Construct b -symplectic toric manifolds from toric manifolds
CCONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROMTORIC MANIFOLDS MINGYANG LI
Abstract.
In [GMPS1], Guillemin et al. proved a Delzant-type the-orem which classifies b -symplectic toric manifolds. More generally, in[GMPS2] they proved a similar convexity result for general Hamiltoniantorus action on b -symplectic manifolds. In this paper we provide a newway to construct b -symplectic toric manifolds from usual toric manifolds.Conversely, through this way we can also decompose a b -symplectic toricmanifolds to usual toric manifolds. Finally we will try to prove that thiskind of decomposition is useful. Contents
1. Introduction 12. Preliminaries 22.1. b -manifolds, b -tangent and b -cotangent bundles 22.2. b -symplectic manifolds 42.3. Torus Hamiltonian actions on b -symplectic manifolds 52.4. b -moment maps and b -Delzant polytopes 63. The b -Delzant classification 84. Construction from toric manifolds 94.1. Observation 104.2. Toric submanifolds 124.3. Local model for toric manifolds 154.4. Cutting and gluing 185. Decomposition to several toric manifolds 225.1. symplectic structures 245.2. toric structures 246. Applications 256.1. Homology of b -toric manifold 25References 291. Introduction
It is a very famous and beautiful result that the image of the moment mapof a compact symplectic toric manifold is a rational convex polytope, which
Date : December 3, 2019. a r X i v : . [ m a t h . S G ] D ec MINGYANG LI we call Delzant polytope. Moreover, the image of the moment map classifiescompact symplectic toric manifolds up to equivariant diffeomorphisms. See[GS1], [GS2] and [D]. More recently, in [GMPS1] Guillemin et al. generalizethis result to b -symplectic toric manifold, which is a symplectic manifoldexcept a hypersurface and admits a Hamiltonian toric action with respectto this singular symplectic form. In this generalized b -symplectic toric case,because of the fact that the symplectic form is singular along some hypers-ufaces, the definition of moment map is also adjusted a little, and the imageof the new moment map will be so called b -Delzant polytope. In [GMPS1],they constructed corresponding b -symplectic toric manifold from b -Delzantpolytope by using symplectic cutting.In this paper, we will give a way to construct the b -symplectic toric man-ifold by cutting and gluing several usual symplectic toric manifolds whichbelong to some special class of symplectic toric manifolds. The gluing pro-cess will be done along two hypersurfaces, and such a construction will tellus more topology information about the b -symplectic manifolds. Conversely,using this kind of idea we can also decompose every b -symplectic manifoldsto some special class of usual toric manifolds, which are easily characterizedcombinatorially.This paper is organized as follows: In Section 2 we will go over somebasic definitions and properties of b -symplectic manifolds, in Section 3 wewill review the b -Delzant type theorem proved in [GMPS1]. In Sections 4and 5 we will construct b -symplectic toric manifolds from some special classof toric manifolds, and converse the construction to decompose b -symplectictoric manifolds. And finally in Section 6, we will try to use this kind ofconstruction to prove some properties of the b -symplectic toric manifolds.2. Preliminaries
In this part, we will go over some basic definitions and properties of b -symplectic manifolds. We will also give some basic examples. More detailscan be found in [S] and [GMP]. As for basic Poisson geometry, the readercan refer to [FM].2.1. b -manifolds, b -tangent and b -cotangent bundles. When we aretalking about b -manifold, we mean an oriented smooth manifold M with aclosed smooth hypersurface Z . Definition 1. A b -manifold is a pair ( M, Z ), where M is an orientedsmooth manifold M and Z is a closed smooth hypersurface in M . We callthe hypersurface Z singular hypersurface (It may not be connected). A b -map is a smooth orientation preserving map f : ( M, Z ) −→ ( M (cid:48) , Z (cid:48) )such that f : M → M (cid:48) is smooth orientation preserving, f − ( Z (cid:48) ) = Z and f is transverse to Z (cid:48) . ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 3 The essential difference between b -manifold and a usual manifold is thedefinition of tangent bundle. Assume U is an open subset of the manifold U ⊂ M , we use Γ( U, b T M ) to denote the vector space consisting of vectorfields on U , which are tangent to Z at points of Z . Definition 2.
We define the b -tangent bundle b T M of a b -manifold( M, Z ) to be the vector bundle defined by the locally free sheaf Γ( U, b T M ).The b -cotangent bundle b T ∗ M is defined to be the dual bundle of b T M .By the above definition, if we are at a point p ∈ M \ Z , then the b -tangent space b T p M (cid:39) T p M (cid:39) span { ∂∂x , · · · , ∂∂x n } , where { x , · · · , x n } isa local coordinate system. But if we are at a point p ∈ Z , then b T p M (cid:39) span { f ∂∂f , · · · , ∂∂x n } . Here f is a defining function for the hypersurface Z (we need to require df (cid:54) = 0 at points of Z ) and { f, x , · · · , x n } is a localcoordinate system around p.Similarly we can define b -differential k -forms to be sections of Λ k ( b T ∗ M ).We denote the space of b -differential k -forms as b Ω k ( M ). In general, if wehave a defining function f for the hypersurface Z , then every b -differential k -form can be written as w = α ∧ dff + β , α ∈ Ω k − ( M ) and β ∈ Ω k ( M ) , where α and β may not be unique. And using this we can extend the exteriordifferential operator d to b Ω ∗ ( M ) by defining dw = da ∧ dff + dβ. This helps us to form a chain complex0 → b Ω ( M ) → b Ω ( M ) → · · · → b Ω n ( M ) . The above b -de Rham complex will give us b -cohomology group b H ∗ ( M ).For more details, one can find in [GMP].The b -forms may blow up at points of Z , but we can still integrate com-pactly supported b -forms over M : given a b -form η ∈ b Ω n ( M ), we define theintegral of η over M to be the singular integral b (cid:90) M η = lim (cid:15) → (cid:90) | f | >(cid:15) η, where f is a defining function of the hypersurface Z . This definition doesnot depend on the choice of defining functions. See [S].In this b -manifold case, if we have a local defining function f for thehypersurface Z , then dff will be a b -form which is closed but not exact. Tomake it exact we have to enlarge the space of smooth functions on b -manifold M to include functions such as log | f | . So we have the following definition. MINGYANG LI
Definition 3.
Given a b -manifold ( M, Z ), we define the sheaf of b -functions on M , b C ∞ to be b C ∞ ( U ) = c log | y | + f, where f ∈ C ∞ ( U ), c ∈ R and y is a defining function for U ∩ Z .Such an enlargement will also help us to define b -moment map in thefuture.2.2. b -symplectic manifolds. In this subsection we will discuss b -symplecticmanifolds. Definition 4.
Assume (
M, Z ) is a 2 n dimensional b -manifold. We say ω ∈ b Ω ( M ) is b -symplectic if it is a closed b -form such that ω np (cid:54) = 0for each point p , as an element in b T np M . When a b -manifold ( M, Z ) admitsa b -symplectic form, we say it is a b -symplectic manifold.Here is a very simple example. Example 5.
Consider the Euclidean space R n with coordinates { x , y , · · · , x n , y n } .Assume the hypersurface is { x = 0 } . Then we have a b -symplectic form ω , ω = dx x ∧ dy + · · · + dx n ∧ dy n . Definition 6. A b -map f between two b -symplectic manifolds ( M, Z ) and( M (cid:48) , Z (cid:48) ) is a b -symplectomorphism, if f ∗ ω (cid:48) = ω , where ω and ω (cid:48) is the b -symplectic form of M and M (cid:48) respectively.From the perspective of Poisson geometry we can also study b -symplecticmanifolds. Recall that there is a one-to-one correspondence between Poissonstructures on a manifold and bi-vector fields π ∈ χ ( M ) (where by χ ( M )we mean vector fields on M ), which satisfy the condition [ π, π ] = 0 on amanifold. If we view a bi-vector π ∈ χ ( M ) as a map π : Ω ( M ) −→ χ ( M )which is defined by contraction, then it will be a bundle map between T ∗ M and T M . If it is an isomorphism we can consider its inverse ω : T M → T ∗ M .And this gives us a symplectic form. Conversely given a symplectic structurewe can also construct a Poisson structure just as above (We will get a non-degenerate Poisson structure). In our case, if we do the construction for a b -symplectic form we in fact get a Poisson bi-vector π whose top exteriorproduct as a section of Λ n ( T M ) vanishes transversely on the hypersurface Z , and is nonzero at other points (One can see this easily by Weinstein’sresult on local structure of a Poisson manifold. See [W] and [FM].). So b -symplectic manifolds are in fact a special kind of Poisson manifolds. ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 5 Torus Hamiltonian actions on b -symplectic manifolds. Let T n be a n -dimensional torus and acts on a manifold M . We use t to denote itsLie algebra and t ∗ to denote the dual of its Lie algebra. In the symplecticcase we call it a Hamiltonian action if there exists a equivariant momentmap µ : M → t ∗ such that for every X ∈ t we have, dµ X = − ι X ω. Here we use µ X to denote the function µ X = (cid:104) µ, X (cid:105) , and X is the vectorfield on M corresponding to the infinitesimal action by X , X ( p ) = ddt (exp( tX ) · p ) . In the setting of b -symplectic manifold, we can define Hamiltonian actionsimilarly. Definition 7. A T n action on a b -symplectic manifold ( M, Z, ω ) will becalled
Hamiltonian if for any X , Y ∈ t , we have, • the one form ι X ω is b -exact, i.e. exists a primitive H X ∈ b C ∞ ( M ); • ω ( X , Y ) = 0.Such a Hamiltonian action will be called toric if it is an effective action anddim( T n )= dim( M ). Example 8.
We can consider the following simple example. Let ( S , Z, ω )be the 2-dimensional sphere with standard b -symplectic form ω = dhh ∧ dθ ,with h ∈ [ − , θ ∈ [0 , π ], and the corresponding hypersurface is Z = { h = 0 } . The S action on the sphere will be rotation with respect to theaxis as in the picture below. In another word, will be the flow of the vectorfield ∂∂θ . If we consider the function µ ( h, θ ) = log | h | , then we will have, dµ X = − ι X ω. Hence the action satisfies the definition above and the action is actually aHamiltonian action. Moreover, this is a toric action. The image of the abovefunction log | h | is as in the picture. Figure 1.
The ”moment map” of the S -actions on M .Moreover, if we consider ( T , Z = { θ ∈ { , π }} , ω = dθ sin θ ∧ dθ ), andlet the torus S act like rotation on θ coordinate. Then if we select the MINGYANG LI ”moment map” as log | θ sin θ | , the S action will also be a Hamiltonianaction. The important point is that the torus T does not admit a symplectictoric structure. For details see [GMPS1].2.4. b -moment maps and b -Delzant polytopes. As in [GMPS1], theyshowed that there exists a b -moment map and its image will satisfies somekind of Delzant property. But because of the fact that our b -symplectic formblows up along the hypersurface Z , the image of our b -moment map cannotfall in one Euclidean space like t ∗ as before. We conclude the results andsketch the proofs that appear in [GMPS1]. Definition 9.
Given a Hamiltonian T k action on our b -manifold, for eachconnected component Z (cid:48) , by the hypothesis that the action is Hamiltonian,the b -form ι X ω has a primitive in b C ∞ . This primitive can be written as c log | y | + g in a neighborhood of Z (cid:48) , where y is the local defining function of Z (cid:48) , c ∈ R and g is smooth. The map X (cid:55)→ c is a well-defined homomorphism,and will correspond to an element ν Z (cid:48) ∈ t ∗ . This is called the modularweight of Z (cid:48) . The kernel of the modular weight ν (cid:48) Z will be denoted as t (cid:48) Z .In [GMPS1] they proved that such a modular weight ν Z for toric actionis nonzero. Moreover, it was also proved that for toric action if Z and Z are two adjacent (i.e. adjacent to the same connected component of M \ Z ),then the modular weights of Z and Z satisfy: ν Z = kν Z , for some k < . That is to say, all the modular weights are in the ”same direction”, but fortwo adjacent modular weights, they will have different sign. We will see inthe future this will help us to glue different t ∗ together. See claim 20 in[GMPS1]. Definition 10.
The adjacency graph of a b -manifold ( M, Z ) is a graph G = ( V, E ). It has a vertex for each component of M \ Z and an edge foreach connected component of Z . Definition 11.
Given a Hamiltonian T k action on a b -symplectic mani-fold ( M, Z ), we define the weighted adjacency graph of the b -symplecticmanifold with the Hamiltonian action above to be G = ( G, w ), where G isthe adjacency graph of M and w is a weight function on the set of edges w : E → t ∗ . The weight function w is defined to be, mapping an edge e ∈ E to the modular weight of the corresponding connected component in Z .In the case of toric action, . To help the reader understand those concepts,we consider the b-symplectic toric manifolds S and T mentioned above asexamples. Example 12.
For the sphere case, there will be two vertexes for the twocomponents in M \ Z and only one edge for the only one component in Z .For the torus case, similarly, there will exist two vertexes and two edges. Butin this case it will form a closed loop, as shown in the following pictures. ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 7 Figure 2.
The adjacency graph of S and T .Now, have already defined the weighted adjacency graph of a b -symplecticmanifold with a Hamiltonian action, we can now construct the momentcodomain of a b -symplectic toric manifold. We construct the b -momentcodomain ( R G , Z G , ˆ x ), where ( R G , Z G ) is a b -manifold and ˆ x : R G \Z G → t ∗ is a smooth map, such that: • If G is a single vertex, we define the triple ( R G , Z G , ˆ x ) = ( t ∗ , ∅ , id ); • If G has at least two vertices, then for each e ∈ E , we have thecorresponding modular weight w ( e ). Let t w = ( w ( e )) ⊥ ⊂ t . We willuse modular weights to glue different t ∗ together as said before. Asa set, we define: R G = t ∗ × V (cid:116) t ∗ w × E Z G = t ∗ w × E. and define ˆ x (( x, v )) = x . R G can be endowed with a smooth structure such that ( R G , Z G ) is a b -manifold, which also makes ˆ x is a smooth map. For technical details see[GMPS1]. And this will be the codomain of the moment map. Definition 13.
Given a b -symplectic manifold with a effective Hamiltonian T k action such that the modular weights are all nonzero. Then a b - momentmap for an effective Hamiltonian T k action will be a T k -invariant b-map µ : M → R G such that for any X ∈ t , the function µ X ( p ) = (cid:104) ˆ x ◦ µ ( p ) , X (cid:105) islinear with respect to X , and satisfies: ι X ω = − dµ X It was proved that for any b -symplectic toric manifold there always existsa b -moment map. See theorem 27 in [GMPS1]. Example 14.
We still consider the sphere S , but this time we can considerthe case where there are more than one singular hypersurfaces. The imageof the moment map will be as below. MINGYANG LI
Figure 3.
The b -moment map image of S with many sin-gular hypersurfaces.3. The b -Delzant classification In this section, we will describe the b -Delzant type theorem proved in[GMPS1]. We first need to define what is a rational convex polytope, or a b -Delzant polytope in R G . Definition 15.
We use A X,k,v and B Y,k to denote the following two kindsof hyperplanes in R G , where X ∈ t , Y ∈ t w , k ∈ R and v is a vertex of G . A X,k,v = { ( v, ξ ) |(cid:104) ξ, X (cid:105) = k } ⊂ v × t ∗ ⊂ R G ,B Y,k = { ( v, ξ ) |(cid:104) ξ, Y (cid:105) = k , v a vertex of G } ⊂ R G \Z G = R G . We call the closure of each component in the complement of the hyperplanea half-space in R G .The following picture shows some examples of those hyperplanes. t * t * t * Figure 4.
Examples of hyperplanes.
Definition 16. A b - polytope ∆ in R G is a bounded connected set suchthat each component of ∆ ∩ R G \Z G is an intersection of half-spaces. ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 9 The definition of b -Delzant is similar to the usual definition. For a b -polytope we can talk about vertices, facets and edges as before. Definition 17.
When G is a line, we say a b -polytope is b - Delzant if forevery vertex v of (cid:52) , there exists a lattice basis { u i } of t ∗ such that theedges linking to v can be written as v + tu i for t (cid:62)
0. When G is a circle, a b -polytope is b - Delzant if the polytope ∆ Z = ∆ ∩ t ∗ w is Delzant.The following picture is an example of b -Delzant polytope. In this pictureit shows a b -Delzant polytope in a b -codomain consisting of two t ∗ gluedtogether along some modular weight direction. Now we are able to state the t * t * Figure 5.
Example of b -Delzant polytope. b -Delzant type classification theorem. Theorem 18.
The b -moment map { b -toric manifolds ( M, Z, ω ) } −→ { b -Delzant polytopes } which sends each b -toric manifold to its b -moment map image is a bijection.Here b -toric manifolds are considered up to equivariant b -symplectomorphismsthat preserves the moment map For the proof of injectivity, see [GMPS1]. For the proof of surjectivity,one can simply consider a standard model X ∆ Z × S and then applyingsymplectic cuttings, as it was done in [GMPS1].4. Construction from toric manifolds
Now we come to the main part of the paper. In this part we come up witha way constructing b -toric manifold from the information of the b -Delzantpolytope. This is mainly done by doing surgeries to toric manifolds andgluing them together. Caution.
It should be pointed out that the image of the moment map,i.e., the Delzant polytope is allowed to be transformed under SL(n , Z ) andbe translated by some constants, because the action is T n which is abelian.The later one can also be done for b -moment map image. Observation.
In order to get a feeling for this construction, we firstconsider a simple example. For example, consider Figure 5. One may wonderis there any link between the b -toric manifold corresponding to Figure 5 andthe toric manifold with Delzant polytope as below. Figure 6.
A usual Delzant polytope ∆ in R = Lie( T ) ∗ .We denote the corresponding toric manifolds as X ∆ , its symplectic formas ω and moment map as µ . If we put ourself in the so-called symplecticcoordinates, i.e. (∆ × T , ( x, ξ )) with symplectic form dx ∧ dξ (see, forexample, [Ab1]), it seems that we should blow up the symplectic form ω around the hypersurface l . This can be easily done, one just needs to splitthe polytope to three part as below. Figure 7.
A partition of ∆Now we can simply keep the symplectic form in part ∆ and denote itby ω , replace the symplectic form in ∆ by a b -symplectic form ω withsingular hypersurface l and in part ∆ , if we use x to denote the horizontalcoordinate, x to denote the vertical coordinate, we should replace dx by − dx in the symplectic form. We denote the resulted symplectic form in ∆ by ω . Then simply use partition of unity to add the three forms together,i.e. ˜ ω = ρ ω + ρ ω + ρ ω . But we will find many problems in this kind of operation: • From the symplectic coordinate one still needs to do compactifica-tion to get an actual toric variety. Is this ”hypersurface” l in thesymplectic coordinate actually a hypersurface in X ∆ ? ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 11 • For the same reason as above, we also need to worry about wetherthe reversed symplectic form in ∆ is compatible with our compact-ification.As we shall see, it will be a hypersurface and the first question can besolved. However, the second one is an actual problem, and the following twoexamples will help us to see why. Example 19.
This is a very simple one. Consider the b -toric manifold( S × S , Z = { h = 0 } , ω = dh ∧ dθ + h dh ∧ dθ ), with toric T actionby rotation along the flows generated by ∂∂θ and ∂∂θ . It is easy to seethat the b -moment map should be h + log | h | , and the b -Delzant polytopecorresponding to it should be as the Figure 8.For this b -toric manifold, if we start with the toric manifold ( S × S , ω = dh ∧ dθ + dh ∧ dθ ), doing the pinching process as described above, wecan get the desired b -toric manifold S × S . Indeed, if we split it to threeparts as before, and select the partition function carefully, we can get b -toric S × S . Figure 8. b -Delzant polytope of S × S . Example 20.
This is a negative example due to the second problem. Con-sider the following b -Delzant polytopes corresponding to a b -toric manifold M and M (cid:48) respetively. Figure 9. b -Delzant polytopes of M and M (cid:48) . The corresponding usual Delzant polytope of M is the moment polytopeof Hirzebruch surface H = P ( O (1) ⊕ (cid:39) CP CP chopped at one vertex,where O (1) ⊕ CP and trivial bundle over CP . See for example [C2]. According to [GS3]this will be equivalent to do symplectic blow up at the corresponding fixedpoint. That is to say, the toric manifold corresponds to CP CP CP . And hence, the base manifold of M should be CP CP CP .However, for the same reason M should be the blow up of M (cid:48) at thecorresponding vertex. But this time, due to the fact that the symplecticform blow the −∞ line is reversed, so it should be CP CP CP . Now if the construction above works, we would conclude that CP CP CP (cid:39) CP CP CP . which is absurd.From the example above, we can see that the essential reason for this tobe wrong is that the orientation of one half of it is reversed. Remark 21.
There is something more one can talk about CP CP CP .By computing Seiberg-Witten invatiant in [T] it was showed that it does notadmit any symplectic structure which is compatible with its orientation. Onecan also refer to [Mo]. So it is impossible for it coming from a toric manifold.This also provide us with an example which admits b -symplectic structurebut does not admit a (orientation compatible) symplectic structure.4.2. Toric submanifolds.
In this subsection we deal with the first problemappears in the above subsection. We only care about a special kind ofDelzant polytope, which is useful in our construction.
Definition 22.
Given 2 n dimensional toric manifold ( M, T n ) and a 2 m dimensional symplectic submanifold M (cid:48) , we say M (cid:48) is a toric submanifold of M if there exists a T m ⊂ T n such that the action by T m preserves M (cid:48) . Definition 23.
We say a Delzant polytope ∆ is parallel if it satisfies thecondition that there exists one hyperplane F intersecting ∆, and each 1dimensional facet l in ∆ which intersects F , intersects F orthogonally. Sucha hyperplane will be called parallel hyperplane . We denote a parallelDelzant polytope by (∆ , F ).Here is one example of parallel Delzant polytope. Lemma 24.
Let ∆ F = F ∩ ∆ . Then ∆ F is also Delzant.Proof. Let v be a vertex in ∆ F . Assume ∆ is of dimension n . What weneed to prove is that, there exists a basis k , . . . , k n − of Z n − such thateach edge starting from v , can be written as v + tk i . In fact, for any vertex v by definition there must exist an edge l of ∆ which intersects ∆ at v . Weconsider a vertex of ∆ which is located at l . Then there exists basis of Z n ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 13 Figure 10.
Example of parallel Delzant polytope. k , . . . , k n . One of them must be along l . Now project to F , we get thedesired k i . (cid:3) Theorem 25.
Given a parallel Delzant polytope (∆ , F ) and the associatedtoric manifold X ∆ . Assume its moment map is µ ∆ . Then for any parallelhyperplane F which intersects ∆ ◦ , µ − ( F ∩ ∆) will be a hypersurface of X ∆ .Moreover, this is isomorphic to X ∆ F × S .Proof. We can prove this through the original construction from Delzantpolytope to get toric variety. Recall that the Delzant polytope is defined by (cid:104) x, u i (cid:105) ≥ λ i , for i = 1 , · · · , d , where u i is primitive element in the lattice Z n ,due to the fact that ∆ is Delzant. Then we have an exact sequence,0 −→ n ι −→ R d β −→ R n −→ . Where β is the map e i (cid:55)→ u i , n is the kernel of β and ι is the inclusion.Dualize it will yield a sequence,0 −→ ( R n ) ∗ β ∗ −→ ( R d ) ∗ ι ∗ −→ n ∗ −→ . Then one can consider the usual action of T d on C d by rotation with momentmap J ( z ) = ( | z | , . . . , | z d | )+( λ , . . . , λ d ), and doing symplectic reductionwith respect to 0 level and the N action, where N is the Lie group generatedby n . That is to say, we quotient { z ∈ C d | ι ∗ ◦ J ( z ) = 0 } by N . And thecorresponding toric manifold is X ∆ = { z ∈ C d | ι ∗ ◦ J ( z ) = 0 } /N . See, forexample, [G].However, ι ∗ ◦ J ( z ) = 0 if and only if J ( z ) = β ∗ ( J N ( z )) for some unique J N ( z ) ∈ ( R n ) ∗ by the exact sequence. In fact, [ z ] ∈ { z ∈ C d | ι ∗ ◦ J ( z ) = 0 } /N ,[ z ] (cid:55)→ J N ( z ) is exactly the moment map of X ∆ (cid:39) { z ∈ C d | ι ∗ ◦ J ( z ) = 0 } /N (Note that this map is well-defined). Taking inner product, we will have (cid:104) J ( z ) , e i (cid:105) = (cid:104) β ∗ ( J N ( z )) , e i (cid:105) = (cid:104) J N ( z ) , u i (cid:105) = 12 | z i | + λ i . That is to say, the distance of the moment map image J N ([ z ]) of [ z ] ∈ X ∆ to each codimension 1 facet of ∆ will be (cid:104) J N ( z ) , u i (cid:105) − λ i = 12 | z i | . In the parallel Delzant case, we use the following notation. We use v i , i = 1 , . . . , s to denote the vectors in { u i } that are parallel to the hyperplane F , and we use r j , j = 1 , . . . , t to denote those who are not parallel to F . Arrange { v i } and { r j } such that v , . . . , v s , r , . . . , r t is the same as u , . . . , u d . Now, by our hypothesis, F ∩ ∆ will have positive distance to eachcodimension 1 facet of ∆ corresponding to r j . Hence for those z ∈ J − N ( F ), | z s + j | (cid:54) = 0.Let t ⊥ ∈ R n = Lie T n be a unit vector such that ker t ⊥ in t ∗ is parallel to F . Claim 26. S = { exp kt ⊥ | k ∈ R } acts freely on J − N ( F )This is simple. Because as a set, J − N ( F ∩ ∆ ◦ ) = F ∩ ∆ ◦ × T n , and theaction on it must be free. At the point x ∈ J − N ( F ∩ P ) for some facet P of∆, the isotropy group will be T P ⊂ T n , where T P = exp V and V ⊂ Lie ( T n )is the kernel of the linear space corresponding to the affine space P . It isobvious that t ⊥ (cid:54)∈ V .Without loss of generality, we can do a SL(n , Z ) transformation to make t ⊥ become e d . This is because t ⊥ is a vector along some edges of ∆.Now for any vector e s + j ∈ R d with s + j < d , by the hypothesis, we canfind a s + j,i , i = 0 , . . . , s such that e s + j − a s + j, t ⊥ − a s + j, e − . . . − a s + j,s e s ∈ n . Claim 27.
Because of the assumption we have v , . . . , v s are all parallel to F , the map β can be factored to get R d ⊂ span { v , . . . , v n } R n − R nββ ψ Then ker β ⊂ n and ker β ⊕ span { e s+j − a s+j , t ⊥ − a s+j , e − . . . − a s+j , s e s } equals to n .This claim is trivial, becasue given any element in n we can use span { e s+j − a s+j , t ⊥ − a s+j , e − . . . − a s+j , s e s } to annihilate its e s + j part, and the restjust consists of elements in ker β .Hence, by acting exp t ( e s + j − a s + j, t ⊥ − a s + j, e − . . . − a s + j,s e s ) ∈ N wecan make, z s + j = | z s + j | , for z ∈ J − N ( F ) and s + j < d .That is to say, wecan forget about e s + j for s + j < d . The last component z d is free becausethe t ⊥ action is free and z d (cid:54) = 0. Thus to consider J − N ( F ) /N , assuming π : C d → C s is the projection to the first s components, it suffices to ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 15 consider π ( J − N ( F )) quotient by exp span { v , . . . , v s } . And J − N ( F ) /N =( π ( J − N ( F )) / (exp span { v , . . . , v s } )) × S . Claim 28. π ( J − N ( F )) / (exp span { v , . . . , v s } ) is exactly X ∆ F .After a similar argument in Claim 26, we can conclude that exp span { v , . . . , v s } preserves π ( J − N ( F )), and ( z , . . . , z s ) ∈ π ( J − N ( F )) satisfies the condition ι ∗ ◦ ( ( | z | , . . . , | z s | ) + ( λ , . . . , λ s )) = 0, from the construction we intro-duced above we see this is exactly X ∆ F . (cid:3) Remark 29.
There is a similar version of this theorem in the b -symplecticcase, see [GMPS1]. Remark 30.
Moreover, we can see easily that in this local model X ∆ F × S ,each codimension 2 submanifolds X ∆ F × { θ } is a toric submanifold.4.3. Local model for toric manifolds.
In this section we study morelocal property of a special kind of toric variety ( X ∆ , ∆). ∆ will be morespecial and such toric varieties will be our fundamental building blocks forour construction. Definition 31.
We say a parallel Delzant polytope (∆ , F ) is strict paral-lel , if there exists a parallel hyperplane F (cid:48) for (∆ , F ) such that F (cid:48) ∩ ∆ = F (cid:48) ∩ ∂ ∆. F (cid:48) is called the strict parallel hyperplane .For example, Figure 10 is not strict parallel with respect to the hyperplane F . However, the following picture gives us an example of strict parallelDelzant polytope. Figure 11.
Example of strict parallel Delzant polytope.We will study the local behavior of a strict parallel toric variety (∆ , F )around the facet F ∩ ∆, where F is a strict parallel hyperplane. In the nextsubsection we will cut F ∩ ∆ and then glue two toric manifold along it.From now on, by a SL(n , Z ) transformation, we can always assume thestrict parallel hyperplane is parallel to the t ∗ , . . . , t ∗ n − plane in ( R n ) ∗ =(Lie ( T n )) ∗ . Moreover, we can assume the entire Delzant polytope is above F , because the case where the polytope is under F is exactly the same. Wealso assume the dimension of ∆ is n and the dimension of ∆ F is n − Theorem 32.
Given a strict parallel toric manifold ( X ∆ , F ) , F a strictparallel hyperplane, assume it has moment map µ ∆ . For a suitable smallneighborhood U of ∆ F = ∆ ∩ F in ∆ , µ − ( U ) will be symplectomorphic to X ∆ F × S [ − r, − r + (cid:15) ) . Here S is a sphere with suitable radius r and thecommon symplectic form dh ∧ dθ . The following picture illustrates the essence of this theorem.
Figure 12.
Local model for neighborhood of strict hyperplane.In fact, we can prove that there is a diffeomorphism very easily. The cru-cial point is to prove this is a symplectomorphism. We will apply symplecticcutting method. See [L].
Proof.
Given a strict parallel Delzant polytope, there is a very simple wayto construct the desired toric manifold X ∆ . We consider X ∆ F × S , where S is a sphere with suitable large radius, and is equipped with the commonsymplectic form dh ∧ dθ . Now the Delzant polytope corresponding Delzantpolytope of X ∆ F × S will be ∆ F × [ − r, r ]. Because of the assumption that∆ is strict parallel, up to adding or subtracting some element in ( R ) ∗ , wecan assume the strict parallel hyperplane F = { t ∗ n = − r } , and then one canget the polytope ∆ by cutting ∆ F × [ − r, r ] at the top. As shown in thefollowing picture. Figure 13.
Symplectic cutting.We select positive (cid:15) small enough such that ∆ F × [ − r, − r + (cid:15) ) is not cutunder our cutting process. ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 17 For the sake of simplicity, we can assume that we only cut ∆ F × [ − r, r ]once. The general case can be got by induction.Now, assume the cut is done with the m ∈ R n = Lie ( T n ) direction andat level δ (such a direction will be rational because the polytope we get isDelzant). That is to say, we consider the S = { exp( tm ) | t ∈ R } action on X ∆ F × S × C byexp( tm ) : ( x, z ) (cid:55)→ (exp( tm ) x, exp( − it ) z ), for x ∈ X ∆ F × S and z ∈ C . The moment map for this action will be (cid:104) µ, m (cid:105) − | z | − δ. Here we write µ for the moment map of X ∆ F × S . Take symplectic reductionwith respect to this action we will get the desired symplectic cut. And in thiscase what we get is X ∆ . In fact, the corresponding symplectic reduction isjust {(cid:104) µ, m (cid:105) = | z | + δ } / S . Hence we are actually keep the part (cid:104) µ, m (cid:105) ≥ δ in X ∆ F × S and then compatify the boundary to get the desired toricmanifold.Now we will consider this symplectic reduction process more specifically.We denote the moment map for the S action on X ∆ F × S × C by J ( x, z ) = (cid:104) µ ( x ) , m (cid:105)− | z | − δ , the quotient map by π : J − (0) → X ∆ and the inclusionmap by ι : J − (0) → X ∆ F × S × C . We use ω for the symplectic form on X ∆ F × S and µ for its moment map. Then the symplectic form ω X ∆ on X ∆ will satisfy π ∗ ω X ∆ = ι ∗ ( ω + dx ∧ dy ) , and the moment maps are related by µ ∆ ◦ π = µ ◦ ι. Here by an abuse of notation, we use µ to denote the moment map of the T n action on X ∆ F × S × C by acting on the first two components. Claim 33.
The symplectic form in the neighborhood µ − (∆ F × [ − r, − r + (cid:15) ))of µ − (∆ F ) is unchanged. More specifically, we have a symplectomorphism X ∆ F × S [ − r, − r + (cid:15) ) = µ − (∆ F × [ − r, − r + (cid:15) )) (cid:39) µ − (∆ F × [ − r, − r + (cid:15) )) . Indeed, the first equality in the above is trivial. For the symplecto-morphism, let the point ( x, z ) ∈ J − (0) ⊂ X ∆ F × S × C , with x ∈ µ − (∆ F × [ − r, − r + (cid:15) )) ⊂ X ∆ F × S . Because of the fact that µ ( x ) is awayfrom the cutting hyperplane, we have | z | = (cid:104) µ ( x ) , m (cid:105) − δ >
0. Hence wewill have µ − (∆ F × [ − r, − r + (cid:15) )) = π ( µ − (∆ F × [ − r, − r + (cid:15) )) ∩ J − (0))= ( µ − (∆ F × [ − r, − r + (cid:15) )) ∩ J − (0)) / S = µ − (∆ F × [ − r, − r + (cid:15) )) . So we have a diffeomorphism.Now we only need to examine the sympelctic form. Given two tangentvectors X and Y of µ − (∆ F × [ − r, − r + (cid:15) )) at a point x , by the symplecticcutting construction we will have ( x, z ) ∈ J − (0) ⊂ X ∆ F × S × C . Wecan select z to be real and positive, because of the S action. Now if γ X ( t )is an integral curve in µ − (∆ F × [ − r, − r + (cid:15) )), then we have a lift of theintegral curve in J − (0) ( γ X ( t ) , z X ( t )), with the first component γ X ( t ) andthe second component always real. Similarly we can do this for Y . Hencewe have ω ∆ ( X, Y ) = ω ∆ (cid:18) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( γ X ( t ) , z X ( t )) , ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( γ Y ( t ) , z Y ( t )) (cid:19) = ( ω + dx ∧ dy ) (cid:18) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( γ X ( t ) , z X ( t )) , ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( γ Y ( t ) , z Y ( t )) (cid:19) = ω ( X, Y ) + dx ∧ dy (cid:18) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 z X ( t ) , ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 z Y ( t ) (cid:19) = ω ( X, Y ) . The second term vanishes because z X ( t ) and z Y ( t ) are always real. And thevector fields they generate are always along with ∂∂x .Till now we have already finished the proof. One just needs to selectthe neighborhood U as ∆ F × [ − r, − r + (cid:15) ). And µ − (∆ F × [ − r, − r + (cid:15) )) issymplectomorphic to X ∆ F × S [ − r, − r + (cid:15) ). (cid:3) The below corollary directly follows from the proof above.
Corollary 34.
In the local model proved above, the T n action splits to T n − × S action, where T n − action is the toric action on X ∆ F and S action is the usual action on a sphere, i.e., rotation along the θ parameter. Cutting and gluing.
Have already determined the local structure ofour building blocks, we can now start the cutting and gluing process.4.4.1. cutting.
Our building blocks consist of strict parallel Delzant poly-topes. Now given a strict parallel toric manifold X ∆ , with ∆ strict paralleland F the strict parallel hyperplane. Still, without loss of generality weassume F is parallel to the t ∗ , . . . , t ∗ n − plane in ( R n ) ∗ .If the Delzant polytope ∆ is over F , we cut X ∆ at the bottom by X ∆ F .That is to say, we consider the local model proved in the above subsection, X ∆ F × S [ − r, − r + (cid:15) ). And we cut X ∆ by excluding X ∆ F = X ∆ F × {− r } ⊂ X ∆ F × S [ − r, − r + (cid:15) ) ⊂ X ∆ . ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 19 This open symplectic manifold will be denoted by cut X ∆ . Similarly one cancut X ∆ at the top, if there is a strict hyperplane F such that ∆ is under F . We denote this by cut X ∆ . The following picture illustrates the cuttingprocess for the first case. Figure 14.
Lower cutting.Here the sphere and the cylinder are all equipped with symplectic form dh ∧ dθ .4.4.2. gluing. In this part we glue two open symplectic manifold cut X ∆ and cut X ∆ together, where ∆ i are strict parallel Delzant polytopes, and theyhave the same ∆ F .By the local model above, we are just gluing two cylinders. There are twoways to do it. The first one is gluing the cylinders with the same orientation(recall as symplectic manifolds they all carry an orientation). The other oneis gluing with reversed orientation. Definition 35.
The orientation preserved gluing is defined to be cut X ∆ (cid:116) X ∆ F × S (cid:116) cut X ∆ , with the obvious smooth structure, making the cylinders smoothly pinchedtogether with the same orientation. The orientation reserved gluing is de-fined to be cut X ∆ (cid:116) X ∆ F × S (cid:116) cut X ∆ , with the obvious smooth structure, making the cylinders smoothly pinchedtogether with the opposite orientation.The following picture show the two different kinds of gluing.The same orientation glued manifold will be denoted as cut X ∆ ∪ cut X ∆ and the orientation reversed gluing will be denoted by cut X ∆ ∪ cut X ∆ . Theorem 36. cut X ∆ ∪ cut X ∆ will carry a toric manifold structure. And cut X ∆ ∪ cut X ∆ will carry a b -toric manifold structure.Proof. The same orientation gluing cut X ∆ ∪ cut X ∆ is easy to prove pos-sessing a toric structure. In fact, because of the local model proved in the Figure 15.
Different way of gluing.above part, it is plain to see that the symplectic form on cut X ∆ ∪ cut X ∆ should be chosen as(1) ω = ω ∆ , when the point is in cut X ∆ ,ω ∆ , when the point is in cut X ∆ ,ω ∆ F + dh ∧ dθ, when the point is in the cut loci . Because of the local symplectomorphism above this is well-defined. As forthe Hamiltonian action, by Corollary 32, we directly define the toric actionas the usual action at points of cut X ∆ , cut X ∆ , and we define the actionon the cut loci as T n = T n − × S action on X ∆ F × S . This is obviouslywell-defined, and the action is Hamiltonian with respect to the moment map µ = µ ∆ , when the point is in cut X ∆ ,µ ∆ , when the point is in cut X ∆ , ( µ ∆ F , h ) , when the point is in the cut loci . Here, of course, one should choose µ ∆ i suitably to make the image pinchedtogether at level h .The difficult part is gluing through opposite orientation. In the oppositegluing case, we will apply the partition as the Figure 7.First, by the construction, the smooth structure is clear. For the T n actionon it, we decompose it into T n − × S , as in Corollary 33. Let T n − acts on cut X ∆ ∪ cut X ∆ by the usual action on cut X ∆ and cut X ∆ . However, forthe S action we let it act on cut X ∆ as usual, but act on cut X ∆ reversely.This action will be compatible with the smooth structure. The questionnow is that the moment map image will not be compatible together. Nowwe introduce the partition function, as the following picture shows. ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 21 Figure 16.
The partition.We use U to denote cental part, and U U to denote the shaded open set.Hence U i forms an open covering of cut X ∆ ∪ cut X ∆ . Define the partitionfunctions ρ i compactly supported in U i and satisfy the conditions • ρ + ρ + ρ = 1. • ρ i ≥ b -symplectic form as ω = ρ ω ∆ + ρ ω ∆ + ρ ( ω ∆ F + c h dh ∧ dθ ) . Here c is some nonzero constant. It is easy to verify this is a b -symplecticform (in fact the part in the form that does not involve dh will form ω ∆ F ,and the part involving dh definitely is closed). Note that this b -symplecticform is compatible with the orientation of cut X ∆ ∪ cut X ∆ no matter aboveor under the cut loci.What left is to examine the action defined above is Hamiltonian. By thedecomposition of T n = T n − × S , we only need to verify the existence ofmoment maps for both of them. Let ι T n − : Lie ( T n − ) → Lie ( T n ) ,ι S : Lie ( S ) → Lie ( T n ) . For the T n − action, we can directly use the definition ι ∗ T n − ◦ µ ∆ i same asbefore, since the T n − action is not reversed. As for the S action, in thepart U \ U we have ω = ω ∆ and we simply define the moment map as ι ∗ S ◦ µ ∆ . When it comes to U = X ∆ F × cylinder, by the local model above,the action will be rotation along the cylinder. And to find a moment mapit suffices to find a function of h , f ( h ) such that df ( h ) = ρ dh + ρ dh + ρ c h dh. Such a function obviously exists and by adding some constant, we can make f ( h ) and ι ∗ S ◦ µ ∆ patch together becomes a continuous function. Similarly,by adding some constants to − ι ∗ S ◦ µ ∆ we can also make f ( h ) and − ι ∗ S ◦ µ ∆ path together to be a continuous function (note that there is a negative sign,because the S action is reversed at this part). We denote the function afterpatched together as above by f ( h ). Away from the cut loci it is obviouslysmooth. And around the cut loci, it will take the form f ( h ) = c log | h | + g ( h )for some smooth function g ( h ). Hence ( ι ∗ T n − ◦ µ ∆ i , f ( h )) is a b -map to the b -moment map codomain, satisfying the condition of a b -moment map withrespect to the toric action as above.Till now we can conclude that the theorem holds. (cid:3) In fact, from the proof one can conclude more about the toric and b -toricstructure constructed above. Corollary 37.
For the toric structure constructed above, the Delzant poly-tope correspond to it will be the union of ∆ and ∆ along ∆ F , and thesymplectic form of it is canonical. For the b -toric structure, by the construc-tion the b -symplectic form is not canonical, and the corresponding b -Delzantpolytope is still the union of ∆ and ∆ along ∆ F , with negative infinity hy-perplane at F , and the b -Delzant polytope falls on the b -moment codomaincoming from a weighted adjacent graph with two vertices, one edge and themodular weight − ct ∗ n . Have already established the theorem above, now one can repeat thiskind of cutting and gluing process. Hence it is easy to see that all b -toricmanifolds can be got by cutting and gluing several toric manifolds together. Example 38.
If we cut and glue two Hirzebruch surface H together withDelzant polytopes corresponding to the figure as follow, we will get thecorresponding toric and b -toric manifold shown in the picture 17.The a and b in the picture 17 will depend on our choice of b -symplecticform. Remark 39.
In fact, if we examine the proof of Theorem 25 carefully, theabove cutting and gluing operation can be applied to parallel toric manifoldby cutting at a parallel hypersurface, then gluing along the cut loci. Andonce again this can produce toric or b -toric manifold, depending on thechoice of orientation.5. Decomposition to several toric manifolds
In this section we converse the construction in the above section. Given a b -symplectic manifold we can decompose it into those building blocks. Butthis decomposition, just as the construction, is not canonical.Given a b -toric manifold ( M, Z, ω M ), by Proposition 26 in [D] we havelocal model for a b -toric manifold around Z . That is to say, there will ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 23 Figure 17.
Gluing Hirzebruch surfaces.exists a small neighborhood U of Z such that there exists a equivariant b -symplectomorphism from U to X ∆ F × S × R , with symplectic form ω ∆ F + c h dh ∧ dθ . Here c is some suitable constant. Moreover the singular hyper-surface Z is mapped to X ∆ F × S × { } .Consider a connected component in M \ Z , denoted by X sing . Without lossof generality we can assume that there is only one connected component in Z that is adjacent to X sing and X sing looks like X ∆ F × S × R + in the localmodel. By an abuse of notation we still denote this connected component in Z by Z . We will equip X sing /Z a toric structure (which has many differentchoices).Because of the local model we mentioned above, in X ∩ Z , around Z wewill have a local model for it and the quotient will be as in the followingpicture. Figure 18.
The quotient operation.From the local model, it is easy to give the quotient space X sing /Z asmooth structure, such that around the point [ Z ] in the quotient space X sing /Z it looks like X ∆ F × part of a sphere. Now X sing /Z is a closed smoothmanifold.5.1. symplectic structures. We need to attach the quotient space witha symplectic structure. It turns out that there is not a canonical way togive a symplectic structure and this is why we said the decomposition is notcanonical.Still, we let V be an open neighborhood of X sing \ U , here we use U todenote the local model neighborhood mentioned above (i.e. U symplecto-morphic to the model X ∆ F × S × R ). Futhermore, we assume V ∩ U takesthe form X ∆ F × S × ( δ, + ∞ ), for some δ sufficiently small. In the quotientspace we still use U and V to denote the corresponding open sets. Now, U and V form an open cover of X sing /Z , we define partition functions ρ U and ρ V with respect to this open cover as before. Hence, one can define thefollowing symplectic form on X sing /Zω = ρ V ω M + ρ U ( ω ∆ F + dh ∧ dθ ) . It is quite clear that this is a well-defined symplectic form (the reason isthe same as our first partition operation). Also one can see clearly why thisform is not canonical.5.2. toric structures.
In this part we deal with the torus action. As amatter of fact the original T n action induces a T n action on the quotientspace X sing /Z . Moreover, around the point [ Z ] ∈ X sing /Z this action canbe split to T n − action on X ∆ F and S action on the part of a sphere. Theonly thing left to be checked is Hamiltonian action. Indeed, after we splitto T n − and S , similarly, we only need to check them separately. For T n − ,since in U the symplectic form ω becomes ω = ρ V ( ω ∆ F + c h dh ∧ dθ ) + ρ U ( ω ∆ F + dh ∧ dθ ) , = ω ∆ F + ρ V c h dh ∧ dθ + ρ U dh ∧ dθ, this part of the action is obviously Hamiltonian. For the second part, still,in ( X sing /Z ) \ U this is trivial and in U , we can directly find a function f ( h )such that df ( h ) = ρ V c h dh + ρ U dh. And by adding some suitable constant we can glue the moment map for V part and f ( h ) together.Till now we can conclude the space X sing /Z does posses a toric structure.And from the construction, it is easy to see that the Delzant polytope willbe the intersection of the part of the b -Delzant polytope in (Lie T n ) ∗ whichcorresponds to X sing , with a upper half space defined by some hyperplaneparallel to the t ∗ , . . . , t ∗ n − plane. And the this hyperplane will be strictparallel to the Delzant polytope.The following picture illustrates this situation. ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 25 Figure 19.
Delzant polytope after decomposition.Now, by the decomposition described above, one can decompose a b -symplectic manifold piece by piece to get several toric manifolds (the choiceof symplectic structures on them are not unique but the manifolds and theactions are unique). And it is easy to see that from those decomposedmanifolds, by the gluing technique we can recover the b -symplectic manifold(still, the action and b -manifold are unique but the b -symplectic form is not).From the process of decoposition and construction it is easy to see, if wedecompose a b -toric manifold and then we use the partition functions usedin the decomposition to glue those toric manifolds together to get a b -toricmanifold, we will end up with the exactly same b -toric manifold, with same b -symplectic form, as we started. Example 40.
For example consider X ∆ F × T , where T will be equippedwith b -toric structure with two singular hypersurfaces, as in Example 25 in[D] or Example 8 as above. Decompose this will give us two X ∆ F × S . Andgluing them will return us with X ∆ F × T .6. Applications
In this section we will try to show that this kind of construction anddecomposition are useful.6.1.
Homology of b -toric manifold. In [C2], it was shown that the ho-mology of a toric manifold can be computed perfectly by using Morse theory.Here, we will carry this way to the b -toric case and compute the homologyof a b -toric manifold. It turns out that the homology of b -toric manifold canbe computed easily also.First, it is plain to see that the case X ∆ F × S case is easy to compute.Because we already know the homology of the usual toric manifold ∆ F , byK¨unneth formula one can get the homology of X ∆ F × S .So it boils down to consider the case that the adjacent graph does no forma loop. Select a generic element X ∈ Lie ( T n ), such that its components are independent over Q . Without loss of generality, we assume that there isonly one edge and two vertices in the adjacent graph and hence the b -toricmanifold will be ( M, Z, ω ), where Z has only one component and M \ Z will have only two components. We also assume in the local model the b -moment map in the local model takes the form c log | h | + g ( h ) with c < − ct ∗ n which is along the directionof − t ∗ n ∈ (Lie T n ) ∗ . The proof of the general case will be almost the same.Denote those two components as M and M . Corresponding to this adjacentgraph, define the following 1 dimensional b -manifold as the image of ourMorse function. • As a set, it is defined as R (cid:116) { e } (cid:116) R . • For the smooth structure, we endow a smooth structure on it suchthat the following map is smooth. f ( x ) = exp( xc ) , when x ∈ R , , when x ∈ { e } , − exp( xc ) , when x ∈ R . where c depends on the modular weight.Just as in Section 5 of [GMPS1], such a smooth structure exists. This issimply identifying the −∞ of R and the −∞ of R together to form a point e . This Morse function codomain will be denoted as b M . We introduce anorder ( b M , (cid:108) ) structure on it, by • for a , b ∈ R , a (cid:108) b if a > b . • for a ∈ R ∪ { e } , b ∈ R , a (cid:108) b . • for a ∈ R , b ∈ R ∪ { e } , a (cid:108) b . • for a , b ∈ R , a (cid:108) b if a < b .The Morse function will be defined to be µ X = (cid:104) µ, X (cid:105) , with µ X takes valuein R when the point is in M , µ X takes value in R when the point is in M and µ X takes value in { e } when the point is in the singular hypersurface Z . This function is clearly a b -map and is smooth. We will consider thetwo components M i separately. Note that they can be viewed as open toricmanifolds from our construction in Section 4. Because of the fact that theaction is Hamiltonian the equality dµ X = − ι X ω holds, which implies thatthe critical points of this function can only appear in the fixed point set ofthe action in M i . By our decomposition and construction around a fixedpoint p the symplectic form and action are the same with the usual toricmanifold. So as in [C2] there is a chart such that the symplectic form andthe moment map take the form ω = n (cid:88) k =1 dx k ∧ dy k . ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 27 µ = µ ( p ) − n (cid:88) k =1 λ ( k ) ( x k + y k ) . Here the − λ k ’s are precisely the edge vectors of the fixed point p in the b -Delzant polytope (this can be seen easily once we decompose the b -toricmanifold to toric manifolds). Hence the index at a fixed point p can be readfrom the b -Delzant polytope. That is to say, taking twice the number ofedges which are pointing up with respect to X (i.e. (cid:104)− λ ( k ) , X (cid:105) > p RR Figure 20.
The Morse function.So now let p , . . . , p s denote the fixed points in M and q , . . . , q t denotethe fixed points in M . Moreover, assume they are arranged such that µ X ( p ) (cid:108) . . . (cid:108) µ X ( p s ) ∈ R and µ X ( q ) (cid:108) . . . (cid:108) µ X ( q t ) ∈ R (from the b -Delzantpolytope it is easy to see that it is impossible to have µ X ( p i ) = µ X ( p j ),because our way of selecting X ). If we use + ∞ to denote the positiveinfinity of R , and we use interval to denote interval under the order structure( b M , (cid:108) ) one will have, Theorem 41. (1) Given an interval [ a, b ] ⊂ b M (it is possible for a and b are in differ-ent R i ), such that [ a, b ] does not contain any critical value of µ X , then ( µ X ) − (+ ∞ , b ] has the homotopy type of ( µ X ) − (+ ∞ , a ] .(2) For (cid:15) small enough, ( µ X ) − (+ ∞ , µ X ( p i ) − (cid:15) ] will have the homotopytype of ( µ X ) − (+ ∞ , µ X ( p i ) + (cid:15) ] with a cell of dimension ind p i .(3) For (cid:15) small enough, ( µ X ) − (+ ∞ , µ X ( q j ) + (cid:15) ] will have the homotopytype of ( µ X ) − (+ ∞ , µ X ( q j ) − (cid:15) ] with a cell of dimension ind q j . In fact, in the above argument we have already proved (2) and (3). Theonly thing left is the case of (1) that the interval contains point e . But byour local model in the construction, around the singular hypersurface it issymplectomorphic to X ∆ F × R × S , and the upper half M in this local model is X ∆ F × (0 , + ∞ ) × S . It will be similar for the lower half. If wewrite the generic vector X as ( X (cid:48) , x n ), the Morse function will be (cid:104) µ ∆ F , X (cid:48) (cid:105) + c log | h | x n . Now in the local model, the case is shown as in the picture as follows. � Figure 21. ( µ X ) − ( −∞ , a ] in local model.Without loss of generality, we can assume | a | and | b | are sufficiently large.The set ( µ X ) − (+ ∞ , a ] will be equivalent to say the points satisfy (cid:104) µ ∆ F , X (cid:48) (cid:105) + c log | h | x n ≥ a, in the local model. Notice that a is large and (cid:104) µ ∆ F , X (cid:48) (cid:105) is bounded because X ∆ F is a compact manifold. Hence, there must exist an a (cid:48) < (cid:104) µ ∆ F , X (cid:48) (cid:105) + c log | a (cid:48) | x n ≥ a, in the local model, which implies X ∆ F × ( −∞ , a (cid:48) ] × S ⊂ ( µ X ) − (+ ∞ , a ]. Itfollows that we can define a deformation retract by shrinking ( µ X ) − (+ ∞ , a ]to X ∆ F × ( −∞ , a (cid:48) ] × S in this local model. Indeed, if ( x, t, θ ) ∈ ( µ X ) − (+ ∞ , a ],then for all t (cid:48) < t one has ( x, t (cid:48) , θ ) ∈ ( µ X ) − (+ ∞ , a ]. Hence we can definethe deformation retraction F s ( x, t, θ ) = (cid:0) x, st + (1 − s ) a (cid:48) , θ (cid:1) . for ( x, t, θ ) ∈ ( µ X ) − (+ ∞ , a ].One can do the exactly same thing to ( µ X ) − (+ ∞ , b ] the upper half,and get some b (cid:48) such that ( µ X ) − (+ ∞ , b ] can be deformation retracted to X ∆ F × ( −∞ , b (cid:48) ] × S in this local model. And finally X ∆ F × ( −∞ , b (cid:48) ] × S obviously can be deformation retracted to X ∆ F × ( −∞ , a (cid:48) ] × S in the localmodel. Hence the theorem holds.Notice that the index of a fixed point is always even as we said before,hence it is plain to see that as a CW -complex, it consists of even cells andby considering cellular homology we have ONSTRUCT b -SYMPLECTIC TORIC MANIFOLDS FROM TORIC MANIFOLDS 29 Theorem 42.
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