introduction - ulisboa

SEIDEL’S MORPHISM OF TORIC 4–MANIFOLDS

SILVIA ANJOS AND REMI LECLERCQ

Abstract. Following McDuff and Tolman’s work on toric manifolds [29], we focus on 4–dimensional NEF toric manifolds and we show that even though Seidel’s elements consist ofinfinitely many contributions, they can be expressed by closed formulas. From these formulas,we then deduce the expression of the quantum homology ring of these manifolds as well as theirLandau–Ginzburg superpotential. We also give explicit formulas for the Seidel elements in somenon-NEF cases. These results are closely related to recent work by Fukaya, Oh, Ohta, andOno [16], Gonzalez and Iritani [17], and Chan, Lau, Leung, and Tseng [9]. The main differenceis that in the 4–dimensional case the methods we use are more elementary: they do not relyon open Gromov–Witten invariants nor mirror maps. We only use the definition of Seidel’selements and specific closed Gromov–Witten invariants which we compute via localization. So,unlike Alice⋆, the computations contained in this paper are not particularly pretty but they dostay on their side of the mirror. This makes the resulting formulas directly readable from themoment polytope.

1. Introduction

Let (M,ω) be a closed connected symplectic manifold and let as usual Ham(M,ω) denote itsHamiltonian diffeomorphism group. Under a suitable condition of semipositivity, Seidel defined in[33] a morphism, S, from π1(Ham(M,ω)) to – after a mild generalization due to Lalonde, McDuff,and Polterovich [25] – QH∗(M,ω)×, the group of invertible elements of the quantum homologyof (M,ω). This morphism has been extensively used in order to get information on the topologyof Hamiltonian diffeomorphism groups as well as the quantum homology of symplectic manifolds.It has also been extended in various directions, see the end of the introduction for some of theseextensions related to the present work.

A quantum class lying in the image of S is called a Seidel element. In [29], McDuff and Tolman wereable to specify the structure of the lower order terms of Seidel’s elements associated to Hamiltoniancircle actions whose maximal fixed point component is semifree. When the codimension of this setis 2, their result immediately ensures that if there exists an almost complex structure J on M sothat (M,J) is Fano, i.e so that there are no J–pseudo-holomorphic spheres in M with non-positivefirst Chern number, all the lower order terms vanish. In the presence of J–pseudo-holomorphicspheres with vanishing first Chern number, there is a priori no reason why arbitrarily large multiplecoverings of such objects should not contribute to the Seidel elements. As a matter of fact, McDuffand Tolman exhibited an example of such a phenomenon when (M,J) is a NEF pair, which bydefinition do not admit J–pseudo-holomorphic spheres with negative first Chern number.

In this paper, we focus on the latter case and we show that even though there are indeed infinitelymany contributions to the Seidel elements associated to the Hamiltonian circle actions of a NEF 4–dimensional toric manifold, these quantum classes can still be expressed by explicit closed formulas.More precisely,

• in Theorem 4.4, we list the different contributions which enter into play,

Date: June 30, 2014.2010 Mathematics Subject Classification. Primary 53D45; Secondary 57S05, 53D05.Key words and phrases. symplectic geometry, Seidel morphism, NEF toric manifolds, Gromov–Witten invari-

ants, quantum homology, Landau–Ginzburg superpotential.⋆see Alice’s adventures in Wonderland, by Lewis Carroll (1865).

1

2 SILVIA ANJOS AND REMI LECLERCQ

• and in Theorem 4.5 we deduce the formulas giving Seidel’s elements.

These formulas depend on the topology of the manifold. More precisely, they depend on therelative position of those elements of π2(M) whose first Chern number vanishes and which admitrepresentatives corresponding to facets of the moment polytope. This observation makes theformulas of Theorem 4.5 directly readable from the polytope.

Interest of our approach. This work is closely related to recent work by Fukaya, Oh, Ohta,and Ono [15], Gonzalez and Iritani [17], and Chan, Lau, Leung, and Tseng [9]. Roughly speaking,for toric NEF symplectic manifolds, on one side Fukaya, Oh, Ohta, and Ono showed that quantumhomology is isomorphic to the Jacobian of the open Gromov–Witten invariants generating function,Jac(W open). On the other side, Gonzalez and Iritani expressed the Seidel elements in terms ofBatyrev’s elements via mirror maps. Finally, Chan, Lau, Leung, and Tseng proved that W open

coincides with the Hori–Vafa superpotential. Then by using this open mirror symmetry andthe aforementioned results, they showed that the Seidel elements correspond to simple explicitmonomials in Jac(W open). In the 4–dimensional case, these results are clearly related to ours –see for example the discussion on the Landau–Ginzburg superpotential in Example 1.3 below –,however our approach is somehow more elementary and stays on the symplectic side of the mirror.

We now sketch our approach. The Seidel element of a symplectic manifold (M,ω) associatedto a loop of Hamiltonian diffeomorphisms φ based at identity is defined by counting pseudo-holomorphic sections of (Mφ,Ω) which is a symplectic fibration over S2 with fibre M and whosemonodromy along the equator is given by φ (this construction is called the clutching construction,see §2.2 for more details). To compute Seidel’s elements when (M,ω) is a toric 4–dimensionalsymplectic manifold and φ = Λ is one of the distinguished circle actions, we proceed as follows.

(1) We first prove that (MΛ,Ω) is a toric 6–dimensional symplectic manifold, see Proposition2.11. This allows us to reduce the computation of the Seidel elements to the computationof some 1–point Gromov–Witten invariants, see §4.2.

(2) Then we compute the latter by induction using localization formulas from Spielberg’s[34] or Liu’s [26] for the base cases and the splitting axiom satisfied by Gromov–Witteninvariants for the inductive steps, see §4.4.

(3) Step (2) completely ends the computation up to some particular 0–point Gromov–Witteninvariants which we preliminarily compute using a localization argument, see §4.3.

Application in terms of Seidel’s morphism and quantum homology. As mentioned above,Seidel’s morphism has been extensively studied for its applications. However not many things areknown concerning S itself, for example its injectivity. It is obvious that Seidel’s morphism is trivialfor symplectically aspherical manifolds since these particular manifolds do not admit non-constantpseudo-holomorphic spheres at all. In [33], Seidel showed that for all m ≥ 1 Seidel’s morphismdetects an element of order m + 1 in π1(Ham(CPm, ωst)), with ωst the Fubini–Study symplecticform. In the case of CP2 for example, this makes the Seidel morphism injective. Determining non-trivial elements of the kernel of S in cases when S is not “obviously” trivial would be interesting,for example to test the Seidel-type second order invariants introduced by Barraud and Corneavia their spectral sequence machinery [4]. In order to find such classes, one should first computeall the Seidel elements in specific cases; here are families of examples for which the present workallows such computations.

Example 1.1 (Hirzebruch surfaces). It is well-known that Hirzebruch surfaces F2k are symplec-tomorphic to S2 × S2 endowed with the split symplectic form ω0

µ with area µ ≥ 1 for the first

S2–factor, and with area 1 for the second factor. Recall that F0 is Fano, F2 is NEF, and thatfor all k ≥ 2, F2k admits spheres with negative first Chern number. As we shall see in §5.3, thecomputations we present in this paper allow us not only to compute directly the Seidel elements

1Actually, this first step does not require M to be 4–dimensional.

SEIDEL’S MORPHISM OF TORIC 4–MANIFOLDS 3

associated to the circle actions of F2, but also to compute the Seidel elements associated to thecircle actions of F2k for all k ≥ 2, that is, in the non-NEF cases. We present explicitly the caseof F4.

Similar computations can be made for F2k+1 which can be identified with the 1–point blow-up ofCP2 endowed with its different symplectic forms.

Example 1.2 (2– and 3–point blow-ups of CP2). In the same spirit, consider the symplectic man-ifold obtained from CP2 by performing 2 or 3 blow-ups. It carries a family of symplectic forms ων ,where ν > 0 determines the cohomology class of ων . It is well-known that it is symplectomorphicto Mµ,c1 or Mµ,c1,c2 , respectively the 1– or 2–point blow-up of S2×S2 endowed with the symplecticform ωµ for which the symplectic area of the first factor is µ ≥ 1 and the area of the second factoris 1. Here, c1 and c2 are the capacities of the blow-ups.

In previous works, Pinsonnault [31], and Anjos and Pinsonnault [3] computed the homotopy algebraof the Hamiltonian diffeomorphism groups of Mµ,c1 and Mµ,c1,c2 . In particular they showed thatall the generators of its fundamental group do not depend on the symplectic form nor the sizeof the blow-ups provided that µ > 1. In both cases, all the generators but one can be obtainedas Hamiltonian circle actions associated to a Fano polytope while the last one is associated to aNEF polytope. When µ = 1, the fundamental group of the Hamiltonian diffeomorphism group isgenerated only by the former. So the computations we present here again allow us to compute allthe Seidel elements of the 2– and 3–point blow-ups of CP2, regardless of the symplectic form andsizes of the blow-ups.

Then we turn to quantum homology. Following [29], we deduce from the expression of the Seidelelements described in Theorem 4.5 a presentation of the quantum homology of 4–dimensionalNEF toric manifolds. Now, recall that in [30] Ostrover and Tyomkin showed that the quantumhomology of Fano toric manifolds is isomorphic to a polynomial ring quotiented by relations givenas the derivatives of the well-known Landau–Ginzburg superpotential. This result was generalisedby Chan and Lau in [8] to NEF toric manifolds (see also relation with results of Fukaya, Oh,Ohta, and Ono [15, 14] therein). As an application of our computations we are able to giveexplicit expressions for the potential in the NEF case which can be read directly from the momentpolytope, and obviously can be related with Chan and Lau’s results.

Example 1.3 (4– and 5–point blow-ups of CP2). To illustrate what is explained above, we explicitlycompute the Seidel elements of the 4– and 5–point blow-ups of CP2. Note that these manifoldsare NEF and do not admit any Fano almost complex structure. Then we deduce their quantumhomology and we give the explicit expression of the related Landau–Ginzburg superpotential, see§5.2. Of course, this expression agrees with Chan and Lau’s result [8] and in Remark 5.4 weindicate how.

Extensions and applications. We now discuss some extensions of Seidel’s morphism for whichthere is hope to get explicit information in the setting of and with similar techniques as the onesused in the present work.

Homotopy of Ham in higher degrees. As mentioned above, since [31] and [3] the homotopy algebraof the Hamiltonian diffeomorphism groups of the 2– and 3–point blow-ups of CP2 is completelyunderstood. It would be interesting in this case to compute explicitly some invariants of thehigher-degree homotopy groups generalizing Seidel’s construction: the Floer-theoretic invariantsfor families defined by Hutchings in [21] and the quantum characteristic classes introduced by Save-lyev in [32]. Briefly recall that the former are morphisms π∗(Ham(M,ω)) → End∗−1(QH∗(M,ω))obtained as higher continuation maps in Floer homology. The latter are defined via parametricGromov–Witten invariants and lead to ring morphisms H∗(ΩHam(M,ω),Q) → QH2n+∗(M,ω).Both constructions restrict to the Seidel representation, respectively in degree 1 and 0.


Bulk extension. In this paper, what is called quantum homology should more precisely be referedto as the small quantum homology ring. There is also a notion of big quantum homology ring,obtained by considering not only the usual quantum product but also a family of deformationsvia even-degree cohomology classes of M , see e.g Usher [36] and Fukaya, Oh, Ohta, and Ono [16]for a precise definition. For b ∈ Hev(M), one ends up with QHb

∗(M,ω) isomorphic to QH∗(M,ω)as a vector space but with a twisted product. In [16], the authors extended Seidel’s morphism tomorphisms π1(Ham(M,ω)) → QHb

∗(M,ω)× and generalized in the toric case part of the resultsof McDuff and Tolman [29]. It would be interesting to see which information on the big quantumhomology can be extracted from the present work.

Lagrangian setting. The Seidel morphism has been extended to the Lagrangian setting in worksby Hu and Lalonde [19], and Hu, Lalonde, and Leclercq [20]. Following McDuff and Tolman [29],Hyvrier [22] computed the leading term of the Lagrangian Seidel elements associated to circleactions preserving some given monotone Lagrangian. He showed that when the latter is the realLagrangian of a Fano toric manifold, all lower order terms vanish. It could be interesting to studythe Lagrangian case in NEF toric manifolds, however the preliminary question of the structureof the higher order terms has to be tackled with different techniques than the ones used in [22]since they require the use of almost complex structures which generically lacks regularity. Let usalso mention that Hyvrier’s work as well as such a possible extension provide examples where onecan apprehend the categorical refinement of the Lagrangian Seidel morphism due to Charette andCornea [10].

Organization of the paper. The paper is organized as follows. In Section 2 we review thenecessary backgroundmaterial, that is toric geometry and quantum homology. Section 3 is devotedto the case of toric 4–dimensional NEF manifolds where we specify these notions. In Section 4, weprecisely state the main theorems enumerating all the contributions to the Seidel morphism andthe expression of the Seidel elements (§4.1) and we prove them (§4.2 to §4.4). Finally, we describeexplicit examples and applications mentioned in the introduction in Section 5. In Appendix A wegather additional computations of Seidel’s elements in more cases, completing Theorem 4.5.

Acknowledgments. The authors would like to thank Eveline Legendre who first suggested toconsider the total space of the fibration obtained via the clutching construction as a toric manifold,and Eduardo Gonzalez who suggested to investigate non-NEF Hirzebruch surfaces. They arealso grateful to both of them as well as to Dusa McDuff and Siu-Cheong Lau for enlighteningdiscussions at different stages of this work. They are particularly indebted to Chiu-Chu MelissaLiu for explaining to them the standard arguments appearing in the computations of the 0–pointGromov–Witten invariants of Proposition 4.8. Any mistake is of course the responsibility of theauthors.

The present work is part of the authors’s activities within CAST, a Research Networking Programof the European Science Foundation. The first author is partially funded by FCT/Portugal throughproject PEst-OE/EEI/LA0009/2013. The second author is partially supported by ANR GrantANR-13-JS01-0008-01.

2. Toric manifolds and quantum homology

2.1. Toric geometry: the symplectic viewpoint. Recall that a closed symplectic 2m–dimen-sional manifold (M,ω) is said to be toric if it is equipped with an effective Hamiltonian actionof a m–torus T and with a choice of a corresponding moment map Φ : M → t

∗, where t∗ is the

dual of the Lie algebra t of the torus T . There is a natural integral lattice tZ in t whose elementsH exponentiate to circles ΛH in T , and hence also a dual lattice t

∗Z in t

∗. The image Φ(M) iswell-known to be a convex polytope P , called a Delzant polytope. It is simple (m facets meet ateach vertex), rational (the conormal vectors ηi ∈ t to each facet may be chosen to be primitive


and integral), and smooth (at each vertex v ∈ P the conormals to the m facets meeting at v forma basis of the lattice tZ). We describe them in the following manner:

P = P (κ) := x ∈ t∗|〈ηi, x〉 ≤ κi, i = 1, . . . , n,

where P has n facets D1, . . . , Dn with outward2 primitive integral conormals ηi ∈ tZ and supportconstants κ = (κ1, . . . , κn) ∈ Rn.

Delzant showed in [12] that there is a one–to–one correspondence between toric manifolds andDelzant polytopes given by the map that sends the toric manifold (M,ω, T,Φ) to the polytopeΦ(M). (See [24] and the references therein for more details on this background material.)

2.2. The clutching construction. Let (M,ω) be a closed symplectic manifold and Λ = Λθbe a loop in Ham(M,ω) based at identity. Denote by MΛ the total space of the fibration over CP1

with fiber M which consists of two trivial fibrations over 2–discs, glued along their boundary viaΛ. Namely, we consider CP1 as the union of the two 2–discs

D1 = [1 : z] ∈ CP1 | |z| ≤ 1 and D2 = [z : 1] ∈ CP1 | |z| ≤ 1

glued along their boundary

∂D1 = [1 : e−2iπθ], θ ∈ [0, 1[ = [e2iπθ : 1], θ ∈ [0, 1[ = ∂D2 .

The total space is

MΛ =(

M ×D1

⊔

M ×D2

)

/

∼Λ with (x, [1 : e−2iπθ]) ∼ (Λθ(x), [e2iπθ : 1]).

This construction only depends on the homotopy class of Λ. Moreover, Ω, the family (parameter-ized by S2) of symplectic forms of the fibers, can be “horizontally completed” to give a symplecticform on MΛ, ωΛ,κ = Ω + κ · π∗(ω0) where ω0 is the standard symplectic form on S2 (with area1), π is the projection to the base of the fibration and κ a big enough constant to make ωΛ,κ

non-degenerate. (Once chosen, κ will be omitted from the notation.)

So we end up with the following Hamiltonian fibration :

(M,ω)

// (MΛ, ωΛ)π

// (S2, ω0)

In [29], McDuff and Tolman observed that, when (M,ω) admits a circle action, the idea behindthe clutching construction can be simplified. Indeed, they noticed that when the manifold admitsa circle action Λ with moment map ΦΛ, MΛ can be seen as the quotient of M×S3 by the diagonalaction of S1, e2πiθ · (x, (z1, z2)) = (Λθ(x), (e

2πiθz1, e2πiθz2)).

The symplectic form also has an alternative description in M ×S1 S3. Let α ∈ Ω1(S3) be thestandard contact form on S3 such that dα = χ∗(ω0) where χ : S3 → S2 is the Hopf map and ω0 isthe standard area form on S2 with total area 1. For all c ∈ R, ω+cdα−d(ΦΛα) is a closed 2–formon M × S3 which descends through the projection to a closed 2–form, ωc, on MΛ which extendsΩ. Now, if c > maxΦΛ, ωc is non-degenerate and coincides with ωΛ,κ for some big enough κ.

In the case of a toric symplectic manifold fiber, the same arguments lead to the fact that (MΛ, ωΛ)itself is toric. Fix Λ, a Hamiltonian circle action on M .

Proposition 2.1. Let (M2n, ω, T,Φ) be a toric symplectic manifold with associated Delzant poly-tope P . Denote by (MΛ, ωΛ) the total space resulting from the clutching construction associated tothe homotopy class of the loop of Hamiltonian diffeomorphisms of (M,ω) corresponding to Λ. Λadmits a representative in T given as the exponential of θb where θ ∈ [0, 1] and b ∈ tZ, the latticeof circle subgroups of T .

2It seemed more relevant to follow the same convention as in [29] even though the polytope is often defined byP ′ = x ∈ t

∗|〈η′i, x〉 ≥ −κi, i = 1, . . . , N for inward normals η′

i.


Figure 1. The polytopes associated to the fiber M (left) and to the total spaceMΛ (right)

Then there exist a (n+1)–dimensional torus TΛ ⊂ Ham(MΛ, ωΛ), and a moment map ΦΛ : MΛ →t∗Λ ≃ t

∗ × R such that (MΛ, ωΛ, TΛ,ΦΛ) is a toric symplectic manifold, whose associated Delzant

polytope PΛ and integral lattice tΛZ are given by

PΛ = (x, x0) ∈ (t× R)∗ | x ∈ P, c′ + 〈x, b〉 ≤ x0 ≤ 0 and tΛZ = tZ × Z ⊂ t× R

where c′ > max〈x, b〉, x ∈ P coincides with the constant c appearing in the (alternative) descrip-tion of ωΛ.

Moreover, the outward normals of PΛ, ηΛ, are given in terms of the ones of P, η, as follows:

ηΛ = (ηi, 0), ηi ∈ η ∪ (0, 1), (b,−1) .

The polytopes P and PΛ are illustrated by Figure 1. The upper and lower facets of PΛ correspondto two copies of P , the former horizontal, the latter orthogonal to (b,−1) ∈ t

∗ × R.

Proof. We start with a toric manifold (M,ω, T,Φ) and perform the clutching construction forsome element [Λb] in π1(Ham(M,ω)) which admits a representative in T given as the exponentialof θb with θ ∈ [0, 1] and b ∈ Zn. We denote by Λθ

b the Hamiltonian diffeomorphism at time θ.Notice that this path is generated by an autonomous Hamiltonian (the evaluation of the momentmap Φ on b). Thus, Λb is a 1–parameter subgroup of T which commutes with every element in T .We denote the induced, time-independent Hamiltonian vector field by Xb = ∂θΛ

θb (Λ

θb)

−1.

The total space of the fibration is by definition

MΛ =(

M ×D1

⊔

M ×D2

)

/

∼b with (x, [1 : e−2iπθ]) ∼b (Λθb(x), [e

2iπθ : 1]).

Notice that we have a natural S1–action on MΛ induced by S1–actions on the trivial fibrations

S1 y M ×D1 via e2iπθ′

· (x, [1 : z]) = (x, [1 : ze−2iπθ′

]), and

S1 y M ×D2 via e2iπθ′

· (x, [z : 1]) = (Λθ′

b (x), [ze2iπθ′

: 1])

which coincide on the boundary of the discs, since

e2iπθ′

· (x, [1 : e−2iπθ]) = (x, [1 : e−2iπ(θ′+θ)]) ∼b

(Λθ′+θb (x), [e2iπ(θ

′+θ) : 1]) = e2iπθ′

· (Λθb(x), [e

2iπθ : 1])

for all θ and θ′ in [0, 1[. We can also naturally extend the T –action from M to MΛ since theobvious extensions to the discs

T y M ×D via Λ · (x, [z1 : z2]) = (Λ(x), [z1 : z2])

(with D = D1 or D2) coincide on the boundary of the discs, since

Λ · (x, [1 : e−2iπθ]) = (Λ(x), [1 : e−2iπθ]) ∼b (Λθb(Λ(x)), [e

2iπθ : 1]) = Λ · (Λθb(x), [e

2iπθ : 1])

because Λb commutes with the elements of T .

It is clear that the extension of the action of T preserves the symplectic form ωΛ = Ω + κπ∗ω0

since it only acts non trivially inside each fiber (and thus the horizontal part, κπ∗ω0, is preserved)


by Hamiltonian diffeomorphisms of (M,ω) (and thus the vertical part, Ω, is preserved). Theadditional S1–action also preserves ωΛ since it acts by rotation on CP1 and by Hamiltoniandiffeomorphisms in the fiber, if acting non-trivially at all. Finally, the fundamental group of MΛ

being trivial, we end up with a Hamiltonian (n + 1)–torus, TΛ = T × S1 acting effectively on(MΛ, ωΛ). We denote the moment map by ΦΛ.

In order to determine the associated polytope, we consider the fixed points of the action. It isclear, by definition of the additional S1–action, that the points fixed by TΛ lie in the fibers overthe points [1 : 0] and [0 : 1].

Now, it is not hard to see that the image of these two fibers are contained in hyperplanes in (t×R)∗

respectively normal to (0, 1) and (b,−1). Indeed, let us first consider the 1–parameter subgroupof Ham(MΛ, ωΛ) induced by an element (a, λ) in the Lie algebra of TΛ (with λ = e2iπs). We usethe same conventions than for the data related to M and denote the Hamiltonian diffeomorphismat time θ by Λθ

(a,λ). In view of the definition of MΛ and of the action,

on M ×D1, Λθ(a,λ)(x, [1 : z]) = (Λθ

a(x), [1 : e−2iπsθz]), and

on M ×D2, Λθ(a,λ)(x, [z : 1]) = (Λθ

a Λθsb(x), [e

2iπsθz : 1]).

By identifying TpMΛ with either TxM × T[1:z]D1 or TxM × T[z:1]D2, we get that the verticalcomponent of X(a,λ)(p) is Xa(x) if p = (x, [1 : z]) ∈ M × D1 and Xa+sb(x) if p = (x, [z : 1]) ∈M ×D2.

From this fact, we immediately deduce that the vector field X(b,−1) as well as the 1–form

ωΛ(X(b,−1), · ) vanish on π−1([0 : 1]), where as above π : MΛ → S2 is the projection to the base.

This implies that the restriction to the fiber of 〈ΦΛ, (b,−1)〉 vanishes so that ΦΛ(π−1([0 : 1])) is

normal to (b,−1). The exact same argument shows that ΦΛ(π−1([1 : 0])) is normal to (0, 1).

Moreover, the computation of Φ(a,λ) above also shows that the points (x, x0) ∈ (t×R)∗ which arefixed by the action of T × 1 ⊂ TΛ are such that x is fixed by T . Thus PΛ is given as the subset ofP × R∗ contained between two hyperplanes respectively defined by

y = (x, x0) ∈ P × R|〈y, (0, 1)〉 = c1 and y = (x, x0) ∈ P × R|〈y, (−b, 1)〉 = c2.

Up to global “vertical” translation of the polytope, we can choose c1 = 0 and up to “horizontal”flip we can require c2 < 0. Then we see that PΛ consists of the points (x, x0) ∈ P × R such that0 ≥ x0 and x0 ≥ c2 + 〈x, b〉. For the polytope to be convex, we necessarily have c2 + 〈x, b〉 < 0,that is, −c2 > max〈x, b〉, x ∈ P. We thus get the description of PΛ stated in Proposition 2.1with c′ = −c2.

The fact that c′ coincides with c can be checked by computing the volume of MΛ directly withωΛ and then via the description of the polytope. Finally, the statement concerning the outwardnormals is obvious in view of the description of PΛ.

2.3. Toric geometry: the algebraic viewpoint. We now briefly review toric varieties sincewe will use this viewpoint extensively. Good basic references are Cox–Katz [11] and Batyrev [5].There is also a good summary of the definition and some properties of smooth toric varieties inSpielberg [35]. In what follows we mainly use his notation.

Letm > 0 be a positive integer, tZ = Zm be them–dimensional integral lattice and t∗Z = Hom(tZ,Z)

be its dual space. Moreover, let t = tZ⊗ZR and t∗ = t

∗Z ⊗ZR be the R–scalar extensions of tZ and

t∗Z, respectively.

A convex subset σ ⊂ t is called a regular k–dimensional cone if there exists a Z– basis η1, . . . , ηk, . . . ,ηm of tZ such that the cone σ is generated by η1, . . . , ηk. The vectors η1, . . . , ηk ∈ tZ are the integralgenerators of σ. If σ′ is a (proper) face of σ, we will write σ′ ≺ σ. A finite system Σ = σ1, . . . σsof regular cones in t is called a regular n–dimensional fan of cones if any face σ′ of a cone σ ∈ Σis in the fan and any intersection of two cones σ1, σ2 ∈ Σ is again in the fan. A fan Σ is called acomplete fan if t = ∪iσi. The k–skeleton Σ(k) of the fan Σ is the set of all k–dimensional cones in


Σ. A subset of the 1–skeleton of a fan is called a primitive collection of Σ if it is not the set ofgenerators of a cone in Σ, while any of its proper subset is. We will denote the set of primitivecollections of Σ by P .

Suppose the 1–skeleton of Σ is given by η1, . . . , ηd. Let z1, . . . , zd be a set of coordinates in Cd

and let ι : Cd → tZ ⊗Z C be a linear map such that ι(zi) = ηi. For each primitive collectionP = ηi1 , . . . , ηip we define a (d− p)–dimensional affine subspace in Cd by

A(P ) := (z1, . . . , zd) ∈ Cd | zi1 = . . . = zip = 0.

Moreover, we define the set U(Σ) to be the open algebraic subset of Cd given by

U(Σ) = Cd\⋃

p∈P

A(P ).

The map ι : Cd → tC induces a map between tori (C∗)d → (C∗)m that we will also call ι. LetD(Σ) := ker(ι : (C∗)d → (C∗)m) to be the kernel of this map, an (d −m)–dimensional subtorus.Then the quotient

XΣ := U(Σ)/D(Σ)

is called the toric manifold associated to Σ. Note that there is a torus of dimension m actingon XΣ. Moreover, Delzant [12] showed that if XΣ is a projective simplicial toric variety then itcan be constructed as a symplectic quotient and therefore it is endowed with a symplectic form ω(it is also endowed with an action of a n–dimensional torus). From the moment polytope of thissymplectic toric manifold it is possible to recover the fan Σ. However, as explained in a very clearway in [7, Part B], changing the cohomology class of the symplectic form corresponds to changingthe lengths of the edges of the polytope. The size of the faces of a polytope cannot be recoveredfrom the fan, where only the combinatorics of the faces is encoded. Hence, the fan does not givethe cohomology class of the symplectic form.

Standard results about toric manifolds explain how to obtain the cohomology ring of the toricvariety XΣ. Assume the moment map Φ : XΣ → t

∗ is chosen so that each of its components ismean-normalised. Let PΣ ⊂ t

∗ be the image of the moment map. Let D1, . . . , Dn be the facets ofP (the codimension–1 faces), and let η1, . . . , ηn ∈ tZ denote the outward primitive integral normalvectors. Let C be the set of subsets I = i1, . . . , ik ⊆ 1, . . . , n such that Di1 ∩ . . . ∩Dik 6= ∅.Consider the two following ideals in Q[Z1, . . . , Zn], where Zi ∈ H∗(XΣ).

Lin(Σ) =⟨

∑

(x, ηi)Zi | x ∈ t∗Z

⟩

and SR(Σ) = 〈Zi1 . . . Zik | i1, . . . , ik /∈ C〉.

The ideal Lin(Σ) is generated by linear relations and the ideal SR(Σ) is called the Stanley–Reisnerideal. A subset I ⊆ 1, . . . , n is called primitive if I is not in C but every proper subset is.Clearly,

SR(Σ) = 〈Zi1 . . . Zik | i1, . . . , ik ⊆ 1, . . . , n is primitive〉.

The map which sends Zi to the Poincare dual of Φ−1(Di) (which we shall also denote by Zi ∈H2(XΣ)) induces an isomorphism

(1) H∗(XΣ;Q) ∼= R[Z1, . . . , Zn]/〈Lin(Σ) + SR(Σ)〉.

Moreover, there is a natural isomorphism between H2(XΣ;Z) and the set of tuples (a1, . . . , an) ∈Zn such that

∑

aiηi = 0, under which the pairing between such an element of H2(XΣ;Z) andZi is ai. The linear functional ηi is constant on Di and let ηi(Di) denote its value. Under theisomorphism of (1) we have

(2) [ω] =∑

i

ηi(Di)Zi and c1(XΣ) =∑

i

Zi.

Dually, let R(Σ) ⊂ Zn be the subgroup of Zn defined by

R(Σ) := λ = (λ1, . . . , λn) ∈ Zn |λ1η1 + . . .+ λnηn = 0 ∼= Zn−k.

Then the group R(Σ) is canonically isomorphic to H2(XΣ;Z).


2.4. Small quantum homology and Gromov–Witten invariants. The small quantum coho-mology ring of a symplectic manifold (M,ω) is, additively, the usual cohomology with coefficientsin a suitable ring Π that is chosen to allow for the definition of a new ring structure. The quantumproduct is the sum of the usual cup product in the cohomology of M with certain correction termsarising from certain 3–point Gromov–Witten invariants. The adjective small means that only asmall part of the information contained in these invariants is used in its definition, while the struc-ture of the big quantum cohomology incorporates all the genus zero Gromov–Witten invariantsand may also include all the information on the higher genus invariants (see more details in [28]).

Although it is often convenient to work with cohomology, sometimes it is useful to think in termsof homology because the link with geometry is more direct. We will use coefficients in the ringΠ := Πuniv[q, q−1] where q is a variable of degree 2 and Πuniv is a generalised Laurent series ringin a variable of degree 0:

Πuniv :=

∑

κ∈R

rκtκ | rκ ∈ Q, #κ > c | rκ 6= 0 < ∞, ∀c ∈ R

.

Correspondingly, quantum cohomology has coefficients in the dual ring

Π := Πuniv[q, q−1]

where q is as before and

Πuniv :=

∑

κ∈R

rκtκ | rκ ∈ Q, #κ < c | rκ 6= 0 < ∞, ∀c ∈ R

.

Therefore we define

QH∗(M ; Π) = H∗(M,Q)⊗Q Π, and QH∗(M ; Π) = H∗(M,Q)⊗Q Π

where this ring is Z–graded in a way such that

deg(a⊗ qdtκ) = deg(a) + 2d

where a ∈ H∗(M) or a ∈ H∗(M).

The quantum product in homology is a deformation of the intersection product that is defined asthe Poincare dual of the quantum cup product. Thus the quantum intersection product

a ∗ b ∈ QHi+j−dimM (M ; Π), for a ∈ Hi(M), b ∈ Hj(M)

has the forma ∗ b =

∑

B∈HS2 (M ;Z)

(a ∗ b)B ⊗ q−c1(B)t−ω(B),

where HS2 (M ;Z) is the image of π2(M) under the Hurewicz map. The homology class (a ∗ b)B ∈

Hi+j−dimM+2c1(B)(M) is defined by the requirement that

(a ∗ b)B ·M c = GWMB,3(a, b, c) for all c ∈ H∗(M).

In this formula GWMB,3(a, b, c) ∈ Q denotes the Gromov–Witten invariant that counts the number

of spheres in M in class B that meet cycles representing the classes a, b, c ∈ H∗(M). The product∗ is extended to QH∗(M) by linearity over Π, and is associative. It also respects the Z–gradingand gives QH∗(M) the structure of a graded commutative ring, with unit [M ].

Gromov–Witten invariants can also be interpreted as homomorphisms

GWMA,k : H∗(M ;Q)⊗k ⊗H∗(M0,k;Q) −→ Q

GWMA,k(a1, . . . , ak;β) =

∫

M0,k(A;J)

ev∗1a1 ∪ . . . ev∗kak ∪ π∗PD(β),

where M0,k(A; J) is the compactified moduli space of J–holomorphic spheres with k markedpoints in M representing the homology class A. This moduli space carries a natural evaluationmap ev to Mk and a natural projection, the forgetful map, to the moduli space M0,k of stable


curves of genus 0 with k marked points. Geometrically, one can think of this invariant as countingthe number of J–holomorphic spheres with k marked points that pass, at the i–th marked point,through a cycle Xi ⊂ M Poincare dual to ai, and such that the image of the curve under theprojection π is restricted to a cycle Y ⊂ M0,k representing the class β.

Let us recall that in general GWMA,k(a1, . . . , ak) is the homomorphism

GWMA,k : H∗(M ;Q)⊗k → Q , (a1, . . . , ak) 7→ GWM

A,k(a1, . . . , ak; [M0,k])

so that when k = 3, GWMA,3(a1, a2, a3) = GWM

A,3(a1, a2, a3; [pt]).

For easy reference, we gather here the properties of Gromov–Witten invariants which will be usedexplicitly at several places in the computations of Section 4.

Proposition 2.2. Let (M,ω) be a semipositive compact symplectic manifold, A ∈ H2(M ;Z),k ≥ 1, and a1, . . . , ak ∈ H∗(M). Then the following properties hold.

(Divisor) If (A, k) 6= (0, 3) and deg(ak) = 2 then

GWMA,k(a1, . . . , ak) = GWM

A,k−1(a1, . . . , ak−1) ·

∫

A

ak .

(Zero) If k 6= 3 then GWM0,k = 0. If k = 3 then

GWM0,3(a1, a2, a3) =

∫

M

a1 ∪ a2 ∪ a3 .

(Splitting) If k = 4 then GWMA,4(a1, . . . , a4; [pt]) is equal to

∑

A=A0+A1

∑

ν,µ

GWMA0,3(a0, a1, eν) g

νµ GWMA0,3(eµ, a2, a3)

where (eν)ν is a basis of H∗(M), gνµ are the coefficients of the cup-product matrix: gνµ =∫

Meν ∪ eµ, and gνµ the coefficients of its inverse.

Remark 2.3. NEF manifolds are obviously semipositive. The first two properties are extractedfrom [28, Proposition 7.5.6]. The third property is the particular case of [28, Theorem 7.5.10]

for the invariants GWMA,4(a1, . . . , a4; [pt]) = GW

M,1,2,3,4A,4 (a1, . . . , a4) (see [28, Remark 7.5.1.(vi)])

when k = 4.

3. Toric 4–dimensional NEF manifolds

Now we restrict ourselves to the case of toric 4–dimensional NEF manifolds. We explain the con-struction of MΛ and its properties including its cohomology ring. This will play a very importantrole in the next section.

3.1. Toric and homological data. We consider a 4–dimensional toric manifold (M,ω, T,Φ) andits moment 2–dimensional Delzant polytope P . Assume it has n facets. Let v1, . . . , vn denote theoutward primitive integral normal vectors and let Λi denote the circle action corresponding to vi,that is, Λi is the circle action whose moment map is given by Φi := 〈vi,Φ(·)〉.

We pick a ω–tame almost complex structure J and denote by c1(M) the first Chern class of(TM, J). We assume that (M,J) is NEF, that is 〈c1(M), B〉 ≥ 0 for every class B ∈ H2(M) witha J–pseudo-holomorphic representative. Moreover, we consider the particular case when there areat most 2 (consecutive) facets corresponding to spheres with vanishing first Chern number andassume their normal vectors are vi0 and vi0+1.

Next, using the clutching construction described in Section 2.2, we construct the manifold MΛi0

associated to the loop Λi0 which we will denote simply by MΛ in order to simplify the notation.As we showed in Proposition 2.1, MΛ is a toric manifold with moment map ΦΛ. The moment


image is a 3–dimensional polytope PΛ with n + 2 facets which we denote by D1, . . . , Dn, Db, Dt

with corresponding outward primitive integral normal vectors η1, . . . , ηn, ηb, ηt. D1, . . . , Dn arethe vertical facets of PΛ “coming from” the facets of P , while Db and Dt are respectively thebottom and top facets. Equivalently, ηb and ηt denote the normals to Db and Dt, respectively,while η1, . . . , ηn are induced by the normal vectors v1, . . . , vn. We can assume that the normalvectors satisfy the following relations:

ηi0−1 = −e2 ηi0+3 = α3e1 + β3e2 ηb = −e3ηi0 = −e1 . . . ηt = e3 − e1ηi0+1 = e2 − 2e1 ηi0+j = αje1 + βje2ηi0+2 = 2e2 − 3e1 . . .

where the vectors e1, e2, e3 form the canonical basis of Z3. Indeed, the expressions for ηb and ηt arean immediate consequence of the clutching construction while the other relations follow from [2,Lemma 6.3]. Without loss of generality we can assume that ηi0+j = ηj with j = 1, . . . , n − 2,ηi0−1 = ηn−1 and ηi0 = ηn. The vertical facets of PΛ and the corresponding outward normals arerepresented in Figure 2.

Di0−1

ηn−1

Di0

ηn

Di0+1

η1

Di0+2

η2

Figure 2. Vertical facets of the polytope PΛ and their outward normals.

The manifold MΛ is 6–dimensional, hence its fan Σ lives in the lattice Z3. Then the 1–dimensionalcones of the fan Σ are generated by the vectors ηi defined above. The set of primitive collectionsof the fan Σ is given by the following set:

P = η1, η3, . . . , η1, ηn−1, η2, η4, . . . , η2, ηn, η3, η5, . . . , η3, ηn, ηn−2, ηn, ηb, ηt.

From the description of the set of primitive collections, it is easy to get the set of maximal conesin Σ. Next we list some 3–dimensional cones (the ones that are going to be relevant for ourcomputations):

σ1 = 〈ηn−2, ηn−1, ηb〉 σ5 = 〈η2, η3, ηb〉 σ9 = 〈η1, η2, ηt〉σ2 = 〈ηn−1, ηn, ηb〉 σ6 = 〈ηn−2, ηn−1, ηt〉 σ10 = 〈η2, η3, ηt〉σ3 = 〈η1, ηn, ηb〉 σ7 = 〈ηn−1, ηn, ηt〉σ4 = 〈η1, η2, ηb〉 σ8 = 〈η1, ηn, ηt〉

From (1) it follows that the cohomology ring of MΛ is given by the following isomorphism:

H∗(MΛ;Q) ∼= Q[Z1, . . . , Zn, Zb, Zt]/〈Lin(Σ) + SR(Σ)〉


where SR(Σ) is the Stanley–Riesner ideal of Σ and Lin(Σ) is the ideal generated by the linearrelations. The former is generated by the set of primitive collections:

Z1Z3, . . . , Z1Zn−1, Z2Z4, . . . , Z2Zn, Z3Z5, . . . , Z3Zn, . . . ,

Zn−3Zn−1, Zn−3Zn, Zn−2Zn and ZbZt,

while the ideal Lin(Σ) is generated by the following three elements:

Zn + 2Z1 + 3Z2 + α3Z3 + . . .+ αn−2Zn−2 + Zt,(3)

Zn−1 − Z1 − 2Z2 + β3Z3 + . . .+ βn−2Zn−2, and(4)

Zt − Zb.(5)

In view of the relations (3)–(5), Z2, Zn−1, and Zt are linear combinations of the others, so that theset Z1, Z3, . . . , Zn−2, Zn, Zb is a basis of the degree 2 part of the cohomology ring. The degree2 homology H2(MΛ;Z) can be identified with the group R(Σ) ⊂ Zn+2 given by:

R(Σ) := λ = (λ1, . . . , λn+2) ∈ Zn+2 |λ1η1 + . . .+ λn+2ηn+2 = 0,

where we identify ηb, ηt with ηn+1, ηn+2 respectively. Using the definition of the vectors ηi it iseasy to see that we can define a basis for the degree 2 homology λ1, λ3, . . . , . . . , λn−2, λn, λbwhich is dual to the basis of the degree 2 cohomology, that is, Zi(λj) = 1 if i = j and 0 otherwise.More precisely, the generators are given by

λ1 =

(

1,−2

3, 0, . . . , 0,−

1

3, 0, 0, 0

)

,

λj =

(

0,1

3αj , 0, . . . , 0, 1, 0, . . . , 0,

2

3αj + βj , 0, 0, 0

)

, j = 3, . . . , n− 2,

λn =

(

0,−1

3, 0, . . . , 0,−

2

3, 1, 0, 0

)

and λb =

(

0,−1

3, 0, . . . , 0,−

2

3, 0, 1, 1

)

where the entree 1 in λj is located at the j–th entrance.

Consider now, for example, the invariant 2–sphere Vσ2∩σ3, connecting the fixed points correspond-

ing to σ2 and σ3. Since σ2 = 〈ηn−1, ηn, ηb〉 and σ3 = 〈η1, ηn, ηb〉, the homology class of Vσ2∩σ3is

Poincare dual to ZnZb. Hence the primitive relations yield

〈Z1, Vσ2∩σ3〉 = Z1ZnZb, 〈Z3, Vσ2∩σ3

〉 = 0, . . . 〈Zn−2, Vσ2∩σ3〉 = 0,

〈Zn, Vσ2∩σ3〉 = Z2

nZb = −2Z1ZnZb, 〈Zb, Vσ2∩σ3〉 = 0.

Since Z1, Z3, . . . , Zn−2, Zn, Zb is dual to λ1, λ3, . . . , . . . , λn−2, λn, λb, this implies that Vσ2∩σ3=

λ1 − 2λn. For another example, consider the homology class of Vσ3∩σ4which is Poincare dual to

Z1Zb and

〈Z1, Vσ3∩σ4〉 = Z2

1Zb = −2Z1ZnZb, 〈Z3, Vσ3∩σ4〉 = 0, . . .

〈Zn−2, Vσ3∩σ4〉 = 0, 〈Zn, Vσ2∩σ3

〉 = Z1ZnZb, 〈Zb, Vσ3∩σ4〉 = 0.

so Vσ3∩σ4= λn − 2λ1. Calculations of the homology classes of the other invariant spheres are

similar. Moreover, it is not hard to check that the ones not identified in the diagram of Figure 3include contributions of generators λi distinct from λ1, λn, and λb.

Let Ai ∈ H∗(M ;Q) with i = 1, . . . , n denote the homology class of the pre–image under themoment map Φ of the i–th facet of the polytope P of M . Since MΛ is the total space of afibration with fiber M , these homology classes can be identified with some invariant 2–spheres inMΛ, Vσj∩σk

. More precisely, we have Ai0 = λ1 − 2λn, Ai0+1 = λn − 2λ1. Let3 Amax = λb − λn =

Vσ3∩σ8= Vσ2∩σ7

. Since

c1(MΛ) = Z1 + . . .+ Zn + Zb + Zt,

3The notation Amax is due to the fact that this represents the homology class of a section of MΛ through pointson the maximal fixed set of the action (prior to the clutching construction).


æ æ

ææ

æ

æ æ

æ æ

æ æ

σ6 σ1

σ7 σ2

σ8 σ3

σ9 σ4

σ10 σ5

λb − λn

λb − λn

λb − 2λ1

λ1 − 2λnλ1 − 2λn

λn − 2λ1λn − 2λ1

Figure 3. Diagram representing some invariant 2–spheres of the toric manifoldMΛ and their homology classes.

where c1(MΛ) is the first Chern class of the tangent bundle ofMΛ, it follows easily that 〈c1(MΛ), λ1〉= 〈c1(MΛ), λn〉 = 0 and 〈c1(MΛ), λb〉 = 1. Therefore we have 〈c1(MΛ), Ai0〉 = 〈c1(MΛ), Ai0+1〉 =0 and 〈c1(MΛ), Amax〉 = 1.

As we shall see in Section 4.1, in order to compute certain Gromov–Witten invariants we will needto know some more information about the ring structure of the cohomology of MΛ, namely certainrelations satisfied by the coefficients of the cup-product matrixG = (gνµ)νµ, with gνµ =

∫

MΛeν∪eµ

(for some basis (eν)ν of the cohomology ring), and its inverse, G−1 = (gνµ)νµ.

By noticing that the cohomology of MΛ is non-zero only in even degrees, that the degree 0 anddegree 6 groups are 1–dimensional (respectively generated by 1 and the fundamental class of MΛ,[MΛ]), and that gνµ 6= 0 only if the degrees of eν and eµ sum up to 6, it is easy to see that, assoon as (eν)ν is ordered so that the degree increases, G decomposes as

0 0 0 1

00 B

0

BT 00 0

1 0 0 0

with B the matrix composed of the (gνµ)|eν |=2,|eµ|=4.

Now, let us specify the basis. Recall that the set Z1, Z3, . . . , Zn−2, Zn, Zb is a basis of the degree2 part of the cohomology. The degree 4 part of the cohomology consists of all products ZiZj

and ZiZb with 1 ≤ i ≤ j ≤ n and i, j 6= 2, n − 1. In view of the relations coming from SR(Σ),Z1Zj = 0 for 3 ≤ j ≤ Zn−2. Then, multiplying (3) and (4) by Z1 immediately leads to the relationsZ1Zn+2Z2

1+3Z1Z2+Z1Zb = 0 and Z21 +2Z1Z2 = 0 respectively , so that Z2

1 = −2Z1Zn−2Z1Zb.Hence, for i = 1, only Z1Zn and Z1Zb need to be considered. For i = n, analogous argumentslead to the fact that we only have to consider ZnZb. (In particular, multiplying (3) by Zn gives


Z2n = −2Z1Zn − ZnZb.) Hence, we can explicitly write some part of B:

Z1Zn Z1Zb ZnZb . . . . . .Z1 0 −2 1 0 — 0Z3 0 0 0| 0 0 0

Zn−2 0 0 0Zn −1 1 −2Zb 1 0 0

Indeed, the vanishing terms come from the relations given by the ideal SR(Σ), while the non-zeroterms can be computed using the definition. For example, since Z1Zb is Poincare dual to λn− 2λ1

as we saw above, it follows that Z1ZnZb is given by∫

MΛ

Z1ZnZb = Zn(λn − 2λ1) = 1.

Using this computation together with the relations given by the ideals SR(Σ) and Lin(Σ) we canobtain the other non-vanishing terms.

In order to simplify the notation, we will denote gνµ and gνµ by using the indices of the corre-sponding elements eν and eµ. For example, for eν = Z1 and eµ = ZnZb, gνµ will be denotedg1,nb and gνµ will be denoted g1,nb. Of course G and G−1 are symmetric so that gνµ = gµν andgνµ = gµν . Moreover, note that by commutativity of the cup-product, permuting the indices doesnot change the value g1,nb = gb,1n = gn,1b. However, this fails for the coefficients of G−1.

Since G−1G = 1, we get relations between the coefficients of G and G−1 by multiplying particularlines of G−1 with columns of G. For example,

∑

ν

g1b,νgν,1b = 1 ⇐⇒ −2g1b,1 + g1b,n = 1

∑

ν

g1b,νgν,1n = 0 ⇐⇒ −g1b,n + g1b,b = 0

∑

ν

g1b,νgν,nb = 0 ⇐⇒ g1b,1 − 2g1b,n = 0

which lead to the fact that g1b,b = g1b,n = − 13 and g1b,1 = − 2

3 . By using the lines of G−1

corresponding to Z1Zn, ZnZb, and again the columns of G corresponding to Z1Zn, Z1Zb, andZnZb, we get some more coefficients of the matrix G−1:

g1n,b = 1g1n,n = g1n,1 = 0

and

gnb,1 = − 13

gnb,n = gnb,b = − 23

To ease the reading later on, we gather in the next lemma the result of these computations.

Lemma 3.1 (Some coefficients of G−1).

g1b,1 = − 23

g1b,n = g1b,b = − 13

,

g1n,b = 1g1n,n = g1n,1 = 0

, and

gnb,1 = − 13

gnb,n = gnb,b = − 23

.

3.2. Gromov–Witten invariants. We now compute some Gromov–Witten invariants of MΛ

using Spielberg’s machinery from [35]. Note that Liu proved a more general result in [26], howeversince we only need to compute genus–0 Gromov–Witten invariants we will use Spielberg’s formulasand notation.

We need to know the weights of the torus action at the different charts (as explained in [34] and[35] each cone gives a chart of the toric manifold). More precisely, as in [35, Lemma 6.10], letσi, σj be two 3–dimensional cones in the fan Σ and let ηk1

, ηk2be the generators of the common

face τ = σi ∩ σj such that

σi = 〈ηk1, ηk2

, ητ(i)〉 and σj = 〈ηk1, ηk2

, ητ(j)〉,


where ητ(i) is the generator of σi which is not a generator of σj and vice-versa for the normal

vector ητ(j). Let ω1, . . . , ωn+2 be the weights of a diagonal action of (C∗)n+2 on Cn+2 with respectto the standard basis. Then the induced C∗–action on the invariant 2–sphere Vσi∩σj

has weightωσiσj

at the point Vσigiven by

ωσiσj

:=n+2∑

ℓ=1

〈ηℓ, u3〉ωℓ

where u1, u2, u3 is a basis of t∗Z dual to

ηk1, ηk2

, ητ(i)

.

For our calculations it will be convenient to have the following weight table at hand (let ωt := ωn+1

and ωb := ωn+2 in the table below):

σ2 = 〈ηn−1, ηn, ηb〉 σ7 = 〈ηn−1, ηn, ηt〉ωσ2σ3

= a1 ωσ7σ8

= a1ωσ2σ1

= a2 + ωt ωσ7σ6

= a2 + ωb

ωσ2σ7

= ωb − ωt ωσ7σ2

= ωt − ωb

σ3 = 〈η1, ηn, ηb〉 σ8 = 〈η1, ηn, ηt〉ωσ3σ2

= −a1 ωσ8σ7

= −a1ωσ3σ4

= 2a1 + a2 + ωt ωσ8σ9

= 2a1 + a2 + ωb

ωσ3σ8


= ωt − ωb

σ4 = 〈η1, η2, ηb〉 σ9 = 〈η1, η2, ηt〉ωσ4σ3

= −2a1 − a2 − ωt ωσ9σ8

= −2a1 − a2 − ωb

ωσ4σ5

= 3a1 + 2a2 + 2ωt ωσ9σ10

= 3a1 + 2a2 + 2ωb

ωσ4σ9


= ωt − ωb

where the a1, a2 ∈ Z are linear functions on the weights ω1, . . . , ωn. Now we are ready to begincalculating Gromov–Witten invariants of this manifold. We will compute first the invariant

GWMΛ

Amax+Ai0+Ai0+1,1

(Z1Zb)

which will be needed later in the proof of Theorem 4.6. We use the formula in [35, Theorem 7.8]and its simplified version in [35, Corollary 8.4]. Since the marked point has to lie in the cone σ3

or σ4, we need to consider the graphs which contain one of these cones and which represent theclass Amax +Ai0 +Ai0+1. It follows that we should consider the following graphs:

æ æ æ æ

æ æ æ æ

æ æ æ æ

æ ææ

æ

æ ææ

æ

σ4σ4

σ4

σ3

σ3

σ3

σ3

σ3

σ2

σ2

σ2

σ7

σ7

σ7

σ8

σ8

σ8

σ8

σ9

σ9(2)

(1)

(3)

(4)

(5)

Therefore Spielberg’s formula gives the following computation

GWMΛ

Amax+Ai0+Ai0+1,1

(Z1Zb) = (1) + . . .+ (5) = −(a1 + a2 + ωt)(a2 + ωt)

a1(a1 + ωb − ωt)

−(a1 + a2 + ωt)(a1 + a2 + ωb)

(ωb − ωt)2−

(a1 + 2(ωb − ωt))(a1 + a2 + ωt)(a1 + a2 + ωb)

(ωb − ωt)2(a1 + ωb − ωt)

+(a1 + a2 + ωb)

2

(ωb − ωt)2+

(a1 + a2 + ωt)2(a1 + ωb − ωt)

(ωb − ωt)2= 1 .

We can compute the invariant

GWMΛ

Amax+Ai0+Ai0+1,1

(ZnZb)


in a similar way. In this case the marked point lies in the cone σ2 or σ3 so we need to considerthe same graphs as in the computation above plus the following graph:

æ æ æ æ

σ2 σ7 σ8 σ9(6)

The formula now gives for GWMΛ

Amax+Ai0+Ai0+1,1

(ZnZb) = (1) + . . .+ (6):

GWMΛ

Amax+Ai0+Ai0+1,1

(ZnZb) =(a1 + a2 + ωt)(a2 + ωt)(a2 − ωb + 2ωt)

a1(ωb − ωt)(a1 + ωb − ωt)

+(a1 + a2 + ωt)(a1 + a2 + ωb)(2a1 + a2 − 2ωb + 3ωt)

(ωb − ωt)2(a1 + ωt − ωb)

+(a1 + a2 + ωt)(a1 + a2 + ωb)(2a1 + a2 + ωb)

(ωb − ωt)2(a1 + ωb − ωt)−

(a1 + a2 + ωb)2(2a1 + a2 + ωt)

a1(ωb − ωt)2

−(a1 + a2 + ωt)

2(2a1 + a2 − ωb + 2ωt)

a1(ωb − ωt)2+

(a1 + a2 + ωb)(a2 + ωb)(a2 + ωt)

a1(ωt − ωb)(a1 + ωt − ωb)= −2.

Again, for easy reference, we gather the results of these two computations in the following lemma.

Lemma 3.2 (Gromov–Witten invariants).

GWMΛ

Amax+Ai0+Ai0+1,1

(Z1Zb) = 1 and GWMΛ

Amax+Ai0+Ai0+1,1

(ZnZb) = −2 .

4. Seidel morphism in the NEF case

In this section we explain how to compute the Seidel morphism in the NEF case.

4.1. The Seidel morphism. This morphism has been defined by Seidel in [33]. Recall fromsection 2.2 that, starting from any symplectic manifold (M,ω) and a loop of Hamiltonian dif-feomorphisms Λ ⊂ Ham(M,ω), one can construct a Hamiltonian fibration over S2 with fiberM :

(M,ω) → (MΛ, ωΛ) → (S2, ω0)

where ωΛ = Ω + κ · π∗(ω0) for some big enough κ. Then, as mentioned above, on can defineSeidel’s morphism by counting pseudo-holomorphic section classes in HS

2 (MΛ;Z), with respect tosome arbitrary choice of such a section. This choice has been made canonical in [25].

Since we are interested in the case of toric symplectic manifolds which has been thoroughly studiedin [29], we only extract from it the relevant information which will be used in this paper. Inparticular, we work under the NEF assumption meaning that we only need to consider almostcomplex structures for which there are no pseudo-holomorphic spheres with 〈c1(M), B〉 < 0.

Notation 4.1. Since the first Chern class of M (and of M only) is extensively used in whatfollows, we will denote c1(M) by c1 and 〈c1(M), B〉 by c1(B).

First, since there are some slight changes of notation, let us emphasize the fact that we denotedthe maximal fixed set component of the action of Λ by D and the normalized moment map by Φ(respectively denoted by Fmax and ΛK in [29]). Notice that D is semifree and that dimD = 2.Secondly, concerning the choice of the section mentioned above, in the toric case it is convenientto choose σmax = x×D1∪Λ x×D2 (see the description of the clutching construction §2.2) forany fixed point of the S1–action x lying in D. If we let Amax = [σmax] ∈ HS

2 (M ;Z) then all thecontributions to the Seidel morphism come from the section classes Amax+B with B ∈ HS

∗ (M ;Z)and are determined by counting Gromov–Witten invariants in the classes Amax + B, see e.g [29,Definition 2.4]. Lastly, by [29, Lemma 2.2] the sum of the weights which appear in the formulagiving the Seidel morphism, as part of the exponent of the q variable, is mmax = −1.

We now state the main results from [29] which will be used below.


Theorem 4.2 (Theorem 1.10 and Lemma 3.10 of [29]). We denote by Φmax the value of themoment map on D and by [D] ∈ HS

2 (M ;Z) the homology class of D. The Seidel element associatedto the circle action Λ is

S(Λ) = [D]⊗ qtΦmax +∑

B∈HS2(M ;Z)>0

aB ⊗ q1−c1(B)tΦmax−ω(B)

where HS2 (M ;Z)>0 consists of the spherical classes of symplectic area ω(B) > 0 and aB ∈

H∗(M ;Z) is the contribution of the section class Amax + B given by aB ·M c = GWMΛ

Amax+B,1(c)

for all homology classes c ∈ H∗(M ;Z). Moreover,

(i) If aB 6= 0 either c1(B) = 0 and aB ∈ H2(M ;Z) or c1(B) = 1 and aB ∈ H4(M ;Z).

(ii) If aB 6= 0 then B intersects D.

(iii) If c1(B′) ≥ 1 for all J–holomorphic spheres B′ which intersect D, then all the lower order

terms vanish.

(iv) If c1(B′) ≥ 1 for all J–holomorphic spheres B′ which intersect D but are not included in D,

then aB 6= 0 ⇒ c1(B) = 0.

Remark 4.3. Item (i) above reads: If aB 6= 0 then c1(B) = 0 and |aB| = 2. Indeed, when M is4–dimensional, |aB| = 4 means that aB has to be a multiple of the fundamental class [M ], howeverthis case can easily be ruled out. (See for example the end of the proof of [29, Theorem 1.10].)

Item (ii) is [29, Lemma 3.10] and shows that, even though the formula above might containinfinitely many terms, computing the Seidel morphism is somehow “local” (that is, one does notneed to know the whole polytope).

Recall the notation we introduced in §3: We consider the case when the polytope P , associatedto M , admits n facets, D1, . . .Dn, where the indices are seen mod n4. For any n–tuple a =(a1, . . . , an), we denote by Aa =

∑

i∈[1,n] aiAi the homology class of the union of (possibly multiply

covered) spheres in M whose projection to P is given by Da = ∪i∈[1,n]Di.

Thus Theorem 4.2, combined with Remark 4.3, implies that the Seidel element is given by

S(Λi0) = Ai0 ⊗ qtΦmax +∑

a

aAa⊗ qtΦmax−ω(Aa)

where aAa6= 0 if and only if

(1) Da is connected and intersects Di0 ,

(2) c1(Aa) = 0 (i.e, by NEF condition, for all i so that ai 6= 0, c1(Ai) = 0).

In Theorem 4.4 below, we compute each contribution aAain the case of polytopes where any Da

satisfying (1) and (2) contains at most two facets corresponding to spheres with vanishing firstChern number. Moreover, we also compute in Appendix A the Seidel morphism in some caseswhere there are three such facets.

Theorem 4.4. Assume n ≥ 3. The following homology classes have non trivial contributions toS(Λi0):

(1) Ai0 contributes by aAi0= Ai0 .

(2) If c1(Ai0 ) = 0,

(2a) then kAi0 (with k > 0) contributes by akAi0= Ai0 ,

4Recall that we denoted the facets of PΛ by the same letter. By this we imply that the vertical facet Di

(1 ≤ i ≤ n) in PΛ comes from the facet Di in P after clutching.


(2b) and if c1(Ai0+1) = 0, then kAi0 + lAi0+1 (with k ≥ 0 and l > 0) contributes and its

contribution is akAi0+lAi0+1

=

Ai0 if k ≥ l−Ai0+1 otherwise.

(3) If c1(Ai0 ) 6= 0,

(3a) if c1(Ai0+1) = 0, then kAi0+1 (with k > 0) contributes by akAi0+1= −Ai0+1,

(3b) if c1(Ai0+1) = 0 and c1(Ai0+2) = 0, then kAi0+1 + lAi0+2 (with k > 0 and l > 0)also contributes, and its contribution is

akAi0+1+lAi0+2=

−Ai0+1 if k ≥ lAi0+2 otherwise.

(3c) if c1(Ai0−1) = 0 and c1(Ai0+1) = 0, then kAi0−1 and lAi0+1 (with k > 0 andl > 0) also contribute, and their respective contributions are akAi0−1

= −Ai0−1 andalAi0+1

= −Ai0+1.

Moreover, in each case, if the facets immediately next to the one mentioned in the theorem cor-respond to spheres with non-vanishing first Chern number, then these are the only non-trivialcontributions.

As a corollary, we can compute the image by Seidel’s morphism of the homotopy class of a loopof Hamiltonian diffeomorphisms induced by the toric action in these cases.

Theorem 4.5. Let (M,ω) be a closed toric 4–dimensional NEF symplectic manifold. We denoteby Ai (i = 1, . . . , n) the pre–image by the moment map Φ of the i–th facet of the polytope Passociated to M .

(1) If c1(Ai0 ), c1(Ai0−1) and c1(Ai0+1) are all non-zero, then S(Λ) = Ai0 ⊗ qtΦmax .

(2) If c1(Ai0 ) = 0

(2a) but c1(Ai0−1) and c1(Ai0+1) are non-zero, then

S(Λ) = Ai0 ⊗ qtΦmax

1− t−ω(Ai0)

(2b) and c1(Ai0+1) = 0 but c1(Ai0−1) and c1(Ai0+2) non-zero, then

S(Λ) =

[

Ai0 ⊗ qtΦmax

1− t−ω(Ai0)−Ai0+1 ⊗ q

tΦmax−ω(Ai0+1)

1− t−ω(Ai0+1)

]

·1

1− t−ω(Ai0)−ω(Ai0+1)

(3) If c1(Ai0 ) 6= 0, then

(3a) if c1(Ai0+1) = 0 and c1(Ai0−1), c1(Ai0+2) non-zero, then

S(Λ) = Ai0 ⊗ qtΦmax −Ai0+1 ⊗ qtΦmax−ω(Ai0+1)

1− t−ω(Ai0+1)

(3b) if c1(Ai0+1) = c1(Ai0+2) = 0 but c1(Ai0−1) and c1(Ai0+3) non-zero, then

S(Λ) = Ai0 ⊗ qtΦmax −Ai0+1 ⊗ qtΦmax−ω(Ai0+1)

1− t−ω(Ai0+1)

−

(

Ai0+1 ⊗ qtΦmax

1− t−ω(Ai0+1)−Ai0+2 ⊗ q

tΦmax−ω(Ai0+2)

1− t−ω(Ai0+2)

)

·t−ω(Ai0+1)−ω(Ai0+2)

1− t−ω(Ai0+1)−ω(Ai0+2)

(3c) if c1(Ai0−1) = c1(Ai0+1) = 0, c1(Ai0−2) and c1(Ai0+2) non-zero, then

S(Λ) = Ai0 ⊗ qtΦmax −Ai0−1 ⊗ qtΦmax−ω(Ai0−1)

1− t−ω(Ai0−1)−Ai0+1 ⊗ q

tΦmax−ω(Ai0+1)

1− t−ω(Ai0+1).


We start by proving Theorem 4.5 from Theorem 4.4. The proof of the latter is postponed to thenext subsection since it is much more involving.

Proof of Theorem 4.5. It is a staigthforward consequence of Theorems 4.2 and 4.4.

(1): By Theorem 4.4, only Ai0 contributes and its contribution is of the form S(Λ) = Ai0 ⊗qtΦmax .

(2a): Here Ai0 and its iterations induce the only non-trivial contributions. The contribution ofkAi0 being Ai0 ⊗ q−c1(kAi0

)tΦmax−ω(kAi0), we get the result by summing over k (starting

at k = 0):

S(Λ) = Ai0 ⊗ q tΦmax

(

∞∑

k=0

(t−ω(Ai0))k

)

= Ai0 ⊗ qtΦmax

1− t−ω(Ai0)

(3a): This case is similar to (2a) except that we sum the contributions of all the kAi0+1’s startingat k = 1 (thus, the new −ω(Ai0+1) as power of t).

(3c): This case is similar to (3a) (but for both Ai0−1 and Ai0+1).

Now we turn to (3b). The first two terms coincide with the sum of the contributions induced byAi0 and kAi0+1. However, we also have to count the contributions of kAi0+1 + lAi0+2. As before,we can see that

−Ai0+1 ⊗ qtΦmax−ω(Ai0+1)−ω(Ai0+2)

(1− t−ω(Ai0+1))(1− t−ω(Ai0+1)−ω(Ai0+2))

=

∞∑

k=1,l=0

ak(Ai0+1+Ai0+2)+lAi0+1⊗ qtΦmax−(k+l)ω(Ai0+1)−kω(Ai0+2)

which sums the contributions of k(Ai0+1 + Ai0+2) + lAi0+1 (with k ≥ 1 and l ≥ 0), that is, thecontributions of all terms of the form kAi0+1 + lAi0+2 with k ≥ l ≥ 1. In the same way,

Ai0+2 ⊗ qtΦmax−2ω(Ai0+2)−ω(Ai0+1)

(1− t−ω(Ai0+2))(1 − t−ω(Ai0+1)−ω(Ai0+2))

=

∞∑

k,l=1

ak(Ai0+1+Ai0+2)+lAi0+2⊗ qtΦmax−kω(Ai0+1)−(k+l)ω(Ai0+2)

which sums the contributions of all terms of the form kAi0+1 + lAi0+2 with k < l. Thus theformula given for the case (3b) is indeed the sum of all non-trivial contributions.

Finally, let us look at (2b). First decompose

1

1− t−ω(Ai0)−ω(Ai0+1)

= 1 +t−ω(Ai0

)−ω(Ai0+1)

1− t−ω(Ai0)−ω(Ai0+1)

and by replacing, we check that

S(Λ) =

[

Ai0 ⊗ qtΦmax

1− t−ω(Ai0)−Ai0+1 ⊗ q

tΦmax−ω(Ai0+1)

1− t−ω(Ai0+1)

]

·1

1− t−ω(Ai0)−ω(Ai0+1)

= Ai0 ⊗ qtΦmax

1− t−ω(Ai0)−Ai0+1 ⊗ q

tΦmax−ω(Ai0+1)

1− t−ω(Ai0+1)

+Ai0 ⊗ qtΦmax−ω(Ai0

)−ω(Ai0+1)

(1− t−ω(Ai0))(1− t−ω(Ai0

)−ω(Ai0+1))

−Ai0+1 ⊗ qtΦmax−2ω(Ai0+1)−ω(Ai0

)

(1− t−ω(Ai0+1))(1− t−ω(Ai0)−ω(Ai0+1))

Now the first term counts the contributions of all terms of the form kAi0 (as in (2a) above), thesecond term counts the contributions of kAi0+1 (or Ai0 +kAi0+1, see above) and then the last two


count (as for (3b) but with Ai0 playing the role of Ai0+1 and Ai0+1 playing the role of Ai0+2) allthe contributions of the terms of the form kAi0 + lAi0+1 (with k and l both non-zero).

4.2. Proof of Theorem 4.4. The proof is more or less a case-by-case proof and we focus onCase (2b), since all the difficulties which one might encounter are already present and since themethods used to compute the Gromov–Witten invariants are the same.

We need to determine the class aB of Theorem 4.2 where B = kAi0 + lAi0+1 ∈ H∗(M ;Z). Recallthat this class is determined by the requirement that

aB · c = GWMΛ

Amax+B,1(c), for all c ∈ H∗(M).

In the notation for the Gromov–Witten invariant we can either use the homology class c or itsPoincare dual. Let us define Bk,l := Amax + kAi0 + lAi0+1. Now we claim that in order to provethe theorem in Case (2b) it is sufficient to compute the following Gromov–Witten invariants.

Theorem 4.6. For any k, l ∈ N0 we have

GWMΛ

Bk,l,1(Z1Zn) =

−1, k ≥ l0, k < l

, GWMΛ

Bk,l,1(Z1Zb) =

1, k ≥ l2, k < l

and

GWMΛ

Bk,l,1(ZnZb) =

−2, k ≥ l−1, k < l

where Z1, Zn, Zb ∈ H2(MΛ;Q) are defined in Section 3.1.

Since the proof of this theorem is quite long and technical, we postpone it to §4.3 and §4.4, andwe first finish the proof of Theorem 4.4 by proving the claim.

The class aB is a linear combination of the homology classes of the pre-images, under the momentmap Φ of the facets of the polyope P = Φ(M), that is,

(6) aB =

n∑

i=1

aiAi,

where ai ∈ Z. Since the dimension of the vector space HS2 (M ;Z) is n − 2, we can assume that

two of the coefficients ai vanish. The following lemma shows that we can choose the coefficientsai0+2 = ai0+3 = 0.

Lemma 4.7. All the classes Ai are linear combinations of the basis elements λ1, λ3, . . . , λn−2, λn,defined in Section 3.

Proof. It is known from the diagram of Figure 3 that Ai0 = λ1 − 2λn and Ai0+1 = λn − 2λ1

which gives λ1 = − 13Ai0 −

23Ai0+1 and λn = − 2

3Ai0 −13Ai0+1. Recall that ηi = αie1 + βie2 where

i = 1, . . . , n. Let γi,j := αiβj −αjβi. Note that γi,j 6= 0 if j = i+1 because two consecutive facetshave distinct normals, and γi,j 6= 0 if j 6= i + 1 because the polytope is convex. The argumentused in Section 3 to write Ai0 and Ai0+1 as linear combinations of the basis elements λi can berepeated for all remaining A′

is:

Ai0−1 = λn + γ−1n−1,n−2 λn−2,

Ai0−2 = γn−1,n−3 γ−1n−2,n−1 γ

−1n−2,n−3 λn−2 + γ−1

n−2,n−3 λn−3,

Ai0−3 = γ−1n−3,n−4 λn−4 + γn−4,n−2 γ

−1n−4,n−3 γ

−1n−3,n−2 λn−3 + γ−1

n−2,n−3 λn−2,

...

Ai0+4 = γ−14,3 λ3 + γ3,5 γ

−14,3 γ

−15,4 λ4 + γ−1

5,4 λ5,

Ai0+3 = γ2,4 γ−13,2 γ

−14,3 λ3 + γ−1

4,3 λ4, Ai0+2 = λ1 + γ−13,2 λ3.


Since λn = − 23Ai0 −

13Ai0+1 it follows from the first equation that

λn−2 = γn−1,n−2(Ai0−1 +2

3Ai0 +

1

3Ai0+1).

Substituting this in the second equation we find an expression of λn−3 as a linear combination ofAi0−2, Ai0−1, Ai0 and Ai0+1. Going around the polytope we easily see that we can, recursively,determine an expression of each λi as a linear combination of the A′

is with i 6= i0 + 2, i0 + 3.In particular, we obtain expressions for λ3 and λ4 which implies, by the last two equations, thatAi0+2 and Ai0+3 are linear combinations of the remaining A′

is.

Therefore, from now on, we assume ai0+2 = ai0+3 = 0 in the linear combination (6). Now recallthat

aB · c = GWMΛ

Amax+B,1(PD(c))

for c ∈ H2(M ;Z). If c does not contain Ai0−1, Ai0 , Ai0+1, Ai0+2 then clearly the Gromov–Witten

invariant GWMΛ

Amax+B,1(PD(c)) vanishes when B = kAi0 + lAi0+1. Therefore

0 = GWMΛ

Amax+B,1(PD(Ai0+3)) = aB · Ai0+3 = ai0+4

because ai0+2 = ai0+3 = 0, which implies that

0 = GWMΛ

Amax+B,1(PD(Ai0+4)) = aB ·Ai0+4 = ai0+5.

Repeating the process around the polytope we get

0 = GWMΛ

Amax+B,1(PD(Ai0−2)) = aB · Ai0−2 = ai0−1,

so that all the coefficients vanish except ai0 , ai0+1. That is, we obtain aB = ai0Ai0 + ai0+1Ai0+1

for some ai0 , ai0+1 ∈ Z when B = kAi0 + lAi0+1. Since PD(Ai0 ) = ZnZb and PD(Ai0+1) = Z1Zb

it follows from Theorem 4.6 that if k ≥ l then

− 2 = GWMΛ

σBk,l,1(ZnZb) = aB · Ai0 = (ai0Ai0 + ai0+1Ai0+1) ·Ai0 = −2ai0 + ai0+1,

1 = GWMΛ

σBk,l,1(Z1Zb) = aB ·Ai0+1 = (ai0Ai0 + ai0+1Ai0+1) ·Ai0+1 = ai0 − 2ai0+1.

We conclude that ai0 = 1, ai0+1 = 0 and aB = Ai0 in this case. If k < l then we obtain

−1 = GWMΛ

σBk,l,1(ZnZb) = −2ai0 + ai0+1 and 2 = GWMΛ

σBk,l,1(Z1Zb) = ai0 − 2ai0+1.

Therefore, in this case, ai0 = 0, ai0+1 = −1 and aB = −Ai0+1. This concludes the proof ofTheorem 4.4(2b).

4.3. An intermediate result. Before giving the proof of Theorem 4.6, we first need an inter-mediate result about some particular 0–point Gromov–Witten invariants. Recall that the 0–pointinvariant GWMΛ

0 (A) in the nonzero class A ∈ H2(MΛ) is defined by

GWMΛ

0 (A) :=1

h(A)3GWMΛ

A,3(h, h, h)

where h ∈ H2(MΛ) is such that h(A) =∫

Ah 6= 0. From now on we will suppress the indication of

the number of marked points when that number is clear from the context and the expression forthe Gromov–Witten invariant.

Proposition 4.8. Let k and l be non-negative integers. Then

GWMΛ(kAi0 + lAi0+1) =

−1

k3if l = 0,

−1

l3if k = 0,

−1

k3if k = l,

0 otherwise.


Proof. A good reference for what follows is [26]. We begin the proof with some preliminaries aboutmoduli spaces of stable curves. Let M0,n(CP

1, k) denote the moduli space of genus 0, n–pointed,

degree k stable maps to CP1. Suppose that k > 0. Let p : M0,1(CP1, k) → M0,0(CP

1, k) be the

universal curve, and let ev : M0,1(CP1, k) → CP1 be the evaluation map at the marked point.

The map p is just the forgetting morphism, that is, the natural morphism that forgets the extramarked point. Consider the following short exact sequence over CP1:

0 → OCP1 → OCP1(1)⊕OCP1(1) → OCP1(2) = TCP1 → 0,

which induces the short exact sequence

0 → T ∗CP1 = OCP1(−2) → OCP1(−1)⊕OCP1(−1) → OCP1 → 0.

Given a genus 0, 0–pointed, degree k > 0 stable map u : C → CP1, we have a short exact sequenceof vector bundles over the domain C:

(7) 0 → u∗OCP1(−2) → u∗OCP1(−1)⊕ u∗OCP1(−1) → OC → 0,

where u∗OCP1(−2) is a degree −2k line bundle on C, and u∗OCP1(−1) is a degree −k line bundleon C. We have

H0(C, u∗OCP1(−2)) = 0 and H0(C, u∗OCP1(−1)⊕ u∗OCP1(−1)) = 0.

The long exact sequence of the cohomology associated to (7) becomes

0 → H0(C,OC) → H1(C, u∗OCP1(−2)) → H1(C, u∗OCP1(−1)⊕ u∗OCP1(−1)) → 0,

where dimC H1(C, u∗OCP1(−2)) = 2k − 1 and

dimC H1(C, u∗OCP1(−1)⊕ u∗OCP1(−1)) = 2k − 2.

Next we define the bundles

Ek := p∗ev∗OCP1(−2) and Vk := p∗ev

∗(OCP1(−1)⊕OCP1(−1))

over M0,0(CP1, k). Then Ek is a vector bundle of rank 2k − 1 whose fiber over [u : C → CP1] is

H1(C, u∗OCP1(−2)) and Vk is a vector bundle of rank 2k − 2 whose fiber over [u : C → CP1] isH1(C, u∗OCP1(−1)⊕ u∗OCP1(−1)). We have the following short exact sequence of vector bundlesover M0,0(CP

1, k)

0 → OM → Ek → Vk → 0,

where OM is the trivial line bundle. Therefore,

(8) e(Ek) = c2k−1(Ek) = 0, e(Vk) = c2k−2(Vk) = c2k−2(Ek),

where e(Ek) and ci(Ek) denote, respectively, the Euler class and the i-th Chern class of the vectorbundle Ek.

M0,0(CP1, k) is a smooth Deligne–Mumford stack of dimension 2k − 2.

We have the following result about the vector bundle Vk.

Theorem 4.9 (Manin [27]).∫

[M0,0(CP1,k)]

e(Vk) =1

k3.

In the case we are studying we have a toric fibration π : MΛ → CP1 where the total space is atoric manifold of (real) dimension 6 and each fiber is diffeomorphic to the toric surface M . Usingthe previous notation, we want to compute

GW(kAi0) =

∫

[M0,0(MΛ,kAi0)]vir

1.

Next we will show that indeed

GW(kAi0 ) = −1

k3.


We first introduce some notation. We have

H∗C∗(point;Z) = H∗(BC∗;Z) = H∗(CP∞;Z) = Z[u],

where u = c1(OCP∞(−1)) is the first Chern class of the tautological line bundle over BC∗ = CP∞.Let Lmu denote the C∗–equivariant line bundle over a point given by the 1–dimensional C∗–representation t 7→ tm. Then

(c1)C∗(Lmu) = mu ∈ H2C∗(point;Z) = Z[u].

Let C∗ acts on CP1 by t · [x, y] = [tx, y]. The two C∗ fixed points in CP1 are 0 = [0 : 1] and∞ = [1 : 0]. We have

(c1)C∗(T0CP1) = u, (c1)C∗(T∞CP1) = −u.

There is a unique lifting of this C∗–action to MΛ such that the action on π−1(0), the fiber ofπ : MΛ → CP1 over 0 = [0 : 1], is trivial. This C∗–action induces a C∗–action on M0,0(MΛ, kAi0).We have

M0,0(MΛ, kAi0)C∗

= F0 ∪ F∞

where F0 = F∞ = M0,0(CP1, k). F0 and F∞ consist of maps to the fibers π−1(0) and π−1(∞),

respectively.

By virtual localization [18],∫

[M0,0(MΛ,kAi0)]vir

1 =

∫

[F0]vir

1

eC∗(NvirF0

)+

∫

[F∞]vir

1

eC∗(NvirF∞

).

Suppose that u : C → CP1 represents a point ξ in F0. As explained in [26], then the tangentspace T 1

ξ and the obstruction space T 2ξ at the moduli point ξ ∈ M0,0(MΛ, kAi0) fit in the tangent-

obstruction exact sequence:

0 → Ext0(ΩC ,OC) →H0(C, u∗TMΛ) → T 1ξ

→ Ext1(ΩC ,OC) → H1(C, u∗TMΛ) → T 2ξ → 0

where

• Ext0(ΩC ,OC) is the space of infinitesimal automorphisms of the domain C,

• Ext1(ΩC ,OC) is the space of infinitesimal deformations of the domain C,

• H0(C, u∗TMΛ) is the space of infinitesimal deformations of the map u, and

• H1(C, u∗TMΛ) is the space of obstructions to deforming the map u.

Equivalently,

0 → Ext0(ΩC ,OC) →H0(C, u∗TCP1)⊕ Lu → T 1ξ

→ Ext1(ΩC ,OC) → H1(C, u∗O(−2)) → T 2ξ → 0 .

So∫

[F0]vir

1

eC∗(NvirF0

)=

∫

M0,0(CP1,k)

e(Ek)

e(Lu)=

∫

M0,0(CP1,k)

0

u= 0 .

Suppose that u : C → CP1 represents a point ξ in F∞. The tangent space T 1ξ and the obstruction

space T 2ξ for M0,0(MΛ, kAi0) at this point fits in the following long exact sequence

0 → Ext0(ΩC ,OC) →H0(C, u∗TMΛ) → T 1ξ

→ Ext1(ΩC ,OC) → H1(C, u∗TMΛ) → T 2ξ → 0 ,

or equivalently,

0 → Ext0(ΩC ,OC) →H0(C, u∗TCP1)⊕ L−u → T 1ξ

→ Ext1(ΩC ,OC) → H1(C, u∗O(−2))⊗ L−u → T 2ξ → 0 .


So∫

[F∞]vir

1

eC∗(NvirF∞

)=

∫

M0,0(CP1,k)

e(Ek ⊗ L−u)

u

where

e(Ek ⊗ L−u) =

2k−1∑

i=0

(−u)ic2k−1−i(Ek) = −ue(Vk) +

2k−1∑

i=2

c2k−1−i(Ek)(−u)i

by (8). So∫

[F∞]vir

1

eC∗(NvirF∞

)= −

∫

M0,0(CP1,k)

e(Vk) = −1

k3.

Therefore,

GW(kAi0 ) = −1

k3.

This finishes the proof of the first two cases in the proposition.

For the third and fourth cases we use a similar argument, but now we consider genus 0, 1–pointed,stable maps u : C → CP1 × CP1 of degree k to the first sphere and of degree l to the secondsphere. We denote this moduli space by M0,1(CP

1 × CP1, (k, l)). This is a Deligne–Mumfordstack of dimension 2k + 2l.

In this case we should consider the evaluation map

ev : M0,2(CP1 × CP1, (k, l)) → CP1 × CP1

and the forgetful map

p : M0,2(CP1 × CP1, (k, l)) → M0,1(CP

1 × CP1, (k, l)),

given by forgetting the second marked point. We have the following short exact sequence overCP1 × CP1:

0 →OCP1(−2)×OCP1(−2) →

→ (OCP1(−1)⊕OCP1(−1))× (OCP1(−1)⊕OCP1(−1)) → OCP1 ×OCP1 → 0 .

Given a genus 0, 1–pointed, degree (k, l) stable map u : C → CP1 × CP1, we have a short exactsequence of vector bundles over the domain C:

0 → u∗(OCP1(−2)×OCP1(−2)) →

→ u∗((OCP1(−1)⊕OCP1(−1))× (OCP1(−1)⊕OCP1(−1))) → u∗(OCP1 ×OCP1) → 0

where u∗(OCP1(−2)× OCP1(−2)) and u∗((OCP1(−1)⊕OCP1(−1)) × (OCP1(−1)⊕OCP1(−1))) aredegree −2k − 2l bundles on C. In a similar way to the previous case we define the bundles

Ek,l := p∗ev∗(OCP1(−2)×OCP1(−2)) and

Vk,l := p∗ev∗((OCP1(−1)⊕OCP1(−1))× (OCP1(−1)⊕OCP1(−1)))

overM0,1(CP1×CP1, (k, l)). Now Ek,l and Vk,l have rank 2k+2l−2 and 2k+2l−4, respectively. In

this case we have the following short exact sequence of vector bundles over M0,1(CP1×CP1, (k, l))

0 → OM → Ek,l → Vk,l → 0 ,

where again OM is the trivial bundle. So the Euler classes now are given by

e(Ek,l) = c2k+2l−2(Ek,l) = 0, e(Vk,l) = c2k+2l−4(Vk,l) = c2k+2l−4(Ek,l) .

We consider the same C∗–action defined as before by t · [x, y] = [tx, y] on CP1, with the fixed points0 = [0 : 1] and ∞ = [1 : 0], and its unique lifting to a C∗–action to MΛ such that the action onthe fiber over 0, π−1(0), is trivial. This action induces a C∗–action on M0,0(MΛ, kAi0 + Ai0+1).Analogously to the first case we have

M0,0(MΛ, kAi0 + lAi0+1)C∗

= F0 ∪ F∞


where F0 and F∞ consist of maps to the fibers π−1(0) and π−1(∞), respectively, but these mapsnow are given by F0 = F∞ = M0,1(CP

1 × CP1, (k, l)).

Again, by virtual localization [18],

∫

[M0,0(MΛ,kAi0+lAi0+1)]vir

1 =

∫

[F0]vir

1

eC∗(NvirF0

)+

∫

[F∞]vir

1

eC∗(NvirF∞

).

However, in this case, since dimM0,1(CP1 × CP1, (k, l)) = 2k + 2l and both Euler classes e(Ek,l)

and e(Ek,l ⊗ L−u) have smaller degree than this dimension we conclude that both integrals

∫

[F0]vir

1

eC∗(NvirF0

)=

∫

M0,1(CP1×CP1,(k,l))

e(Ek)

e(Lu)and

∫

[F∞]vir

1

eC∗(NvirF∞

)=

∫

M0,1(CP1×CP1,(k,l))

e(Ek ⊗ L−u)

u

vanish, unless k = l when we can reduce the calculation of the Gromov–Witten invariant to thefirst case by considering curves in class k(Ai0 +Ai0+1).

4.4. Proof of Theorem 4.6. We are now ready to prove Theorem 4.6 which will conclude theproof of Theorem 4.4.

We use an induction argument. First note that using Spielberg’s results in [34] or [35] we caneasily compute the value of the three Gromov–Witten invariants of Theorem 4.6 for the basecases k = 0, 1 and l = 0, 1 (see Lemma 3.2 for the computation of some of these invariants).Now we assume they hold for k − 1 and l − 1 and we will prove they also hold for k and l.Because [M ] · [σ] = 1 for any section class σ, the divisor axiom for Gromov–Witten invariants (see

Proposition 2.2) implies that the 1–point invariant GWMΛ

Amax+B,1(c) equals the more usual 3–point

invariant GWMΛ

Amax+B,3([M ], [M ], c). It follows easily, from the fan description of the manifold MΛ

in Section 3, that PD([M ]) = Zb. Therefore we need to compute the Gromov–Witten invariants

GWMΛ

Amax+B,3(Zb, Zb, Z)

where Z ∈ H4(M ;Z). Indeed the degrees must satisfy the equation

2 degZb + degZ = 2N + 2c1(Bk,l) + 2m− 6

where dimMΛ = 2N = 6, degZb = 2, c1(Bk,l) = 1 and m = 3 is the number of marked points.

The main idea of the proof is to use the axioms of the Gromov–Witten invariants and, in particular,the splitting axiom to compute the following three invariants:

GWMΛ

Bk,l,4(Z1, Zb, Zb, Z1; [pt]), GWMΛ

Bk,l,4(Z1, Zb, Zb, Zn; [pt]), and GWMΛ

Bk,l,4(Zn, Zb, Zb, Zn; [pt]).

Using the partition S0 = 1, 2, S1 = 3, 4 of the index set 1, 2, 3, 4 and setting

A0 = Amax + k0Ai0 + l0Ai0+1

A1 = k1Ai0 + l1Ai0+1or

A0 = k0Ai0 + l0Ai0+1

A1 = Amax + k1Ai0 + l1Ai0+1


one can apply the splitting axiom (see Proposition 2.2) to the following invariant:

GWMΛ

Bk,l,4(Z1, Zb, Zb, Z1; [pt]) =

∑

A0+A1=Bk,l

GWMΛ

A0,3(Z1, Zb, eν) g

νµ GWMΛ

A1,3(eµ, Zb, Z1)

= GWMΛ

Amax,3(Z1, Zb, eν) g

νµ GWMΛ

kAi0+lAi0+1,3

(eµ, Zb, Z1)

+ GWMΛ

Bk,l,3(Z1, Zb, eν) g

νµ GWMΛ

0,3 (eµ, Zb, Z1)

+∑

1≤k0+l0≤k+l−1

GWMΛ

Bk0,l0,3(Z1, Zb, eν) g

νµ GWMΛ

k1Ai0+l1Ai0+1,3

(eµ, Zb, Z1)

+ GWMΛ

0,3 (Z1, Zb, eν) gνµ GWMΛ

Bk,l,3(eµ, Zb, Z1)

+ GWMΛ

kAi0+lAi0+1,3

(Z1, Zb, eν) gνµ GWMΛ

Amax,3(eµ, Zb, Z1)

+∑

1≤k0+l0≤k+l−1

GWMΛ

k0Ai0+l0Ai0+1,3

(Z1, Zb, eν) gνµ GWMΛ

Bk1,l1t,3(eµ, Zb, Z1)

= 2GWMΛ

Bk,l,3(Z1, Zb, eν) g

νµ GWMΛ

0,3 (eµ, Zb, Z1)(9)

where the last equality follows from the divisor axiom (see Proposition 2.2) together with the factthat

∫

Ai0

Zb =∫

Ai0+1Zb = 0. Moreover,

∫

Bk,lZ1 = Z1(Amax+kAi0+ lAi0+1) = k−2l,

∫

Bk,lZb = 1

and by the zero axiom (see Proposition 2.2):

GWMΛ

0,3 (eµ, Zb, Z1) =

∫

MΛ

eµ ∪ Zb ∪ Z1 =

−2 if eµ = Z1,1 if eµ = Zn,0 otherwise.

So one gets that (9) leads to

(10) GWMΛ

Bk,l,4(Z1, Zb, Zb, Z1; [pt]) = 2(k − 2l)

∑

ν

|eν |=4

GWMΛ

Bk,l,1(eν) (g

νn − 2gν1) .

Remark 4.10. From the diagram of Figure 3 it follows that GWMΛ

Bk,l,1(eν) is non-zero only if

the class eν is Poincare dual to one of the following homology classes: Ai0−1, Ai0 , Ai0+1, Ai0+2,Amax, or Amax+Ai0+1, since the marked point should lie in one of the following cones: σ2, σ3, σ4,σ7, σ8, or σ9. Their Poincare duals are the classes Zn−1Zb, ZnZb, Z1Zb, Z2Zb, Z1Zn, and Z1Z2,respectively. Note that the only ones that belong to the basis of the cohomology are Z1Zb, ZnZb,and Z1Zn. Therefore, at most three terms appear in the summation in Equation (10) above andthe coefficients can be computed thanks Lemma 3.1.

In the case of Equation (10), we end up with

(11) GWMΛ

Bk,l,4(Z1, Zb, Zb, Z1; [pt]) = 2(k − 2l)GWMΛ

Bk,l,1(Z1Zb) .

On the other hand, using the partition S0 = 1, 4, S1 = 2, 3, the same Gromov–Witten invariantis given by the following expression

(12) GWMΛ

Bk,l,4(Z1, Zb, Zb, Z1; [pt]) = GWMΛ

0,3 (Z1, Z1, eν) gνµ GWMΛ

Bk,l,3(eµ, Zb, Zb)

+∑

k0,l01≤k0+l0≤k+l

GWMΛ

k0Ai0+l0Ai0+1,3

(Z1, Z1, eν) gνµ GWMΛ

Bk1,l1,3(eµ, Zb, Zb).

Since (again by the zero axiom):

GWMΛ

0,3 (Z1, Z1, eν) =

∫

MΛ

Z21 ∪ eµ =

4 if eµ = Z1,−2 if eµ = Zb,0 otherwise

and

∫

k0Ai0+l0Ai0+1

eν =

k0 − 2l0 if eν = Z1,l0 − 2k0 if eν = Zn,

0 otherwise


it follows from the divisor axiom that (12) is equal to

(13) =∑

µ

|eµ|=4

(4g1µ − 2gbµ)GWMΛ

Bk,l,1(eµ)

+∑

µ

∑

1≤k0+l0≤k+l

(k0− 2l0)2GW(k0Ai0 + l0Ai0+1) ((k0 − 2l0)g

1µ+(l0− 2k0)gnµ)GWMΛ

Bk1,l1,1(eµ)

where GW(k0Ai0 + l0Ai0+1) denotes the 0–point invariant in class k0Ai0 + l0Ai0+1. These werecomputed in Proposition 4.8. In order to simplify the expression, we will denote them by GW0.We will also omit the index 1 indicating the number of marked points for the various 1–pointGromov–Witten invariants appearing in what remains of the proof.

In view of Remark 4.10 above, equation (13) actually reads

=− 2GWMΛ

Bk,l(Z1Zb)− 2GWMΛ

Bk,l(Z1Zn)

+∑

1≤k0+l0≤k+l

(k0 − 2l0)2GW0

[

((k0 − 2l0)g1µ + (l0 − 2k0)g

nµ)GWMΛ

Bk1,l1(eµ)

]

.

Then, using Proposition 4.8, we separate the summation on k0 and l0 in three distinct summations:k0 = 0, l0 = 0, and k0 = l0:

= −2GWMΛ

Bk,l(Z1Zb) +

l∑

l0=1

4l30

(

−1

l30

)

(gnµ − 2g1µ)GWMΛ

Bk1,l1(eµ)

− 2GWMΛ

Bk,l(Z1Zn) +

k∑

k0=1

k30

(

−1

k30

)

(g1µ − 2gnµ)GWMΛ

Bk1,l1(eµ)

+

min(k,l)∑

k0=1

k30

(

1

k30

)

(gnµ + g1µ)GWMΛ

Bk1,l1(eµ).

Using again the coefficients of G−1 given by Lemma 3.1 one can simplify it in order to obtain thefollowing expression

(14) = −2GWMΛ


Bk,l(Z1Zn)− 4

l∑

l0=1

GWMΛ

Bk,l1(Z1Zb)

−k∑

k0=1

GWMΛ

Bk1,l(ZnZb)−

min(k,l)∑

k0=1

[

GWMΛ

Bk1,l1(Z1Zb) + GWMΛ

Bk1,l1(ZnZb)

]

.

Applying the induction hypotheses we can simplify even further this expression. However we needto consider two different cases:

(a) If k ≥ l then (14) is equal to

=− 2GWMΛ


Bk,l(Z1Zn)− 4l−

l−1∑

k1=0

(−1)−k−1∑

k1=l

(−2)−l∑

k0=1

(−2 + 1)

=− 2GWMΛ


Bk,l(Z1Zn)− 4l+ 2k.

(15)

Finally, setting (11)=(15) and using Lemma 3.1 one more time one obtains the followingequation

2(k − 2l)GWMΛ

Bk,l(Z1Zb) + 2GWMΛ

Bk,l(Z1Zb) + 2GWMΛ

Bk,l(Z1Zn) = 2k − 4l.

That is, for general k, l ∈ N we have the following equation

(16) (k − 2l+ 1)GWMΛ

Bk,l(Z1Zb) + GWMΛ

Bk,l(Z1Zn) = k − 2l.


(b) If k < l then (14) is equal to

= −2GWMΛ


Bk,l(Z1Zn)− 4

k∑

l1=0

1− 4

l−1∑

l1=k+1

2−k∑

k0=1

(−1)−k∑

k0=1

(−1 + 2)

= −2GWMΛ


Bk,l(Z1Zn) + 4k − 8l + 4.

(17)

Setting (10)=(17) we now get the following equation

(18) (k − 2l+ 1)GWMΛ

Bk,l(Z1Zb) + GWMΛ

Bk,l(Z1Zn) = 2k − 4l+ 2.

The two remaining Gromov–Witten invariants, namely,

GWMΛ

Bk,l,4(Z1, Zb, Zb, Zn; [pt]) and GWMΛ

Bk,l,4(Zn, Zb, Zb, Zn; [pt])

can be computed in a similar way. Since the method is exactly the same, that is, to apply thesplitting axiom together with an induction argument, we leave the computation to the interestedreader and we simply give the four resulting equations. The first invariant produces the followingtwo equations:

(a) If k ≥ l then

(19) (l − 2k − 1)GWMΛ

Bk,l(Z1Zb) + (k − 2l)GWMΛ

Bk,l(ZnZb)−GWMΛ

Bk,l(Z1Zn) = 5l− 4k.

(b) If k < l then

(20) (l − 2k − 1)GWMΛ

Bk,l(Z1Zb) + (k − 2l)GWMΛ

Bk,l(ZnZb)−GWMΛ

Bk,l(Z1Zn) = 4l − 5k − 2.

The computation of GWMΛ

Bk,l,4(Zn, Zb, Zb, Zn; pt) gives two more equations:

(a) If k ≥ l then

(21) (2l − 4k − 1)GWMΛ

Bk,l(ZnZb) + 2GWMΛ

Bk,l(Z1Zn) = 8k − 4l.

(b) If k < l then

(22) (2l− 4k − 1)GWMΛ

Bk,l(ZnZb) + 2GWMΛ

Bk,l(Z1Zn) = 4k − 2l + 1.

In order to conclude the computation we consider two linear systems: one given by the equations(16), (19), (21), corresponding to the case k ≥ l and the other given by the equations (18), (20),(22) corresponding to the case k < l. The unknowns of these linear systems are the Gromov–

Witten invariants we are looking for, namely, GWMΛ

Bk,l(Z1Zb), GWMΛ

Bk,l(Z1Zn), and GWMΛ

Bk,l(ZnZb).

The unique solutions of these systems give us the desired result.

5. Applications and explicit examples

In this section we show some applications of our results and illustrate their relevance with someparticular examples. More precisely, in Section 5.1 we show how to obtain an expression forthe Landau–Ginzburg superpotential from the moment polytope of a NEF toric 4–manifold. InSection 5.2 we compute the Seidel elements, the quantum homology ring and the Landau–Ginzburgsuperpotential for two examples of NEF toric surfaces, namely CP2 blown–up at 4 or 5 points.Finally, in Section 5.3 we show how we can use the Fano and NEF computations to obtain explicitexpressions of Seidel elements for some particular non-NEF manifolds, namely the Hirzebruchsurfaces F2k or F2k−1 with k ≥ 2. As an example, we compute them explicitly for F4.


5.1. The Landau–Ginzburg potential. Let us recall some notation. Consider a torus T withLie algebra t and lattice tZ. Let (M,ω) be a smooth toric 2m–manifold with moment map Φ : M →t∗ and with moment polytope P . Let D1, . . . , Dn be the facets of P , Ai = [Φ−1(Di)] ∈ H2(M ;Z)the homology class of the pre-image of the facetDi, and let η1, . . . , ηn denote the outward primitiveintegral normal vectors to the facets (note that only in this section we use upper indices to labeleach normal vector while we use lower indices to label the components of each vector). Themoment polytope can be given by

P = x ∈ t∗|〈x, ηj〉 ≤ κj, for j = 1, . . . , n

where κj ∈ R. Given any face of P , let Dj1 , . . . , Djℓ be the facets that intersect to form this face.The dual cone is the set of elements in t which can be written as a positive linear combination ofηj1 , . . . , ηjℓ . As explained in [29, Section 5.1], every vector in t lies in the dual cone of a uniqueface of P . Therefore, given any subset I = i1, . . . , ik ⊆ 1, . . . n there is a unique face of P sothat ηi1 + . . .+ ηik lies in its dual cone. Let Dj1 , . . . , Djℓ be the facets that intersect to form thisunique face. Then there exist unique positive integers c1, . . . , cℓ so that

ηi1 + . . .+ ηik − c1ηj1 − . . .− cℓη

jℓ = 0.

Batyrev showed that if I is primitive the sets I and J = j1, . . . , jℓ are disjoint. Moreover, ifβI ∈ H2(M ;Z) is the class corresponding to the above relation, then by (2) one obtains

c1(βI) = k − c1 − . . .− cℓ ,(23)

ω(βI) = ηi1(Di1) + . . .+ ηik(Dik)− c1ηj1(Dj1)− . . .− cℓη

jℓ(Djℓ)

= κi1 + . . .+ κik − c1κj1 − . . .− cℓκjℓ .(24)

For each ηi define Φηi

: M → R to be the composition of the moment map Φ : M → t∗ with the

linear functional ηi ∈ t. Thus Φηi

is the moment map for the circle action Λi with Φmax = Φ−1(Di).

Assume S∗(Λi) = yi ⊗ q−1t−ηi(Di) ∈ QHev(M, Π)×. In [29] the authors show the following result.

Proposition 5.1. Let QH∗(M) denote the small quantum cohomology of the toric manifold(M,ω). The map Θ which sends Zi to the Poincare dual of Φ−1(Di) induces an isomorphism

Q[Z1, . . . , Zn]⊗ Π/(Lin(P ) + SRY (P )) ∼= QH∗(M),

where the ideal Lin(P ) is generated by the linear relations

Lin(P ) =⟨

∑

(x, ηj)Zj | x ∈ t∗Z

⟩

and the ideal SRY (P ) is given by

SRY (P ) =⟨

Yi1 . . . Yik − Y c1j1

. . . Y cℓjℓ

⊗ qc1(βI)tω(βI)| I = i1, . . . , ik is primitive⟩

,

where

(25) Yi = Zi + higher order terms,

is a lift of the Seidel element yi in Q[Z1, . . . , Zn]⊗ Π, such that Θ(Yi) = yi.

As McDuff and Tolman explain in [29], in general it is not possible to find Yi without priorknowledge of the ring structure on QH∗(M) but, in special cases, we can indeed describe Yi. Inthe Fano case the higher terms vanish and we may take Yi = Zi. In the NEF case there mightbe higher order terms in the Seidel elements yi, however, from [29, Theorem 1.10] we know thatthe lifts Yi of yi are determined by some linear combination of the Zi which is unique up to theadditive relations Lin(P ) (see [29, Example 5.4] for more details).


5.1.1. Fano case. In this case the Landau–Ginzburg superpotential is given by

W =

n∑

j=1

zηj

tκj

where for η = (η1, . . . , ηm) ∈ Zm the term zη is the monomial zη1

1 . . . zηmm .

We now recall part of the results obtained by Ostrover and Tyomkin in the Fano case, see [30](or [6] for a summary). Their main motivation to study properties of the quantum homologyalgebra, namely semi–simplicity or the existence of a field as a direct summand, was previousworks by Entov and Polterovich on Calabi quasi-morphisms and symplectic quasi-states (see [13]and references therein), in which the algebraic structure of the quantum homology plays a keyrole. Much progress was made since then in this area by many authors, namely McDuff, Oh,Usher, and Fukaya, Oh, Ohta, and Ono. One of the main results by Ostrover and Tyomkin is thefollowing.

Theorem 5.2 ([30]). If (M,ω) is a symplectic Fano manifold, then

QH∗(M,ω) ∼= Π[z±1 , . . . , z±m]/JW as Π–algebras

and in particular

QH0(M,ω) ∼= Πuniv[z±1 , . . . , z±m]/JW as Πuniv–algebras

where JW is the ideal generated by all partial derivatives of W .

The main idea of the proof in [30] of this theorem is to consider the natural homomorphism

Ψ : Q[Z1, . . . , Zn]⊗ Π → Π[z±1 , . . . , z±m]

such that SRY (P ) is in the kernel of Ψ and the image of the additive relations give the ideal JW .In this case the homomorphism is defined by

Ψ(Zj) = qzηj

tκj

and it is easy to see that Ψ satisfies the desired properties. Indeed, as we saw above, in the Fanocase we may set Yi = Zi hence

SRY (P ) =⟨

Zi1 . . . Zik − Zc1j1

. . . Zcℓjℓ

⊗ qc1(βI)tω(βI) | I = i1, . . . , ik is primitive⟩

and

Ψ(Zi1 . . . Zik − Zc1j1

. . . Zcℓjℓ

⊗ qc1(βI)tω(βI))

= qkzηi1. . . zη

iktκi1

+...+κik − qc1+...+cℓzc1ηj1. . . zcℓη

jℓtc1κj1

+...+cℓκjℓ ⊗ qc1(βI)tω(βI)

= qkzηi1+...+ηik

tκi1+...+κik − qkzη

i1+...+ηiktκi1

+...+κik = 0

by (23) and (24). Therefore the ideal SRY (P ) is in the kernel of Ψ.

The image of the additive relations is the following

Ψ

n∑

j=1

(x, ηj)Zj

= q

n∑

j=1

(x, ηj)zηj

tκj .

On the other hand, we have

qzi∂W

∂zi= qzi

n∑

j=1

ηji zηj1

1 . . . zηji−1

i . . . zηjn

n tκj = q

n∑

j=1

ηji zηj

tκj .

Note that if x = xi is the i–th vector of the canonical base in Rn then we have (x, ηj) = ηji andone obtains the desired result.


5.1.2. NEF case. In this subsection we give the explicit expression of the Landau–Ginzburg su-perpotential when M is a NEF 4–dimensional toric manifold for which at most 2 of the homologyclasses Ai = [Φ−1(Di)] of the pre-image of the facets Di have vanishing first Chern number. It isvery easy to see from the proof of the next proposition that the result generalizes to any numberof classes (corresponding to facets of the polytope) with Chern number zero, the expressions getmore complicated as we increase the number of such classes. Moreover, Theorem 5.2 still holdsfor these cases.

Proposition 5.3. If M is a NEF toric 4–manifold and Ai = [Φ−1(Di)] where Di is a facet of themoment polytope then the Landau–Ginzburg superpotential is given by the following expression:

(1) if c1 only vanishes on the class Ak then

W =

n∑

j=1

zηj

tκj + zηk

tκk+1+κk−1−κk ,

(2) if c1 only vanishes on the classes Ak−1 and Ak then

W =n∑

j=1

zηj

tκj + zηk

tκk+1+κk−1−κk + zηk−1

tκk+κk−2−κk−1

+ zηk

tκk+1+κk−2−κk−1 + zηk−1

tκk+1+κk−2−κk .

Proof. Case (1): in this case the Seidel elements are given by Theorem 4.5. We have

S(Λj) = Aj ⊗ qtκj if j 6= k − 1, k, k + 1,

S(Λk) = Ak ⊗ qtκk

1− t−ω(Ak),

S(Λk−1) = Ak−1 ⊗ qtκk−1 − Ak ⊗ qtκk−1−ω(Ak)

1− t−ω(Ak),

S(Λk+1) = Ak+1 ⊗ qtκk+1 −Ak ⊗ qtκk+1−ω(Ak)

1− t−ω(Ak).

If S∗ denotes the Seidel morphism in cohomology then we have

S∗(Λj) = Zj ⊗ q−1t−κj if j 6= k − 1, k, k + 1,

S∗(Λk) = Zk ⊗ q−1 t−κk

1− tω(Ak),

S∗(Λk−1) =

(

Zk−1 − Zk ⊗tω(Ak)

1− tω(Ak)

)

⊗ q−1t−κk−1 ,

S∗(Λk+1) =

(

Zk+1 − Zk ⊗tω(Ak)

1− tω(Ak)

)

⊗ q−1t−κk+1 .

Thus in equation (25) we may take

Yj = Zj if j 6= k − 1, k, k + 1, Yk = Zk ⊗1

1− tω(Ak),

Yk−1 = Zk−1 − Zk ⊗tω(Ak)

1− tω(Ak), Yk+1 = Zk+1 − Zk ⊗

tω(Ak)

1− tω(Ak)

where ω(Ak) = κk+1 + κk−1 − 2κk. In this case, the definition of the homomorphism Ψ is suchthat

(26) ∀ 1 ≤ j ≤ n, Ψ(Yj) = qzηj

tκj


so one obtains

Ψ(Zj) = qzηj

tκj if j 6= k − 1, k, k + 1 ,

Ψ(Zk) = qzηk

tκk(1− tω(Ak)) = qzηk

tκk − qzηk


Ψ(Zk−1) = qzηk−1

tκk−1 + qzηk


Ψ(Zk+1) = qzηk+1

tκk+1 + qzηk


It is clear, by definition of Ψ and the proof in the Fano case that SRY (P ) is in the kernel of thehomomorphism. Computing the image of the additive relations gives

Ψ(

n∑

j=1

(x, ηj)Zj

)

= q

n∑

j=1

(x, ηj)zηj

tκj − q(x, ηk)zηk

tκk+1+κk−1−κk

+ q(x, ηk−1)zηk

tκk+1+κk−1−κk + q(x, ηk+1)zηk


In order to obtain the derivatives of the potential we need

(x, ηk−1) + (x, ηk+1)− (x, ηk) = (x, ηk), that is, ηk−1 + ηk+1 = 2ηk,

which holds, if dimM = 4 and c1(Ak) = 0, as noticed already in Section 3.1 (it follows from [2,Lemma 6.3]).

Case (2): In this case Theorem 4.5 gives the following:

Yj = Zj if j 6= k − 2, k − 1, k, k + 1 ,

Yk−2 = Zk−2 − Zk−1 ⊗tω(Ak−1)

1− tω(Ak−1)− Zk−1 ⊗

tω(Ak−1)+ω(Ak)

(1 − tω(Ak−1))(1− tω(Ak−1)+ω(Ak))

+ Zk ⊗tω(Ak−1)+2ω(Ak)

(1 − tω(Ak))(1− tω(Ak−1)+ω(Ak)),

Yk−1 =

(

Zk−1 ⊗1

1− tω(Ak−1)− Zk ⊗

tω(Ak)

1− tω(Ak)

)

1

1− tω(Ak−1)+ω(Ak),

Yk =

(

Zk ⊗1

1− tω(Ak)− Zk−1 ⊗

tω(Ak−1)

1− tω(Ak−1)

)

1

1− tω(Ak−1)+ω(Ak),

Yk+1 = Zk+1 − Zk ⊗tω(Ak)

1− tω(Ak)− Zk ⊗

tω(Ak−1)+ω(Ak)

(1− tω(Ak))(1− tω(Ak−1)+ω(Ak))

+ Zk−1 ⊗tω(Ak)+2ω(Ak−1)

(1 − tω(Ak−1))(1 − tω(Ak−1)+ω(Ak)).

Therefore, using the same argument, that is, if we define Ψ such that it satisfies (26) then weobtain

Ψ(Zj) = qzηj

tκj if j 6= k − 2, k − 1, k, k + 1,

Ψ(Zk) = qzηk

tκk(1− tκk+1,k+κk−1,k) + qzηk−1

tκk−2(tκk,k−1 − tκk+1,k),


tκk−1(1 − tκk,k−1+κk−2,k−1) + qzηk

tκk+1(tκk−1,k − tκk−2,k−1),


tκk−2 + qzηk

tκk+1,k−1+κk−2 + qzηk−1

tκk−2(tκk,k−1 + tκk+1,k),

Ψ(Zk+1) = qzηk+1

tκk+1 + qzηk−1

tκk+1,k+κk−2 + qzηk

tκk+1(tκk−1,k + tκk−2,k−1)

where κi,j = κi − κj. Again, it is clear that SRY (P ) is in the kernel of the homomorphismand it is not hard to check that the image of the additive relations gives the derivatives of thesuperpotential, under the assumptions that ηk−1 + ηk+1 = 2ηk and ηk−2 + ηk = 2ηk−1.


5.2. NEF examples: The case of a blow–up of CP2 at 4 or 5 points. In this section, as anapplication of our results, we compute explicitly the small quantum cohomology (and homology)of the manifold obtained from CP2 by performing 4– and 5–blow-ups. Note that these manifoldsadmit NEF structures, but do not admit Fano complex structures. Since the computations aresimilar, we show the full computations for the former case and only give the final result for thelatter case. As already noticed in Example 1.2 the symplectic manifold Xℓ obtained from CP2 byperforming ℓ ≥ 2 blow-ups is symplectomorphic to the (ℓ− 1)–point blow-up of S2 × S2 endowedwith the symplectic form ωµ for which the symplectic area of the first factor is µ and the area of thesecond factor is 1 (see [3, Section 2.1] for more details). Let ci, i = 1, . . . , ℓ be the capacities of theblow-ups. Let B, F ∈ H2(X4) be the homology classes defined by B = [S2 × p], F = [p × S2]and let Ei ∈ H2(X4) be the exceptional class corresponding to the blow-up of capacity ci. ConsiderX4 endowed with the standard action of the torus T = S1 × S1 for which the moment polytope isgiven by

P =

(x1, x2) ∈ R2 | 0 ≤ x2 ≤ µ, x2 + x1 ≤ µ− c3, −1 ≤ x1 ≤ 0, c1 ≤ x2 − x1 ≤ µ+ 1− c2

so the primitive outward normals to P are as follows:

η1 = (0, 1), η2 = (1, 1), η3 = (1, 0), η4 = (1,−1), η5 = (0,−1), η6 = (−1, 0), and η7 = (−1, 1).

The normalised moment map Φ : X4 → R2 is given by

Φ(z1, . . . , z7) =(

−1

2|z3|

2 + ǫ1,−1

2|z1|

2 + µ− ǫ2

)

,

where

ǫ1 =c31 + 3c22 − c32 + c33 − 3µ

3(c21 + c22 + c23 − 2µ)and ǫ2 =

c31 − c32 − c33 + 3c22µ+ 3c23µ− 3µ2

3(c21 + c22 + c23 − 2µ).

Moreover, the homology classes Ai = [Φ−1(Di)] of the pre-images of the corresponding facets Di

are: A1 = F −E2 − E3, A2 = E3, A3 = B − E1 − E3, A4 = E1, A5 = F −E1, A6 = B −E2, andA7 = E2. Let Λi be the circle action associated to ηi. Since the complex structure on X4 is NEFand T–invariant, it follows from Theorem 4.5 that the Seidel elements associated to these actionsare given by the following expressions

S(Λ1) = (F − E2 − E3)⊗ qtµ−ǫ2

1− tc2+c3−1,

S(Λ2) = E3 ⊗ qtµ−c3+ǫ1−ǫ2 − (F − E2 − E3)⊗ qtµ+c2−1+ǫ1−ǫ2

1− tc2+c3−1− (B − E1 − E3)⊗ q

tc1+ǫ1−ǫ2

1− tc1+c3−µ,

S(Λ3) = (B − E1 − E3)⊗ qtǫ1

1− tc1+c3−µ,

S(Λ4) = E1 ⊗ qtǫ1+ǫ2−c1 − (B − E1 − E3)⊗ qtǫ1+ǫ2+c3−µ

1− tc1+c3−µ,

S(Λ5) = (F − E1)⊗ qtǫ2 , S(Λ6) = (B − E2)⊗ qt1−ǫ1 ,

S(Λ7) = E2 ⊗ qtµ+1−c2−ǫ1−ǫ2 − (F − E2 − E3)⊗ qtµ+c3−ǫ1−ǫ2

1− tc2+c3−1.

Therefore we have

S∗(Λ1) = Z1 ⊗ q−1 tǫ2−µ

1− t1−c2−c3, S∗(Λ3) = Z3 ⊗ q−1 t−ǫ1

1− tµ−c1−c3,

S∗(Λ2) = Z2 ⊗ q−1tc3−µ−ǫ1+ǫ2 − Z1 ⊗ q−1 t1−µ−c2−ǫ1+ǫ2

1− t1−c2−c3− Z3 ⊗ q−1 tǫ2−ǫ1−c1

1− tµ−c1−c3,

S∗(Λ4) = Z4 ⊗ q−1tc1−ǫ1−ǫ2 − Z3 ⊗ q−1 tµ−c3−ǫ1−ǫ2

1− tµ−c1−c3,

S∗(Λ5) = Z5 ⊗ q−1t−ǫ2 , S∗(Λ6) = Z6 ⊗ q−1tǫ1−1,

S∗(Λ7) = Z7 ⊗ q−1tc2−µ−1+ǫ1+ǫ2 − Z1 ⊗ q−1 tǫ1+ǫ2−µ−c3

1− t1−c2−c3.


Thus in equation (25) we may take

Y1 = Z1 ⊗1

1− t1−c2−c3, Y2 = Z2 − Z1 ⊗

t1−c2−c3

1− t1−c2−c3− Z3 ⊗

tµ−c1−c3

1− tµ−c1−c3,

Y3 = Z3 ⊗1

1− tµ−c1−c3, Y4 = Z4 − Z3 ⊗

tµ−c1−c3

1− tµ−c1−c3, Y5 = Z5,

Y6 = Z6, and Y7 = Z7 − Z1 ⊗t1−c2−c3

1− t1−c2−c3.

There are fourteen primitive sets: 1, 3, 1, 4, 1, 5, 1, 6, 2, 4, 2, 5, 2, 6, 2, 7, 3, 5,3, 6, 3, 7, 4, 6, 4, 7, 5, 7. Let t2,3 = 1− t1−c2−c3 and t1,3 = 1− tµ−c1−c3 . The correspond-ing multiplicative relations for QH∗(X4), that is, the generators of the ideal SRY (P ) defined inProposition 5.1, can be written as follows

Z1Z3 = Z2 ⊗ qtc3t2,3t1,3 − Z1 ⊗ qt1−c2t1,3 − Z3 ⊗ qtµ−c1t2,3,

Z1Z4t1,3 = Z1Z3 ⊗ tµ−c1−c3 + Z3 ⊗ qtµ−c1t2,3,

Z1Z5 = 1⊗ q2tµt2,3,

Z1Z6 = Z7 ⊗ qtc2t2,3 − Z1 ⊗ qt1−c3 ,

Z2Z4t2,3t1,3 = Z3(Z2 + Z3 + Z4)⊗ tµ−c1−c3t2,3 + Z1Z4 ⊗ t1−c2−c3t1,3 − Z1Z3 ⊗ t1+µ−c1−c2−2c3 ,

Z2Z5t2,3t1,3 = Z1Z5 ⊗ t1−c2−c3t1,3 + Z3Z5 ⊗ tµ−c1−c3t2,3 + Z3 ⊗ qtµ−c3t2,3,

Z2Z6t2,3t1,3 = Z1Z6 ⊗ t1−c2−c3t1,3 + Z3Z6 ⊗ tµ−c1−c3t2,3 + Z1 ⊗ qt1−c3t1,3,

Z2Z7t2,3t1,3 = Z1(Z1 + Z2 + Z7)⊗ t1−c2−c3t1,3 + Z3Z7 ⊗ tµ−c1−c3t2,3 − Z1Z3 ⊗ t1+µ−c1−c2−2c3 ,

Z3Z5 = Z4 ⊗ qtc1t1,3 − Z3 ⊗ qtµ−c3 ,

Z3Z6 = 1⊗ q2tt1,3,

Z3Z7t2,3 = Z1Z3 ⊗ t1−c2−c3 + Z1 ⊗ qt1−c2t1,3,

Z4Z6t1,3 = Z5 ⊗ qt1−c1t1,3 + Z3Z6 ⊗ tµ−c1−c3 ,

Z4Z7t2,3t1,3 = Z1Z4 ⊗ t1−c2−c3t1,3 + Z3Z7 ⊗ tµ−c1−c3t2,3

− Z3Z1 ⊗ qt1+µ−c1−c2−2c3 + 1⊗ q2tµ+1−c1−c2t2,3t1,3,

Z5Z7t2,3 = Z1Z5 ⊗ t1−c2−c3 + Z6 ⊗ qtµ−c2t2,3.

(27)

where we should also take in account the additive relations Z6 = Z1 + 2Z2 + Z3 − Z5 and Z7 =−Z1 − Z2 + Z4 + Z5. It follows from Proposition 5.1 that QH∗(X4) is isomorphic as a ring toQ[Z1, . . . , Zn] ⊗ Π/I where I is the ideal generated by the relations above. We can describe theresult also in terms of homology. For that consider the homology classes Ai = [Φ−1(Di)] ∈ H2(X4).The classes Ai are additive generators of H2(X4) and multiplicative generators of QH∗(X4). Notethat for many applications, namely to the study of quasi-states or Calabi quasi-morphims, itis sufficient to know the graded component QH2m(M) of degree 2m of the quantum homologyalgebra QH∗(M) of a manifold M of dimension 2m, which is an algebra over the field Πuniv.Moreover QH4(X4) is generated, as a subring of QH∗(X4), by the elements qAi. These generatorsare Ei⊗ q, where i = 1, 2, 3, (F −E1)⊗ q, (B−E2)⊗ q, (F −E2−E3)⊗ q, and (B−E1−E3)⊗ q.In what follows in order to simplify notation we shall drop the sign ∗ for the quantum product.The multiplicative relations (27) translated to homology together with the additive relations givea complete description of the Πuniv–algebra QH4(X4). More precisely, we obtain

QH4(X4) ∼= Πuniv[u, v]/J

where u = (F − E2 − E3) ⊗ q(1 − tc2+c3−1)−1, v = (B − E1 − E3) ⊗ q(1 − tc1+c3−µ)−1, and J isthe ideal generated by the two following relations:

u2tµ(v + tc2−1)(1 + vtc3) = v(1 + vtc1), and v2t(u + tc1−µ)(1 + utc3) = u(1 + utc2).(28)


It follows from Proposition 5.3 (1) that the Landau–Ginzburg superpotential is given in thisexample by

W = z2tµ + z1z2t

µ−c3 + z1 + z1z−12 t−c1 + z−1

2 + z−11 t+ z−1

1 z2tµ+1−c2

+ z1tµ−c1−c3 + z2t

µ+1−c2−c3 .

Therefore we have∂W

∂z1= z2t

µ−c3 + 1 + z−12 t−c1 − z−2

1 t− z−21 z2t

µ+1−c2 + tµ−c1−c3 ,

∂W

∂z2= tµ + z1t

µ−c3 − z1z−22 t−c1 − z−2

2 + z−11 tµ+1−c2 + tµ+1−c2−c3 .

Passing to homology, simplifying the expressions and setting u = z−12 t−µ and v = z−1

1 we obtainrelations (28) again, as we wish.

Similar arguments give an explicit description of the quantum homology Πuniv–algebra QH4(X5).Moreover, we have

QH4(X5) ∼= Πuniv[u, v]/J

where again u = (F − E2 − E3)⊗ q(1 − tc2+c3−1)−1, v = (B − E1 − E3)⊗ q(1− tc1+c3−µ)−1 andJ is now the ideal generated by the two following relations:

u2tµ(v + tc2−1)(1 + vtc3) = (1 + vtc1)(v + tc4−1),

v2t(u+ tc1−µ)(1 + utc3) = (1 + utc2)(u+ tc4−µ).

In this case the Landau–Ginzburg superpotential is given by

W = z2tµ + z1z2t

µ−c3 + z1 + z1z−12 t−c1 + z−1

2 + z−11 z−1

2 t1−c4 + z−11 t

+ z−11 z2t

µ+1−c2 + z1tµ−c1−c3 + z2t

µ+1−c2−c3 + z−11 tµ+1−c2−c4 + z−1

2 t1−c1−c4 .

Remark 5.4. Note that these results agree with the results of Chan and Lau in [8]. Our NEFmanifolds X4 and X5 coincide with the surfaces X7 and X10, respectively, described in the Ap-pendix A of their paper. We obtain the same expressions for the potential after making a changeof variable. More precisely, if we replace z2 by z1z

−12 t−c1 , keep the same variable z1 and let

q1 = tµ−c1−c3 , q2 = tµ−c2 , q3 = tc2 , q4 = t1−c2−c3 and q5 = tc3 in the potential for X7, we obtainour expression for the potential of X4. Similarly, making the same change of variable for X10 andletting q1 = tµ−c1−c3 , q2 = tc4 , q3 = tµ−c2−c4 , q4 = tc2 , q5 = t1−c2−c3 and q6 = tc3 we see againthat the two expressions for the potential agree.

5.3. Non-NEF examples. Particularly interesting examples which are relevant for our studyare the Hirzebruch surfaces. We use the conventions and the description adopted in [3] for thesesurfaces.

We first focus on “even” Hirzebruch surfaces. Let T 4 ⊂ U(4) act in the standard way on C4.Given an integer k ≥ 0, the action of the subtorus T 2

2k := (2ks + t, t, s, s) is Hamiltonian withmoment map

(z1, . . . , z4) 7→ (2k|z1|2 + |z3|

2 + |z4|2, |z1|

2 + |z2|2).

The toric Hirzebruch surface Fµ2k, 0 ≤ k ≤ ℓ (where ℓ < µ ≤ ℓ + 1 and ℓ ∈ N) is defined as the

symplectic quotient C4//T 22k at the regular value (µ + k, 1) endowed with the residual action of

the torus T (2k) := (0, u, v, 0) ⊂ T 4. The image P (2k) of the moment map Φ2k is the convex hullof

(0, 0), (1, 0), (1, µ+ k), (0, µ− k).

This surface can be identified with the symplectic manifold M0µ = (S2 × S2, ω0

µ) where ω0µ is the

split symplectic form with area µ ≥ 1 for the first S2–factor, and with area 1 for the second factor.

The Kahler isometry group of Fµ2k is N(T 2

2k)/T22k where N(T 2

2k) is the normalizer of T 22k in

U(4). There is a natural isomorphism5 N(T0)/T20 ≃ SO(3) × SO(3), while for k ≥ 1, we have

5In the untwisted case, we assume µ > 1 so that the permutation of the two S2 factors is not an isometry.


N(T 22k)/T2k ≃ S1 × SO(3). The restrictions of these isomorphisms to the maximal tori are given

in coordinates by

(u, v) 7→ (−u, v) ∈ T (0) := S1 × S1 ⊂ SO(3)× SO(3), and

(u, v) 7→ (u, ku+ v) ∈ T (2k) := S1 × S1 ⊂ S1 × SO(3).

These identifications imply that the moment polytope associated to the maximal tori T (2k) =S1 × S1 is the image of P (2k) under the transformation C2k ∈ GL(2,Z) given by

C0 =

(

−1 00 1

)

and C2k =

(

1 0−k 1

)

.

Let x2k and y2k represent the circle actions whose moment maps are, respectively, the first and thesecond component of the moment map associated to T (2k). We will also denote by x2k, y2k thegenerators in π1(T (2k)). It follows from the classification of 4–dimensional Hamiltonian S1–spacesgiven by Karshon in [23] that the generators x2k, y2k satisfy the relations x2k = kx2 +(k− 1)x0

and y2k = kx0 + y0. Since F0 is Fano and F2 is NEF we can obtain from our results the Seidelelements associated to x0, y0, and x2. Note that for all k ≥ 2, F2k is non-NEF, that is, admitsspheres with negative first Chern number. However, our computations allow us to compute theSeidel elements associated to the circle actions of F2k.

In particular, we can give explicit expressions for the Seidel elements associated to F46. Since F0 is

Fano it is easy to check that the Seidel elements associated to the circle actions x0 and y0 are given

by S(x0) = B⊗ qt12 and S(y0) = F ⊗ qt

µ2 , where B = [S2×p], F = [p×S2] ∈ H2(F0) (see [29,

Example 5.7]). From this case we can also obtain the following products in the quantum homologyring: F ∗ F = 1 ⊗ q−2t−µ, B ∗ B = 1 ⊗ q−2t−1, F ∗ B = p and deduce the remaining productsfrom these ones. For the toric manifold F2 the moment polytope associated to the maximal toriT (2) (before normalizing the moment map) is

(x1, x2) ∈ R2 | 0 ≤ x1 ≤ 1, x2 + x1 ≥ 0, x2 − x1 ≤ µ− 1

so the primitive outward normals are

η1 = (1, 0), η2 = (−1,−1), η3 = (−1, 0), and η4 = (−1, 1).

Therefore the normalised moment map is given by

Φ(z1, z2, z3, z4) =

(

−1

2|z1|

2 +1

2− ǫ,−

1

2|z1|

2 −1

2|z4|

2 +µ+ 1

2

)

,

where ǫ = 16µ . If Λi denotes the circle action corresponding to ηi then Theorem 4.5 implies that

the Seidel elements associated to these actions are given by

S(Λ1) = (B + F )⊗ qt12−ǫ, S(Λ3) = (B − F )⊗ q

t12+ǫ

1− t1−µand

S(Λ2) = S(Λ4) = F ⊗ qtµ2+ǫ − (B − F )⊗ q

t1−µ2+ǫ

1− t1−µ.

Since x2 = Λ1, S(x2) = S(Λ1) and it follows that for the non-NEF toric manifold F4 the Seidelelements associated to the circle actions x4 and y4 are given by

S(x4) = S(x2)2 ∗ S(x0) = (B + 2F )⊗ qt

12−2ǫ +B ⊗ qt

32−µ−2ǫ,

S(y4) = S(x0)2 ∗ S(y0) = S(y0) = F ⊗ qt

µ2 ,

because S(x0)2 = 1. Recall that the moment polytope in the case of F4 is identified with the

region

(x1, x2) ∈ R2 | 0 ≤ x1 ≤ 1, x2 + 2x1 ≥ 0, x2 − 2x1 ≤ µ− 2

6Notice that this implies that the symplectic form is such that there exists a pseudo-holomorphic representativein the class B − 2F or, equivalently, µ > 2.


and the primitive outward normals are given by

η1 = (1, 0), η2 = (−2,−1), η3 = (−1, 0), and η4 = (−2, 1).

If Γi denotes the circle action corresponding to ηi then it is clear that Γ1 = x4, so

S(Γ1) = qt12−2ǫ ⊗ ((B + 2F ) +B ⊗ qt1−µ).

Since η1 + η3 = 0 it follows that S(Γ3) = S(x4)−1 =

(

S(x2)−1)2

∗ S(x0)−1. It is not hard to see

that

S(x2)−1 = (B − F )⊗ q

t12+ǫ

1− t1−µ

therefore we obtain

S(Γ3) =qt

12+2ǫ

(1− t1−µ)2⊗[

(B − 2F ) +B ⊗ qt1−µ]

.

Finally, since η4 = 2η3 + (0, 1), η2 = 2η3 + (0,−1), and S(y4) = S(y4)−1 it follows that S(Γ2) =

S(Γ4) = S(Γ3)2 ∗ S(y4), hence

S(Γ2) = S(Γ4) =qt

µ2+4ǫ

(1 − t1−µ)4[

F ⊗ (1− t1−µ)2 − 4t1−µ((B − 2F ) +B ⊗ qt1−µ)]

It follows that in equation (25) we may take

Y1 = Z1 + (Z3 + Z2 + Z4)⊗ tµ−1, Y3 =1

(1− tµ−1)2(Z3 + (Z3 + Z2 + Z4)⊗ tµ−1),

Y2 =1

(1− tµ−1)2(Z2 − 4tµ−1Y3), Y4 =

1

(1 − tµ−1)2(Z4 − 4tµ−1Y3).

Note that since the ring structure on the quantum homology is known we can check that indeedthe choice of the elements Yi satisfy the equations induced by the primitive relations, that is, theelements

Y1Y3 − 1⊗ q2t and Y2Y4 − (Y3)4 ⊗ q−2tµ−2

are generators of the ideal SRY . In order to have a potential W such that the isomorphismin Theorem 5.2 holds we need that the homomorphism Ψ, inducing the isomorphism, satisfiesequations (26). Recall that the generators of the ideal SRY should be in the kernel of Ψ and theimage of the additive relations give the derivatives of the potential.

Ψ(Y1) = qz1t ⇔ Ψ(Z1) + Ψ(Z2 + Z3 + Z4)tµ−1 = qz1t(29)

Ψ(Y2) = qz−21 y−1 ⇔ Ψ(Z2)− 4t1−µΨ(Y3) = qz−2

1 y−1(1− tµ−1)2(30)

Ψ(Y3) = qz−11 ⇔ Ψ(Z3) + Ψ(Z2 + Z3 + Z4)t

µ−1 = qz−11 (1− tµ−1)2(31)

Ψ(Y4) = qz−21 ytµ−2 ⇔ Ψ(Z4)− 4t1−µΨ(Y3) = qz−2

1 ytµ−2(1− tµ−1)2(32)

Since the additive relations are Z1 − Z3 − 2Z2 − 2Z4 = 0 and Z4 − Z2 = 0 it follows fromequations (29), (30), (31) and (32) that the derivatives of the potential W are given by thefollowing expressions:

qz1∂W

∂z1= Ψ(Z1)−Ψ(Z3)− 2Ψ(Z2)− 2Ψ(Z4)

= qz1t− qz−11 (1 − tµ−1)2 − 16qz−1

1 tµ−1 − 2(qz−21 z−1

2 + qz−21 z2t

µ−2)(1 − tµ−1)2,

qz2∂W

∂z2= Ψ(Z4)−Ψ(Z2) = (qz−2

1 z2tµ−2 − qz−2

1 z−12 )(1− tµ−1)2

Therefore the potential is given by

(33) W = z1t+ (z−11 + z−2

1 z−12 + z−2

1 z2tµ−2)(1 − tµ−1)2 + 16z−1

1 tµ−1.


Remark 5.5. In this non-NEF example we see that the number of terms corresponding to thequantum corrections in the Landau–Ginzburg superpotential is still finite. In the formalism of[8] and [9] the primitive rays of the fan (or the interior normal vectors of the polytope) are givenby v1 = (1, 0), v2 = (0, 1), v3 = (−1,−4), and v4 = (0,−1) and the polytope is defined by thefollowing inequalities

x1 ≥ 0, x2 ≥ 0, 4t1 + t2 − x1 − 4x2 ≥ 0 and t1 − x2 ≥ 0,

where the tl’s are positive numbers. Let ql = exp(−tl) be the Kahler parameters. Then thepotential is given by

W = z1(1− 2q1q2 + q21q22) + z2 +

q41q2z1z42

(1− 2q1q2 + q21q22) +

q1z2

(1 + 14q1q2 + q21q22).

In this expression z1 and z2 correspond to z−21 z−1

2 and z1t, respectively, in equation (33) whileq1 = t and q2 = tµ−2. Moreover, if [9, Conjecture 6.7] holds then we can obtain the open Gromov–Witten invariants of F4 from our computation of the potential. In particular we see that theremust be some negative open Gromov–Witten invariants which does never happen in the NEF case.

We conclude that, even in this non-NEF example, although there are infinitely many contributionsto the Seidel elements associated to the Hamiltonian circle actions, these quantum classes can stillbe expressed by explicit closed formulas. It is clear that as we increase the value of k the expressionsfor the Seidel elements corresponding to the circle actions x2k, y2k in F2k are going to be harder towrite explicitly. However, from the work of Abreu and McDuff in [1] we know that the generatorsof the fundamental group of the symplectomorphism group of M0

µ are given by x0, y0 and x2,so our computations allow us to give a complete description of the Seidel representation for themanifold M0

µ for all µ ≥ 1.

Remark 5.6. “Odd” Hirzebruch surfaces admit a similar description: Fµ2k−1, 1 ≤ k ≤ ℓ, is

defined as the symplectic quotient C4//T 22k−1 at the value (µ + k, 1). The moment polytope

P (2k − 1) := Φ2k−1(F2k−1) of the residual action of the 2–torus (0, u, v, 0) is the convex hull of(0, 0), (1, 0), (1, µ+k), (0, µ−k+1). This surface can be identified with the symplectic manifoldM1

µ = (CP2#CP2, ω1µ) where the symplectic area of the exceptional divisor is µ > 0 and the area

of the projective line is µ+ 1. Similar computations can be made for F2k−1, since F1 is Fano andwe can show that x2k−1 = y2k−1 = (k − 1)x1 + ky1, using Karshon’s classification of Hamiltoniancircle actions.

Appendix A. Additional computations of Seidel’s elements

We gather here results of computations of Seidel’s elements in three other particular cases (com-plementary to Theorem 4.5). They correspond to the cases where three consecutive facets havevanishing first Chern number. In order to ease the reading, we denote the weights ω(Ai) by ωi.

(2c) If c1(Ai0) = c1(Ai0+1) = c1(Ai0+2) = 0 but c1(Ai0−1) and c1(Ai0+3) are non-zero, then

S(Λ) =

[(

Ai0 ⊗ qtΦmax

1− t−ωi0−Ai0+1 ⊗ q

tΦmax−ωi0+1

1− t−ωi0+1

)

·1

1− t−ωi0−ωi0+1

−

(

Ai0+1 ⊗ qtΦmax

1− t−ωi0+1−Ai0+2 ⊗ q

tΦmax−ωi0+2

1− t−ωi0+2

)

·t−ωi0+1−ωi0+2

1− t−ωi0+1−ωi0+2

]

·1

1− t−ωi0−ωi0+1−ωi0+2


(2d) If c1(Ai0 ) = c1(Ai0−1) = c1(Ai0+1) = 0 but c1(Ai0+2) and c1(Ai0−2) are non-zero, then

S(Λ) =

[(

Ai0 ⊗ qtΦmax

1− t−ωi0−Ai0−1 ⊗ q

tΦmax−ωi0−1

1− t−ωi0−1

)

·1

1− t−ωi0−ωi0−1

+

+

(

Ai0 ⊗ qtΦmax

1− t−ωi0−Ai0+1 ⊗ q

tΦmax−ωi0+1

1− t−ωi0+1

)

·1

1− t−ωi0−ωi0+1

−Ai0 ⊗ qtΦmax

1− t−ωi0

]

·1

1− t−ωi0−ωi0−1−ωi0+1

(3d) If c1(Ai0+1) = c1(Ai0+2) = c1(Ai0+3) = 0 but c1(Ai0), c1(Ai0+4) and c1(Ai0−1) are non-zero,then

S(Λ) = Ai0 ⊗ qtΦmax −Ai0+1 ⊗ qtΦmax−ωi0+1

1− t−ωi0+1

−

(

Ai0+1 ⊗ qtΦmax

1− t−ωi0+1−Ai0+2 ⊗ q

tΦmax−ωi0+2

1− t−ωi0+2

)

·t−ωi0+1−ωi0+2

1− t−ωi0+1−ωi0+2

−

(

Ai0+1 ⊗ qtΦmax

1− t−ωi0+1−Ai0+2 ⊗ q

tΦmax−ωi0+2

1− t−ωi0+2

)

·

t−ωi0+1−ωi0+2−ωi0+3

(1− t−ωi0+1−ωi0+2−ωi0+3)(1 − t−ωi0+1−ωi0+2)

+

(

Ai0+2 ⊗ qtΦmax

1− t−ωi0+2−Ai0+3 ⊗ q

tΦmax−ωi0+3

1− t−ωi0+3

)

·

t−ωi0+1−2ωi0+2−2ωi0+3

(1− t−ωi0+2−ωi0+3)(1 − t−ωi0+1−ωi0+2−ωi0+3)

References

[1] M. Abreu and D. McDuff, Topology of symplectomorphism groups of rational ruled surfaces, J. Amer.Math. Soc. 13 (2000), 971–1009 (electronic).

[2] M. Abreu and R. Sena–Dias, Scalar–flat Kahler metrics on non–compact symplectic toric 4–manifolds,Ann. Global Anal. Geom. 41 (2012), 209–239.

[3] S. Anjos and M. Pinsonnault, The homotopy Lie algebra of symplectomorphism groups of some blow-ups

of the projective plane, Math. Z. 275 (2013), 245–292.[4] J.-F. Barraud and O. Cornea, Higher order Seidel invariants for loops of hamiltonian isotopies, in prepa-

ration.[5] V. Batyrev, Quantum cohomology rings of toric manifolds, Asterisque 218 (1993), 9–34.[6] M. Borman, Quasi-states, quasi-morphisms and the moment map, Int. Math. Res. Notices 2013(11) (2013),

2497–2533.[7] A. Cannas da Silva, Symplectic Toric Manifolds. In Symplectic Geometry of Integrable Hamiltonian Sys-

tems, M. Castellet ed., Advanced Courses in Mathematics, CRM Barcelona (2003), 85–173.

[8] K. Chan, and S.-C. Lau, Open Gromov-Witten invariants and superpotentials for semi-Fano toric sur-

faces, Int. Math. Res. Not. (2013), doi: 10.1093/imrn/rnt050.[9] K. Chan, S.-C. Lau, N C Leung, and H.-H. Tseng, Open Gromov–Witten invariants, mirror maps, and

Seidel’s representations for toric manifolds, arXiv:1209.6119 (2012).[10] F. Charette and O. Cornea, Categorification of Seidel’s representation, arXiv:1307.7235 (2013).[11] D. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Math. Surveys and Monographs vol 68,

AMS, Providence (1999).[12] T. Delzant, Hamiltoniens periodiques et image convexe de l’application moment, Bulletin de la Societe

Mathematique de France 116 (1988), 315–339.[13] M. Entov and L. Polterovich, Symplectic quasi-states and semi-simplicity of quantum homology. In Toric

topology, vol. 460 of Contemp. Math., Amer. Math. Soc., Providence, RI (2008), 47–70.[14] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Floer theory on compact toric manifolds. I, Duke

Math. J. 151 (2010), 23–174.[15] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Floer theory and mirror symmetry on compact

toric manifolds, arXiv:1009.1648 (2010).[16] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Spectral invariants with bulks, quasi-morphisms, and La-

grangian Floer theory, arXiv:1105.5123 (2011).


[17] E. Gonzalez and H. Iritani, Seidel elements and mirror transformations, Selecta Math. (N.S.) 18 (2012),no. 3, 557–590.

[18] T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), 487–518.[19] S. Hu and F. Lalonde, A relative Seidel morphism and the Albers map, Trans. amer. Math. Soc. 362

(2010), 1135–1168.[20] S. Hu, F. Lalonde, and R. Leclercq, Homological Lagrangian monodromy, Geom. Topol. 15 (2011), 1617–

1650.[21] M. Hutchings, Floer homology for families. I, Algebr. Geom. Topol. 8 (2008), 435–492.[22] C. Hyvrier, Lagrangian circle actions, arXiv:1307.8196 (2013).[23] Y. Karshon, Periodic Hamiltonian Flows on Four Dimensional Manifolds. Mem. Amer. Math. Soc. 141,

no. 672 (1999).[24] Y. Karshon, L. Kessler, and M. Pinsonnault, A compact symplectic four-manifold admits only finitely

many inequivalent toric actions, J. Symplectic Geom. 5 (2007), 139–166.[25] F. Lalonde, D. McDuff, and L. Polterovich, Topological rigidity of Hamiltonian loops and quantum ho-

mology, Invent. Math. 135 (1999), 369–385.[26] C.–C. M. Liu, Localization in Gromov–Witten theory and orbifold Gromov–Witten theory. In Handbook of

Moduli, Volume II, Adv. Lect. Math., (ALM) 25, International Press and Higher Education Press (2013),353–425.

[27] Y. Manin, Generating functions in algebraic geometry and sums over trees. In The moduli space of curves,R. Dijkgraaf, C. Faber, and G. van der Geer, eds., Birkhauser (1995), 401–417.

[28] D. McDuff and D. Salamon, J–holomorphic Curves and Symplectic Topology, Amer. Mat. Soc., Provi-dence, RI (2004).

[29] D. McDuff and S. Tolman, Topological properties of Hamiltonian circle actions, Int. Math. Res. Papers(2006), doi:10.1155/IMRP/2006/72826.

[30] Y. Ostrover and I. Tyomkin, On the quantum homology algebra of toric Fano manifolds, Selecta Math.(N.S.) 15 (2009), 121–149.

[31] M. Pinsonnault, Symplectomorphism groups and embeddings of balls into rational ruled surfaces, Compos.Math. 144 (2008), 787–810.

[32] Y. Savelyev, Quantum characteristic classes and the Hofer metric, Geom. Topol. 12 (2008), 2277–2326.[33] P. Seidel, π1 of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct.

Anal. 7 (1997), 1046–1095.[34] H. Spielberg, Une formule pour les invariants de Gromov–Witten des varietes toriques, Ph.D. thesis

(1999).[35] H. Spielberg, The Gromov–Witten invariants of symplectic toric manifolds, arXiv: math/0006156v1

(2000).[36] M. Usher, Deformed Hamiltonian Floer theory, capacity estimates, and Calabi quasimorphisms, Geom.

Topol. 15 (2011), 1313–1417.

Sılvia Anjos: Center for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics De-

partment, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

E-mail address: [email protected]

Remi Leclercq: Universite Paris-Sud, Departement de Mathematiques, Bat. 425, 91400 Orsay, France

E-mail address: [email protected]

introduction - ulisboa

Documents