# Characteristic Number Associated to Mass Linear Pairs

Authors: Andrés Viña
Journal: ISRN Geometry
 (2011)

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## Abstract

LetΔ be a Delzant polytope in ℝn and b∈ℤn. 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(MΔ), the Hamiltonian group of the symplectic manifold defined by Δ, we determine loops in Ham(MΔ) that define infinite cyclic subgroups in π1(Ham(MΔ)) when Δ satisfies any of the following conditions: (i) it is the trapezium associated with a Hirzebruch sur-face, (ii) it is a Δp bundle over Δ1, and (iii) Δ is the truncated simplex associated with the one point blowup of ℂPn.


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## Body

## 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 loopψ 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: (1.1)I(ψ)∶=∫Ec1(VTE)cn,
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 (1.2)I:ψ∈π1(Ham(N,Ω))⟼I(ψ)∈R
is an ℝ-valued group homomorphism [4].Our purpose is to study this characteristic number whenN 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.LetT be the torus (U(1))n and Δ=Δ(n,k) the polytope in 𝔱* with m facets defined by (1.3)Δ(n,k)=⋂j=1m{x∈t*:〈x,nj〉≤kj},
where kj∈ℝ and the nj∈𝔱 are the outward conormals to the facets. The facet defined by the equation 〈x,nj〉=kj will be denoted Fj, and we put Cm(Δ) for the mass center of the polytope Δ.In [3], the chamber 𝒞Δ of Δ is defined as the set of k′∈ℝm such that the polytope Δ′:=Δ(n,k′) is analogous to Δ; that is, the intersection ∩j∈JFj is nonempty if and only if ∩j∈JFj′≠∅ 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 (1.4)k∈CΔ⟼〈Cm(Δ(k)),b〉∈R
is linear. That is, (1.5)〈Cm(Δ(k)),b〉=∑jRjkj+C,
where Rj and C are constant.Let us assume thatΔ is a Delzant polytope [6]. We shall denote by (MΔ,ωΔ,μΔ) the toric manifold determined by Δ (μΔ:M→𝔱* being the corresponding moment map). Given b, an element in the integer lattice of 𝔱, we shall write ψb for the loop of Hamiltonian diffeomorphisms of (MΔ,ωΔ) defined by b through the toric action. We will let I(Δ;b) for the characteristic number I(ψb). 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 groupG of the translations defined by the elements of 𝔱* acts freely on 𝒞Δ. We put r:=m-n for the dimension of the quotient 𝒞Δ/G. Thus, r is the number of effective parameters which characterize the polytopes in 𝒞Δ 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∈𝒞Δ.(b)
(Δ,b) is a mass linear pair as in (1.5), with ∑jRj=0.In [7], by direct computation, we proved the equivalence between the vanishing of I(k;b) on 𝒞Δ 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 aΔp bundle over Δ1 [3], and(iii)
Δ is the truncated simplex associated with the one point blowup of ℂPn.On the other hand, whenΔ 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 ∑jRj=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 groupsHam(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 ψb to generate an infinite cyclic subgroup in π1(Ham(MΔ,ωΔ)) 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 polytopeΔ and  b an element in the integer lattice of  𝔱. If  r≤2 and  (Δ,b) is not mass linear, then  ψb generates an infinite cyclic subgroup in  π1(Ham(MΔ(k),ωΔ(k))), for all  k∈𝒞Δ.In the proof of Theorem1.1, a formula for the characteristic number I(ψb) obtained in [9] plays a crucial role. This formula gives I(ψb) in terms of the integrals, on the facets of the polytope, of the normalized Hamiltonian function corresponding to the loop ψb (see (2.13)). From this expression for I(ψb), we will deduce a relation between the directional derivative of map (1.4) along the vector (1,…,1) of ℝm, 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 Section2, 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 kj (Proposition 2.4).In Section3, 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 𝒞Δ. We also express ∑jRj in terms of the displacement of the center of mass Cm(Δ(k)) produced by the change kj→kj+1 (Proposition 3.7).Section4 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 ∑jRj=0.


## 2. A Characteristic Number

Let us suppose that the polytopeΔ defined in (1.3) is a Delzant polytope in 𝔱*. Following [10], we recall some points of the construction of (MΔ,ωΔ,μΔ) from the polytope Δ. We put T̃:=(S1)m-n. The ni determine weights wj∈𝔱̃*, j=1,…,m for a T̃-action on ℂm. Then moment map for this action is (2.1)J:z∈Cm⟼J(z)=π∑j=1m|zj|2wj∈t̃*.
The ki define a regular value σ for J, and the manifold MΔ is the following orbit space (2.2)MΔ={z∈Cm:π∑j=1m|zj|2wj=σ}T̃,
where the relation defined by T̃ is (2.3)(zj)≃(zj′)iff there isy∈t̃such thatzj′=zje2πi〈wj,y〉forj=1,…,m.Identifying𝔱̃* with ℝr, σ=(σ1,…,σr) and each σa is a linear combination of the kj.Given a facetF of Δ, 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 j∈{1,…,n}.If we writeza=ρaeiθa, then the symplectic form can be written on {[z]∈M:za≠0,for alla}(2.4)ωΔ=12∑i=1ndρi2∧dφi,
with φi an angular variable, linear combination of the θa.The action ofT=(S1)n on MΔ(2.5)(α1,…,αn)[z1,…,zm]:=[α1z1,…,αnzn,zn+1,…,zm]
endows MΔ with a structure of toric manifold. Identifying 𝔱* with ℝn, the moment map μΔ:MΔ→𝔱=ℝn is defined by (2.6)μΔ([z])=π(ρ12,…,ρn2)+(d1,…,dn),
where the constants di are linear combinations of the kj and (2.7)imμΔ=Δ.
The facet F=Fj of Δ is the image by μΔ of the submanifold (2.8)Dj={[z1,…,zm]∈MΔ∣zj=0}.We writexi∶=πρi2, then (2.9)∫MΔ(ωΔ)n=n!∫Δdx1⋯dxn.Letb 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 (2.10)f=〈μΔ,b〉+constant,∫MΔf(ωΔ)n=0.
That is, f=〈μΔ,b〉-〈Cm(Δ),b〉, where (2.11)〈Cm(Δ),b〉=∫M〈μΔ,b〉(ωΔ)n∫M(ωΔ)n.
Moreover, (2.12)∫MΔ〈μΔ,b〉(ωΔ)n=n!∫Δ∑i=1nbixidx1⋯dxn.An expression for the value of the invariantI(ψb) in terms of integrals of the Hamiltonian function has been obtained in Section 4 of [9] (see also [11, 12]) (2.13)I(Δ;b)∶=I(ψb)=-n∑FfacetN(F),
where the contribution N(F) of the above facet F=Fj (with j=1,…,n) is (2.14)Nj∶=N(F)=(n-1)!∫Fjfdx1⋯dxĵ⋯dxn=(n-1)!(∫Fj〈μΔ,b〉dx1⋯dxĵ⋯dxn-〈Cm(Δ),b〉∫Fjdx1⋯dxĵ⋯dxn),
with dx1⋯dxĵ⋯dxn∶=dx1⋯dxj-1dxj+1⋯dxn.GivenΔ=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, (2.15)MΔ′=MΔ,ωΔ′=ωΔ,μΔ′=μΔ+a.
But the normalized Hamiltonians f and f′ corresponding to the action of b on MΔ and MΔ′ 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.
Ifa is an arbitrary vector of 𝔱*, then  I(k;b)=I(k′;b), for  kj′=kj+〈a,nj〉,  j=1,…,m.By Proposition2.1, we can assume that all dj in (2.6) are zero for the determination of I(k;b).The following lemma is elementary.Lemma 2.2.
If(2.16)Sn(τ)∶={(x1,…,xn)∈Rn∣∑i=1nxi≤τ,0≤xj,∀j},
then
(2.17)∫Sn(τ)f(x1,…,xn)dx1⋯dxn={τnn!iff=1cτn+c(n+c)!iff=xic,c=1,2τn+2(n+2)!iff=xixj,i≠j.More general, ifc1,…,cn∈ℝ>0, we put (2.18)Sn(c,τ)∶={(x1,…,xn)∈Rn∣∑i=1ncixi≤τ,0≤xj,∀j},
then (2.19)∫Sn(c,τ)dx1⋯dxn=1n!∏i=1nτci,∫Sn(c,τ)xjdx1⋯dxn=1(n+1)!τcj∏i=1nτci.
Thus, in the particular case that Δ=Sn(c,τ), the integral ∫MΔ(ωΔ)n is a monomial of degree n in τ, and ∫MΔ〈μΔ,b〉(ωΔ)n is a monomial of degree n+1.We return to the general case in whichΔ is the polytope defined in (1.3). Its vertices are the solutions to (2.20)〈x,nja〉=kja,a=1,…,n;
hence, the coordinates of any vertex of Δ are linear combinations of the kj.A hyperplane inℝn through a vertex (x10,…,xn0) of Δ is given by an equation of the form (2.21)〈x,n〉=〈x0,n〉=∶κ.
Thus, the independent term κ is a linear combination (l. c.) of the kj. Moreover, the coordinates of the common point of n hyperplanes (2.22)〈x,ñi〉=κi,
with κi l. c. of, the kj are also l. c. of the kj.By drawing hyperplanes through vertices ofΔ (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)
eachβS is the transformed of a simplex Sn(b,τ) by an element of the group of Euclidean motions in ℝn.(b)
Forα≠β, αS∩βS is a subset of the border of αS.(c)
⋃ββS=Δ.Thus, by construction, each facet ofβS is contained in a hyperplane π of the form 〈x,n〉=κ, with κ l. c. of the kj.On the other hand, the hyperplaneπ is transformed by an element of SO(n) in an hyperplane 〈x,n′〉=κ. If 𝒯 is a translation in ℝn which applies Sn(b,τ) onto βS, then this transformation maps (0,…,0) in a vertex a=(a1,…,an) of βS. So, the translation 𝒯 transforms π in 〈x,n〉=κ+〈a,n〉=∶κ′. As each aj is a l. c. of the kj, so is κ′. Hence, any element of the group of Euclidean motions in ℝn which maps Sn(b,τ) onto βS transforms the hyperplane π(2.23)〈x,n′〉=κ′,
with κ′ a l. c. of the kj.Let assume that(R𝒯a)(S(b,τ))=βS, with R∈SO(n) and 𝒯a the translation defined by a. Then the oblique facet of S(b,τ), contained in the hyperplane ∑bixi=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 kj). The argument of the preceding paragraph applied to R-1 and 𝒯-a proves that τ is a l. c. of the kj. Hence, by (2.19), the integral (2.24)∫βSdx1⋯dxn=∫Sn(b,τ)dx1⋯dxn
is a monomial of degree n of a l. c. of the kj. Thus, (2.25)∫M(ωΔ)n=∑β∫βSdx1⋯dxn,
is a homogeneous polynomial of degree n of the kj.Similarly,(2.26)∫MΔ〈μΔ,b〉(ωΔ)n
is a homogeneous polynomial of degree n+1 of the kj. Analogous results hold for (2.27)∫Fjdx1⋯dxĵ⋯dxn,∫Fj〈μΔ,b〉dx1⋯dxĵ⋯dxn.From formulas (2.9)–(2.14) together with the preceding argument, it follows the following proposition.Proposition 2.3.
Given a Delzant polytopeΔ, if   b belongs to the integer lattice of   𝔱, then  I(k;b) is a rational function of the  kj, for  k∈𝒞Δ.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  kj of degree  n, when  k∈𝒞Δ.We will use the following simple lemma in the proof of Theorem1.1.Lemma 2.5.
Ifk̂j=skj for  j=1,…,m, with  s∈ℝ, then  Cm(Δ(n,k̂))=sCm(Δ(n,k)).Proof.
The vertices ofΔ(n,k) are the solutions of (2.20), and the vertices of Δ(n,k̂) are the solutions of 〈x,nja〉=skja, 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 Theorem1.1

Let us assume that the polytopeΔ defined by (1.3) is Delzant and let k be an element of 𝒞Δ. 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.Letb be an element in the integer lattice of 𝔱. We put (3.1)A(k)∶=∫M(k)〈μ(k),b〉(ω(k))n,B(k)∶=∫M(k)(ω(k))n.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 xj=0. If we make an infinitesimal variation of the facet F(k)j, by means of the translation defined by kj→kj+ϵ (keeping unchanged the other ki), then the volume of Δ(k) changes according to (3.2)1n!B(k)⟶1n!B(k)+ϵ∫F(k)jdx1⋯dxĵ⋯dxn+O(ϵ2).
We write dXj for dx1⋯dxĵ⋯dxn. Thus, (3.3)∂B(k)∂kj=n!∫F(k)jdXj,∂A(k)∂kj=n!∫F(k)j〈μ(k),b〉dXj.
So, by (2.11), (3.4)∂∂kj〈Cm(Δ(k)),b〉=n!(B(k))2(B(k)∫F(k)j〈μ(k),b〉dXj-A(k)∫F(k)jdXj).
From (2.13) and (2.14), it follows (3.5)∑j=1m∂∂kj〈Cm(Δ(k)),b〉=-1B(k)I(k;b).
Thus, we have proved the following proposition.Proposition 3.1.
I(k;b)=0 for all  k∈𝒞Δ if and only if   ∑j=1m(∂/∂kj)〈Cm(Δ(k)),b〉=0, for all  k∈𝒞Δ.Next, we will parametrize the quotient𝒞Δ/G (of classes of polytopes in 𝒞Δ module translation) defined in Section 1.After a possible renumbering, we may assume that the intersection of facetsF1,…,Fn is a vertex of Δ. Thus, the conormals n1,…,nn are linearly independent in 𝔱. So, given k∈𝒞Δ, there is a unique v∈𝔱*, such that (3.6)〈v,ni〉+ki=0,i=1,…,n.
(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(k) depends linearly on the ki; that is, 〈v(k),c〉 is a linear function of k1,…,kn, for all c∈𝔱.Ifm-n=2, we write (3.7)λ=kn+1+〈v(k),nn+1〉,τ=km+〈v(k),nm〉,
where v(k) the element in 𝔱* defined by (3.6). From the linearity of v with respect to the ki, it follows that λ and τ are linear combinations of k1,…,km.The polytope in𝒞Δ defined by (k1′=0,…,kn′=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, (3.8)Δ0(λ,τ)=Δ(k)+v(k).Letb an element in the integer lattice of 𝔱, we define the function g by (3.9)g(λ,τ)∶=〈Cm(Δ0(λ,τ)),b〉.
The function g is defined on the pairs (λ,τ) such that (0,…,0,λ,τ)∈𝒞Δ. By Lemma 2.5, it follows (3.10)g(sλ,sτ)=sg(λ,τ),
for any real number s such that (sλ,sτ) belongs to the domain of g. This property implies that (3.11)g=λ∂g∂λ+τ∂g∂τ.Theorem 3.2.
IfI(k;b)=0, for all  k∈𝒞Δ and  r=2, then  〈Cm(Δ(k)),b〉=∑jRjkj, with  Rj constant (i.e.,  (Δ,b) is a mass linear pair) and  ∑jRj=0.Proof.
We setf(k1,…,km)∶=〈Cm(Δ(k)),b〉. It follows from (3.8) that
(3.12)f(k)=g(λ,τ)-〈v(k),b〉.
By the hypothesis and Proposition3.1,
(3.13)∑j=1m∂f∂kj=0.
Since(3.14)∑j=1m∂f∂kj=∂g∂λ∑j=1m∂λ∂kj+∂g∂τ∑j=1m∂τ∂kj-〈∂v∂kj,b〉,
from (3.13), we deduce
(3.15)p∂g∂λ+q∂g∂τ-t=0,
where p, q, t stand for the following constants
(3.16)p=∑j=1m∂λ∂kj,q=∑j=1m∂τ∂kj,t=〈∂v∂kj,b〉.
Sinceqλ-pτ and tτ-qg are first integrals of (3.15), the general solution of this equation is
(3.17)g(λ,τ)=(tq)τ+Φ(qλ-pτ),
where Φ is a derivable function of one variable.
It follows from (3.11) and (3.17) that
(3.18)Φ(u)=uΦ′(u).
Thus, Φ(u)=αu, with α constant. We have for f(3.19)f(k)=(bq)τ+α(qλ-pτ)-〈v(k),b〉.
In other words, f is a linear function of the kj; that is, f(k)=∑jRjkj, with Rj constant. From (3.13), it follows ∑jRj=0.Remark 3.3.
The proof of Theorem3.2 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 kj.
On the other hand, the proof of this theorem does not admit an adaptation to the caser>2. In fact, the corresponding function Φ would be a function of r-1 variables Φ(u1,…,ur-1). The equation which corresponds to (3.18) in this case would be
(3.20)Φ=∑i=1r-1ui∂Φ∂ui.
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)
(3.21)I(k;b)=-B(k)∑jRj,
for all k∈𝒞Δ. 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∈𝒞Δ if and only if  ∑jRj=0.Proof of Theorem1.1.
It is a direct consequence of Proposition3.4, Theorem 3.2, and the remark above.We will deduce a sufficient condition for a Delzant polytopeΔ to admit mass linear functions. We write (3.22)Cṁ(Δ(k))∶=ddϵ|ϵ=0Cm(Δ(k+ϵ̌)),
with ϵ̌=(ϵ,…,ϵ).Proposition 3.5.
If all pointsCṁ(Δ(k)), for  k∈𝒞Δ, belong to a hyperplane of  (ℝn)* with a conormal vector in  ℤn and  r≤2, then  Δ admits a mass linear function.Proof.
Letb∈ℤn be a conormal vector to the hyperplane, then
(3.23)0=〈Cṁ(Δ(k)),b〉=〈∑j∂∂kjCm(Δ(k)),b〉.
By (3.5), I(k,b)=0; Theorem 3.2 applies and (Δ,b) is a mass linear pair.Proposition 3.6.
LetΔ be a Delzant polytope, such that  k=0 belongs to the closure of  𝒞Δ. If  r≤2, a necessary condition for the vanishing of  I(k;b) on  𝒞Δ is
(3.24)〈ddϵ|ϵ=0Cm(Δ(ϵ̌)),b〉=0.Proof.
IfI(k;b) vanishes on 𝒞Δ, then (Δ,b) is a linear pair, by Theorem 1.1. Thus, 〈Cm(Δ(k)),b〉=∑jRjkj+C, on 𝒞Δ. So, given k∈𝒞Δ and ϵ small enough
(3.25)〈Cm(Δ(k+ϵ̌)),b〉=∑jRjkj+ϵ∑jRj+C.
By Theorem 1.1, ∑jRj=0. Thus, for any k∈𝒞Δ,
(3.26)ddϵ|ϵ=0〈Cm(Δ(k+ϵ̌)),b〉=0.
Taking the limit as k→0,
(3.27)0=lim⁡k→0ddϵ|ϵ=0〈Cm(Δ(k+ϵ̌)),b〉=〈ddϵ|ϵ=0Cm(Δ(ϵ̌)),b〉.Next, we will describe a geometric interpretation of the number∑jRj. Given an arbitrary Delzant polytope Δ. If a is a vector of 𝔱*, then (3.28)Cm(Δ(k′))=Cm(Δ(k))+a,
if kj′=kj+〈a,nj〉.We will denote byd the element of 𝔱* defined by the following relation (3.29)Cm(Δ(k̃))=Cm(Δ(k))+d,
with k̃j=kj+1 for all j.From (3.28) and (3.29), we have (3.30)Cm(Δ(kj+〈d,nj〉))=Cm(Δ(kj))+d=Cm(Δ(k̃j=kj+1)).Now, we assume that(Δ,b) is a mass linear pair. From (1.5), it follows (3.31)〈Cm(Δ(kj+〈d,nj〉)),b〉=∑Rjkj+∑Rj〈d,nj〉+C,〈Cm(Δ(kj))+d,b〉=∑Rjkj+〈d,b〉+C,〈Cm(Δ(k̃)),b〉=∑Rjkj+∑Rj+C.
These formulas allow us to state the following proposition that gives an interpretation of the sum ∑jRj in terms of the variation of Cm(Δ(k)) with the kj.Proposition 3.7.
Let(Δ,b) be a mass linear pair as in (1.5). Then,
(3.32)∑jRj〈d,nj〉=〈d,b〉=∑jRj,d being the element of 𝔱* defined by (3.29).


## 4. Examples

In this section, we will deduce the particular form which adopts Theorem1.2 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

Givenr∈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 (4.1){z∈C4:|z1|2+r|z2|2+|z4|2=τ/π,|z2|2+|z3|2=λ/π}T,
where the equivalence defined by 𝕋=(S1)2 is given by (4.2)(a,b)⋅(z1,z2,z3,z4)=(az1,arbz2,bz3,az4),
for (a,b)∈(S1)2.The manifoldN equipped with the following (U(1))2 action (4.3)(ϵ1,ϵ2)[zj]=[ϵ1z1,ϵ2z2,z3,z4],
is a toric manifold. The corresponding moment polytope Δ is the trapezium in ℝ2 with vertices (4.4)P1=(0,0),P2=(0,λ),P3=(τ,0),P4=(σ,λ).
That is, N is the toric manifold MΔ determined by the trapezium Δ.As the conormals to the facets ofΔ 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 𝒞Δ are on the straights (4.5)-x=k1,-y=k2,y=k3,x+ry=k4.
The vertices of Δ(k) are the points (4.6)(-k1,-k2),(-k1,k3),(k4-rk3,k3),(k4+rk2,-k2).
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 (4.7)τ=k4+rk2+k1,λ=k3+k2.
So, (4.8)Cm(Δ(k))=Cm(Δ)+(-k1,-k2).
Moreover, the mass center of Δ is (4.9)Cm(Δ)=(3τ2-3rτλ+r2λ23(2τ-rλ),3λτ-2rλ23(2τ-rλ)).The chamber𝒞Δ 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 𝒞Δ. From (4.8), together with (4.7) and (4.9), it follows (4.10)Cm(Δ(ϵ̌))=(r2ϵ12,-rϵ6),
where ϵ̌=(ϵ,ϵ,ϵ,ϵ). By Proposition 3.6, if I(k;b) with b=(b1,b2)∈ℤ2 vanishes on the chamber 𝒞Δ, then rb1-2b2=0.On the other hand, from (4.9) and (4.8), it follows (4.11)〈Cm(Δ(k)),b〉=(3τ2-3rτλ+r2λ2)b1+(3λτ-2rλ2)b23(2τ-rλ)-k1b1-k2b2.
By (4.7), expression (4.11) is linear in the ki if and only if (4.12)(3τ2-3rτλ+r2λ2)b1+(3λτ-2rλ2)b23(2τ-rλ)
is linear in τ, λ. That is, if and only if there exist constants A, B such that for al τ, λ(4.13)(3τ2-3rτλ+r2λ2)b1+(3λτ-2rλ2)b2=3(2τ-rλ)(Aτ+Bλ).
From this relation, it follows the above condition rb1=2b2. In this case (4.11) reduces to (4.14)〈Cm(Δ(k),b〉=-b12k1+b12k4.Comparing (1.5) with (4.14), we obtain R1=-R4=-b1/2, R2=R3=0; so, ∑jRj=0. That is, the condition ∑jRj=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 ∑jRj=0.By Theorem1.2, we have the following corollary.Corollary 4.2.
Ifrb1≠2b2, then ψb generates an infinite cyclic subgroup in π1(Ham(MΔ,ωΔ)).Remark 4.3.
We denote byϕt the following isotopy of MΔ(4.15)ϕt[z]=[e2πitz1,z2,z3,z4],
where ϕ is a loop in the Hamiltonian group of MΔ. By ϕ′, we denote the Hamiltonian loop
(4.16)ϕt′[z]=[z1,e2πitz2,z3,z4].
In Theorem 8 of [11], we proved that I(ϕ′)=(-2/r)I(ϕ). If b=(b1,b2)∈ℤ2, then
(4.17)I(ψb)=b1I(ϕ)+b2I(ϕ′)=(b1-(2r)b2)I(ϕ).
That is, I(ψb)=0 if and only if rb1=2b2, which is in agreement with Proposition 4.1 and Theorem 1.1.


### 4.2.Δp Bundle over Δ1

Given the integerp>1, as McDuff and Tolman in [3], we consider the following vectors in ℝp+1: (4.18)ni=-ei,i=1,…,p,np+1=∑i=1pei,np+2=-ep+1,np+3=ep+1-∑i=1paiei,
where e1,…,ep+1 is the standard basis of ℝp+1 and ai∈ℤ. We write (4.19)a∶=(a1,…,ap)∈Zp,A∶=∑i=1pai,a⋅a=∑i=1pai2.Letλ, τ be real positive numbers with λ+ai>0, for i=1,…,p. In this subsection, we will consider the polytope Δ in (ℝp+1)* defined by the above conormals nj and the following kj: (4.20)k1=⋯=kp=kp+2=0,kp+1=τ,kp+3=λ.This polytope will be also denoted byΔ0(λ,τ). It is a Δp 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 polytopeΔ is a Delzant polytope. The manifold (2.2) is in this case (4.21)MΔ={z∈Cp+3:∑i=1p+1|zi|2=τ/π,-∑j=1paj|zj|2+|zp+2|2+|zp+3|2=λ/π}≃,
where (zj)≃(zj′) if and only if there are α,β∈U(1) such that (4.22)zj′=αβ-ajzj,j=1,…,p,zp+1′=αzp+1,zk′=βzk,k=p+2,p+3.
Thus, MΔ is the total space of the fibre bundle ℙ(L1⊕⋯⊕Lp⊕ℂ)→ℂP1, where Lj is the holomorphic line bundle over ℂP1 with Chern number aj.The symplectic form (2.4) is (4.23)ωΔ=12(σ1+⋯+σp+σp+2),
where σk=dρk2∧dφk.And the moment map(4.24)μΔ([z])=(x1,…,xp,xp+2),
where xi:=πρi2.Proposition 4.4.
The coordinatesx̅j of  Cm(Δ0(λ,τ)) are given by
(4.25)x̅k=τp+2λ(p+2)+τ(A+ak)λ(p+1)+τA,fork=1,…,p,(4.26)x̅p+2=12(p+1)(p+2)λ2+2(p+2)Aλτ+(a⋅a+A2)τ2(p+2)((p+1)λ+Aτ).Proof.
Since the points[z]∈MΔ satisfy |zp+2|2≤λ/π+∑j=1paj|zj|2, by (2.9) and Lemma 2.2, we have
(4.27)∫MΔ(ωΔ)p+1=(p+1)!∫Sp(τ)(λ+∑j=1pajxj)=(p+1)!(λτpp!+τp+1A(p+1)!).
Similarly, for k=1,…,p(4.28)∫MΔxk(ωΔ)p+1=(p+1)!(λτp+1(p+1)!+τp+2(p+2)!∑j≠kaj+2τp+2ak(p+2)!).
The kth coordinate of Cm(Δ), x̅k, is the quotient of (4.28) by (4.27); that is,
(4.29)x̅k=τp+2λ(p+2)+τ(A+ak)λ(p+1)+τA.
For thep+2-coordinate of Cm(Δ), we need to calculate ∫Mxp+2(ωΔ)p+1. By Lemma 2.2,
(4.30)1(p+1)!∫Mxp+2(ωΔ)p+1=12∫Sp(τ)(λ+∑j=1pajxj)2=12(λ2τpp!+2Aλτp+1(p+1)!+(a⋅a+A2)τp+2(p+2)!).
Formula (4.26) is a consequence of (4.27) together with (4.30).The translation in(ℝp+1)* defined by the vector (-k1,…,-kp,-kp+2) transforms the hyperplanes 〈x,np+3〉=λ and 〈x,np+1〉=τ in (4.31)〈x,np+3〉=λ-kp+2+∑j=1pajkj,〈x,np+1〉=τ-∑j=1pkj,
respectively.LetΔ(k) be a polytope with k=(k1,…,kp+3) generic in the chamber 𝒞Δ. From (4.31), it follows that Δ(k) is the image of the polytope Δ0(λ,τ) by the translation determined by (-k1,…,-kp,-kp+2), whenever (4.32)kp+2-∑j=1pajkj+kp+3=λ,∑j=1pkj+kp+1=τ.
In this case, (4.33)Cm(Δ(k))=Cm(Δ0(λ,τ))-(k1,…,kp,kp+2).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 (4.34)ϵ-∑j=1pajϵ+ϵ=(2-A)ϵ
and τ by (p+1)ϵ, and finally take into account (4.33). These operations give (4.35)x̅j(Δ(ϵ̌))=ϵ2(p+2)((p+1)aj-A),j=1,…,p,x̅p+2(Δ(ϵ̌))=ϵ4(p+2)(-A2+(p+1)(a⋅a)).Givenb=(b1,…,bp,b)≡(b̂,ḃ), with b̂=(b1,…,bp,0) and ḃ=(0,…,0,b),(4.36)〈ddϵ|ϵ=0Cm(Δ(ϵ̌)),b〉=14(p+2)((p+1)(2a⋅b̂-ba⋅a)-A(2B+bA)),
where a·b̂=∑j=1pajbj and B=∑j=1pbj.By Proposition3.6, we have the following theorem.Theorem 4.5.
LetΔ be the Δp bundle over Δ1 defined by (4.18) and (4.20). Given b=(b̂,ḃ)∈ℤp+1,  if
(4.37)(p+1)(2a⋅b̂-ba⋅a)-A(2B+bA)≠0,
then ψb defines an infinite cyclic subgroup in the fundamental group π1(Ham(MΔ,ωΔ)).It is straightforward to check that(4.38)(p+1)(2a⋅b̂-ba⋅a)-A(2B+bA)=0
is also a sufficient condition for (Δ,b) to be a mass linear pair.Since(4.39)〈Cm(Δ0(λ,τ)),b〉=〈Cm(Δ0(λ,τ)),b̂〉+〈Cm(Δ0(λ,τ)),ḃ〉,
if (4.38) holds, using (4.25) and (4.26), one obtains (4.40)〈Cm(Δ0(λ,τ)),b〉=bλ2+(b2(a⋅a+A2)(p+2)A+(a⋅b̂+AB)(p+2)A)τ.
By (4.33), for k∈𝒞Δ, (4.41)〈Cm(Δ(k)),b〉=〈Cm(Δ0(λ,τ)),b〉-∑j=1pbjkj-bkp+2,
with λ and τ given by (4.32).Ifb=b̂, the condition (4.38) reduces to (p+1)a·b̂=AB and (4.42)〈Cm(Δ(k)),b〉=(a⋅b̂+AB)(p+2)A(∑j=1pkj+kp+1)-∑j=1pbjkj.
Hence, 〈Cm(Δ(k)),b〉=∑Rjkj, where (4.43)Rj=(a⋅b̂+AB)(p+2)A-bj,j=1,…,p,Rp+1=(a⋅b̂+AB)(p+2)A,Rp+2=Rp+3=0.
So, (4.44)∑j=1p+3Rj=(p+1)a⋅b̂-AB(p+2)A=0.A similar calculation for the caseb=ḃ shows that the corresponding ∑jRj vanishes. That is, we have the following proposition.Proposition 4.6.
LetΔ be a Δp bundle over Δ1.  If (Δ,b) is a mass linear pair, then ∑jRj=0.Forp=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 (4.45)3(a1b1+a2b2)=(a1+a2)(b1+b2).
Or in terms of the γi(4.46)a1γ1+a2γ2=0.
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 ofℂPn

In this subsection,Δ≡Δ0(λ,τ) will be (4.47)Δ={(x1,…,xn)∈Rn∣∑i=1nxi≤τ,0≤xi,xn≤λ},
where τ,λ∈ℝ>0 and σ∶=τ-λ>0. That is, Δ is the polytope obtained truncating the simplex Sn(τ), defined in Lemma 2.2, by a “horizontal” hyperplane through the point (0,…,0,λ). The manifold MΔ associated with Δ is the one point blowup of ℂPn.The mass center of the simplexSn(τ) is the point (4.48)Cm(Sn(τ))=τn+1w,
with w=(1,…,1).As the volume ofSn(τ) is τn/n!, it follows from (4.48) (4.49)(τn-σn)Cm(Δ)=τnτn+1w-σn(σn+1w+λen).
That is, (4.50)Cm(Δ)=1τn-σn((τn+1-σn+1n+1)w-λσnen).Givenk=(k1,…,kn+2)∈𝒞Δ, the facets of Δ(k) are in the following hyperplanes: (4.51)-xj=kj,j=1,…,n;∑i=1pxk=kn+1;xn+1=kn+2.
As in the preceding subsections, (4.52)Δ(k)=Δ0(λ,τ)-(k1,…,kn),
provided that λ=kn+kn+2 and τ=∑i=1n+1ki.The pair(Δ,b=(b1,…,bn)) is mass linear if and only if there exist A,B,C∈ℝ such that (4.53)∑j=1n-1bjτn+1-σn+1n+1+bn(τn+1-σn+1n+1-(τ-σ)σn)=(Aτ+Bσ+C)(τn-σn),
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
(4.54)bn=1n∑j=1n-1bj.From Theorem1.2 together with Proposition 4.7, it follows the following theorem.Theorem 4.8.
Ifb=(b1,…,bn)∈ℤn and  ∑j=1n-1bj≠nbn, then  ψb generates an infinite cyclic subgroup in  π1(Ham(MΔ,ωΔ)).Fork∈𝒞Δ, by (4.52) (4.55)〈Cm(Δ(k)),b〉=〈Cm(Δ0(λ,τ)),b〉-∑j=1nbjkj.
If (Δ,b) is a mass linear pair, by (4.50) and Proposition 4.7, we have 〈Cm(Δ0(λ,τ)),b〉=bnτ. Thus, (4.56)〈Cm(Δ(k)),b〉=∑j=1n+1Rjkj,
where Rj=bn-bj, for j=1,…,n and Rn+1=bn. Hence, we have the following proposition.Proposition 4.9.
LetΔ be the polytope obtained by truncating the standard n-simplex Sn(τ) by a horizontal hyperplane. If (Δ,b) is a mass linear pair, then ∑jRj=0.Remark 4.10.
Whenn=3, the toric manifold MΔ is
(4.57)MΔ={z∈C5:|z1|2+|z2|2+|z3|2+|z5|2=τ/π,|z3|2+|z4|2=λ/π}T,
where the action of 𝕋=(U(1))2 is defined by
(4.58)(a,b)(z1,z2,z3,z4,z5)=(az1,az2,abz3,bz4,az5),
for a,b∈U(1).
We consider the following loops in the Hamiltonian group of(MΔ,ωΔ):
(4.59)ψt[z]=[z1e2πit,z2,z3,z4,z5],ψt′[z]=[z1,z2e2πit,z3,z4,z5],ψ̃t[z]=[z1,z2,z3e2πit,z4,z5].
In [9] (Remark 4.10), we gave formulas that relate the characteristic numbers associated with these loops
(4.60)I(ψ)=I(ψ′)=(-13)I(ψ̃).
So, for b=(b1,b2,b3)∈ℤ3,
(4.61)I(ψb)=(b1+b2-3b3)I(ψ).
By Proposition 4.7, the vanishing of I(ψb) 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.


## 4.1. Hirzebruch Surfaces

Givenr∈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 (4.1){z∈C4:|z1|2+r|z2|2+|z4|2=τ/π,|z2|2+|z3|2=λ/π}T,
where the equivalence defined by 𝕋=(S1)2 is given by (4.2)(a,b)⋅(z1,z2,z3,z4)=(az1,arbz2,bz3,az4),
for (a,b)∈(S1)2.The manifoldN equipped with the following (U(1))2 action (4.3)(ϵ1,ϵ2)[zj]=[ϵ1z1,ϵ2z2,z3,z4],
is a toric manifold. The corresponding moment polytope Δ is the trapezium in ℝ2 with vertices (4.4)P1=(0,0),P2=(0,λ),P3=(τ,0),P4=(σ,λ).
That is, N is the toric manifold MΔ determined by the trapezium Δ.As the conormals to the facets ofΔ 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 𝒞Δ are on the straights (4.5)-x=k1,-y=k2,y=k3,x+ry=k4.
The vertices of Δ(k) are the points (4.6)(-k1,-k2),(-k1,k3),(k4-rk3,k3),(k4+rk2,-k2).
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 (4.7)τ=k4+rk2+k1,λ=k3+k2.
So, (4.8)Cm(Δ(k))=Cm(Δ)+(-k1,-k2).
Moreover, the mass center of Δ is (4.9)Cm(Δ)=(3τ2-3rτλ+r2λ23(2τ-rλ),3λτ-2rλ23(2τ-rλ)).The chamber𝒞Δ 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 𝒞Δ. From (4.8), together with (4.7) and (4.9), it follows (4.10)Cm(Δ(ϵ̌))=(r2ϵ12,-rϵ6),
where ϵ̌=(ϵ,ϵ,ϵ,ϵ). By Proposition 3.6, if I(k;b) with b=(b1,b2)∈ℤ2 vanishes on the chamber 𝒞Δ, then rb1-2b2=0.On the other hand, from (4.9) and (4.8), it follows (4.11)〈Cm(Δ(k)),b〉=(3τ2-3rτλ+r2λ2)b1+(3λτ-2rλ2)b23(2τ-rλ)-k1b1-k2b2.
By (4.7), expression (4.11) is linear in the ki if and only if (4.12)(3τ2-3rτλ+r2λ2)b1+(3λτ-2rλ2)b23(2τ-rλ)
is linear in τ, λ. That is, if and only if there exist constants A, B such that for al τ, λ(4.13)(3τ2-3rτλ+r2λ2)b1+(3λτ-2rλ2)b2=3(2τ-rλ)(Aτ+Bλ).
From this relation, it follows the above condition rb1=2b2. In this case (4.11) reduces to (4.14)〈Cm(Δ(k),b〉=-b12k1+b12k4.Comparing (1.5) with (4.14), we obtain R1=-R4=-b1/2, R2=R3=0; so, ∑jRj=0. That is, the condition ∑jRj=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 ∑jRj=0.By Theorem1.2, we have the following corollary.Corollary 4.2.
Ifrb1≠2b2, then ψb generates an infinite cyclic subgroup in π1(Ham(MΔ,ωΔ)).Remark 4.3.
We denote byϕt the following isotopy of MΔ(4.15)ϕt[z]=[e2πitz1,z2,z3,z4],
where ϕ is a loop in the Hamiltonian group of MΔ. By ϕ′, we denote the Hamiltonian loop
(4.16)ϕt′[z]=[z1,e2πitz2,z3,z4].
In Theorem 8 of [11], we proved that I(ϕ′)=(-2/r)I(ϕ). If b=(b1,b2)∈ℤ2, then
(4.17)I(ψb)=b1I(ϕ)+b2I(ϕ′)=(b1-(2r)b2)I(ϕ).
That is, I(ψb)=0 if and only if rb1=2b2, which is in agreement with Proposition 4.1 and Theorem 1.1.


## 4.2.Δp Bundle over Δ1

Given the integerp>1, as McDuff and Tolman in [3], we consider the following vectors in ℝp+1: (4.18)ni=-ei,i=1,…,p,np+1=∑i=1pei,np+2=-ep+1,np+3=ep+1-∑i=1paiei,
where e1,…,ep+1 is the standard basis of ℝp+1 and ai∈ℤ. We write (4.19)a∶=(a1,…,ap)∈Zp,A∶=∑i=1pai,a⋅a=∑i=1pai2.Letλ, τ be real positive numbers with λ+ai>0, for i=1,…,p. In this subsection, we will consider the polytope Δ in (ℝp+1)* defined by the above conormals nj and the following kj: (4.20)k1=⋯=kp=kp+2=0,kp+1=τ,kp+3=λ.This polytope will be also denoted byΔ0(λ,τ). It is a Δp 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 polytopeΔ is a Delzant polytope. The manifold (2.2) is in this case (4.21)MΔ={z∈Cp+3:∑i=1p+1|zi|2=τ/π,-∑j=1paj|zj|2+|zp+2|2+|zp+3|2=λ/π}≃,
where (zj)≃(zj′) if and only if there are α,β∈U(1) such that (4.22)zj′=αβ-ajzj,j=1,…,p,zp+1′=αzp+1,zk′=βzk,k=p+2,p+3.
Thus, MΔ is the total space of the fibre bundle ℙ(L1⊕⋯⊕Lp⊕ℂ)→ℂP1, where Lj is the holomorphic line bundle over ℂP1 with Chern number aj.The symplectic form (2.4) is (4.23)ωΔ=12(σ1+⋯+σp+σp+2),
where σk=dρk2∧dφk.And the moment map(4.24)μΔ([z])=(x1,…,xp,xp+2),
where xi:=πρi2.Proposition 4.4.
The coordinatesx̅j of  Cm(Δ0(λ,τ)) are given by
(4.25)x̅k=τp+2λ(p+2)+τ(A+ak)λ(p+1)+τA,fork=1,…,p,(4.26)x̅p+2=12(p+1)(p+2)λ2+2(p+2)Aλτ+(a⋅a+A2)τ2(p+2)((p+1)λ+Aτ).Proof.
Since the points[z]∈MΔ satisfy |zp+2|2≤λ/π+∑j=1paj|zj|2, by (2.9) and Lemma 2.2, we have
(4.27)∫MΔ(ωΔ)p+1=(p+1)!∫Sp(τ)(λ+∑j=1pajxj)=(p+1)!(λτpp!+τp+1A(p+1)!).
Similarly, for k=1,…,p(4.28)∫MΔxk(ωΔ)p+1=(p+1)!(λτp+1(p+1)!+τp+2(p+2)!∑j≠kaj+2τp+2ak(p+2)!).
The kth coordinate of Cm(Δ), x̅k, is the quotient of (4.28) by (4.27); that is,
(4.29)x̅k=τp+2λ(p+2)+τ(A+ak)λ(p+1)+τA.
For thep+2-coordinate of Cm(Δ), we need to calculate ∫Mxp+2(ωΔ)p+1. By Lemma 2.2,
(4.30)1(p+1)!∫Mxp+2(ωΔ)p+1=12∫Sp(τ)(λ+∑j=1pajxj)2=12(λ2τpp!+2Aλτp+1(p+1)!+(a⋅a+A2)τp+2(p+2)!).
Formula (4.26) is a consequence of (4.27) together with (4.30).The translation in(ℝp+1)* defined by the vector (-k1,…,-kp,-kp+2) transforms the hyperplanes 〈x,np+3〉=λ and 〈x,np+1〉=τ in (4.31)〈x,np+3〉=λ-kp+2+∑j=1pajkj,〈x,np+1〉=τ-∑j=1pkj,
respectively.LetΔ(k) be a polytope with k=(k1,…,kp+3) generic in the chamber 𝒞Δ. From (4.31), it follows that Δ(k) is the image of the polytope Δ0(λ,τ) by the translation determined by (-k1,…,-kp,-kp+2), whenever (4.32)kp+2-∑j=1pajkj+kp+3=λ,∑j=1pkj+kp+1=τ.
In this case, (4.33)Cm(Δ(k))=Cm(Δ0(λ,τ))-(k1,…,kp,kp+2).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 (4.34)ϵ-∑j=1pajϵ+ϵ=(2-A)ϵ
and τ by (p+1)ϵ, and finally take into account (4.33). These operations give (4.35)x̅j(Δ(ϵ̌))=ϵ2(p+2)((p+1)aj-A),j=1,…,p,x̅p+2(Δ(ϵ̌))=ϵ4(p+2)(-A2+(p+1)(a⋅a)).Givenb=(b1,…,bp,b)≡(b̂,ḃ), with b̂=(b1,…,bp,0) and ḃ=(0,…,0,b),(4.36)〈ddϵ|ϵ=0Cm(Δ(ϵ̌)),b〉=14(p+2)((p+1)(2a⋅b̂-ba⋅a)-A(2B+bA)),
where a·b̂=∑j=1pajbj and B=∑j=1pbj.By Proposition3.6, we have the following theorem.Theorem 4.5.
LetΔ be the Δp bundle over Δ1 defined by (4.18) and (4.20). Given b=(b̂,ḃ)∈ℤp+1,  if
(4.37)(p+1)(2a⋅b̂-ba⋅a)-A(2B+bA)≠0,
then ψb defines an infinite cyclic subgroup in the fundamental group π1(Ham(MΔ,ωΔ)).It is straightforward to check that(4.38)(p+1)(2a⋅b̂-ba⋅a)-A(2B+bA)=0
is also a sufficient condition for (Δ,b) to be a mass linear pair.Since(4.39)〈Cm(Δ0(λ,τ)),b〉=〈Cm(Δ0(λ,τ)),b̂〉+〈Cm(Δ0(λ,τ)),ḃ〉,
if (4.38) holds, using (4.25) and (4.26), one obtains (4.40)〈Cm(Δ0(λ,τ)),b〉=bλ2+(b2(a⋅a+A2)(p+2)A+(a⋅b̂+AB)(p+2)A)τ.
By (4.33), for k∈𝒞Δ, (4.41)〈Cm(Δ(k)),b〉=〈Cm(Δ0(λ,τ)),b〉-∑j=1pbjkj-bkp+2,
with λ and τ given by (4.32).Ifb=b̂, the condition (4.38) reduces to (p+1)a·b̂=AB and (4.42)〈Cm(Δ(k)),b〉=(a⋅b̂+AB)(p+2)A(∑j=1pkj+kp+1)-∑j=1pbjkj.
Hence, 〈Cm(Δ(k)),b〉=∑Rjkj, where (4.43)Rj=(a⋅b̂+AB)(p+2)A-bj,j=1,…,p,Rp+1=(a⋅b̂+AB)(p+2)A,Rp+2=Rp+3=0.
So, (4.44)∑j=1p+3Rj=(p+1)a⋅b̂-AB(p+2)A=0.A similar calculation for the caseb=ḃ shows that the corresponding ∑jRj vanishes. That is, we have the following proposition.Proposition 4.6.
LetΔ be a Δp bundle over Δ1.  If (Δ,b) is a mass linear pair, then ∑jRj=0.Forp=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 (4.45)3(a1b1+a2b2)=(a1+a2)(b1+b2).
Or in terms of the γi(4.46)a1γ1+a2γ2=0.
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 ofℂPn

In this subsection,Δ≡Δ0(λ,τ) will be (4.47)Δ={(x1,…,xn)∈Rn∣∑i=1nxi≤τ,0≤xi,xn≤λ},
where τ,λ∈ℝ>0 and σ∶=τ-λ>0. That is, Δ is the polytope obtained truncating the simplex Sn(τ), defined in Lemma 2.2, by a “horizontal” hyperplane through the point (0,…,0,λ). The manifold MΔ associated with Δ is the one point blowup of ℂPn.The mass center of the simplexSn(τ) is the point (4.48)Cm(Sn(τ))=τn+1w,
with w=(1,…,1).As the volume ofSn(τ) is τn/n!, it follows from (4.48) (4.49)(τn-σn)Cm(Δ)=τnτn+1w-σn(σn+1w+λen).
That is, (4.50)Cm(Δ)=1τn-σn((τn+1-σn+1n+1)w-λσnen).Givenk=(k1,…,kn+2)∈𝒞Δ, the facets of Δ(k) are in the following hyperplanes: (4.51)-xj=kj,j=1,…,n;∑i=1pxk=kn+1;xn+1=kn+2.
As in the preceding subsections, (4.52)Δ(k)=Δ0(λ,τ)-(k1,…,kn),
provided that λ=kn+kn+2 and τ=∑i=1n+1ki.The pair(Δ,b=(b1,…,bn)) is mass linear if and only if there exist A,B,C∈ℝ such that (4.53)∑j=1n-1bjτn+1-σn+1n+1+bn(τn+1-σn+1n+1-(τ-σ)σn)=(Aτ+Bσ+C)(τn-σn),
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
(4.54)bn=1n∑j=1n-1bj.From Theorem1.2 together with Proposition 4.7, it follows the following theorem.Theorem 4.8.
Ifb=(b1,…,bn)∈ℤn and  ∑j=1n-1bj≠nbn, then  ψb generates an infinite cyclic subgroup in  π1(Ham(MΔ,ωΔ)).Fork∈𝒞Δ, by (4.52) (4.55)〈Cm(Δ(k)),b〉=〈Cm(Δ0(λ,τ)),b〉-∑j=1nbjkj.
If (Δ,b) is a mass linear pair, by (4.50) and Proposition 4.7, we have 〈Cm(Δ0(λ,τ)),b〉=bnτ. Thus, (4.56)〈Cm(Δ(k)),b〉=∑j=1n+1Rjkj,
where Rj=bn-bj, for j=1,…,n and Rn+1=bn. Hence, we have the following proposition.Proposition 4.9.
LetΔ be the polytope obtained by truncating the standard n-simplex Sn(τ) by a horizontal hyperplane. If (Δ,b) is a mass linear pair, then ∑jRj=0.Remark 4.10.
Whenn=3, the toric manifold MΔ is
(4.57)MΔ={z∈C5:|z1|2+|z2|2+|z3|2+|z5|2=τ/π,|z3|2+|z4|2=λ/π}T,
where the action of 𝕋=(U(1))2 is defined by
(4.58)(a,b)(z1,z2,z3,z4,z5)=(az1,az2,abz3,bz4,az5),
for a,b∈U(1).
We consider the following loops in the Hamiltonian group of(MΔ,ωΔ):
(4.59)ψt[z]=[z1e2πit,z2,z3,z4,z5],ψt′[z]=[z1,z2e2πit,z3,z4,z5],ψ̃t[z]=[z1,z2,z3e2πit,z4,z5].
In [9] (Remark 4.10), we gave formulas that relate the characteristic numbers associated with these loops
(4.60)I(ψ)=I(ψ′)=(-13)I(ψ̃).
So, for b=(b1,b2,b3)∈ℤ3,
(4.61)I(ψb)=(b1+b2-3b3)I(ψ).
By Proposition 4.7, the vanishing of I(ψb) 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.



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*Source: 642834-2011-10-17.xml*