+𝜖 (keeping unchanged the other 𝑘_𝑖), then the volume of Δ(𝑘) changes according to 1𝑛!𝐵^(𝑘)⟶1𝑛!𝐵^(𝑘)+𝜖_{𝐹(𝑘)𝑗}𝑑𝑥₁⋯𝑑𝑥_𝑗⋯𝑑𝑥_𝑛+𝑂𝜖².(3.2) We write 𝑑𝑋^𝑗 for 𝑑𝑥₁⋯𝑑𝑥_𝑗⋯𝑑𝑥_𝑛. Thus, 𝜕𝐵_(𝑘)𝜕𝑘_𝑗=𝑛!_{𝐹(𝑘)𝑗}𝑑𝑋^𝑗,𝜕𝐴^(𝑘)𝜕𝑘_𝑗=𝑛!_{𝐹(𝑘)𝑗}𝜇_(𝑘),𝐛𝑑𝑋^𝑗.(3.3) So, by (2.11), 𝜕𝜕𝑘_𝑗⟨Cm(Δ(𝑘)),𝐛⟩=𝑛!𝐵_(𝑘)²𝐵^(𝑘)_{𝐹(𝑘)𝑗}𝜇_(𝑘),𝐛𝑑𝑋^𝑗−𝐴_(𝑘)_{𝐹(𝑘)𝑗}𝑑𝑋^𝑗.(3.4) From (2.13) and (2.14), it follows ^𝑚_𝑗=1𝜕𝜕𝑘_𝑗⟨Cm(Δ(𝑘)),𝐛⟩=−1𝐵_(𝑘)𝐼(𝑘;𝐛).(3.5) Thus, we have proved the following proposition.

Proposition 3.1. 𝐼(𝑘;𝐛)=0 for all 𝑘∈𝒞_Δ if and only if   ∑^𝑚𝑗=1(𝜕/𝜕𝑘_𝑗)⟨Cm(Δ(𝑘)),𝐛⟩=0, for all 𝑘∈𝒞_Δ.

Next, we will parametrize the quotient 𝒞_Δ/𝐺 (of classes of polytopes in 𝒞_Δ module translation) defined in Section 1.

After a possible renumbering, we may assume that the intersection of facets 𝐹₁,…,𝐹_𝑛 is a vertex of Δ. Thus, the conormals 𝐧₁,…,𝐧_𝑛 are linearly independent in 𝔱. So, given 𝑘∈𝒞_Δ, there is a unique 𝑣∈𝔱^∗, such that ⟨𝑣,𝐧_𝑖⟩+𝑘_𝑖=0,𝑖=1,…,𝑛.(3.6) (Expressing the 𝐧_𝑖 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(𝑘) depends linearly on the 𝑘_𝑖; that is, ⟨𝑣(𝑘),𝐜⟩ is a linear function of 𝑘₁,…,𝑘_𝑛, for all 𝐜∈𝔱.

If 𝑚−𝑛=2, we write 𝜆=𝑘_𝑛+1+𝑣(𝑘),𝐧_𝑛+1,𝜏=𝑘_𝑚+⟨𝑣(𝑘),𝐧_𝑚⟩,(3.7) where 𝑣(𝑘) the element in 𝔱^∗ defined by (3.6). From the linearity of 𝑣 with respect to the 𝑘_𝑖, it follows that 𝜆 and 𝜏 are linear combinations of 𝑘₁,…,𝑘_𝑚.

The polytope in 𝒞_Δ defined by (𝑘^1=0,…,𝑘^𝑛=0,𝜆,𝜏) will be denoted by Δ₀(𝜆,𝜏). It is the result of the translation of Δ(𝑘) by the vector 𝑣(𝑘); that is, Δ₀(𝜆,𝜏)=Δ(𝑘)+𝑣(𝑘).(3.8)

Let 𝐛 an element in the integer lattice of 𝔱, we define the function 𝑔 by 𝑔(𝜆,𝜏)∶=CmΔ₀(𝜆,𝜏),𝐛.(3.9) The function 𝑔 is defined on the pairs (𝜆,𝜏) such that (0,…,0,𝜆,𝜏)∈𝒞_Δ. By Lemma 2.5, it follows 𝑔(𝑠𝜆,𝑠𝜏)=𝑠𝑔(𝜆,𝜏),(3.10) for any real number 𝑠 such that (𝑠𝜆,𝑠𝜏) belongs to the domain of 𝑔. This property implies that 𝑔=𝜆𝜕𝑔𝜕𝜆+𝜏𝜕𝜕𝑔𝜏.(3.11)

Theorem 3.2. If 𝐼(𝑘;𝐛)=0, for all 𝑘∈𝒞_Δ and 𝑟=2, then ⟨Cm(Δ(𝑘)),𝐛⟩=∑_𝑗𝑅_𝑗𝑘_𝑗, with 𝑅_𝑗 constant (i.e., (Δ,𝐛) is a mass linear pair) and ∑_𝑗𝑅_𝑗=0.

Proof. We set 𝑓(𝑘₁,…,𝑘_𝑚)∶=⟨Cm(Δ(𝑘)),𝐛⟩. It follows from (3.8) that 𝑓(𝑘)=𝑔(𝜆,𝜏)−⟨𝑣(𝑘),𝐛⟩.(3.12)
By the hypothesis and Proposition 3.1, ^𝑚_𝑗=1𝜕𝑓𝜕𝑘_𝑗=0.(3.13)
Since ^𝑚_𝑗=1𝜕𝑓𝜕𝑘_𝑗=𝜕𝑔𝜕𝜆^𝑚_𝑗=1𝜕𝜆𝜕𝑘_𝑗+𝜕𝑔𝜕𝜏^𝑚_𝑗=1𝜕𝜏𝜕𝑘_𝑗−𝜕𝑣𝜕𝑘_𝑗,𝐛,(3.14) from (3.13), we deduce 𝑝𝜕𝑔𝜕𝜆+𝑞𝜕𝜕𝑔𝜏−𝑡=0,(3.15) where 𝑝, 𝑞, 𝑡 stand for the following constants 𝑝=^𝑚_𝑗=1𝜕𝜆𝜕𝑘_𝑗,𝑞=^𝑚_𝑗=1𝜕𝜏𝜕𝑘_𝑗,𝑡=𝜕𝑣𝜕𝑘_𝑗,𝐛.(3.16)
Since 𝑞𝜆−𝑝𝜏 and 𝑡𝜏−𝑞𝑔 are first integrals of (3.15), the general solution of this equation is 𝑔(𝜆,𝜏)=𝑡𝑞𝜏+Φ(𝑞𝜆−𝑝𝜏),(3.17) where Φ is a derivable function of one variable.
It follows from (3.11) and (3.17) that Φ(𝑢)=𝑢Φ^(𝑢).(3.18) Thus, Φ(𝑢)=𝛼𝑢, with 𝛼 constant. We have for 𝑓𝑓(𝑘)=𝑏𝑞𝜏+𝛼(𝑞𝜆−𝑝𝜏)−⟨𝑣(𝑘),𝐛⟩.(3.19) In other words, 𝑓 is a linear function of the 𝑘_𝑗; that is, 𝑓(𝑘)=∑_𝑗𝑅_𝑗𝑘_𝑗, with 𝑅_𝑗 constant. From (3.13), it follows ∑_𝑗𝑅_𝑗=0.

Remark 3.3. The proof of Theorem 3.2 can be adapted to the simpler case when 𝑟=1. In this case, the function 𝑔(𝜆)=⟨Cm(Δ₀(𝜆)),𝐛⟩ satisfies 𝑝(d𝑔/d𝜆)−𝑡=0 and 𝑔(𝑠𝜆)=𝑠𝑔(𝜆). So, 𝑔(𝜆)=(𝑡/𝑝)𝜆 and 𝑓(𝑘)=(𝑡/𝑝)𝜆+⟨𝑣(𝑘),𝐛⟩ is a linear map of the variables 𝑘_𝑗.
On the other hand, the proof of this theorem does not admit an adaptation to the case 𝑟>2. In fact, the corresponding function Φ would be a function of 𝑟−1 variables Φ(𝑢₁,…,𝑢_𝑟−1). The equation which corresponds to (3.18) in this case would be Φ=^𝑟−1_𝑖=1𝑢_𝑖𝜕Φ𝜕𝑢_𝑖.(3.20) But this condition does not imply the linearity of Φ.
When (Δ,𝐛) is a mass linear pair as in (1.5), by (3.5) 𝐼(𝑘;𝐛)=−𝐵_(𝑘)_𝑗𝑅_𝑗,(3.21) for all 𝑘∈𝒞_Δ. From (3.21), we deduce the following proposition.

Proposition 3.4. Let (Δ,𝐛) be a mass linear pair. 𝐼(𝑘;𝐛)=0 for all 𝑘∈𝒞_Δ if and only if ∑_𝑗𝑅_𝑗=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 Δ to admit mass linear functions. We write Cṁ(Δ(𝑘))∶=dd𝜖|||_𝜖=0Cm(Δ(𝑘+𝜖̌)),(3.22) with 𝜖̌=(𝜖,…,𝜖).

Proposition 3.5. If all points Cṁ(Δ(𝑘)), for 𝑘∈𝒞_Δ, belong to a hyperplane of (ℝ^𝑛)^∗ with a conormal vector in ℤ^𝑛 and 𝑟≤2, then Δ admits a mass linear function.

Proof. Let 𝐛∈ℤ^𝑛 be a conormal vector to the hyperplane, then 0=Cṁ(Δ(𝑘)),𝐛=_𝑗𝜕𝜕𝑘_𝑗Cm(Δ(𝑘)),𝐛.(3.23) By (3.5), 𝐼(𝑘,𝐛)=0; Theorem 3.2 applies and (Δ,𝐛) is a mass linear pair.

Proposition 3.6. Let Δ be a Delzant polytope, such that 𝑘=0 belongs to the closure of 𝒞_Δ. If 𝑟≤2, a necessary condition for the vanishing of 𝐼(𝑘;𝐛) on 𝒞_Δ is dd𝜖|||_𝜖=0Cm(Δ(𝜖̌)),𝐛=0.(3.24)

Proof. If 𝐼(𝑘;𝐛) vanishes on 𝒞_Δ, then (Δ,𝐛) is a linear pair, by Theorem 1.1. Thus, ⟨Cm(Δ(𝑘)),𝐛⟩=∑_𝑗𝑅_𝑗𝑘_𝑗+𝐶, on 𝒞_Δ. So, given 𝑘∈𝒞_Δ and 𝜖 small enough ⟨Cm(Δ(𝑘+𝜖̌)),𝐛⟩=_𝑗𝑅_𝑗𝑘_𝑗+𝜖_𝑗𝑅_𝑗+𝐶.(3.25) By Theorem 1.1, ∑_𝑗𝑅_𝑗=0. Thus, for any 𝑘∈𝒞_Δ, dd𝜖|||_𝜖=0⟨Cm(Δ(𝑘+𝜖̌)),𝐛⟩=0.(3.26) Taking the limit as 𝑘→0, 0=lim_𝑘→0dd𝜖|||_𝜖=0⟨Cm(Δ(𝑘+𝜖̌)),𝐛⟩=dd𝜖|||_𝜖=0Cm(Δ(𝜖̌)),𝐛.(3.27)

Next, we will describe a geometric interpretation of the number ∑_𝑗𝑅_𝑗. Given an arbitrary Delzant polytope Δ. If 𝑎 is a vector of 𝔱^∗, then CmΔ𝑘^=Cm(Δ(𝑘))+𝑎,(3.28) if 𝑘^𝑗=𝑘_𝑗+⟨𝑎,𝐧_𝑗⟩.

We will denote by 𝑑 the element of 𝔱^∗ defined by the following relation CmΔ𝑘̃=Cm(Δ(𝑘))+𝑑,(3.29) with 𝑘̃_𝑗=𝑘_𝑗+1 for all 𝑗.

From (3.28) and (3.29), we have CmΔ𝑘_𝑗+𝑑,𝐧_𝑗=CmΔ𝑘_𝑗+𝑑=CmΔ𝑘̃_𝑗=𝑘_𝑗+1.(3.30)

Now, we assume that (Δ,𝐛) is a mass linear pair. From (1.5), it follows CmΔ𝑘_𝑗CmΔ𝑘_𝑗+𝑑,𝐧_𝑗,𝐛=𝑅_𝑗𝑘_𝑗+𝑅_𝑗𝑑,𝐧_𝑗+𝐶,+𝑑,𝐛=𝑅_𝑗𝑘_𝑗+⟨𝑑,𝐛⟩+𝐶,CmΔ𝑘̃,𝐛=𝑅_𝑗𝑘_𝑗+𝑅_𝑗+𝐶.(3.31) These formulas allow us to state the following proposition that gives an interpretation of the sum ∑_𝑗𝑅_𝑗 in terms of the variation of Cm(Δ(𝑘)) with the 𝑘_𝑗.

Proposition 3.7. Let (Δ,𝐛) be a mass linear pair as in (1.5). Then, _𝑗𝑅_𝑗𝑑,𝐧_𝑗=⟨𝑑,𝐛⟩=_𝑗𝑅_𝑗,(3.32)𝑑 being the element of 𝔱^∗ defined by (3.29).

4. Examples

In this section, we will deduce the particular form which adopts Theorem 1.2 when Δ 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 Δ(𝑘) and the condition for (Δ,𝐛) to be a mass linear pair. We will dedicate a subsection to each type.

4.1. Hirzebruch Surfaces

Given 𝑟∈ℤ_>0 and 𝜏,𝜆∈ℝ_>0 with 𝜎∶=𝜏−𝑟𝜆>0, in [7], we considered the Hirzebruch surface 𝑁 determined by these numbers. 𝑁 is the quotient 𝑧∈ℂ⁴∶||𝑧₁||²+𝑟||𝑧₂||²+||𝑧₄||²=𝜏/𝜋,||𝑧₂||²+||𝑧₃||²=𝜆/𝜋𝕋,(4.1) where the equivalence defined by 𝕋=(𝑆¹)² is given by (𝑎,𝑏)⋅𝑧₁,𝑧₂,𝑧₃,𝑧₄=𝑎𝑧₁,𝑎^𝑟𝑏𝑧₂,𝑏𝑧₃,𝑎𝑧₄,(4.2) for (𝑎,𝑏)∈(𝑆¹)².

The manifold 𝑁 equipped with the following (U(1))² action 𝜖₁,𝜖₂𝑧_𝑗=𝜖₁𝑧₁,𝜖₂𝑧₂,𝑧₃,𝑧₄,(4.3) is a toric manifold. The corresponding moment polytope Δ is the trapezium in ℝ² with vertices 𝑃₁=(0,0),𝑃₂=(0,𝜆),𝑃₃=(𝜏,0),𝑃₄=(𝜎,𝜆).(4.4) That is, 𝑁 is the toric manifold 𝑀_Δ determined by the trapezium Δ.

As the conormals to the facets of Δ are the vectors 𝐧₁=(−1,0), 𝐧₂=(0,−1), 𝐧₃=(0,1), and 𝐧₄=(1,𝑟), the facets of a generic polytope Δ(𝑘) in 𝒞_Δ are on the straights −𝑥=𝑘₁,−𝑦=𝑘₂,𝑦=𝑘₃,𝑥+𝑟𝑦=𝑘₄.(4.5) The vertices of Δ(𝑘) are the points −𝑘₁,−𝑘₂,−𝑘₁,𝑘₃,𝑘₄−𝑟𝑘₃,𝑘₃,𝑘₄+𝑟𝑘₂,−𝑘₂.(4.6) Thus, the translation in the plane 𝑥, 𝑦 defined by (−𝑘₁,−𝑘₂) transforms the trapezium determined by the vertices (4.4) in Δ(𝑘) if 𝜏=𝑘₄+𝑟𝑘₂+𝑘₁,𝜆=𝑘₃+𝑘₂.(4.7) So, Cm(Δ(𝑘))=Cm(Δ)+−𝑘₁,−𝑘₂.(4.8) Moreover, the mass center of Δ is Cm(Δ)=3𝜏²−3𝑟𝜏𝜆+𝑟²𝜆²3(2𝜏−𝑟𝜆),33𝜆(𝜏2𝜏−−2𝑟𝑟𝜆𝜆²).(4.9)

The chamber 𝒞_Δ consists of the points (𝑘₁,…,𝑘₄) such that 𝜏−𝑟𝜆>0, with 𝜏 and 𝜆 given by (4.7). So, the point 𝑘=0 belongs to the closure of 𝒞_Δ. From (4.8), together with (4.7) and (4.9), it follows Cm(Δ(𝜖̌))=𝑟²𝜖12,−6𝑟𝜖,(4.10) where 𝜖̌=(𝜖,𝜖,𝜖,𝜖). By Proposition 3.6, if 𝐼(𝑘;𝐛) with 𝐛=(𝑏₁,𝑏₂)∈ℤ² vanishes on the chamber 𝒞_Δ, then 𝑟𝑏₁−2𝑏₂=0.

On the other hand, from (4.9) and (4.8), it follows ⟨Cm(Δ(𝑘)),𝐛⟩=3𝜏²−3𝑟𝜏𝜆+𝑟²𝜆²𝑏₁+3𝜆𝜏−2𝑟𝜆²𝑏₂3(2𝜏−𝑟𝜆)−𝑘¹𝑏₁−𝑘₂𝑏₂.(4.11) By (4.7), expression (4.11) is linear in the 𝑘_𝑖 if and only if 3𝜏²−3𝑟𝜏𝜆+𝑟²𝜆²𝑏₁+3𝜆𝜏−2𝑟𝜆²𝑏₂3(2𝜏−𝑟𝜆)(4.12) is linear in 𝜏, 𝜆. That is, if and only if there exist constants 𝐴, 𝐵 such that for al 𝜏, 𝜆3𝜏²−3𝑟𝜏𝜆+𝑟²𝜆²𝑏₁+3𝜆𝜏−2𝑟𝜆²𝑏₂=3(2𝜏−𝑟𝜆)(𝐴𝜏+𝐵𝜆).(4.13) From this relation, it follows the above condition 𝑟𝑏₁=2𝑏₂. In this case (4.11) reduces to ⟨Cm(Δ(𝑘),𝐛⟩=−𝑏¹2𝑘¹+𝑏¹2𝑘⁴.(4.14)

Comparing (1.5) with (4.14), we obtain 𝑅₁=−𝑅₄=−𝑏₁/2, 𝑅₂=𝑅₃=0; so, ∑_𝑗𝑅_𝑗=0. That is, the condition ∑_𝑗𝑅_𝑗=0 holds for all the mass pairs (Δ,𝐛) when Δ is the polytope associated to a Hirzebruch surface. Hence, we have following proposition.

Proposition 4.1. (Δ,𝐛) is a mass linear pair if and only if 𝑟𝑏₁=2𝑏₂. Moreover, in this case ∑_𝑗𝑅_𝑗=0.

By Theorem 1.2, we have the following corollary.

Corollary 4.2. If 𝑟𝑏₁≠2𝑏₂, then 𝜓_𝐛 generates an infinite cyclic subgroup in 𝜋₁(Ham(𝑀_Δ,𝜔_Δ)).

Remark 4.3. We denote by 𝜙_𝑡 the following isotopy of 𝑀_Δ𝜙_𝑡[𝑧]=𝑒^{2𝜋𝑖𝑡}𝑧₁,𝑧₂,𝑧₃,𝑧₄,(4.15) where 𝜙 is a loop in the Hamiltonian group of 𝑀_Δ. By 𝜙^, we denote the Hamiltonian loop 𝜙^𝑡[𝑧]=𝑧₁,𝑒^{2𝜋𝑖𝑡}𝑧₂,𝑧₃,𝑧₄.(4.16) In Theorem 8 of [11], we proved that 𝐼(𝜙^)=(−2/𝑟)𝐼(𝜙). If 𝐛=(𝑏₁,𝑏₂)∈ℤ², then 𝐼𝜓_𝐛=𝑏₁𝐼(𝜙)+𝑏₂𝐼𝜙^=𝑏₁−2𝑟𝑏²𝐼(𝜙).(4.17) That is, 𝐼(𝜓_𝐛)=0 if and only if 𝑟𝑏₁=2𝑏₂, which is in agreement with Proposition 4.1 and Theorem 1.1.

4.2. Δ_𝑝 Bundle over Δ₁

Given the integer 𝑝>1, as McDuff and Tolman in [3], we consider the following vectors in ℝ^𝑝+1: 𝐧_𝑖=−𝑒_𝑖,𝑖=1,…,𝑝,𝐧_𝑝+1=^𝑝_𝑖=1𝑒_𝑖,𝐧_𝑝+2=−𝑒_𝑝+1,𝐧_𝑝+3=𝑒_𝑝+1−^𝑝_𝑖=1𝑎_𝑖𝑒_𝑖,(4.18) where 𝑒₁,…,𝑒_𝑝+1 is the standard basis of ℝ^𝑝+1 and 𝑎_𝑖∈ℤ. We write 𝐚∶=𝑎₁,…,𝑎_𝑝∈ℤ^𝑝,𝐴∶=^𝑝_𝑖=1𝑎_𝑖,𝐚⋅𝐚=^𝑝_𝑖=1𝑎^2𝑖.(4.19)

Let 𝜆, 𝜏 be real positive numbers with 𝜆+𝑎_𝑖>0, for 𝑖=1,…,𝑝. In this subsection, we will consider the polytope Δ in (ℝ^𝑝+1)^∗ defined by the above conormals 𝐧_𝑗 and the following 𝑘_𝑗: 𝑘₁=⋯=𝑘_𝑝=𝑘_𝑝+2=0,𝑘_𝑝+1=𝜏,𝑘_𝑝+3=𝜆.(4.20)

This polytope will be also denoted by Δ₀(𝜆,𝜏). It is a Δ_𝑝 bundle on Δ₁ (see [3]). When 𝑝=2, Δ=Δ₀(𝜆,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,𝜆), (𝜏,0,𝜆+𝑎₁𝜏), and (0,𝜏,𝜆+𝑎₂𝜏) (see Figure 1).

Figure 1 

Δ2 bundle over Δ1.

We assume that the above polytope Δ is a Delzant polytope. The manifold (2.2) is in this case 𝑀_Δ=𝑧∈ℂ^𝑝+3∶∑^𝑝+1𝑖=1||𝑧_𝑖||²=𝜏/𝜋,−∑^𝑝𝑗=1𝑎_𝑗||𝑧_𝑗||²+||𝑧_𝑝+2||²+||𝑧_𝑝+3||²=𝜆/𝜋≃,(4.21) where (𝑧_𝑗)≃(𝑧^𝑗) if and only if there are 𝛼,𝛽∈𝑈(1) such that 𝑧^𝑗=𝛼𝛽^−𝑎𝑗𝑧_𝑗,𝑗=1,…,𝑝,𝑧^𝑝+1=𝛼𝑧_𝑝+1,𝑧^𝑘=𝛽𝑧_𝑘,𝑘=𝑝+2,𝑝+3.(4.22) Thus, 𝑀_Δ is the total space of the fibre bundle ℙ(𝐿₁⊕⋯⊕𝐿_𝑝⊕ℂ)→ℂ𝑃¹, where 𝐿_𝑗 is the holomorphic line bundle over ℂ𝑃¹ with Chern number 𝑎_𝑗.

The symplectic form (2.4) is 𝜔_Δ=12𝜎¹+⋯+𝜎_𝑝+𝜎_𝑝+2,(4.23) where 𝜎_𝑘=𝑑𝜌^2𝑘∧𝑑𝜑_𝑘.

And the moment map 𝜇_Δ([𝑧])=𝑥₁,…,𝑥_𝑝,𝑥_𝑝+2,(4.24) where 𝑥_𝑖∶=𝜋𝜌^2𝑖.

Proposition 4.4. The coordinates 𝑥_𝑗 of Cm(Δ₀(𝜆,𝜏)) are given by 𝑥_𝑘=𝜏𝑝+2𝜆(𝑝+2)+𝜏𝐴+𝑎^𝑘𝜆(𝑝+1)+𝜏𝐴,for𝑘=1,…,𝑝,(4.25)𝑥_𝑝+2=12(𝑝+1)(𝑝+2)𝜆²+2(𝑝+2)𝐴𝜆𝜏+𝐚⋅𝐚+𝐴²𝜏²(𝑝+2)((𝑝+1)𝜆+𝐴𝜏).(4.26)

Proof. Since the points [𝑧]∈𝑀_Δ satisfy |𝑧_𝑝+2|²≤𝜆/𝜋+∑^𝑝𝑗=1𝑎_𝑗|𝑧_𝑗|², by (2.9) and Lemma 2.2, we have _𝑀Δ𝜔_Δ^𝑝+1=(𝑝+1)!_{𝑆𝑝(𝜏)}𝜆+^𝑝_𝑗=1𝑎_𝑗𝑥_𝑗=(𝑝+1)!𝜆𝑝𝜏!^𝑝+𝜏^𝑝+1𝐴(𝑝+1)!.(4.27) Similarly, for 𝑘=1,…,𝑝_𝑀Δ𝑥_𝑘𝜔_Δ^𝑝+1=(𝑝+1)!𝜆𝜏^𝑝+1(𝑝+1)!+(𝑝𝜏+2^𝑝+2)!_𝑗≠𝑘𝑎_𝑗+2𝜏^𝑝+2𝑎_𝑘(𝑝+2)!.(4.28) The 𝑘th coordinate of Cm(Δ), 𝑥_𝑘, is the quotient of (4.28) by (4.27); that is, 𝑥_𝑘=𝜏𝑝+2𝜆(𝑝+2)+𝜏𝐴+𝑎^𝑘𝜆(𝑝+1)+𝜏𝐴.(4.29)
For the 𝑝+2-coordinate of Cm(Δ), we need to calculate ∫_𝑀𝑥_𝑝+2(𝜔_Δ)^𝑝+1. By Lemma 2.2, 1(𝑝+1)!_𝑀𝑥_𝑝+2𝜔_Δ^𝑝+1=12_{𝑆𝑝(𝜏)}𝜆+^𝑝_𝑗=1𝑎_𝑗𝑥_𝑗²=12𝜆²𝜏^𝑝𝑝!+2(𝑝𝐴+𝜆𝜏1)!^𝑝+1+𝐚⋅𝐚+𝐴²𝜏^𝑝+2(𝑝+2)!.(4.30) Formula (4.26) is a consequence of (4.27) together with (4.30).

The translation in (ℝ^𝑝+1)^∗ defined by the vector (−𝑘₁,…,−𝑘_𝑝,−𝑘_𝑝+2) transforms the hyperplanes ⟨𝑥,𝐧_𝑝+3⟩=𝜆 and ⟨𝑥,𝐧_𝑝+1⟩=𝜏 in 𝑥,𝐧_𝑝+3=𝜆−𝑘_𝑝+2+^𝑝_𝑗=1𝑎_𝑗𝑘_𝑗,𝑥,𝐧_𝑝+1=𝜏−^𝑝_𝑗=1𝑘_𝑗,(4.31) respectively.

Let Δ(𝑘) be a polytope with 𝑘=(𝑘₁,…,𝑘_𝑝+3) generic in the chamber 𝒞_Δ. From (4.31), it follows that Δ(𝑘) is the image of the polytope Δ₀(𝜆,𝜏) by the translation determined by (−𝑘₁,…,−𝑘_𝑝,−𝑘_𝑝+2), whenever 𝑘_𝑝+2−^𝑝_𝑗=1𝑎_𝑗𝑘_𝑗+𝑘_𝑝+3=𝜆,^𝑝_𝑗=1𝑘_𝑗+𝑘_𝑝+1=𝜏.(4.32) In this case, Cm(Δ(𝑘))=CmΔ₀(𝜆,𝜏)−𝑘₁,…,𝑘_𝑝,𝑘_𝑝+2.(4.33)

According to (4.32), the coordinates of the mass center Cm(Δ(𝜖̌)), with 𝜖̌=(𝜖,…,𝜖), can be obtained substituting in (4.25) and in (4.26) 𝜆 by 𝜖−^𝑝_𝑗=1𝑎_𝑗𝜖+𝜖=(2−𝐴)𝜖(4.34) and 𝜏 by (𝑝+1)𝜖, and finally take into account (4.33). These operations give 𝑥_𝑗(Δ(𝜖̌))=𝜖2(𝑝+2)(𝑝+1)𝑎^𝑗−𝐴,𝑗=1,…,𝑝,𝑥_𝑝+2(Δ(𝜖̌))=𝜖4(𝑝+2)−𝐴²+(𝑝+1)(𝐚⋅𝐚).(4.35)

Given 𝐛=(𝑏₁,…,𝑏_𝑝,𝑏)≡(𝐛̂,𝐛̇), with 𝐛̂=(𝑏₁,…,𝑏_𝑝,0) and 𝐛̇=(0,…,0,𝑏),dd𝜖|||_𝜖=0Cm(Δ(𝜖̌)),𝐛=4(𝑝1+2)(𝑝+1)2𝐚⋅𝐛̂−𝑏𝐚⋅𝐚−𝐴(2𝐵+𝑏𝐴),(4.36) where 𝐚⋅𝐛̂=∑^𝑝𝑗=1𝑎_𝑗𝑏_𝑗 and 𝐵=∑^𝑝𝑗=1𝑏_𝑗.

By Proposition 3.6, we have the following theorem.

Theorem 4.5. Let Δ be the Δ_𝑝 bundle over Δ₁ defined by (4.18) and (4.20). Given 𝐛=(𝐛̂,𝐛̇)∈ℤ^𝑝+1, if (𝑝+1)2𝐚⋅𝐛̂−𝑏𝐚⋅𝐚−𝐴(2𝐵+𝑏𝐴)≠0,(4.37) then 𝜓_𝐛 defines an infinite cyclic subgroup in the fundamental group 𝜋₁(Ham(𝑀_Δ,𝜔_Δ)).

It is straightforward to check that (𝑝+1)2𝐚⋅𝐛̂−𝑏𝐚⋅𝐚−𝐴(2𝐵+𝑏𝐴)=0(4.38) is also a sufficient condition for (Δ,𝐛) to be a mass linear pair.

Since CmΔ₀(𝜆,𝜏),𝐛=CmΔ₀(𝜆,𝜏),𝐛̂+CmΔ₀(𝜆,𝜏),𝐛̇,(4.39) if (4.38) holds, using (4.25) and (4.26), one obtains CmΔ₀(𝜆,𝜏),𝐛=𝑏2𝜆+2𝑏𝐚⋅𝐚+𝐴²(𝑝+2)𝐴+𝐚(𝑝⋅𝐛+̂+2)𝐴𝐴𝐵𝜏.(4.40) By (4.33), for 𝑘∈𝒞_Δ, ⟨Cm(Δ(𝑘)),𝐛⟩=CmΔ₀(𝜆,𝜏),𝐛−^𝑝_𝑗=1𝑏_𝑗𝑘_𝑗−𝑏𝑘_𝑝+2,(4.41) with 𝜆 and 𝜏 given by (4.32).

If 𝐛=𝐛̂, the condition (4.38) reduces to (𝑝+1)𝐚⋅𝐛̂=𝐴𝐵 and ⟨Cm(Δ(𝑘)),𝐛⟩=𝐚⋅𝐛̂+𝐴𝐵(𝑝+2)𝐴^𝑝_𝑗=1𝑘_𝑗+𝑘_𝑝+1−^𝑝_𝑗=1𝑏_𝑗𝑘_𝑗.(4.42) Hence, ⟨Cm(Δ(𝑘)),𝐛⟩=∑𝑅_𝑗𝑘_𝑗, where 𝑅_𝑗=𝐚⋅𝐛̂+𝐴𝐵(𝑝+2)𝐴−𝑏^𝑗,𝑗=1,…,𝑝,𝑅_𝑝+1=𝐚⋅𝐛̂+𝐴𝐵(𝑝+2)𝐴,𝑅^𝑝+2=𝑅_𝑝+3=0.(4.43) So, ^𝑝+3_𝑗=1𝑅_𝑗=(𝑝+1)𝐚⋅𝐛̂−𝐴𝐵(𝑝+2)𝐴=0.(4.44)

A similar calculation for the case 𝐛=𝐛̇ shows that the corresponding ∑_𝑗𝑅_𝑗 vanishes. That is, we have the following proposition.

Proposition 4.6. Let Δ be a Δ_𝑝 bundle over Δ₁. If (Δ,𝐛) is a mass linear pair, then ∑_𝑗𝑅_𝑗=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=𝛾₃−𝛾₁, 𝑏₂=𝛾₃−𝛾₂. In this case, condition (4.38) reduces to 3𝑎₁𝑏₁+𝑎₂𝑏₂=𝑎₁+𝑎₂𝑏₁+𝑏₂.(4.45) Or in terms of the 𝛾_𝑖𝑎₁𝛾₁+𝑎₂𝛾₂=0.(4.46) This is a necessary and sufficient condition for (Δ,𝐛) to be mass linear. This result is the statement of Lemma 4.8 in [3].

4.3. One Point Blowup of ℂ𝑃^𝑛

In this subsection, Δ≡Δ₀(𝜆,𝜏) will be Δ=𝑥₁,…,𝑥_𝑛∈ℝ^𝑛∣^𝑛_𝑖=1𝑥_𝑖≤𝜏,0≤𝑥_𝑖,𝑥_𝑛≤𝜆,(4.47) where 𝜏,𝜆∈ℝ_>0 and 𝜎∶=𝜏−𝜆>0. That is, Δ is the polytope obtained truncating the simplex 𝑆_𝑛(𝜏), defined in Lemma 2.2, by a “horizontal” hyperplane through the point (0,…,0,𝜆). The manifold 𝑀_Δ associated with Δ is the one point blowup of ℂ𝑃^𝑛.

The mass center of the simplex 𝑆_𝑛(𝜏) is the point Cm𝑆_𝑛(𝜏)=𝜏𝑛+1𝑤,(4.48) with 𝑤=(1,…,1).

As the volume of 𝑆_𝑛(𝜏) is 𝜏^𝑛/𝑛!, it follows from (4.48) (𝜏^𝑛−𝜎^𝑛)Cm(Δ)=𝜏^𝑛𝜏𝑛+1𝑤−𝜎^𝑛𝑛+𝜎1𝑤+𝜆𝑒^𝑛.(4.49) That is, Cm(Δ)=1𝜏^𝑛−𝜎^𝑛𝜏^𝑛+1−𝜎^𝑛+1𝑛+1𝑤−𝜆𝜎^𝑛𝑒_𝑛.(4.50)

Given 𝑘=(𝑘₁,…,𝑘_𝑛+2)∈𝒞_Δ, the facets of Δ(𝑘) are in the following hyperplanes: −𝑥_𝑗=𝑘_𝑗,𝑗=1,…,𝑛;^𝑝_𝑖=1𝑥_𝑘=𝑘_𝑛+1;𝑥_𝑛+1=𝑘_𝑛+2.(4.51) As in the preceding subsections, Δ(𝑘)=Δ₀(𝜆,𝜏)−𝑘₁,…,𝑘_𝑛,(4.52) provided that 𝜆=𝑘_𝑛+𝑘_𝑛+2 and 𝜏=∑^𝑛+1𝑖=1𝑘_𝑖.

The pair (Δ,𝐛=(𝑏₁,…,𝑏_𝑛)) is mass linear if and only if there exist 𝐴,𝐵,𝐶∈ℝ such that ^𝑛−1_𝑗=1𝑏_𝑗𝜏^𝑛+1−𝜎^𝑛+1𝑛+1+𝑏^𝑛𝜏^𝑛+1−𝜎^𝑛+1𝑛+1−(𝜏−𝜎)𝜎^𝑛=(𝐴𝜏+𝐵𝜎+𝐶)(𝜏^𝑛−𝜎^𝑛),(4.53) for all 𝜏, 𝜎 “admissible”. A simple calculation proves the following proposition.

Proposition 4.7. The pair (Δ,𝐛) is mass linear if and only if 𝑏_𝑛=1𝑛^𝑛−1_𝑗=1𝑏_𝑗.(4.54)

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

Theorem 4.8. If 𝐛=(𝑏₁,…,𝑏_𝑛)∈ℤ^𝑛 and ∑^{𝑛−1𝑗=1}𝑏_𝑗≠𝑛𝑏_𝑛, then 𝜓_𝐛 generates an infinite cyclic subgroup in 𝜋₁(Ham(𝑀_Δ,𝜔_Δ)).

For 𝑘∈𝒞_Δ, by (4.52) ⟨Cm(Δ(𝑘)),𝐛⟩=CmΔ₀(𝜆,𝜏),𝐛−^𝑛_𝑗=1𝑏_𝑗𝑘_𝑗.(4.55) If (Δ,𝐛) is a mass linear pair, by (4.50) and Proposition 4.7, we have ⟨Cm(Δ₀(𝜆,𝜏)),𝐛⟩=𝑏_𝑛𝜏. Thus, ⟨Cm(Δ(𝑘)),𝐛⟩=^𝑛+1_𝑗=1𝑅_𝑗𝑘_𝑗,(4.56) where 𝑅_𝑗=𝑏_𝑛−𝑏_𝑗, for 𝑗=1,…,𝑛 and 𝑅_𝑛+1=𝑏_𝑛. Hence, we have the following proposition.

Proposition 4.9. Let Δ be the polytope obtained by truncating the standard 𝑛-simplex 𝑆_𝑛(𝜏) by a horizontal hyperplane. If (Δ,𝐛) is a mass linear pair, then ∑_𝑗𝑅_𝑗=0.

Remark 4.10. When 𝑛=3, the toric manifold 𝑀_Δ is 𝑀_Δ=𝑧∈ℂ⁵∶||𝑧₁||²+||𝑧₂||²+||𝑧₃||²+||𝑧₅||²=𝜏/𝜋,||𝑧₃||²+||𝑧₄||²=𝜆/𝜋𝕋,(4.57) where the action of 𝕋=(U(1))² is defined by (𝑎,𝑏)𝑧₁,𝑧₂,𝑧₃,𝑧₄,𝑧₅=𝑎𝑧₁,𝑎𝑧₂,𝑎𝑏𝑧₃,𝑏𝑧₄,𝑎𝑧₅,(4.58) for 𝑎,𝑏∈U(1).
We consider the following loops in the Hamiltonian group of (𝑀_Δ,𝜔_Δ): 𝜓_𝑡[𝑧]=𝑧₁𝑒^{2𝜋𝑖𝑡},𝑧𝜓₂,_𝑡𝑧₃,𝑧₄,𝑧₅,𝜓^𝑡[𝑧]=𝑧₁,𝑧₂𝑒^{2𝜋𝑖𝑡},𝑧₃,𝑧₄,𝑧₅,[𝑧]=𝑧₁,𝑧₂,𝑧₃𝑒^{2𝜋𝑖𝑡},𝑧₄,𝑧₅.(4.59) In [9] (Remark 4.10), we gave formulas that relate the characteristic numbers associated with these loops 𝐼(𝜓)=𝐼𝜓^=−13𝐼𝜓.(4.60) So, for 𝐛=(𝑏₁,𝑏₂,𝑏₃)∈ℤ³, 𝐼𝜓_𝐛=𝑏₁+𝑏₂−3𝑏₃𝐼(𝜓).(4.61) By Proposition 4.7, the vanishing of 𝐼(𝜓_𝐛) in (4.61) is equivalent to the fact that (Δ,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 Tecnología, Grant FPA2009-11061.