Remarks on Lagrangian intersections in toric manifolds

We consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. Our remarks address the fact that one can use simple cartesian product and symplectic reduction considerations to go from basic examples to much more sophisticated ones. We show in particular how rigidity results for the above Lagrangian intersection problems in weighted projective spaces can be combined with these considerations to prove analogous results for all monotone toric symplectic manifolds. We also discuss non-monotone and/or non-Fano examples, including some with a continuum of non-displaceable torus orbits.


Introduction
Let (M 2n , ω) be a toric symplectic manifold, i.e. a symplectic manifold equipped with an effective Hamiltonian T n -action generated by a moment map where the moment polytope P is defined by (1) i (x) := x, ν i + a i ≥ 0 , i = 1, . . . , d. Here the a i 's are real numbers, each vector ν i ∈ Z n is the primitive integral interior normal to the facet F i of the polytope P and d is the number of facets of P .
Denote by τ : M → M the canonical anti-symplectic involution, characterized by µ • τ = µ, and let R := M τ denote its fixed point set. R is a Lagrangian manifold, often called the real part of M . Given x ∈ int(P ), let T x := µ −1 (x) denote the corresponding T n -orbit, a Lagrangian torus in M . Since µ • τ = µ, we also have that τ (T x ) = T x and (T x ) τ = T x ∩ R. This last set, the real part of a regular T n -orbit, is discrete with 2 n points.
In this case, the Lagrangian torus fiber over the origin 0 ∈ P will be called the centered or special torus fiber. It is the unique monotone torus fiber of the monotone toric symplectic manifold (M 2n , ω). Without any loss of generality, we will usually assume that λ = 1 (as we already did in Example 1.1).
In this context of toric symplectic manifolds, it is natural to consider the following Lagrangian intersection rigidity question: given x ∈ int(P ), does there exist ψ ∈ Ham(M, ω) such that ψ(T x ) ∩ T x = ∅ or ψ(T x ) ∩ R = ∅ or ψ(R) ∩ R = ∅ ? Our remarks in this paper concern the first two of these Lagrangian intersection problems and will show how simple cartesian product (Section 2) and symplectic reduction (Section 3) considerations can be used to go from basic examples to more sophisticated ones.
is the normal to a facet of P then −ν ∈ Z n is also the normal to a facet of P , then Remark 1.6. A particular interesting example that fits both (i) and (ii) of this theorem is M = CP 2 3CP 2 , equipped with a monotone symplectic form (cf. § 4.5).
With the help of another basic example, i.e. the total space of the line bundle O(−1) → CP 1 (cf. Section 5), our remarks can also be used to prove interesting non-displaceability results on certain non-monotone and/or non-Fano examples, such as: For all non-displaceable torus fibers in these examples, we can also use our remarks to obtain an appropriate optimal lower bound for the number of transversal intersection points, which for these Lagrangian 2-tori is 4.
Remark 1.10. Regarding this last example, McDuff shows in [18] that it gives rise, under a repeated wedge construction, to a monotone symplectic toric 12-manifold with a continuous interval of Lagrangian torus fibers that cannot be displaced by her method of probes [17]. Hence, as she points out, these fibers "may perhaps be non-displaceable by Hamiltonian isotopies, even though, according to [14], their Floer homology vanishes". Although we can use the remarks in this paper to prove non-displaceability of the relevant torus fibers in the non-Fano toric symplectic 4-manifold (cf. § 6.2), so far we have not been able to do the same for the corresponding monotone toric symplectic 12-manifold.

First Remark: Cartesian Product
This first remark, on cartesian products, is motivated by Alston's result in part (iii) of Theorem 1.3 and can be used to remove its dimension restriction, i.e. prove Theorem 1.4. We will also use it in combination with our second remark, on symplectic reduction.
2.1. Combinatorial Floer Invariant. Let P ⊂ (R n ) * be a moment polytope defined by is the primitive integral normal to the facet F i of the polytope P , and d is the number of facets of P . We will say that P is even if d is even.
Consider the linear map ∂ P : CF n → CF n defined on basis elements by Proposition 2.4. If P = P × P , with P and P even integral polytopes, then HF (P ) = HF (P ) · HF (P ) .
Then P = P × P ⊂ R n × R n = R n , n = n + n , has normal vectors Hence, the linear map ∂ P on CF n = CF n ⊗ CF n is given by The result of the proposition follows by a standard application of the Künneth formula in this context.
Using this proposition and the fact that P × P always has an even number of facets, we can define the Floer invariant of any integral polytope. 2.6. Relation with Lagrangian Floer Homology. Suppose now that P is a Fano Delzant polytope and let M P denote its associated smooth Fano toric variety. This means that when defining P ⊂ (R n ) * by we can assume that a 1 = · · · = a d = 1 and (M P , ω P ) is monotone: [ω P ] = 2πc 1 (M P ).
Denote by R P the Lagrangian real part of (M P , ω P ) and by T P the Lagrangian torus over 0 ∈ int(P ). Assuming that the Lagrangian Floer homology HF (R P , T P ; Z 2 ) is well defined, we have the following theorem.
Proof of the Corollary. If P is a Fano Delzant polytope, then P ×P is an even Fano Delzant polytope to which Theorem 2.7 applies. Suppose that .
This contradicts Theorem 2.7. As before, we are assuming that HF (R P ×P , T P ×P ; Z 2 ) is well defined.
Remark 2.9. The meaning of HF (R P , T P ; Z 2 ) being well defined depends on the technical tools one is willing to use. If one uses only a basic version of Lagrangian Floer homology, then the condition that the minimal Maslov number of R P is bigger than two has to be enforced. Under this assumption, it can be proved as in [1,8,11,19] that, for the standard complex structure J on M P , the boundary of the moduli space of holomorphic strips with dimension one is given by broken strips and holomorphic disks with Maslov index two and boundary in T P . Moreover, the linearized Cauchy-Riemann operator D u∂J is surjective for all J-holomorphic strips u of Maslov index one or two. It follows from this that HF (R P , T P ; Z 2 ) is well defined whenever d is even. As a matter of fact, if d is even then the holomorphic disks with index two and boundary in T P counted by ∂ 2 P cancel out when we work with Z 2 coefficients; this is the key point in the proof of Proposition 2.2. In particular, HF (R P ×P , T P ×P ; Z 2 ) is well defined whatever is the parity of d. Indeed, the minimal Maslov number of R P ×P equals the minimal Maslov number of R P .
With this hypothesis on R P the set of examples that are covered by Corollary 2.8 is a bit restrictive. If one uses a sophisticated version of Lagrangian Floer homology, such as the one developed by Fukaya, Oh, Ohta and Ono in [14,15,16], then Corollary 2.8 covers a lot more ground (see [2]).
Remark 2.10. Even with only basic Lagrangian Floer homology, HF (R P ×P , T P ×P ; Z 2 ) is indeed well defined when P is a simplex, i.e. M P = CP n , and Corollary 2.8 can be combined with a combinatorial computation of HF (P ×P ) to prove Theorem 1.4, hence removing the dimension restriction of Alston's result. We will not present that combinatorial computation here since (i) it is presented in [2] and (ii), as we will see, our remark on symplectic reduction can also be used to easily remove this restriction.
The proof of Theorem 2.7 is a simple combination of the following two ingredients: (i) the characterization by Cho [8] and Cho-Oh [11] of the holomorphic discs on Fano toric manifolds, that are relevant for the differential on the Lagrangiam Floer complex, as Blaschke products; (ii) the existence of global homogeneous coordinates on any smooth toric variety M P , not just on CP n (see §4.4 of Cox [12]).
We will now discuss some details of (ii), and how they contribute to the proof of Theorem 2.7. From the well known quotient representation defining homogeneous coordinates on M P , where homogeneous here is with respect to the action of the torus G P . Note that the canonical anti-holomorphic involution τ P : M P → M P is given in these homogeneous coordinates as Let m 1 , . . . , m be the integral points in P , i.e.
and consider the map where It turns out that φ induces a well defined embedding If one restricts this map to (C * ) n ⊂ M P , one gets an embedding given in the standard coordinates of (C * ) n as . Any such intersection point can be written in homogeneous coordinates as As Alston [1] does in the case of CP n , using the work of Cho [8] and Cho-Oh [11], the characterization of relevant holomorphic discs as Blaschke products shows that the Lagrangian Floer differential is given in homogeneous coordinates by We can now use the maps φ and ϕ, defined by (3) and (4) On the other hand, if Hence, 2.11. Examples.
In this case R P is a meridian circle and T P is the equator. P has normals ν 1 = 1 and ν 2 = −1. CF is a 2-dimensional vector space with basis e 1 = (1) and e 2 = (−1). The differential ∂ P is given by Hence, HF (P ) = 2 .
Proposition 2.13. If P is an even polytope such that whenever ν is a normal −ν is also a normal, i.e. such that P is symmetric, then HF (P ) = 2 n .
Proof. As in the example above, we always have ∂ P = 0 under these conditions. Remark 2.14. Under the assumption that HF (R P , T P ; Z 2 ) is well defined, it follows from this proposition that if P is symmetric then In fact, as we will see in Proposition 4.6, one can easily use our remark on symplectic reduction to remove the assumption and give another proof of this result.
Example 2.15. When n = 2 there are exactly two even Fano polytopes with the property of Proposition 2.13: the Fano square, corresponding to CP 1 × CP 1 , and the Fano hexagon, corresponding to CP 2 blown up at 3 points.
Example 2.16. Let P ⊂ R 2 be the Fano simplex corresponding to M P = CP 2 . Since P has 3 facets, an odd number, we will consider the even Fano Delzant polytope P × P ⊂ R 4 , whose 6 facets have normals and ν 6 = (0, 0, −1, −1) . This means that ∂ P ×P : CF 4 → CF 4 is given by With this explicit formula it is not hard to check that

Hence,
HF (P ) = √ 4 = 2 and applying Corollary 2.8 we conclude that for any ψ ∈ Ham(CP 2 ). This estimate is known to be optimal (see for example the end of section 5 in [2]).

Second Remark: Symplectic Reduction
Here we will state some elementary general facts in the particular context of symplectic toric manifolds.
Let (M ,ω) be a symplectic toric manifold of dimension 2N withT-action generated by a moment mapμ :M →P ⊂ (R N ) * .
As before, given x ∈ int(P ), letT x :=μ −1 (x) denote the correspondingT-orbit, a Lagrangian torus inM , and letR denote the real part ofM , i.e. the Lagrangian submanifold given by the fixed point set of the canonical anti-symplectic involutionτ :M →M , characterized byμ •τ =μ. Recall that Moreover, (T x )τ =T x ∩R and this set, the real part of a regularT-orbit, is discrete with 2 N points. Let K ⊂T be a subtorus of dimension N − n determined by an inclusion of Lie algebras The moment map for the induced action of K onM is Let c ∈μ K (M ) ⊂ (R N −n ) * be a regular value and assume that K acts freely on the level set Z :=μ −1 K (c) ⊂M . Then, the reduced space (M := Z/K, ω) is a symplectic toric manifold of dimension 2n with T :=T/K-action generated by a moment map where π is the quotient projection and the vertical arrow on the right is the inclusion Recall that the reduced symplectic form ω is characterized by π * ω =ω| Z . Note that given T x := µ −1 (x), with x ∈ int(P ) ⊂ int(P ), we have that Moreover,τ (Z) = Z, Zτ = Z ∩R andτ induces the canonical anti-symplectic involution τ : M → M via π •τ = τ • π . Let p ∈ R := M τ . Then a simple counting argument shows that (π −1 (p) ∩R) = 2 N −n .
Proof. Given a time dependent hamiltonian h t : M → R leth t :M → R be any smooth extension toM of h t • π : Z → R. The Hamiltonian flow generated byh t has the desired properties.

Remark 3.3.
McDuff 's method of probes [17] can be seen as a particular case of (ii).
To prove (ii), suppose thatq ∈ψ(T x ) ∩T x and letp =ψ −1 (q) ∈T x . Then p = π(p), q = π(q) ∈ T x and ψ(p) = ψ(π(p)) = π(ψ(p)) = π(q) = q which implies that q ∈ ψ(T x ) ∩ T x . To prove (iii), note that since T x and ψ(T x ) intersect transversely at r points in M , say p 1 , . . . , p r ∈ M , we have thatT x andψ(T x ) intersect in Z in a Morse-Bott way along the r orbits of the subtorus K ⊂T given by π −1 (p 1 ), . . . , π −1 (p r ) ⊂M . Standard equivariant neighborhood theorems in symplectic geometry imply that a sufficiently small neigh-borhoodŨ ⊂M of each of these isotropic tori is K-equivariantly symplectomorphic to V 1 × V 2 ⊂ (R 2n , ω st ) × (T * K, ω can ), where V 1 ⊂ R 2n is a neighborhood of the origin, V 2 ⊂ T * K is a neighborhood of the 0-section, and ω st = du ∧ dv = n j=1 du j ∧ dv j and ω can = −dλ can are the usual symplectic forms on R 2n and T * K respectively. Moreover, we can identify V 1 with a neighborhood U ⊂ M of the point in ψ(T x ) T x under consideration and require that Let ϕ ∈ Ham c (V 2 , ω can ) be such that (ϕ(0-section) (0-section)) = 2 N −n (one can clearly construct such optimal displacing Hamiltonians supported in arbitrarily small neighborhoods of the 0-section in T * K). We can then consider id × ϕ : V 1 × V 2 → V 1 × V 2 , extend as the identity toM and compose withψ to obtain a Hamiltonian that perturbs the relevant intersection K-orbit into 2 N −n transversal intersection points. By doing that at each of the r points in ψ(T x ) T x we obtain the desiredφ ∈ Ham(M ,ω).   Ham(M, ω) . Remark 3.5. This idea of using symplectic reduction to prove intersection properties of Lagrangian submanifolds was used by Tamarkin in [20]. It is also present in the work of Borman [6] on reduction properties of quasi-morphisms and quasi-states (see also [7]).

Symplectic Reduction Construction of Toric Manifolds.
Recall that any symplectic toric manifold (M 2n , ω) can be constructed as a symplectic reduction of where d is the number of facets of the corresponding polytope P ⊂ (R n ) * . This reduction is with respect to the natural action of a subtorus K ⊂T = T d of dimension d − n, whose Lie algebra Lie(K) ⊂ R d = Lie(T d ) is determined as the kernel of the linear map where {e 1 , . . . , e d } is the canonical basis of R d and ν 1 , . . . , ν d ∈ Z n ⊂ R n are the primitive integral interior normals to the facets of the moment polytope P . When K = K 1 ×K 2 ⊂ T d , correspondng to a splitting Lie(K) = Lie(K 1 )×Lie(K 2 ) ⊂ R d , recall that the principle of reduction in stages tells us that, at appropriate level sets, reduction with respect to the action of K ⊂ T d is equivalent to -first reducing with respect to K 1 ⊂ T d , obtaining a symplectic manifold (M 1 , ω 1 ) with Hamiltonian action of T d /K 1 , -then reducing (M 1 , ω 1 ) with respect to K 2 ⊂ T d /K 1 . This principle will be used repeatedly in the applications considered in the next sections. It is also the main ingredient in the proof of the following proposition, which in turn will be used in the proof of Theorem 1.8 (cf. Proposition 4.9 in § 4.7). Proposition 3.7. Let (M 2n , ω) be a symplectic toric manifold and ν 1 , . . . , ν d ∈ Z n the primitive integral interior normals to the facets of its moment polytope P ⊂ (R n ) * . Let m 1 , . . . , m d ∈ N be such that Then (M 2n , ω) can be obtained as a symplectic reduction of the weighted projective space CP(m 1 , . . . , m d ).
Proof. (M 2n , ω) can be obtained as the symplectic reduction of (R 2d , dx ∧ dy) by the action of a subtorus K ⊂ T d with Lie(K) = ker β, where the linear map β is given by (5). This means in particular that which together with (6) implies that K can be written as K = K 1 × K 2 with Lie(K 1 ) = span{(m 1 , . . . , m d )}. Since the weighted projective space CP(m 1 , . . . , m d ) is obtained as the symplectic reduction of (R 2d , dx ∧ dy) by the action of K 1 , one can use the principle of reduction in stages to conclude the proof.

First Application: Monotone Cases
We will use the following results stated in Theorem 1.3: Definition 4.1. Recall from Example 1.2 that any monotone toric symplectic manifold has a unique monotone torus fiber, called the centered or special torus fiber, which is the Lagrangian torus orbit over the "center" of its moment polytope. A symplectic reduction of a monotone toric symplectic manifold at a level containing its centered torus fiber, i.e. through the special "center" of its moment polytope, will be called a centered symplectic reduction.
Note that the fact that this estimate is known to be optimal for CP 2 implies that Alston's bound for CP 3 is also optimal.

Application 2. Let (M 2n
, ω) be a monotone toric manifold, R its real part and T its special centered torus fiber. Let ν 1 , . . . , ν d ∈ Z n denote the primitive integral interior normals to the facets of the moment polytope of M .  It can also be obtained as a centered symplectic reduction of CP 2 × CP 2 . Since our Floer combinatorial invariant of CP 2 × CP 2 is 4, this gives the same bound: (ψ(T ) R) ≥ 4 2 2 = 1 for any ψ ∈ Ham(M ).
However, if one sees M as a centered symplectic reduction of CP 1 ×CP 1 ×CP 1 , cf. Figure 2, one improves the bound to (ψ(T ) R) ≥ 2 3 2 = 4 for any ψ ∈ Ham(M ), which is optimal and coincides with the value of the Floer combinatorial invariant of the hexagon. Let us describe the details of this reduction construction. The cube at the left of Figure 2, corresponding to the moment polytope of a monotone CP 1 × CP 1 × CP 1 , can be described by the following inequalites: The monotone CP 2 3CP 2 can be obtained from this toric manifold by centered symplectic reduction with respect to the circle S 1 ⊂ T 3 determined by the Lie algebra vec- The quotient 2-torus T 3 /S 1 acts on the reduced manifold and, with respect to its Lie algebra basis given by (1, 0, 0), (0, 1, 0) ∈ Lie(T 3 /S 1 ) ∼ = R 3 /{(−1, −1, 1)}, the resulting moment polytope is described by the above inequalities with x 3 = x 1 + x 2 , i.e. the hexagon at the right of Figure 2.
In fact, the monotone CP 2 3CP 2 is just a particular case of the following more general proposition.
Proposition 4.6. Let (M 2n , ω) be a monotone toric manifold, R its real part and T its special centered torus fiber. Suppose that the corresponding moment polytope P ⊂ R n is symmetric, i.e. if ν ∈ Z n is the interior normal to a facet of P then −ν is also the interior normal to a facet of P . Then (ψ(T ) R) ≥ 2 n for any ψ ∈ Ham(M ) and this bound is optimal.
Proof. The fact that the polytope P is symmetric implies that M can be obtained as a symplectic reduction of the product of d copies of CP 1 , where 2d is the number of facets of P . The fact that M is monotone implies that all the CP 1 's have the same area and that this is a centered symplectic reduction. Hence, we get from Corolary 3.4 that Since (T R) = 2 n the bound is indeed optimal. 4.7. Application 4. Let (M 2n , ω) be a compact monotone symplectic toric manifold and T its special centered torus fiber. Denote by ν 1 , . . . ν d ∈ Z n the primitive integral interior normals to the facets of its Delzant polytope P ⊂ (R n ) * . Lemma 4.8. There exists k ∈ {1, . . . , d} such that Proof. Since P is the moment polytope of a compact toric manifold, the support of its associated fan is the whole R n . In particular, every lattice vector ν ∈ Z n belongs to a cone of the fan determined by some vertex of P , which means that it can be written as a non-negative integral linear combination of the primitive integral interior normals to the n facets that meet at that vertex. The Lemma follows by taking Proposition 4.9. On any compact monotone toric symplectic manifold (M, ω) the special centered Lagrangian torus fiber T is non-displaceable. Moreover, Proof. Using the previous Lemma, and after a possible re-ordering of the normals, we can assume that This condition and Proposition 3.7 imply that the standard symplectic reduction construction of M from C d factors through the weighted projective space CP(1, m 1 , . . . , m d−1 ). The monotone condition implies that this factorization goes through the special centered nondisplaceable torus fiber of this weighted projective space (cf. Theorem 1.7) and we can apply Corollary 3.4.
As a particular example, consider the monotone M = CP 2 CP 2 , i.e. the monotone blow-up of CP 2 at one point, with polytope P ⊂ (R 2 ) * given by (1, 1)).

Second Application: non-monotone Fano cases
For the applications in this section we will assume that some form of the following general result is true: • If T i or the pair (T i , R i ) have HF = 0 or are non-displaceable in (M i , ω i ), i = 1, 2, then the same is true for the corresponding T 1 × T 2 and ( Remark 5.1. For Lagrangian torus orbits in toric symplectic manifolds, the set-up of Woodward [22] applies and proves a result of this form (cf. [21]).
Moreover, in some of the applications below we will also use the following result: • In the total space of the line bundle O(−1) → CP 1 , the special torus T sitting over the origin in the polygon on the right side of Figure 3 is non-displaceable. This has been proved by Woodward (cf. Example 1.3 in [22]) and can also be seen as a consequence of a result of Cho in [9] (cf. polygon on the left side of Figure 3).  Figure 4 illustrates how one can obtain two non-displaceable torus fibers when the exceptional divisor is small, i.e. smaller than monotone. On the left, one thinks of M as a symplectic reduction of CP 2 × CP 1 , with CP 2 "smaller" than CP 1 , to show that the torus fiber over the origin is non-displaceable. On the right, one thinks of M as a symplectic reduction of O(−1) × CP 2 to show that a torus fiber "close" to the exceptional divisor is non-displaceable. Figure 5 illustrates how one can obtain one non-displaceable torus fiber when the exceptional divisor is big, i.e. bigger than monotone. One thinks again of M as a symplectic reduction of CP 2 × CP 1 , but now with CP 2 "bigger" than CP 1 . Figure 5. CP 2 CP 2 as reduction of CP 2 × CP 1 , now with CP 2 "bigger" than CP 1 .
Note that the monotone case with just one non-displaceable torus fiber over the special central point can be obtained as a limit of any of these.

Application 6. A very similar idea applies to
i.e. the equal blow-up of CP 2 at two points which can also be thought of as the blow-up of CP 1 × CP 1 at one point. One recovers the results of Fukaya-Oh-Ohta-Ono [14,15] and Woodward [22] illustrated in Figures 6 and 7. Figure 6 illustrates how one can obtain two non-displaceable torus fibers in the big twopoint blow-up of CP 2 , which is equivalent to a small one-point blow-up of CP 1 × CP 1 . On the left, one thinks of M as a symplectic reduction of CP 2 × CP 1 × CP 1 , with "big" CP 2 , to show that the torus fiber over the origin is non-displaceable. On the right, one thinks of M as a symplectic reduction of O(−1) × CP 2 to show that a torus fiber "close" to the blown-up point on CP 1 × CP 1 is non-displaceable.  On the left, one thinks of M as a symplectic reduction of CP 2 × CP 1 × CP 1 , with "small" CP 2 and large CP 1 's, to show that a torus fiber "close" to the origin is non-displaceable. On the right, one thinks of M as a symplectic reduction of O(−1) × CP 1 × CP 1 to show that there is a non-displaceable torus fiber "close" to each blown-up point on CP 2 .
Again, note that the monotone case with just one non-displaceable torus fiber over the special central point can be obtained as a limit of any of these. It could also be obtained using Proposition 4.9, as was illustrated in § 4.7 for the monotone one-point blow-up 5.4. Application 7. Here we will use the same idea to understand an example Fukaya, Oh, Ohta and Ono [14,15], presented in Figure 8 (see Example 10.3 in [16]). The symplectic manifold is M = CP 2 2CP 2 with blow-ups of different sizes, one smaller than monotone and the other bigger than monotone, and they obtain a closed interval of non-displaceable torus fibers. This can also be obtained by considering M as the symplectic reduction of O(−1) × CP 1 × CP 1 (or the compact (CP 2 CP 2 ) × CP 1 × CP 1 ) as shown in Figure 8. The details are as follows. Consider (CP 2 2CP 2 , ω α ) given by the Delzant polytope P α ⊂ (R 2 ) * determined by the following inequalities: The non-displaceable torus fibers are over the points with coordinates (−α + λ, −α) for 0 < λ < 3α/2 ( Figure 8 corresponds to α = λ = 1/4). To prove that for each such pair of real numbers α and λ, we consider O(−1) × CP 1 × CP 1 with moment polytope given by the cartesian product of the following polytopes: -the one for the O(−1) factor is given by the inequalities having a non-displaceable torus fiber over the point with coordinates x 1 = −α + λ and x 2 = −α. -the one for the first CP 1 factor is given by the inequalities having a non-displaceable torus fiber over the point with coordinate x 3 = −α.
-the one for the second CP 1 factor is given by the inequalities having a non-displaceable torus fiber over the point with coordinate x 4 = −2α + λ. We can now do symplectic reduction with respect to the 2-torus T 2 ⊂ T 4 determined by the Lie algebra vectors (0, −1, 1, 0), (−1, −1, 0, 1) ∈ R 4 = Lie(T 4 ) at the level given by to obtain the polytope P α with non-displaceable torus fiber over the point with coordinates x 1 = −α + λ and x 2 = −α. 5.5. Application 8. One can use the same idea to understand non-displaceable torus fibers on M = CP 2 3CP 2 for all possible sizes of blown-up points. Figure 9 illustrates the case of three small size blow-ups, where one gets four non-displaceable torus fibers. The center fiber is obtained by seeing M as the symplectic reduction of CP 2 × CP 1 × CP 1 × CP 1 , as in the left side of Figure 10, while each of the off-center fibers is obtained by seeing M as the symplectic reduction of O(−1) × CP 1 × CP 1 × CP 1 , as in the right side of Figure 10.
Note that we can also get a closed interval of non-displaceable torus fibers for M = CP 2 3CP 2 , e.g. by blowing up the example in Application 6 at the lower right corner of

Third Application: non-Fano cases
Here we will use the basic non-displaceability result for the central torus fiber of a weighted projective space stated in Theorem 1.7 and show how it implies non-displaceability of at least one torus fiber on any Hirzebruch surface, a result of Fukaya-Oh-Ohta-Ono [14] (see Example 10.1 in [16]), and a continuous interval of non-displaceable torus fibers on a particular non-Fano toric surface considered by McDuff [18].  6.1. Application 9. Consider the weighted projective space CP(1, 1, k), the symplectic quotient of C 3 by the S 1 -action with weights (1, 1, k), with moment polytope P k ⊂ R 2 given by x 1 + 1 ≥ 0 , x 2 + 1 ≥ 0 and − x 1 − kx 2 + 1 ≥ 0 . The special torus sitting over the origin is non-displaceable. Figure 11 illustrates the k = 2 case. Now let H k := P(O(−k) ⊕ C) → CP 1 be a Hirzebruch surface, with 2 ≤ k ∈ N. Each of these Hirzebruch surfaces can be seen as a centered symplectic reduction of CP(1, 1, k) × Figure 11. CP(1, 1, 2). CP 1 and that implies at least one non-displaceable torus fiber on any H k . Figure 12 illustrates the k = 2 case. 6.2. Application 10. Here we will consider the non-Fano toric symplectic 4-manifold described by McDuff in § 2.1 of [18]. We already pointed out its potential significance in Remark 1.10.