Let be a Delzant polytope in and . Let denote the symplectic fibration over determined by the pair . Under certain hypotheses, we prove the equivalence between the fact that is a mass linear pair (McDuff and Tolman, 2010) and the vanishing of a characteristic number of . Denoting by , the Hamiltonian group of the symplectic manifold defined by , we determine loops in that define infinite cyclic subgroups in when satisfies any of the following conditions: (i) it is the trapezium associated with a Hirzebruch sur-face, (ii) it is a bundle over , and (iii) is the truncated simplex associated with the one point blowup of .
1. Introduction
Let be a closed connected symplectic -manifold. By , we denote the Hamiltonian group of [1, 2]. Associated with a loop in , there exist characteristic numbers which are invariant under deformation of . 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 , whose definition will be recalled below. By proving the nonvanishing of for certain loops, we will deduce the existence of infinity cyclic subgroups of , when is a toric manifold. The vanishing of the invariant on particular loops in 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 in determines a Hamiltonian fibre bundle with standard fibre , via the clutching construction. Various characteristic numbers for the fibre bundle have been defined in [4]. These numbers give rise to topological invariants of the loop . In this paper, we will consider only the following characteristic number:
where is the vertical tangent bundle of , and is the coupling class of the fibration [1, 5]. depends only on the homotopy class of the loop . Moreover, the map
is an -valued group homomorphism [4].
Our purpose is to study this characteristic number when is a toric manifold and is a 1-parameter subgroup of defined by the toric action. The referred 1-parameter subgroup is determined by an element 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 with the fact that 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 be the torus and the polytope in with facets defined by
where and the are the outward conormals to the facets. The facet defined by the equation will be denoted , and we put for the mass center of the polytope .
In [3], the chamber of is defined as the set of such that the polytope is analogous to ; that is, the intersection is nonempty if and only if for any . When we consider only polytopes which belong to the chamber of a fixed polytope, we delete the in the notation introduced in (1.3).
Further, McDuff and Tolman [3] introduced the concept of mass linear pair. Given the polytope and , the pair is mass linear if the map
is linear. That is,
where and are constant.
Let us assume that is a Delzant polytope [6]. We shall denote by the toric manifold determined by ( being the corresponding moment map). Given , an element in the integer lattice of , we shall write for the loop of Hamiltonian diffeomorphisms of defined by through the toric action. We will let for the characteristic number . When we consider only polytopes in the chamber of a given polytope, we will write instead of for in this chamber.
The group of the translations defined by the elements of acts freely on . We put for the dimension of the quotient . Thus, is the number of effective parameters which characterize the polytopes in considered as “physical bodies.”
We will prove the following theorem.
Theorem 1.1. Let be a pair consisting of a Delzant polytope in and an element in the integer lattice of . If , the following statements are equivalent: (a), for all .(b) is a mass linear pair as in (1.5), with .
In [7], by direct computation, we proved the equivalence between the vanishing of on and the fact that is a mass linear pair, when satisfies any of the following conditions:(i)it is the trapezium associated with a Hirzebruch surface,(ii)it is a bundle over [3], and(iii) is the truncated simplex associated with the one point blowup of .
On the other hand, when is any of these polytopes (i)–(iii), the number is equal to 2; thus, from Theorem 1.1 and the result of [7], it follows that condition is satisfied by all the mass linear pairs . 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 is known only for some symplectic manifolds [8], the invariant allows us to identify nontrivial elements in . As is a group homomorphism, from Theorem 1.1, we deduce that a sufficient condition for to generate an infinite cyclic subgroup in is that the above condition (b) does not hold for . More precisely, we have the following consequence of Theorem 1.1.
Theorem 1.2. Given the Delzant polytope and an element in the integer lattice of . If and is not mass linear, then generates an infinite cyclic subgroup in , for all .
In the proof of Theorem 1.1, a formula for the characteristic number obtained in [9] plays a crucial role. This formula gives in terms of the integrals, on the facets of the polytope, of the normalized Hamiltonian function corresponding to the loop (see (2.13)). From this expression for , we will deduce a relation between the directional derivative of map (1.4) along the vector of , the Euclidean volume of and (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 when is a linear pair and varies in the chamber of ; we prove that is a homogeneous polynomial of the (Proposition 2.4).
In Section 3, we prove Theorem 1.1. In Proposition 3.5, a sufficient geometric condition for the Delzant polytope to admit a mass linear pair is given. For a Delzant polytope , Proposition 3.6 gives a necessary condition for the vanishing of on . We also express in terms of the displacement of the center of mass produced by the change (Proposition 3.7).
Section 4 concerns the form which Theorem 1.2 adopts, when 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 is a mass linear pair, then .
2. A Characteristic Number
Let us suppose that the polytope defined in (1.3) is a Delzant polytope in . Following [10], we recall some points of the construction of from the polytope . We put . The determine weights , for a -action on . Then moment map for this action is
The define a regular value for , and the manifold is the following orbit space
where the relation defined by is
Identifying with , and each is a linear combination of the .
Given a facet of , we choose a vertex of . After a possible change in numeration of the facets, we can assume that intersect at . In this numeration, , for some .
If we write , then the symplectic form can be written on
with an angular variable, linear combination of the .
The action of on
endows with a structure of toric manifold. Identifying with , the moment map is defined by
where the constants are linear combinations of the and
The facet of is the image by of the submanifold
We write , then
Let be an element in the integer lattice of . The normalized Hamiltonian of the circle action generated by is the function determined by
That is, , where
Moreover,
An expression for the value of the invariant in terms of integrals of the Hamiltonian function has been obtained in Section 4 of [9] (see also [11, 12])
where the contribution of the above facet (with ) is
with .
Given , we consider the polytope obtained from by the translation defined by a vector of . As we said, we write and for the corresponding characteristic numbers. According to the construction of the respective toric manifolds,
But the normalized Hamiltonians and corresponding to the action of on and are equal. Thus, it follows from (2.13) that . More precisely, we have the evident proposition.
Proposition 2.1. If is an arbitrary vector of , then , for , .
By Proposition 2.1, we can assume that all in (2.6) are zero for the determination of .
The following lemma is elementary.
Lemma 2.2. If
then
More general, if , we put
then
Thus, in the particular case that , the integral is a monomial of degree in , and is a monomial of degree .
We return to the general case in which is the polytope defined in (1.3). Its vertices are the solutions to
hence, the coordinates of any vertex of are linear combinations of the .
A hyperplane in through a vertex of is given by an equation of the form
Thus, the independent term is a linear combination (l. c.) of the . Moreover, the coordinates of the common point of hyperplanes
with l. c. of, the are also l. c. of the .
By drawing hyperplanes through vertices of (or more generally, through points which are the intersection of hyperplanes as (2.22)), we can obtain a family of subsets of such that(a)each is the transformed of a simplex by an element of the group of Euclidean motions in .(b)For , is a subset of the border of .(c).
Thus, by construction, each facet of is contained in a hyperplane of the form , with l. c. of the .
On the other hand, the hyperplane is transformed by an element of in an hyperplane . If is a translation in which applies onto , then this transformation maps in a vertex of . So, the translation transforms in . As each is a l. c. of the , so is . Hence, any element of the group of Euclidean motions in which maps onto transforms the hyperplane
with a l. c. of the .
Let assume that , with and the translation defined by . Then the oblique facet of , contained in the hyperplane , is the image by of a facet of , which in turn is contained in a hyperplane of (2.23) ( being a l. c. of the ). The argument of the preceding paragraph applied to and proves that is a l. c. of the . Hence, by (2.19), the integral
is a monomial of degree of a l. c. of the . Thus,
is a homogeneous polynomial of degree of the .
Similarly,
is a homogeneous polynomial of degree of the . Analogous results hold for
From formulas (2.9)–(2.14) together with the preceding argument, it follows the following proposition.
Proposition 2.3. Given a Delzant polytope , if belongs to the integer lattice of , then is a rational function of the , for .
Analogously, we have the following proposition.
Proposition 2.4. If is mass linear pair, then is a homogeneous polynomial in the of degree , when .
We will use the following simple lemma in the proof of Theorem 1.1.
Lemma 2.5. If for , with , then .
Proof. The vertices of are the solutions of (2.20), and the vertices of are the solutions of , with . Thus, the vertices of are those of multiplied by .
The lemma also follows from the fact that (2.25) and (2.26) are homogeneous polynomials of degree and , respectively.
Let us assume that the polytope defined by (1.3) is Delzant and let be an element of . We denote by , , and , the manifold, the symplectic structure and the moment map (resp.) determined by . The facets of will be denoted by .
Let be an element in the integer lattice of . We put
By (2.9), is the Euclidean volume of the polytope . Given a facet , we can assume that (see third paragraph of Section 2). So, is defined by the equation . If we make an infinitesimal variation of the facet , by means of the translation defined by (keeping unchanged the other ), then the volume of changes according to
We write for . Thus,
So, by (2.11),
From (2.13) and (2.14), it follows
Thus, we have proved the following proposition.
Proposition 3.1. for all if and only if , for all .
Next, we will parametrize the quotient (of classes of polytopes in module translation) defined in Section 1.
After a possible renumbering, we may assume that the intersection of facets is a vertex of . Thus, the conormals are linearly independent in . So, given , there is a unique , such that
(Expressing the in terms of a basis of and in the dual basis, (3.6) is a compatible and determined system of linear equations for the coordinates of .) Moreover depends linearly on the ; that is, is a linear function of , for all .
If , we write
where the element in defined by (3.6). From the linearity of with respect to the , it follows that and are linear combinations of .
The polytope in defined by will be denoted by . It is the result of the translation of by the vector ; that is,
Let an element in the integer lattice of , we define the function by
The function is defined on the pairs such that . By Lemma 2.5, it follows
for any real number such that belongs to the domain of . This property implies that
Theorem 3.2. If , for all and , then , with constant (i.e., is a mass linear pair) and .
Proof. We set It follows from (3.8) that
By the hypothesis and Proposition 3.1,
Since
from (3.13), we deduce
where , , stand for the following constants
Since and are first integrals of (3.15), the general solution of this equation is
where is a derivable function of one variable. It follows from (3.11) and (3.17) that
Thus, , with constant. We have for
In other words, is a linear function of the ; that is, , with constant. From (3.13), it follows .
Remark 3.3. The proof of Theorem 3.2 can be adapted to the simpler case when . In this case, the function satisfies and . So, and is a linear map of the variables . On the other hand, the proof of this theorem does not admit an adaptation to the case . In fact, the corresponding function would be a function of variables . The equation which corresponds to (3.18) in this case would be
But this condition does not imply the linearity of . When is a mass linear pair as in (1.5), by (3.5)
for all . From (3.21), we deduce the following proposition.
Proposition 3.4. Let be a mass linear pair. for all if and only if .
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 to admit mass linear functions. We write
with .
Proposition 3.5. If all points , for , belong to a hyperplane of with a conormal vector in and , then admits a mass linear function.
Proof. Let be a conormal vector to the hyperplane, then
By (3.5), ; Theorem 3.2 applies and is a mass linear pair.
Proposition 3.6. Let be a Delzant polytope, such that belongs to the closure of . If , a necessary condition for the vanishing of on is
Proof. If vanishes on , then is a linear pair, by Theorem 1.1. Thus, , on . So, given and small enough
By Theorem 1.1, . Thus, for any ,
Taking the limit as ,
Next, we will describe a geometric interpretation of the number . Given an arbitrary Delzant polytope . If is a vector of , then
if .
We will denote by the element of defined by the following relation
with for all .
Now, we assume that is a mass linear pair. From (1.5), it follows
These formulas allow us to state the following proposition that gives an interpretation of the sum in terms of the variation of with the .
Proposition 3.7. Let be a mass linear pair as in (1.5). Then,
being the element of defined by (3.29).
4. Examples
In this section, we will deduce the particular form which adopts Theorem 1.2 when 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 and the condition for to be a mass linear pair. We will dedicate a subsection to each type.
4.1. Hirzebruch Surfaces
Given and with , in [7], we considered the Hirzebruch surface determined by these numbers. is the quotient
where the equivalence defined by is given by
for .
The manifold equipped with the following action
is a toric manifold. The corresponding moment polytope is the trapezium in with vertices
That is, is the toric manifold determined by the trapezium .
As the conormals to the facets of are the vectors , , , and , the facets of a generic polytope in are on the straights
The vertices of are the points
Thus, the translation in the plane , defined by transforms the trapezium determined by the vertices (4.4) in if
So,
Moreover, the mass center of is
The chamber consists of the points such that , with and given by (4.7). So, the point belongs to the closure of . From (4.8), together with (4.7) and (4.9), it follows
where . By Proposition 3.6, if with vanishes on the chamber , then .
On the other hand, from (4.9) and (4.8), it follows
By (4.7), expression (4.11) is linear in the if and only if
is linear in , . That is, if and only if there exist constants , such that for al ,
From this relation, it follows the above condition . In this case (4.11) reduces to
Comparing (1.5) with (4.14), we obtain , ; so, . That is, the condition holds for all the mass pairs when is the polytope associated to a Hirzebruch surface. Hence, we have following proposition.
Proposition 4.1. is a mass linear pair if and only if . Moreover, in this case .
Corollary 4.2. If , then generates an infinite cyclic subgroup in .
Remark 4.3. We denote by the following isotopy of
where is a loop in the Hamiltonian group of . By , we denote the Hamiltonian loop
In Theorem 8 of [11], we proved that . If , then
That is, if and only if , which is in agreement with Proposition 4.1 and Theorem 1.1.
4.2. Bundle over
Given the integer , as McDuff and Tolman in [3], we consider the following vectors in :
where is the standard basis of and . We write
Let , be real positive numbers with , for . In this subsection, we will consider the polytope in defined by the above conormals and the following :
This polytope will be also denoted by . It is a bundle on (see [3]). When , is the prism whose base is the triangle of vertices , , and and whose ceiling is the triangle determined by , , and (see Figure 1).
Figure 1: bundle over .
We assume that the above polytope is a Delzant polytope. The manifold (2.2) is in this case
where if and only if there are such that
Thus, is the total space of the fibre bundle , where is the holomorphic line bundle over with Chern number .
Proof. Since the points satisfy , by (2.9) and Lemma 2.2, we have
Similarly, for
The th coordinate of , , is the quotient of (4.28) by (4.27); that is,
For the -coordinate of , we need to calculate . By Lemma 2.2,
Formula (4.26) is a consequence of (4.27) together with (4.30).
The translation in defined by the vector transforms the hyperplanes and in
respectively.
Let be a polytope with generic in the chamber . From (4.31), it follows that is the image of the polytope by the translation determined by , whenever
In this case,
According to (4.32), the coordinates of the mass center , with , can be obtained substituting in (4.25) and in (4.26) by
and by , and finally take into account (4.33). These operations give
Given , with and ,
where and .
By Proposition 3.6, we have the following theorem.
Theorem 4.5. Let be the bundle over defined by (4.18) and (4.20). Given , if
then defines an infinite cyclic subgroup in the fundamental group .
It is straightforward to check that
is also a sufficient condition for to be a mass linear pair.
Since
if (4.38) holds, using (4.25) and (4.26), one obtains
By (4.33), for ,
with and given by (4.32).
If , the condition (4.38) reduces to and
Hence, , where
So,
A similar calculation for the case shows that the corresponding vanishes. That is, we have the following proposition.
Proposition 4.6. Let be a bundle over . If is a mass linear pair, then .
For , let be the following linear combination of the conormal vectors with . By (4.18), with , . In this case, condition (4.38) reduces to
Or in terms of the
This is a necessary and sufficient condition for to be mass linear. This result is the statement of Lemma 4.8 in [3].
4.3. One Point Blowup of
In this subsection, will be
where and . That is, is the polytope obtained truncating the simplex , defined in Lemma 2.2, by a “horizontal” hyperplane through the point . The manifold associated with is the one point blowup of .
The mass center of the simplex is the point
with .
As the volume of is , it follows from (4.48)
That is,
Given , the facets of are in the following hyperplanes:
As in the preceding subsections,
provided that and .
