HHAMILTONIAN S -MANIFOLDS ARE UNIRULED DUSA MCDUFF
Abstract.
The main result of this note is that every closed Hamiltonian S manifoldis uniruled, i.e. it has a nonzero Gromov–Witten invariant one of whose constraintsis a point. The proof uses the Seidel representation of π of the Hamiltonian groupin the small quantum homology of M as well as the blow up technique recentlyintroduced by Hu, Li and Ruan. It applies more generally to manifolds that havea loop of Hamiltonian symplectomorphisms with a nondegenerate fixed maximum.Some consequences for Hofer geometry are explored. An appendix discusses thestructure of the quantum homology ring of uniruled manifolds. Contents
1. Introduction 12. The main argument 72.1. Uniruled manifolds and their pointwise blowups 72.2. The Seidel representation 102.3. Proof of the main results 173. Gromov–Witten invariants and blowing up 213.1. Relative invariants of genus zero 213.2. Applications of the decomposition formula 243.3. Blowing down section invariants. 283.4. Identities for descendent classes 344. Special cases 39Appendix A. The structure of QH ∗ ( M ) for uniruled M Introduction
A projective manifold is said to be projectively uniruled if there is a holomorphicrational curve through every point. Projective manifolds with this property form an
Date : August 4, 2007, revised May 6 2008.2000
Mathematics Subject Classification.
Key words and phrases. symplectically uniruled, Hamiltonian S -actions, quantum homology, rela-tive Gromov–Witten invariants, Seidel representation.partially supported by NSF grant DMS 0604769. a r X i v : . [ m a t h . S G ] J u l DUSA MCDUFF important class of varieties in birational geometry since they do not have minimalmodels but rather give rise to Fano fiber spaces; see for example Kollar [18, Ch IV].One way to translate this property into the symplectic world is to call a sym-plectic manifold (
M, ω ) (symplectically) uniruled if there is a nonzero genus zeroGromov–Witten invariant of the form (cid:10) pt, a , . . . , a k (cid:11) Mk,β where β (cid:54) = 0. Here k ≥ (cid:54) = β ∈ H ( M ), and a i ∈ H ∗ ( M ), and we consider the invariant where the k markedpoints are allowed to vary freely. Because Gromov–Witten invariants are preservedunder symplectic deformation, a symplectically uniruled manifold ( M, ω ) has a J -holomorphic rational curve through every point for every J that is tamed by somesymplectic form deformation equivalent to ω . Further justification for this definitionis given in the foundational paper by Hu–Li–Ruan [15]. They show that ( M, ω ) issymplectically uniruled whenever there is any nontrivial genus zero Gromov–Witteninvariant (cid:10) τ i pt, τ i a , . . . , τ i k a k (cid:11) Mk,β with β (cid:54) = 0, where the i j ≥ They also show that the uniruled property is preservedunder symplectic blowing up and down, and is therefore preserved by symplectic bira-tional equivalences.We will call a symplectic manifold (
M, ω ) strongly uniruled if there is a nonzeroinvariant (cid:10) pt, a , a (cid:11) M ,β . (Since one can always add marked points with insertions givenby divisors, this is the same as requiring there be some nonzero invariant with k ≤ P through everypoint there is a nonzero invariant (cid:10) pt, a , a (cid:11) Mβ . It is not clear that the same is true in thesymplectic category. Two questions are included here. Firstly, there is a question aboutthe behavior of Gromov–Witten invariants: if there is some nonzero k -point invariantwith a point insertion must there be a similar nonzero 3-point invariant? Secondly,there is a more geometric question. Suppose that M is covered by J -holomorphic 2-spheres either for one ω -tame J or for a significant class of J . Must M then be uniruled?Hamiltonian S -manifolds are a good test case here since, when J is ω -compatible and S -invariant, there is an S -invariant J sphere through every point (given by the orbitof a gradient flow trajectory of the moment map with respect to the associated metric g J .)In this note we show that every Hamiltonian S -manifold is uniruled. This is obviouswhen n = 1 and is well known for n = 2 since, by Audin [2] and Karshon [17], the only4-dimensional Hamiltonian S -manifolds are blow ups of rational or ruled surfaces. Ourproof in higher dimensions relies heavily on the approach used by Hu–Li–Ruan [15] toanalyse the Gromov–Witten invariants of a blow up. Descendent insertions τ i are defined in the discussion following equation (3.2). AMILTONIAN S -MANIFOLDS ARE UNIRULED 3 The first step is to argue as follows. By Lemma 2.18 we may blow M up along itsmaximal and minimal fixed point sets F max , F min until these two are divisors. Thenconsider the gradient flow of the (normalized) moment map K with respect to an S -invariant metric g J constructed from an ω -compatible invariant almost complexstructure J . The S -orbit of any gradient flow line is J -holomorphic. If α is the classof the S -orbit of a flow line from F max to F min then c ( α ) = 2, and because there isjust one of these spheres through a generic point of M one might naively think thatthe Gromov–Witten invariant (cid:10) pt, F min , F max (cid:11) Mα is 1. However, there could be othercurves in class α that cancel this one. Almost the only case in which one can be surethis does not happen is when the S action is semifree, i.e. no point in M has finitestabilizer: see Proposition 4.3. If the order of the stablizers is at most 2, then one canalso show that ( M, ω ) must be strongly uniruled: see Proposition 4.2. However it is notclear whether (
M, ω ) must be strongly uniruled when the isotropy has higher order.To deal with the general case, we use the Seidel representation of π (Ham( M, ω ))in the group of multiplicative units of the quantum homology ring QH ∗ ( M ). Tomake the argument work, we blow up once more at a point in F max , obtaining an S -manifold called ( (cid:102) M , (cid:101) ω ). By Proposition 2.14 the Seidel element S ( (cid:101) γ ) ∈ QH n ( (cid:102) M ) ofthe resulting S action (cid:101) γ on (cid:102) M involves the exceptional divisor E on (cid:102) M . Although weknow very little about the structure of QH ∗ ( M ), the fact ( M, ω ) is not uniruled impliesby Proposition 2.5 that the part of QH ∗ ( (cid:102) M ) that does not involve E forms an ideal.Moreover, the quotient of QH ∗ ( (cid:102) M ) by this ideal has an understandable structure. Usingthis, we show that the invertibility of S ( (cid:101) γ ) implies that certain terms in the inverseelement S ( (cid:101) γ − ) cannot vanish. This tells us that certain section invariants of thefibration (cid:101) P (cid:48) → S defined by the loop (cid:101) γ − cannot vanish. The homological constraintshere involve E . The final step is to show that these invariants can be nonzero only if( M, ω ) is uniruled. Hence the original manifold is as well, by the blow down result ofHu–Li–Ruan [15, Thm 1.1]. Their blowing down argument does not give control on thenumber of insertions; a 3-point invariant might blow down to an invariant with moreinsertions. Hence we cannot conclude that (
M, ω ) is strongly uniruled.This argument does not use the properties of F min nor does it use much about thecircle action. We do need to assume that the loop γ = { φ t } has a fixed maximalsubmanifold ; i.e. that there is a nonempty (but possibly disconnected) submanifold F max such that at each time t the generating Hamiltonian K t for γ takes its maximumon F max in the strict sense that if x max ∈ F max then K t ( x ) ≤ K t ( x max ) for all x ∈ M with equality iff x ∈ F max . Further we need γ to restrict to an S action near F max ; i.e.in appropriate coordinates ( z , . . . z k ) normal to F max we assume that near F max the Although we describe this here in terms of genus zero Gromov–Witten invariants, another wayto think of it is as the transformation induced on Hamiltonian Floer homology by continuation alongnoncontractible Hamiltonian loops. Suppose that P → S is a bundle with fiber ( M, ω ) with Hamiltonian structure group. Then thefiberwise symplectic form ω extends to a symplectic form Ω on P . Gromov–Witten invariants of P inclasses β ∈ H ( P ; Z ) that project to the positive generator of H ( S ; Z ) are called section invariants ,while those whose class lies in the image of H ( M ; Z ) are called fiber invariants . DUSA MCDUFF generating Hamiltonian K t has the form const − (cid:80) m i | z i | for some positive integers m i .In these circumstances, we shall say that γ is a circle action near its maximum. Note that this action is effective (i.e. no element other than the identity acts trivially)iff gcd( m , . . . , m k ) = 1.Our main result is the following. Theorem 1.1.
Suppose that
Ham( M ) contains a loop γ with a fixed maximum nearwhich γ is an effective circle action. Then ( M, ω ) is uniruled. Corollary 1.2.
Every Hamiltonian S -manifold is uniruled. Remark 1.3.
Since our proofs involve the blow up of M they work only when n := dim M ≥
2. However Theorem 1.1 is elementary when n = 1. For then, if M (cid:54) = S ,the group Ham M is contractible, as is each component of the group G of elementsin Ham M that fix x . Therefore the homomorphism π ( G ) → π (Sp(2 , R )) given bytaking the derivative at x is trivial. This implies that there is no loop γ in Ham M that is a nonconstant circle action near x .In fact, in this case there is also no nonconstant loop that is the identity neara fixed maximum. For ω is exact on M (cid:114) { x } so that the Calabi homomorphismCal : Ham c ( M (cid:114) { x } ) → R is well defined on the group Ham c of compactly supportedHamiltonian symplectomorphisms of M (cid:114) { x } ; cf. [31, Ch. 10.3]. In particularCal( φ ) := (cid:90) (cid:16)(cid:90) M (cid:0) H t ( x ) − H t ( x ) (cid:1) ω (cid:17) dt, is independent of the choice of path φ Ht in Ham c with time 1 map φ . But if γ were anonconstant loop in Ham c with fixed maximum at x we could arrive at φ = id by apath for which this integral is negative.If all we know is that γ has a fixed maximum F max then for each x ∈ F max thelinearized flow A t ( x ) in the 2 k -dimensional normal space N Fx := T x M/T x ( F max ) isgenerated by a family of nonpositive quadratic forms. We shall say that F max is non-degenerate if these quadratic forms are everywhere negative definite. In this case, thelinearized flow is a so-called positive loop in the symplectic group Sp(2 k ; R ), i.e. a loopgenerated by a family of negative definite quadratic forms; cf. Lalonde–McDuff [20]. If2 k ≤ k ; R ) are in fact homotopic through a family of positive loops. (In principlethis should hold in all dimensions, but the details have been worked out only in lowdimensions.) By Lemma 2.22 below, in the 4-dimensional case one can then homotop γ so that it is an effective local circle action near its maximum. Thus we find: Proposition 1.4.
Suppose that dim M ≤ and the loop γ has a nondegenerate maxi-mum at x max . Then γ can be homotoped so that it is a nonconstant circle action nearits maximum, that when dim M = 4 can be assumed effective. Thus ( M, ω ) is uniruled. Slimowitz’s result also implies that every S action on C k with strictly positiveweights ( m , . . . , m k ) is homotopic through positive loops to an action with positiveweights ( m (cid:48) , m (cid:48) , m . . . , m k ) where m (cid:48) , m (cid:48) are mutually prime. (Here we do not change AMILTONIAN S -MANIFOLDS ARE UNIRULED 5 the action on the last ( k −
2) coordinates.) Since actions are effective iff their weightsat any fixed point are mutually prime we deduce:
Corollary 1.5.
Theorem 1.1 holds for any loop that is a circle action near its maxi-mum.
These results have implications for Hofer geometry. It is so far unknown whichelements in π (Ham( M )) can be represented by loops that minimize the Hofer length.By Lalonde–McDuff [19, Prop. 2.1] such a loop has to have a fixed maximum andminimum x max , x min , i.e. points such that the generating Hamiltonian K t satisfies theinequalities K t ( x min ) ≤ K t ( x ) ≤ K t ( x max ) for all x ∈ M and t ∈ [0 , t .If fixed extrema exist, then the S -orbit of a path from x max to x min forms a 2-sphere, and it is easy to check that the integral of ω over this 2-sphere is just (cid:107) K (cid:107) := (cid:82) (cid:0) K t ( x max ) − K t ( x min ) (cid:1) dt ; cf.[32, Ex.9.1.11]. Hence such a loop cannot exist on asymplectically aspherical manifold ( M, ω ). However, so far no examples are known ofsymplectically aspherical manifolds with nonzero π (Ham( M )).It is easy to construct nonzero elements in π (Ham( M )) for manifolds that are notuniruled. For example, by [29, Prop 1.10] one could take M to be the two point blowup of any symplectic 4-manifold. Hence, if one could better understand the situationwhen the fixed points are degenerate, it might be possible to exhibit manifolds forwhich π (Ham) is nontrivial but where no nonzero element in this group has a lengthminimizing representative. This question is the subject of ongoing research. Remark 1.6. (i) It is not clear whether a Hamiltonian S -manifold ( M, ω ) always hasa blow up that is strongly uniruled. However, as we show in Proposition A.4 there is notmuch difference between the uniruled and the strongly uniruled conditions. Moreoverby Corollary 4.5 they coincide in the case when H ∗ ( M ; Q ) is generated by H ( M ; Q ).(ii) It is essential in Corollary 1.2 that the loop is Hamiltonian; it is not enough that ithas fixed points. For example, the semifree symplectic circle action constructed in [26]has fixed points but these all have index and coindex equal to 2. Hence the first Chernclass c vanishes on π ( M ) and M cannot be uniruled for dimensional reasons.(iii) One might wonder why we work with the blow up (cid:102) M rather than with M itself.One answer is seen in results such as Lemma 2.8 and Propositions 2.14 and 2.19. Thesemake clear that the exceptional divisor in (cid:102) M forms a visible marker that allows us toshow that certain Gromov–Witten invariants of M do not vanish. The point is thateven though we cannot calculate QH ∗ ( (cid:102) M ) in general, the results of § QH ∗ ( (cid:102) M ) / I provided that ( M, ω ) is not uniruled.As always, the problem is that it is very hard to calculate Gromov–Witten invariantsfor a general symplectic manifold; if one has no global information about the manifold,the contribution to an invariant of a known J -holomorphic curve could very well becancelled by some other unseen curve. The Seidel representation is one of the very fewgeneral tools that can be used to show that certain invariants of an arbitrary symplecticmanifold cannot vanish and hence it has found several interesting applications; cf. DUSA MCDUFF
Seidel [36] and Lalonde–McDuff [21]. It is made by counting section invariants ofHamiltonian bundles P with fiber M and base S . It is fairly clear that the existenceof fiber invariants of the blow up bundle (cid:101) P that involve the exceptional divisor E shouldimply that ( M, ω ) is uniruled. A crucial step in our argument is Lemma 2.20 whichstates that the existence of certain section invariants of P also implies that ( M, ω ) isuniruled.Another way of getting global information on Gromov–Witten invariants is to usesymplectic field theory. This approach was recently taken by J. He [9] who shows thatsubcritical manifolds are strongly uniruled, thus proving a conjecture of Biran andCieliebak in [3, § P n is subcritical while P × P is not.The main proofs are given in §
2. They use some properties of absolute and relativeGromov–Witten invariants established in §
3. Conditions that imply (
M, ω ) is stronglyuniruled are discussed in §
4. The appendix treats general questions about the structureof the quantum homology ring.
Remark 1.7 (Technical underpinnings of the proof) . The main technical tool used inthis paper is the decomposition rule for relative genus zero Gromov–Witten invariants.Since we use this with descendent absolute insertions, it is useful to have this in astrong form in which the virtual moduli cycles in question are at least C . To date themost relevant references for this result are Li–Ruan [23, Thm 5.7] and Hu–Li–Ruan [15, § § M that are S -invariant and these are almost never regular unless the action is semifree.Another important ingredient of the proof is the Seidel representation of π of theHamiltonian group Ham( M, ω ), which is proved for general symplectic manifolds in [27, § Acknowledgements
Many thanks to Tian-Jun Li, Yongbin Ruan, Rahul Pandhari-pande and Aleksey Zinger for useful discussions about relative Gromov–Witten invari-ants. I also thank the first two named above as well as Jianxun Hu for showing meearly versions of their paper [15], and Mike Chance and the referees for various helpful
AMILTONIAN S -MANIFOLDS ARE UNIRULED 7 comments. Finally I wish to thank MSRI for their May 2005 conference on Gromov–Witten invariants and Barnard College and Columbia University for their hospitalityduring the final stages of my work on this project.2. The main argument
Because the main argument is somewhat involved, we shall begin by outlining it. In § QH ∗ ( (cid:102) M ) of the one pointblow up of a manifold ( M, ω ) that is not uniruled. By Proposition 2.5 this conditionon (
M, ω ) implies that the subspace of QH ∗ ( (cid:102) M ) spanned by the homology classes in H ∗ ( M (cid:114) pt ) forms an ideal I . Moreover, the quotient R := QH ∗ ( (cid:102) M ) / I decomposes asthe sum of two fields (Lemma 2.7), and we can work out an explicit formula for theinverse u − of any unit u in R .The unit in question is the Seidel element S ( (cid:101) γ ) ∈ QH ∗ ( (cid:102) M ) of the Hamiltonian loop (cid:101) γ on (cid:102) M . Usually it is very hard to calculate S ( (cid:101) γ ). However, when the maximum fixedpoint set (cid:101) F max of (cid:101) γ is a divisor, one can calculate its leading term (Corollary 2.12).Moreover, because the final blow up is at a point of F max we show in Proposition 2.14that this leading term has nontrivial image in R . We next use the explicit formula for u − ∈ R to deduce the nonvanishing of certain terms in the Seidel element S ( (cid:101) γ (cid:48) ) forthe inverse loop (cid:101) γ (cid:48) := (cid:101) γ − . But S ( (cid:101) γ (cid:48) ) is calculated from the Gromov–Witten invariantsof the fibration (cid:101) P (cid:48) → P determined by (cid:101) γ (cid:48) . Hence we deduce in Corollary 2.15 thatsome of these invariants do not vanish.Finally we show that these particular invariants can be nonzero only if M is uniruled,so that our initial assumption that ( M, ω ) is not uniruled is untenable. The proof hereinvolves two steps. First, by blowing down the bundle (cid:101) P (cid:48) to P (cid:48) , the bundle determinedby γ (cid:48) := γ − , we deduce the nonvanishing of certain Gromov–Witten invariants in P (cid:48) (Proposition 2.19). Second, we show in Lemma 2.20 that these invariants can benonzero only if ( M, ω ) is uniruled.In this section we prove essentially all results that do not involve comparing theGromov–Witten invariants of a manifold and its blow up. (Those proofs are deferredto § M, ω ) is a closed symplectic manifold of dimension2 n where n ≥ Uniruled manifolds and their pointwise blowups.
Consider the small quan-tum homology QM ∗ ( M ) := H ∗ ( M ) ⊗ Λ of M . Here we use the Novikov ring Λ :=Λ ω [ q, q − ] where q is a polynomial variable of degree 2 and Λ ω denotes the general-ized Laurent series ring with elements (cid:80) i ≥ r i t − κ i , where r i ∈ Q and κ i is a strictlyincreasing sequence that tends to ∞ and lies in the period subgroup P ω of [ ω ], i.e. theimage of the homomorphism I ω : π ( M ) → R given by integrating ω . We assume that[ ω ] is chosen so that I ω is injective. We write the elements of QH ∗ ( M ) as infinite sums (cid:80) i ≥ a i ⊗ q d i t − κ i , where a i ∈ H ∗ ( M ; Q ) =: H ∗ ( M ), the | d i | are bounded and κ i is asbefore. The term a ⊗ q d t κ has degree 2 d + deg a . DUSA MCDUFF
The quantum product a ∗ b of the elements a, b ∈ H ∗ ( M ) ⊂ QH ∗ ( M ) is defined asfollows. Let ξ i , i ∈ I , be a basis for H ∗ ( M ) and write ξ ∗ i , i ∈ I, for the basis of H ∗ ( M )that is dual with respect to the intersection pairing, that is ξ ∗ j · ξ i = δ ij . Then(2.1) a ∗ b := (cid:88) i,β ∈ H ( M ; Z ) (cid:10) a, b, ξ i (cid:11) Mβ ξ ∗ i ⊗ q − c ( β ) t − ω ( β ) , where (cid:10) a, b, ξ i (cid:11) Mβ denotes the Gromov–Witten invariant in M that counts curves in class β through the homological constraints a, b, ξ i . Note that if ( a ∗ b ) β := (cid:80) i (cid:10) a, b, ξ i (cid:11) Mβ ξ ∗ i ,then ( a ∗ b ) β · c = (cid:10) a, b, c (cid:11) Mβ . Further, deg( a ∗ b ) = deg a + deg b − n , and the identityelement is 1l := [ M ]. Lemma 2.1.
The following conditions are equivalent: (i) (
M, ω ) is not strongly uniruled; (ii) Q − := ⊕ i< n H i ( M ) ⊗ Λ is an ideal in the small quantum homology QH ∗ ( M ) ; (iii) pt ∗ a = 0 for all a ∈ Q − , where ∗ is the quantum product.Moreover, if these conditions hold every unit u ∈ QH ∗ ( M ) × in QH ∗ ( M ) has the form u = 1l ⊗ λ + x , where x ∈ Q − and λ is a unit in Λ .Proof. Choose the basis ξ i so that ξ = pt , ξ M = [ M ] =: 1l and all other ξ i lie in H j ( M )for 1 ≤ j < n . Then ξ ∗ = ξ M , and the coefficient of 1l ⊗ q − d t − κ in a ∗ b is (cid:88) β : ω ( β )= κ,c ( β )= d (cid:10) a, b, pt (cid:11) β . Since we assume that I ω : π ( M ) → Γ ω is an isomorphism and κ ∈ Γ ω , there is preciselyone class β such that ω ( β ) = κ . If β = 0 this invariant is the usual intersection product a ∩ b ∩ pt and so always vanishes for a, b ∈ H < n ( M ). Therefore, this coefficient is nonzerofor some such a, b iff ( M, ω ) is strongly uniruled. But this coefficient is nonzero iff Q − is not a subring. Note finally that since Q − has codimension 1, it is an ideal iff it is asubring.This proves the equivalence of (i) and (ii). The other statements are proved bysimilar arguments. (cid:3) Remark 2.2.
It seems very likely that (
M, ω ) is strongly uniruled iff Q − contains nounits. We prove this in the appendix, as well as a generalization to uniruled manifolds,in the case when the odd Betti numbers of M vanish.Now let ( (cid:102) M , (cid:101) ω ) be the 1-point blow up of M with exceptional divisor E . Put ε := E n − , the class of a line in E . The blow up parameter is δ := (cid:101) ω ( ε ), which we assumeto be Q -linearly independent from Γ ω . Later it will be convenient to write Λ e ω =: Λ ω,δ . Lemma 2.3.
Invariants of the form (cid:10) E i , E j , E k (cid:11) f Mpε , < i, j, k < n, p > are nonzero only if p = 1 and i + j + k = 2 n − , and in this case the invariant is − . AMILTONIAN S -MANIFOLDS ARE UNIRULED 9 Proof.
Since c ( ε ) = n − (cid:10) E i , E j , E k (cid:11) f Mpε is nonzero only if n + p ( n −
1) = i + j + k . Since i + j + k ≤ n − p = 1. The result now follows fromthe fact that if J is standard near E then the only ε curves are lines in E . (cid:3) Corollary 2.4.
Let ≤ i, j < n . Then E i ∗ E j = E i ∩ E j = E i + j if i + j < n,E i ∗ E n − i = − pt + E ⊗ q − n +1 t − δ , and E i ∗ E j = E i + j − n +1 ⊗ q − n +1 t − δ if n < i + j < n − . We prove the next result in § Proposition 2.5.
Let ( (cid:102) M , (cid:101) ω ) be the one point blow up of M with exceptional divisor E , and let a, b ∈ H < n ( M ) . If any invariant of the form (cid:10) a, b, E k (cid:11) f M e β , (cid:10) a, E i , E j (cid:11) f M e β , (cid:10) E i , E j , E k (cid:11) f M e β , for (cid:54) = (cid:101) β (cid:54) = pε, i, j, k ≥ is nonzero, then ( M, ω ) is uniruled. Decompose QH ∗ ( (cid:102) M ) additively as I ⊕ E where I is generated as a Λ e ω -module by theimage of H < n ( M ) in H ∗ ( (cid:102) M ) and E is spanned by 1l , E, . . . , E n − = ε . Proposition 2.5implies: Corollary 2.6. If ( M, ω ) is not uniruled, I is an ideal in QH ∗ ( (cid:102) M ) .Proof. Let a, b ∈ H < n ( M ). Then a ∗ b ∈ I iff ( a ∗ b ) e β · E k = (cid:10) a, b, E k (cid:11) f M e β = 0 for all (cid:101) β ∈ H ( (cid:102) M ) and all k ≥
1. Therefore the hypotheses imply that I is a subring. It is anideal because a ∗ E k = 0 for all k ≥ (cid:3) Let E n denote the degree 2 n part of E . This is not a subring of QH ∗ ( (cid:102) M ) since forexample ( E ⊗ q ) ∗ ( ε ⊗ q n − ) = − pt ⊗ q n + E ⊗ qt − δ . Nevertheless if (
M, ω ) is not uniruled, E n can be given the ring structure of the quotient QH n ( (cid:102) M ) / I . By Corollary 2.4 and Proposition 2.5, there is a ring isomorphism E n → Λ ω,δ [ s ] / (cid:0) s n = st − δ (cid:1) given by E ⊗ q (cid:55)→ s. Composing with the quotient map QH n ( (cid:102) M ) → E n ∼ = QH n ( (cid:102) M ) / I we get a homo-morphism Φ E : QH n ( (cid:102) M ) → R := Λ ω,δ [ s ] / (cid:0) s n = st − δ (cid:1) . Lemma 2.7.
The ring R decomposes as the direct sum of two fields R ⊕ R .Proof. Denote e := 1 − s n − t δ and e := s n − t δ . Because se = 0 , se = s we find e i = e i and e e = 0. Moreover R := e R is isomorphic to the field e Λ ω,δ . To seethat R := e R is a field, consider the homomorphism F : R → Λ ω,δ/ ( n − given by s (cid:55)→ t − δ/ ( n − , t (cid:55)→ t. Then F ( e ) = 1 and the kernel of F is e R . Hence F gives an isomorphism between e R and the field Λ ω,δ/ ( n − . (cid:3) Denote X := (cid:8) x ∈ R : x = (cid:88) i ≥ r i s d i t − κ i , r i ∈ Q , ≤ d i ≤ n, κ i > (cid:9) . Then for all x ∈ X the element 1 + x is invertible with inverse 1 − x + x − . . . . Lemma 2.8.
Let u ∈ R be a unit of the form rst κ (1 + x ) where x ∈ X , r (cid:54) = 0 and κ ∈ Λ ω is > δ .(i) If n ≥ then u − = 1 − s n − t δ + 1 r s n − t δ − κ (cid:0) y (cid:1) , for some y ∈ X . (ii) If n = 2 then u − = 1 − st δ + r st δ − κ (cid:0) y (cid:1) for some y ∈ X .Proof. Since e u = e ( e + rst κ (1 + x )) we may write u = e u + e u = e + e (cid:16) s n − t δ + rst κ (1 + x ) (cid:17) = e + e rst κ (1 + x (cid:48) ) where x (cid:48) := x + 1 r s n − t δ − κ . If n > v := e + r e s n − t δ − κ (1 + x (cid:48) ) − . Then e uv = e while e uv = e s n − t δ = e . Hence u − = v . Since e s n − = s n − when n >
2, this has the formrequired in (i). When n = 2 we take v = e + r e st δ − κ (1 + x (cid:48) ) − . (cid:3) Remark 2.9.
Here we restricted the coefficients of QH ∗ ( M ) to Λ ω [ q, q − ] to make thestructure of R as simple as possible. However, in order to define the Seidel representa-tion we will need to use the larger coefficient ring Λ univ [ q, q − ] where Λ univ consists ofall formal series (cid:80) i ≥ r i t − κ i , where κ i ∈ R is any increasing sequence that tends to ∞ .Since R injects into R (cid:48) := Λ univ [ s ] / (cid:0) s n = st − δ (cid:1) the statement in Lemma 2.8 remainsvalid when we think of u as an element of R (cid:48) .2.2. The Seidel representation.
This is a homomorphism S from π (Ham( M, ω ))to the degree 2 n multiplicative units QH n ( M ) × of the small quantum homology ringfirst considered by Seidel in [35]. To define it, observe that each loop γ = { φ t } inHam( M ) gives rise to an M -bundle P γ → P defined by the clutching function γ : P γ := M × D + ∪ M × D − / ∼ where (cid:0) φ t ( x ) , e πit (cid:1) + ∼ (cid:0) x, e πit (cid:1) − . Because the loop γ is Hamiltonian, the fiberwise symplectic form ω extends to a closedform Ω on P , that we can arrange to be symplectic by adding to it the pullback of asuitable form on the base P ; see the proof of Proposition 2.11 below.In the case of a circle action with normalized moment map K : M → R we maysimply take ( P γ , Ω) to be the quotient ( M × S S , Ω c ), where S acts diagonally on S and Ω c pulls back to ω + d (cid:0) ( c − K ) α (cid:1) . Here α is the standard contact form on S normalized so that it descends to an area form on S with total area 1, and c is any AMILTONIAN S -MANIFOLDS ARE UNIRULED 11 constant larger than the maximum K max of K . Points x max , x min in the fixed pointsets F max and F min give rise to sections s max := x max × P and s min := x min × P . Notethat our orientation conventions are chosen so that the integral of Ω over the section s min is larger than that over s max . For example, if M = S and γ is a full rotation, P γ can be identified with the one point blow up of P , and s max is the exceptional divisor.In the following we denote particular sections as s max or s min , while writing σ max , σ min for the homology classes they represent.The bundle P γ → P carries two canonical cohomology classes, the first Chern class c Vert1 of the vertical tangent bundle and the coupling class u γ , the unique class thatextends the fiberwise symplectic class [ ω ] and is such that u n +1 γ = 0 . Then(2.2) S ( γ ) := (cid:88) σ a σ ⊗ q − c Vert1 ( σ ) t − u γ ( σ ) , where σ ∈ H ( P ; Z ) runs over all section classes (i.e. classes that cover the positivegenerator of H ( P ; Z )) and a σ ∈ H ∗ ( M ) is defined by the requirement that(2.3) a σ · M c = (cid:10) c (cid:11) Pσ , for all c ∈ H ∗ ( M ) . (Cf. [32, Def. 11.4.1]. For this to make sense we must use Λ univ instead of Λ ω asexplained in Remark 2.9. Note that we write · M for the intersection product in M .)Further for all b, c ∈ H ∗ ( M )(2.4) S ( γ ) ∗ b = (cid:88) σ b σ ⊗ q − c Vert1 ( σ ) t − u γ ( σ ) , where b σ · M c := (cid:10) b, c (cid:11) Pσ . Remark 2.10.
Lemma 2.1 shows that if (
M, ω ) is not strongly uniruled then S ( γ ) =1l ⊗ λ + x where x ∈ Q − ( M ). Therefore S ( γ ) ∗ pt = pt ⊗ λ , which in turn means that all2-point invariants (cid:10) pt, c (cid:11) Pσ with c ∈ H < n ( M ) must vanish. In other words, a sectioninvariant in P γ with more than a single point constraint must vanish. We shall expandon this theme later in Lemmas 2.20 and 2.21.In general it is very hard to calculate S ( γ ). The following result is essentially dueto Seidel [35, § K t is anormalized generating Hamiltonian for γ = { φ t } iff ω ( ˙ φ t , · ) = − dK t , and (cid:90) M K t ω n = 0 . Further we define(2.5) K max := (cid:90) max x ∈ M K t ( x ) dt. Proposition 2.11.
Suppose that γ has a nonempty maximal submanifold F max and isgenerated by the normalized Hamiltonian K t . Then S ( γ ) := a max ⊗ q m max t K max + (cid:88) β ∈ H ( M ; Z ) , ω ( β ) > a β ⊗ q m max − c ( β ) t K max − ω ( β ) , where m max := − c Vert1 ( σ max ) . Moreover a max · M c = (cid:10) c (cid:11) Pσ max . Proof.
Because γ is assumed to have a fixed maximum, we may write the section classes σ appearing in S ( γ ) as σ max + β where β ∈ H ( M ), and then denote a β := a σ . Thereforeby equation (2.2) the result will follow if we show that:(a) the only class with ω ( β ) ≤ that contributes to the sum is the class β = 0, and(b) − u γ ( σ max ) = K max .Consider the renormalized polar coordinates ( r, t ) on the unit disc D , where t := θ/ π . Define Ω − := ω + εd ( r ) ∧ dt on M × D − for some small constant ε >
0, and setΩ + := ω + (cid:16) κ ( r , t ) d ( r ) − d (cid:0) ρ ( r ) K t (cid:1)(cid:17) ∧ dt, on M × D + , where ρ ( r ) is a nondecreasing function that equals 0 near 0 and 1 near 1. If κ ( r , t ) = ε near r = 1, these two forms fit together to give a closed form Ω on P . Moreover, Ω issymplectic iff κ ( r , t ) − ρ (cid:48) ( r ) K t ( x ) > r, t and x ∈ M . Hence we may supposethat(2.6) ν := (cid:90) s max Ω = πε + (cid:90) D + (cid:0) κ ( r , t ) − ρ (cid:48) ( r ) max x ∈ M K t ( x ) (cid:1) rdrdt is as close to zero as we like.A class β contributes to S ( γ ) iff some invariant (cid:10) c (cid:11) Pσ max + β (cid:54) = 0. In particular, wemust have (cid:82) σ max + β Ω = ν + ω ( β ) > P . Hence we need ω ( β ) ≥ ω ( β ) = 0. Note that Ω defines a connection on P → P whose horizontal spaces are the Ω-orthogonals to the vertical tangent spaces.Further, this connection does not depend on the choice of function κ . Choose an Ω-compatible almost complex structure J on P . We may assume that both the fibersof P → P and the horizontal subspaces are J -invariant. Thus J induces a compactfamily J z , z ∈ P , of ω -tame almost complex structures on the fiber M . Let (cid:126) be theminimum of the energies of all nonconstant J z -holomorphic spheres in M for z ∈ P .Standard compactness results imply that (cid:126) >
0. Note that if we change κ (keeping Ωsymplectic) J is remains Ω-compatible. Hence we may suppose that ν := (cid:82) s max Ω < (cid:126) . We now claim that the only J -holomorphic sections of P with energy ≤ ν are theconstant sections x × P for x ∈ F max . To see this, decompose tangent vectors to P as v + h where v is tangent to the fiber and h is horizontal. Then, because of our choiceof J , at a point ( x, z ) over the fiber at z = ( r, t ) ∈ D + ,Ω( v + h, J v + J h ) = ω ( v, J v ) + Ω( h, J h ) ≥ Ω( h, J h ) ≥ r (cid:0) κ ( r , t ) − ρ (cid:48) ( r ) max x ∈ M K t ( x ) (cid:1) dr ∧ dt ( h, J h ) . Here the first inequality is an equality only along a section that is everywhere horizontal,while the second is an equality only if the section is contained in F max × P . (Recallthat by assumption K t ( x ) < K max for x / ∈ F max .) Similarly, the corresponding integralover D − is ≥ ε with equality only if the section is constant. Comparing with equation(2.6) we see that the energy of a section is ≥ ν with equality only if the section isconstant. This proves the claim. AMILTONIAN S -MANIFOLDS ARE UNIRULED 13 Now suppose that the class β with ω ( β ) = 0 contributes to S ( γ ). Then the class σ max + β of energy ν must be represented by a J -holomorphic stable map consistingof a section, possibly with some other bubble components in the fibers. Since eachbubble has energy ≥ (cid:126) > ν , the stable map has no bubbles and hence by the precedingparagraph must consist of a single constant section in the class σ max . Thus β = 0. Thiscompletes the proof of (a).To prove (b) it suffices to check that the coupling class u γ on P γ is represented bythe form Ω that equals ω on M × D − and ω − d ( ρ ( r ) K t ) ∧ dt, on M × D + . For this to hold we need that (cid:82) P Ω n +10 = 0. But this follows easily from the fact that K t is normalized. (cid:3) Corollary 2.12.
Suppose that the fixed maximum F max of γ is a divisor and that γ isan effective circle action near F max . Then (2.7) S ( γ ) = F max ⊗ q t K max + (cid:88) β ∈ H ( M ; Z ) , ω ( β ) > a β ⊗ q − c ( β ) t K max − ω ( β ) . Proof.
By Proposition 2.11, it remains to check that the contribution a max of the sec-tions in class σ max to S ( γ ) is F max . We saw there that the moduli space of (un-parametrized) holomorphic sections in class σ max can be identified with F max . In partic-ular, it is compact. Because S acts with normal weight − c Vert1 ( σ max ) = − m max = − a max · a := (cid:10) a (cid:11) Pσ max = F max · a for all a ∈ H ∗ ( M ). (cid:3) If (
M, ω ) is not strongly uniruled then, by Lemma 2.1, the unit S ( γ ) must be of theform 1l ⊗ λ + x where x ∈ H < n ( M ) and λ ∈ Λ is also a unit. The previous lemmashows that λ = rt K max − κ + l . o . t . where r ∈ Q is nonzero, κ = ω ( β ) >
0, and l.o.t.denotes lower order terms. We need to estimate the size of κ for the blow up ( (cid:102) M , (cid:101) ω ).In the course of the argument we shall also need to consider the bundles P (cid:48) → P and (cid:101) P (cid:48) → P defined by γ (cid:48) := γ − and its blow up ( (cid:101) γ ) − . We will represent the inverse loop γ − by the path { φ − t } , so that it is generated by the Hamiltonian K (cid:48) t := − K − t . Iffor any normalized Hamiltonian K t we set K min := (cid:82) min x ∈ M K t ( x ) dt, then K (cid:48) max = − K min , and K (cid:48) min = − K max . Similarly, If (cid:101) K t and (cid:101) K (cid:48) t generate the blow up loop (cid:101) γ and (cid:101) γ (cid:48) := (cid:101) γ − then (cid:101) K (cid:48) max = − (cid:101) K min , and (cid:101) K (cid:48) min = − (cid:101) K max . The above remarks imply that the coefficients λ, λ (cid:48) of 1l in the formulas for S ( γ ), S ( γ (cid:48) ) have the form λ = r t K max − κ + l . o . t ., λ (cid:48) = r (cid:48) t K (cid:48) max − κ (cid:48) + l . o . t . for some nonzero r , r (cid:48) ∈ Q . Similarly, we write the coefficients (cid:101) λ, (cid:101) λ (cid:48) of 1l in the formulasfor S ( (cid:101) γ ), S ( (cid:101) γ (cid:48) ) as (cid:101) λ = (cid:101) r t e K max − e κ + l . o . t ., (cid:101) λ (cid:48) = (cid:101) r (cid:48) t e K (cid:48) max − e κ (cid:48) + l . o . t . where (cid:101) r , (cid:101) r (cid:48) are nonzero. Corollary 2.12 implies that if [ ω ] is assumed to be integral then κ ∈ Z because it is a value ω ( β ) of [ ω ] on an integral class β ∈ H ( M ; Z ). Similarly (cid:101) κ is a value of (cid:101) ω on H ( (cid:102) M ; Z ). On the other hand, because the loop γ (cid:48) need not havea fixed maximum κ (cid:48) need not be a value of ω on H ( (cid:102) M ; Z ), although − K (cid:48) max − κ (cid:48) is avalue of the coupling class u γ (cid:48) on H ( P (cid:48) ; Z ).In the next lemma we assume that ( (cid:102) M , (cid:101) ω ) is not strongly uniruled. Hu [14, Thm. 1.2]showed that ( M, ω ) is strongly uniruled only if ( (cid:102)
M , (cid:101) ω ) is also (cf. Lemma 3.3). Henceour hypothesis implies that ( M, ω ) also is not strongly uniruled.
Lemma 2.13.
Let ( (cid:102) M , (cid:101) ω ) be the blow up of ( M, ω ) at a point of F max . Then: (i) K max − K min = (cid:101) K max − (cid:101) K min . (ii) If ( (cid:102) M , (cid:101) ω ) is not strongly uniruled then κ + κ (cid:48) = (cid:101) κ + (cid:101) κ (cid:48) = K max − K min . (iii) If ( (cid:102) M , (cid:101) ω ) is not strongly uniruled, then (cid:101) κ = κ and (cid:101) κ (cid:48) = κ (cid:48) .Proof. Observe that the S action near the point x max ∈ F max in M extends to alocal toric structure in some neighborhood U of x max that preserves the submanifold U ∩ F max . Hence we can form ( (cid:102) M , (cid:101) ω ) by cutting off a neighborhood of the vertex ofthe local moment polytope corresponding to x max . Since we do not cut off the wholeof F max this does not change the length max x ∈ U K t ( x ) − min x ∈ U K t ( x ) of the image ofthe normalized local S -moment map at each t . This proves (i).The identity 1l = S ( γ ) ∗ S ( γ (cid:48) ) = (1l ⊗ λ + x )(1l ⊗ λ (cid:48) + x (cid:48) )implies that λ (cid:48) = λ − . By Lemma 2.1 this is possible only if K max − κ + K (cid:48) max − κ (cid:48) = 0.Therefore κ + κ (cid:48) = K max − K min . (Here we use the fact that K (cid:48) max = − K min .) Asimilar argument shows that (cid:101) κ + (cid:101) κ (cid:48) = (cid:101) K max − (cid:101) K min . In view of (i), this proves (ii).The proof of (iii) is deferred to the end of this section (after Corollary 2.15). (cid:3) We now prove the key result of this section that ties the current ideas to the previ-ously developed algebra.
Proposition 2.14.
Suppose that ( M, ω ) is not uniruled, and that [ ω ] ∈ H ∗ ( M ; Z ) . Sup-pose further that γ has a maximal fixed point set F max that is a divisor, and that γ actsnear F max as an effective circle action. Then there is a unit u ∈ E n := QH n ( (cid:102) M ) / I of the form u = rE ⊗ qt κ + 1l + x, r (cid:54) = 0 , where x = (cid:80) κ i ≥ δ E j i ⊗ q j i t κ − κ i for some < j i < n and κ ∈ Γ ω is positive.Proof. Let ( (cid:102)
M , (cid:101) ω ) be the one point blow up of ( M, ω ) with blow up parameter δ asbefore. Then [ (cid:101) F max ] = [ F max ] − E where E is (the class of) the exceptional divisor AMILTONIAN S -MANIFOLDS ARE UNIRULED 15 and we consider [ F max ] ∈ H n − ( M ) ≡ H n − ( (cid:102) M (cid:114) E ). Denote by (cid:101) K t the normalizedgenerating Hamiltonian for the lift (cid:101) γ of γ to (cid:102) M . By Corollary 2.12 the Seidel elementof (cid:101) γ has the form S ( (cid:101) γ ) = (cid:101) F max ⊗ q t e K max + (cid:88) e β ∈ H ( f M ; Z ) , e ω ( e β ) > a β ⊗ q − c ( e β ) t e K max − e ω ( e β ) = − E ⊗ qt e K max + (cid:88) i E j i ⊗ q j i t e K max − e κ i (mod I ) . (2.8)Denote the coefficient of 1l = E in this formula by (cid:101) λ ∈ Λ univ . Since ( (cid:102) M , (cid:101) ω ) is the blowup of ( M, ω ) it is not uniruled by Hu–Li–Ruan [15, Thm 1.1]. Therefore the fact that S ( (cid:101) γ ) is a unit implies by Lemma 2.1 that (cid:101) λ is also a unit. Write (cid:101) λ = t e K max − e κ ( (cid:101) r + y )where (cid:101) r (cid:54) = 0 and y is a sum of negative powers of t . Lemma 2.13(iii) implies that (cid:101) κ = κ = ω ( β ) > β ∈ H ( M ). Now let u = S ( (cid:101) γ ) ⊗ (cid:101) λ − mod I . This hasthe required form. (cid:3) In the next corollary we denote the exceptional divisor in the fiber (cid:102) M of (cid:101) P (cid:48) → P by E , and write E j for its j -fold intersection product in (cid:102) M considered (where appropriate)as a class in (cid:101) P (cid:48) . Corollary 2.15. (i) Let n ≥ . Then under the conditions of Proposition 2.14 thereis σ ∈ H ( P (cid:48) ; Z ) such that (cid:10) E (cid:11) e P (cid:48) σ − ε (cid:54) = 0 .(ii) Similarly, if n = 2 there is σ ∈ H ( P (cid:48) ; Z ) such that (cid:10) E (cid:11) e P (cid:48) σ − ε (cid:54) = 0 .Proof. (i) By Lemma 2.8 u − has the form 1 − s n − t δ + r s n − t δ − κ (cid:0) . o . t . (cid:1) . But u − = S ( (cid:101) γ (cid:48) ) ⊗ (cid:101) λ mod I . Therefore S ( (cid:101) γ (cid:48) ) = (cid:101) λ − (cid:16) − s n − t δ + 1 r s n − t δ − κ (cid:0) . o . t . (cid:1)(cid:17) (mod I ) . By Lemma 2.13 (iii) (cid:101) λ − is a series in t with highest order term t − e K max + κ . Thereforethe largest κ such that the coefficient of s n − t κ in S ( (cid:101) γ (cid:48) ) is nonzero is κ = − (cid:101) K max + κ + δ − κ = − (cid:101) K max + δ. Since the coefficient of t K (cid:48) max − κ (cid:48) in S ( γ (cid:48) ) is nonzero, it follows from the definition of S in equation (2.7) that there is a section σ (cid:48) of P (cid:48) such that − u γ (cid:48) ( σ (cid:48) ) = K (cid:48) max − κ (cid:48) .We will take this as our reference section, writing all other sections in P (cid:48) in the form σ (cid:48) + β where β ∈ H ( M ; Z ) and those of (cid:101) P (cid:48) as σ (cid:48) + (cid:101) β where (cid:101) β ∈ H ( (cid:102) M ; Z ).Observe that − u e γ (cid:48) ( σ (cid:48) ) = (cid:101) K (cid:48) max − (cid:101) κ (cid:48) = (cid:101) K (cid:48) max − κ (cid:48) = − (cid:101) K max + κ , where the second equality holds by Lemma 2.13 (iii), and the third by Lemma 2.13(ii) and the identity (cid:101) K (cid:48) max = − (cid:101) K min . Since the coefficient of t κ in S ( (cid:101) γ (cid:48) ) is nonzero by hypothesis, there is (cid:101) β ∈ H ( (cid:102) M ; Z ) such that κ = − u e γ (cid:48) ( σ (cid:48) + (cid:101) β ) = − u e γ (cid:48) ( σ (cid:48) ) − (cid:101) ω ( (cid:101) β ) . Therefore (cid:101) ω ( (cid:101) β ) = − u e γ (cid:48) ( σ (cid:48) ) − κ = − (cid:101) K max + κ + (cid:101) K max − δ = κ − δ. Since I e ω is injective (cid:101) β must have the form β − ε for some β ∈ H ( M ). Hence we maywrite the class σ (cid:48) + (cid:101) β as σ (cid:48) + β − ε = σ − ε for some σ = σ (cid:48) + β ∈ H ( P (cid:48) ). Thereforethe coefficient of s n − t κ in S ( (cid:101) γ (cid:48) ) arises from a nonzero invariant of the form (cid:10) E (cid:11) e P (cid:48) σ − ε where σ ∈ H ( P (cid:48) ). (To check the power of E here, note that E · M E n − = − (cid:3) Remark 2.16. (i) By construction − u γ (cid:48) ( σ ) = − K min − κ (cid:48) − κ = − K max = − u γ ( s (cid:48) min ),where s (cid:48) min is the section of P (cid:48) that is blown up to get (cid:101) P (cid:48) . Therefore the classes σ and σ (cid:48) min are equal.(ii) The reader might wonder why we ignored the coefficient of s n − t δ in S ( (cid:101) λ (cid:48) ). But,reasoning as above, one can check that this term gives rise to a nonzero invariant ofthe form (cid:10) E (cid:11) e P (cid:48) σ − ε , where σ = σ (cid:48) . When one blows (cid:101) P (cid:48) down to P (cid:48) this corresponds to the invariant (cid:10) pt (cid:11) Pσ ,which we already know is nonzero. Hence this term gives no new information.We finish this section by proving Lemma 2.13 (iii). For this we need the followingpreliminary result. Lemma 2.17.
Let P, (cid:101) P , P (cid:48) , (cid:101) P (cid:48) be as above. Then: (i) For any section class σ ∈ H ( P ) (cid:10) pt (cid:11) Pσ = (cid:10) pt (cid:11) e Pσ , where on RHS we consider σ ∈ H ( (cid:101) P ) . (ii) Similarly, for any section class σ ∈ H ( P (cid:48) ) (cid:10) pt (cid:11) P (cid:48) σ = (cid:10) pt (cid:11) e P (cid:48) σ , where on RHS we consider σ ∈ H ( (cid:101) P (cid:48) ) .Proof. Hu showed in [14, Thm. 1.5] that if (cid:101) Q is the blow up of Q along an embedded2-sphere C in a class with c ( C ) ≥ σ ∈ H ( Q ) with homological constraints in H ∗ ( Q (cid:114) C ) remain unchanged. (The proofis similar to that of Proposition 2.19 below.) The result now follows because (cid:101) P is theblow up of P along the section s max with c ( s max ) = 1, while (cid:101) P (cid:48) is the blow up of P (cid:48) AMILTONIAN S -MANIFOLDS ARE UNIRULED 17 along a section s (cid:48) min with c ( s (cid:48) min ) = 3 . (Cf. § (cid:101) P (cid:48) .) (cid:3) Proof of Lemma 2.13 (iii).
We need to show that κ = (cid:101) κ and (cid:101) κ (cid:48) = κ (cid:48) . Since κ + κ (cid:48) = (cid:101) κ + (cid:101) κ (cid:48) by part (ii) of Lemma 2.13, it suffices to show that (cid:101) κ ≤ κ and (cid:101) κ (cid:48) ≤ κ (cid:48) .But by equations (2.2) and (2.3), κ is the minimum of ω ( β ) over all classes σ = σ max + β such that (cid:10) pt (cid:11) Pσ (cid:54) = 0. Similarly, (cid:101) κ is the minimum of (cid:101) ω ( (cid:101) β ) over all classes (cid:101) σ = (cid:101) σ max + (cid:101) β such that (cid:10) pt (cid:11) P e σ (cid:54) = 0. But σ max = (cid:101) σ max , and by Lemma 2.17(i) everyclass σ with nonzero invariant in P also has nonzero invariant in (cid:101) P . Hence we musthave (cid:101) κ ≤ κ .Now consider the fibrations P (cid:48) → P , (cid:101) P (cid:48) → P . To compare κ (cid:48) with (cid:101) κ (cid:48) we needto use suitable reference sections that do not change under blow up. Instead of themaximal sections s max , (cid:101) s max we use the corresponding minimal sections s (cid:48) min and (cid:101) s (cid:48) min that represent the classes σ (cid:48) min and (cid:101) σ (cid:48) min . Thus we write every section class σ (cid:48) of P (cid:48) as σ (cid:48) min + β , where β ∈ H ( M ). Note that σ (cid:48) min is taken to (cid:101) σ (cid:48) min under the naturalinclusion H ( P (cid:48) ) → H ( (cid:101) P (cid:48) ).The symplectomorphism from the fiber connect sum P P (cid:48) to the trivial bundletakes the connect sum of the sections s max and s (cid:48) min to the trivial section P × pt andthe connect sum of the coupling classes u γ u γ (cid:48) to the coupling class pr ∗ M ( ω ) of thetrivial bundle P × M pr −→ P . Hence u γ ( s max ) + u γ (cid:48) ( s (cid:48) min ) = pr ∗ M ( ω )( P × pt ) = 0 . Therefore u γ (cid:48) ( s (cid:48) min ) = − u γ ( s max ) = K max . A similar argument applied to the blowupsshows that u e γ (cid:48) ( s (cid:48) min ) = (cid:101) K max .By Proposition 2.11 we can interpret κ (cid:48) as the minimum of K (cid:48) max + u γ (cid:48) ( σ (cid:48) ) over allclasses σ (cid:48) such that (cid:10) pt (cid:11) P (cid:48) σ (cid:48) (cid:54) = 0. Writing σ (cid:48) as σ (cid:48) min + β and using K (cid:48) max = − K min , wefind K (cid:48) max + u γ (cid:48) ( σ (cid:48) min + β ) = K max − K min + ω ( β ) . Hence κ (cid:48) is the minimum of K max − K min + ω ( β ) over the corresponding set B of classes β . Similarly, (cid:101) κ (cid:48) is the minimum of (cid:101) K max − (cid:101) K min + (cid:101) ω ( (cid:101) β ) over a set of classes (cid:101) β that,by Lemma 2.17 (ii) and the fact that σ (cid:48) min = (cid:101) σ (cid:48) min , includes B . Since K max − K min = (cid:101) K max − (cid:101) K min , we find that (cid:101) κ (cid:48) ≤ κ (cid:48) as required. (cid:50) Proof of the main results.
Suppose that γ is a loop of Hamiltonian symplecto-morphisms of ( M, ω ) with a fixed maximum near which it is an effective circle action.A standard Moser argument implies that we can assume that I ω is injective. The nextlemma shows that we can always arrange for the other conditions of Proposition 2.14to hold. Perturb the given form to a suitable class and then average over the S action. Lemma 2.18.
Suppose that the submanifold F max of M is a fixed maximum of theloop γ and that nearby γ is an effective S action. Then we can blow up along F max to achieve an action on some blow up of M for which the new F max is a divisor withnormal weight m = 1 .Proof. Suppose that the weights of the action normal to F max are ( − m , . . . , − m k )where 0 < m ≤ m ≤ · · · ≤ m k . Let (cid:102) M be the blow up of M along F max . Thenthe weights of the induced action normal to the new maximal fixed point set are − m , − ( m i − m ) , . . . , − ( m k − m ), where i is the smallest index such that m i > m .Therefore by repeated blowing up along the maximal fixed point of the moment we canreduce to the case when there is just one normal weight, i.e. F max is a divisor. Thenormal weight is 1 since the action is effective. (cid:3) Corollary 2.15 summarizes the information given to us by the Seidel representation.To make use of it, we need one more result relating the Gromov–Witten invariants of (cid:101) P (cid:48) and P (cid:48) . This is proved in § Proposition 2.19.
Suppose that ( M, ω ) is not uniruled.(i) If n ≥ and (cid:10) E (cid:11) e P (cid:48) σ − ε (cid:54) = 0 for some section class σ ∈ H ( P (cid:48) ; Z ) then (cid:10) τ pt (cid:11) P (cid:48) σ (cid:54) = 0 .(ii) If n = 2 and (cid:10) E (cid:11) e P (cid:48) σ − ε (cid:54) = 0 then for some section s in P (cid:48) at least one of (cid:10) pt, s (cid:11) P (cid:48) σ and (cid:10) τ pt (cid:11) P (cid:48) σ is nonzero. We next show that the situation discussed in the above proposition cannot occur: infact, if the conclusions hold (
M, ω ) must be strongly uniruled.
Lemma 2.20.
If there is an element γ ∈ π (Ham( M )) such that the descendent in-variant (cid:10) τ k pt (cid:11) Pσ (cid:54) = 0 for some k > and some section class σ in P := P γ then ( M, ω ) is strongly uniruled.Proof. As explained in § (cid:10) τ k pt, M, M (cid:11) Pσ = (cid:88) i,α + α = σ (cid:10) τ k − pt, ξ i (cid:11) Pα (cid:10) ξ ∗ i , M, M (cid:11) Pα , where α j ∈ H ( P ) and ξ i runs over a basis for H ∗ ( P ) with dual basis ξ ∗ i . Because thesymplectic manifold ( P, Ω) supports an Ω-tame almost complex structure J such thatthe projection P → P is holomorphic, the only classes in H ( P ) with nontrivial GWinvariants project to nonnegative multiples of the generator of H ( P ). Thus one of theclasses α j is a section class and the other is a fiber class, i.e. a class in H ( M ). But if α is a fiber class then it cannot meet two separate copies of M . Hence any nonzeroterm in this expansion must have section class α . Then α is a fiber class, and so byLemma 3.8 (cid:10) τ k − pt, ξ i (cid:11) Pα = (cid:10) τ k − pt, ξ i ∩ M (cid:11) Mα . But now by repeated applications of equation (3.9) one finds that M is strongly unir-uled; cf. the proof of Lemma 4.8. (cid:3) AMILTONIAN S -MANIFOLDS ARE UNIRULED 19 The following lemma is proved by a similar argument; see § § Lemma 2.21.
Suppose that ( M, ω ) is the blow up of a symplectic -manifold that isnot rational or ruled. Then there is no γ ∈ π (Ham( M )) such that the correspondingfibration P → P has a section s and a section class σ with (cid:10) pt, s (cid:11) Pσ (cid:54) = 0 . Proof of Theorem 1.1.
Suppose first that dim M ≥ M, ω ) is not uniruled.By Lemma 2.18 we may assume that we are in the situation of Proposition 2.14. ThenCorollary 2.15(i) together with Lemma 2.20 implies that (
M, ω ) is strongly uniruled.This contradiction shows that the initial hypothesis must be wrong: in other words,(
M, ω ) is uniruled.When dim M = 4 the argument is similar, except that we use Lemma 2.21 insteadof Lemma 2.20. (cid:50) Finally note that Proposition 1.4 follows from the next lemma.
Lemma 2.22.
Suppose that the loop γ in Ham(
M, ω ) has a nondegenerate fixed max-imum at x max . Suppose also that the linearized flow A t , t ∈ [0 , , at x max is homotopicthrough positive paths to a linear circle action. Then γ is homotopic through Hamil-tonian loops with fixed maximum at x max to a loop γ (cid:48) that is a circle action near x max .Proof. Identify a neighborhood of x max with a neighborhood of { } in R n with itsstandard symplectic form ω . Step 1:
The loop γ =: γ is homotopic through loops γ s = { φ st } with fixed maximumat { } to a loop γ that equals the linear flow A t in some neighborhood of { } . Let γ = { φ t } with generating Hamiltonian H t . By assumption H t = H t + O ( (cid:107) x (cid:107) )where for each t there is a positive definite symmetric matrix Q t such that H t ( x ) = − x T Q t x . We shall choose φ st of the form φ t ◦ g st ◦ h st where • for each s ∈ [0 , g st , t ∈ [0 , , consists of diffeomorphisms with support insome small ball B r := B r (0) such that for all s, t we have dg st (0) = id, g t = id, g t = ( φ t ) − ◦ A t in B r ; • h st has support in the ball B r and is such that h ∗ st ( g ∗ st ω ) = ω . Moreover h t = id on B r .Choose g st satisfying all conditions above and with support in int B r . Since dg st (0) = id , we may write g st ( x ) = x + O ( (cid:107) x (cid:107) ). Hence we may arrange that forsome constants c , c we have (cid:107) g st ( x ) − x (cid:107) ≤ c (cid:107) x (cid:107) and | g ∗ st ω − ω | ≤ c (cid:107) x (cid:107) . Notethat we may assume that these constants c i , as well as all subsequent ones, dependonly on the initial path φ t , i.e. there are constants R > c i such that suitable g st exist for all r < R . Then, reducing R as necessary, c (cid:107) x (cid:107) is arbitrarily small on B r so that we may assume that the forms ω st := g ∗ st ω are all nondegenerate. Let ρ st := ddt ( g ∗ st ω ). Since g st = id for all s near 0 by construction, ρ t = 0. Similarly, ρ t = 0 in B r . Further for some constant c as above | ρ st | ≤ c (cid:107) x (cid:107) . Now, ρ st := dβ st , where for each s, tβ st ( x ) = (cid:90) ρ st ( ∂ r , · )( λx ) dλ, x ∈ B r , and ∂ r is the radial vector field in R n . Hence β t ≡
0, and β t = 0 in B r . Moreover,each 1-form β st ( x ) satisfies | β st ( x ) | ≤ c (cid:107) x (cid:107) and, because ρ st has support in int B r ,is closed and independent of r := (cid:107) x (cid:107) near ∂B r . Therefore, near ∂B r we may write β st = df st where f st is a function on S n − of norm ≤ c r . Therefore we may extend β st by d ( α ( r ) f st ) for a suitable cut off function α so that it has support in B r andstill satisfies an estimate | β st ( x ) | ≤ c (cid:107) x (cid:107) .Now construct h st , t ∈ [0 , , for each fixed s by the usual Moser homotopy method sothat ω st ( ddt h st , · ) = β st .Then h st satisfies the required conditions. Moreover (cid:107) ddt h st (cid:107) ≤ c (cid:107) x (cid:107) . Hence the Hamiltonian G st that generates g st ◦ h st satisfies | G st | ≤ c (cid:107) x (cid:107) for (cid:107) x (cid:107) ≤ r . The generating Hamiltonian for φ st := φ t ◦ g st ◦ h st is H st ( x ) := H t ( x ) + G st ( φ − t ( x )) = H t ( x ) + O ( (cid:107) x (cid:107) ) . Since c is independent of r , we can now choose r so small that x = 0 is still theglobal maximum of H st . (The size of r will depend on the smallest eigenvalues of thematrices Q t and also on the second derivatives of the φ t , i.e. the cubic term in H t .)This completes Step 1.By assumption there is a homotopy of the path { A t } := { A t } to a circle action { A t } through positive paths { A st } t ∈ [0 , . Denote by Q st the corresponding family ofpositive definite matrices. Reparametrize this homotopy with respect to s to stretch itout over the interval s ∈ [1 , N ] for some large N to be chosen later. Note that φ t = A t on B r . Step 2:
There is a sequence r k , k ≥ , satisfying < r k +1 < r k for all k and a finitesequence of homotopies γ s = { φ st } , s ∈ [ k, k + 1] , k = 1 , . . . , N − , of Hamiltonian loopswith support in B r k so that: • for all k ∈ [1 , N − φ k +1 t = A k +1 t on B r k +1 and φ k +1 t = A k t on B r k (cid:114) B r k +1 . • for each s the Hamiltonian loop φ st , t ∈ [0 , , has fixed maximum at { } . If r k and φ k t are given, an obvious modification of the procedure described in Step1 gives suitable r k +1 and φ k +1 t provided that the derivative dds A st , s ∈ [ k, k + 1] , isnot too large. As before, the idea is first to use the linear homotopy A st for s ∈ [ k, k + 1] to construct a smooth interpolation g st between A k +1 t on B r k +1 and A k t ona neighborhood of ∂B r k , and then correct using a Moser homotopy. Then (cid:107) g st ( x ) − A k t ( x ) (cid:107) ≤ c (cid:107) x (cid:107) , where c depends only on dds A st . (Note that c does not depend on r k because the linear maps A st are invariant under homotheties.) But we can arrangethat dds A st is as small as we like by choosing N sufficiently large. In order that theresulting loops φ st have fixed maximum at { } , the permissible size for c (and hence N ) will depend on the smallest eigenvalue of the compact family of matrices Q st . AMILTONIAN S -MANIFOLDS ARE UNIRULED 21 Step 2 completes the proof, since the loop φ (cid:48) := φ Nt satisfies the requirements of thelemma. (cid:3) Gromov–Witten invariants and blowing up
We now prove the results on Gromov–Witten invariants needed above, i.e. Propo-sitions 2.5 and 2.19. In [15, Theorem 5.15] Hu–Li–Ruan establish a correspondencebetween the relative genus g invariants of the blow up ( (cid:102) M , E ) and certain correspond-ing sets of absolute invariants of M . Proposition 2.5 could be proved using a specialcase of this general correspondence. However, instead of quoting their result we shallreprove the parts we need, since we do not need the full force of their results and alsoneed some other related results.We begin this section with a discussion of relative GW invariants since this will beour main tool. For more details see Li–Ruan [23, § § et al. [4, §
10] for relevant compactness results. After stating the decom-position formula, we prove Proposition 3.5, a generalization of Proposition 2.5 thatcharacterizes uniruled manifolds M in terms of properties of the one point blow up (cid:102) M .Finally we prove Proposition 2.19 by using the decomposition formula for sectionclasses of the bundle P → P .3.1. Relative invariants of genus zero.
Consider a pair (
X, D ) where D is a divisorin X , i.e. a codimension 2 symplectic submanifold. The relative invariants count con-nected J -holomorphic curves in the k -fold prolongation X k (defined below) of X . Here J is an ω -tame almost complex structure on X satisfying certain normalization condi-tions along D . In particular D is J -holomorphic, i.e. J ( T D ) ⊂ T D . The invariantsare defined by first constructing a compact moduli space M := M X,Dβ,d ( J ) of genus zero J -holomorphic curves C in class β ∈ H ( X ) as described below and then integratingthe given constraints over the corresponding virtual cycle M [ vir ] . Here d = ( d , . . . , d r )is a partition of d := β · D ≥ X , i.e. equivalenceclasses C = [Σ , u, . . . ] of stable maps to X , that intersect the divisor D at r points withmultiplicities d i . More precisely, each such curve has ( k + 1) levels C i for some k ≥ C in X (cid:114) D and the higher levels C i (sometimes called bubbles) inthe C ∗ -bundle L ∗ D (cid:114) D where L ∗ D → D is the dual of the normal bundle to D . Thewhole curve C therefore lies in a space X k called the k th prolongation of X , which isdefined as follows. Identify L ∗ D (cid:114) D with the complement of the sections D , D ∞ of theruled manifold π Q : Q := P ( C ⊕ L ∗ D ) → D, where the zero section D := P ( C ⊕ { } ) has normal bundle L ∗ D and the infinity section D ∞ := P ( { } ⊕ L ∗ D ) has normal bundle L D . Think of the initial divisor D ⊂ X as theinfinity section D ∞ at level 0. Then the prolongation X k is simply the disjoint unionof X with k copies of Q , but it is useful to think that for each i ≥ D i of the i th level is identified with the infinity section D i − ∞ of the preceding level. The most important divisor in X k is D k ∞ , which carries the relative constraints andhence plays the role of the relative divisor.For each i > i th level curve C i along the zero section D i match with those of C i − along D i − ∞ . The final level C k carries the relative marked points which are mapped to D k ∞ . Assuming there are noabsolute marked points, each level of C is an equivalence class of stable maps[Σ i , u i , y , . . . , y r i , y ∞ , . . . , y r i ∞ ∞ ] , in some class β i , where the internal relative marked points y , . . . , y r i are mapped to D i with multiplicities m i that sum to β i · D i and the marked points y ∞ , . . . , y r i ∞ ∞ aretaken to D i ∞ with multiplicities m i ∞ that sum to β i · D i ∞ . Note that the components C i and C i +1 match along D i ∞ = D i +1 , only if the multiplicities m i ∞ and m i +1 , agree.In symplectic field theory such multilevel curves are called buildings: see [4, § C in M might also have some absolute marked points; these could lie onany of the levels C i but must be disjoint from the relative marked points. We requirefurther that each component C i be stable, i.e. have a finite group of automorphisms.For C this has the usual meaning. However, when i ≥ i curves C i , C (cid:48) i if they lie in the same orbit of the fiberwise C ∗ action; i.e. given representingmaps u i : (Σ i , j ) → Q and u (cid:48) i : (Σ (cid:48) i , j (cid:48) ) → Q , the curves are identified if there is c ∈ C ∗ and a holomorphic map h : (Σ i , j ) → (Σ (cid:48) i , j (cid:48) ) (preserving all marked points) such that u i ◦ h = c u (cid:48) i . Thus L ∗ D (cid:114) D should be thought of as a rubber space.Note that although the whole curve is connected, the individual levels need not bebut should fit together to form a genus zero curve. The homology class β of such acurve is defined to be the sum of the homology class of its principal component withthe projections to D of the classes of its higher levels. When doing the analysis it is best to think that the domains and targets of the curveshave cylindrical ends. However, their indices are the same as those of the correspondingcompactified curves; see [23, Prop 5.3]. Thus the (complex) dimension of the modulispace M X,Dβ,k, ( d ,...,d r ) of genus zero curves in class β with k absolute marked points and r relative ones is(3.1) n + c X ( β ) + k + r − − r (cid:88) i =1 ( d i −
1) = n + c X ( β ) + k + 2 r − − d. Here k + r − d i − d i since, as far as a dimensional countis concerned, what is happened at such a point is that d i of the d := β · D intersectionpoints of the β -curve with D coincide. Hence a level C i cannot consist only of multiply covered curves z (cid:55)→ z k in the fibers C ∗ of L ∗ D (cid:114) D because these are not stable. This is the class of the curve in X obtained by gluing all the levels together. When k = 0 thecurve only has one level and so the relative constraints lie along D ⊂ X . Ionel–Parker [16] work withthe moduli space obtained by closing the space of 1-level curves, which in principle could give slightlydifferent invariants from the ones considered here. AMILTONIAN S -MANIFOLDS ARE UNIRULED 23 Example 3.1.
Let X = P P , the one point blow up of P and set D := E , theexceptional curve. Denote by π : X → P the projection, and fix another section H of π that is disjoint from E . Let J be the usual complex structure and M X,Eλ ( J ; p ) bethe moduli space of holomorphic lines through some point p ∈ H , where λ = [ H ] is theclass of a line. Since λ · E = 0 there are no relative constraints. Then M X,Eλ ( J ; p ) hascomplex dimension 1 and should be diffeomorphic to P . The closure of the ordinarymoduli space of lines in X though p contains all such lines together with one reduciblecurve consisting of the union of the exceptional divisor E with the fiber π − ( π ( p ))through p . But the elements of M X,Eλ ( J ; p ) do not contain components in E . Instead,this component becomes a higher level curve lying in Q = P ( C ⊕ O (1)) that intersects E = P ( C ⊕ { } ) in the point E ∩ π − ( π ( p )) and lies in the class λ Q of the line in Q ∼ = X . Note that modulo the action of C ∗ on Q there is a unique such bubble.Thus the corresponding two-level curve in M X,Eλ ( J ; p ) is a rigid object. Moreover,because λ Q projects to the class ε ∈ H ( E ) of the exceptional divisor, its homologyclass ( λ − ε ) + pr ( λ Q ) is ( λ − ε ) + ε = λ. The constraints for the relative invariants consist of homology classes b j in the divisor D (the relative insertions) together with absolute (possibly descendent) insertions τ i j a j where a j ∈ H ∗ ( X ) and i j ≥
0. We shall denote the (connected) relative genus zeroinvariants by:(3.2) (cid:10) τ i a , . . . , τ i q a q | b , . . . , b r (cid:11) X,Dβ, ( d ,...,d r ) , where a i ∈ H ∗ ( X ) , b i ∈ H ∗ ( D ) , and d := (cid:80) d i = β · D ≥ . (If β · D < β that intersect D to order d := ( d , . . . , d r ) in the b i , i.e the i th relative marked point intersects D toorder d i ≥ b i . Moreover, the insertion τ i occurring at the j th absolute marked point z j means that we add the constraint( c j ) i , where c j is the first Chern class of the cotangent bundle L j to the domain at z j . One can evaluate (3.2) (which in general is a rational number) by integrating anappropriate product of Chern classes over the virtual cycle corresponding to the modulispace of stable J -holomorphic maps that satisfy the given homological constraints andtangency conditions. This cut down virtual cycle has dimension equal to the index ofthe curves C satisfying these incidence conditions; if this dimension does not equal thetotal degree (cid:80) qj =1 i j of the descendent classes the invariant is by definition set equalto 0. For details on how to construct this virtual cycle see for example Li–Ruan [23] orHu–Li–Ruan [15]. (Also cf. Remark 1.7.)We shall need the following information about specific genus zero relative invariants. Lemma 3.2.
Let X = P n , and D = P n − be the hyperplane. Denote by λ the class ofa line. Then: Thus, if z j lies at the m th level, the fiber of L j at C is T ∗ z j (Σ m ). (i) if d > and n > (cid:10) | b , . . . , b r (cid:11) P n , P n − ,dλ,d = 0 , for any b i ∈ H ∗ ( P n − ) . (ii) (Hu [14, §
3] and Gathmann [7, Lemma 2.2])
Let (cid:102) M be a one point blow up withexceptional divisor E and suppose that β ∈ H ( M ) ⊂ H ( (cid:102) M ) so that β · E = 0 . Thenfor any a i ∈ H ∗ ( (cid:102) M (cid:114) E ) , the relative invariant (cid:10) a , . . . , a k | (cid:11) f M,E ,β equals the absoluteinvariant (cid:10) a , . . . , a k (cid:11) f M ,β . (iii) (Hu–Li–Ruan [15, Theorem 6.1]) The -point invariant (3.3) (cid:10) τ k pt | D j (cid:11) P n , P n − ,df, ( d ) , ≤ j ≤ n, d ≥ , is nonzero iff k = nd − j .Proof. (i) holds for dimensional reasons. We shall show that under the given conditionson n and d it is impossible to choose d and the b i so that M P n , P n − dλ,d has formal dimension0. Therefore the invariant vanishes by definition.By equation (3.1) the formal (complex) dimension of the moduli space of genus zerostable maps through the relative constraints b , . . . , b r is n + d ( n + 1) + r − − ( d − r ) + δ − rn, where δ is half the sum of the degrees of the b i . Thus 0 ≤ δ ≤ rn . Since d − r ≥ r >
0, we therefore need ( d − r ) n + n + 2 r + δ = 3, which implies d = r and n = r = 1.Since we assumed n >
1, this is impossible.Hu’s proof of (ii) uses the decomposition formula stated below. (His proof can bereconstructed by arguing as in the proof of Proposition 3.5.) The proof by Gathmannis more elementary but applies only in the case of projective algebraic manifolds. Claim(iii) is much deeper. It is proved by Hu–Li–Ruan using localization techniques. (cid:3)
Applications of the decomposition formula.
Our main tool is the decom-position rule of Li–Ruan [23, Thm 5.7] and (in a slightly different version) of Ionel–Parker [16]. So suppose that the manifold M is the fiber sum of ( X, D ) with (
Y, D + ),where the divisors D := D − and D + are symplectomorphic with dual normal bundles.Since this is the only case we shall need, let us assume that the absolute constraints canbe represented by cycles in M that do not intersect the inverse image of the divisor, i.e.that a i ∈ H ∗ ( X (cid:114) D ) , i ≤ q, and a i ∈ H ∗ ( Y (cid:114) D + ) , q < i ≤ p . For simplicity we assumealso that the map H ( M ) → H ( X ∪ D Y ) is injective. (This hypothesis is satisfiedwhenever H ( D ) = 0.) Further, let b i , i ∈ I, be a basis for H ∗ ( D ) = H ∗ ( D ; Q ) withdual basis b ∗ i for H ∗ ( D ). Then the genus zero decomposition formula has the followingshape: (cid:10) a , . . . , a p (cid:11) Mβ =(3.4) (cid:88) Γ ,d, ( i ,...,i r ) n Γ ,d (cid:10) a , . . . , a q | b i , . . . , b i r (cid:11) Γ ,X,Dβ ,d (cid:10) a q +1 , . . . , a p | b ∗ i , . . . , b ∗ i r (cid:11) Γ ,Y,D + β ,d . AMILTONIAN S -MANIFOLDS ARE UNIRULED 25 Here we sum with rational weights n Γ ,d over all decompositions d of d , all possibleconnected labelled trees Γ, and all possible sets i , . . . , i r of relative constraints. EachΓ describes a possible combinatorial structure for a stable map that glues to give a β -curve. Thus Γ is a disjoint union Γ ∪ Γ , where the graph Γ (resp. Γ ) describesthe part of the curve lying in some X k (resp. some Y (cid:96) ). Also β (resp. β ) is thepart of its label that describes the homology class; the pair ( β , β ) runs through alldecompositions such that the result of gluing the two curves in the prolongations X k and Y k along their intersections with the relative divisors gives a curve in class β. Moreover, there is a bijection between the labels { ( d i , b i ) ∈ N × H ∗ ( D ) } of the relativeconstraints in Γ and those { ( d i , b ∗ i ) ∈ N × H ∗ ( D + ) } in Γ . (These labels are calledrelative “tails” in [15].) Γ i need not be connected; if it is not, we define (cid:10) . . . | . . . (cid:11) Γ i tobe the product of the invariants defined by its connected components.Because the total curve has genus zero, each component of Γ has at most one relativetail in common with each component of Γ . In many cases we will be able to show thatΓ is connected, and hence that Γ has r components, one for each relative constraint.Most of the next result is known: part (i) follows from Theorem 1.2 in Hu [14], whilepart (ii) is very close to [15, Theorem 6.1]. Lemma 3.3.
Let ( (cid:102) M , (cid:101) ω ) be the one point blow up of ( M, ω ) with exceptional divisor E . (i) If ( M, ω ) is strongly uniruled, then ( (cid:102) M , (cid:101) ω ) is also. (ii) If (cid:10) a , a , pt (cid:11) f Mβ (cid:54) = 0 for some a i ∈ H ∗ ( M ) = H ∗ ( (cid:102) M (cid:114) E ) and some β ∈ H ( M ) ⊂ H ( (cid:102) M ) then ( M, ω ) is strongly uniruled.Proof. Consider (i). By hypothesis there is a nonzero invariant (cid:10) a , a , pt (cid:11) Mβ . Thinkof M as the fiber sum of ( (cid:102) M , E ) with ( P n , P n − ) and evaluate this invariant by thedecomposition formula, putting all constraints into (cid:102) M (cid:114) E . It follows that there is anonzero relative invariant (cid:10) a , a , pt | E j , . . . , E j s (cid:11) Γ , f M,E e β,(cid:96) with s ≥ (cid:10) | E n − j +1 , . . . , E n − j s +1 (cid:11) Γ , P n , P n − (cid:96)λ,(cid:96) .Note that (cid:101) β = β − (cid:96)ε , where (cid:96) = (cid:80) (cid:96) i . Since there are no absolute constraints in P n ,Lemma 3.2 (i) implies that s = (cid:96) = 0 and Γ = ∅ . But then Lemma 3.2 (ii) states thatthe above relative invariant equals the absolute invariant (cid:10) a , a , pt (cid:11) f Mβ . This proves (i).Now consider (ii). By Lemma 3.2 (ii) (cid:10) a , a , pt (cid:11) f Mβ = (cid:10) a , a , pt | (cid:11) f M,Eβ (cid:54) = 0. Nowuse the decomposition formula to evaluate (cid:10) a , a , pt (cid:11) Mβ , again putting all the absoluteconstraints in (cid:102) M (cid:114) E . Because there are no absolute constraints in ( P n , P n − ), it followsas before that there are no terms in this formula with Γ (cid:54) = ∅ . Hence (cid:10) a , a , pt (cid:11) Mβ = (cid:10) a , a , pt | (cid:11) f Mβ (cid:54) = 0 as required. (cid:3) Remark 3.4.
The proof of Proposition 3.5 given below can be adapted to show thatthe statement in Lemma 3.3 (ii) holds also in the case (cid:101) β · E = 1. However, it is notclear whether it continues to hold when (cid:101) β · E >
1. It is also not known whether thestrongly uniruled property persists under blow ups along arbitrary submanifolds.We now prove the following version of Proposition 2.5.
Proposition 3.5.
If there is a nonzero invariant (3.5) (cid:10) a , . . . , a p , E j , . . . , E j q (cid:11) f M e β , with a i ∈ H ∗ ( (cid:102) M (cid:114) E ) ∼ = H ∗ ( M ) , q ≥ and (cid:101) β · E > then ( M, ω ) is uniruled. We prove this by the method of Hu–Li–Ruan [15]. Thus we first think of (cid:102) M asthe fiber (or Gompf) sum of ( (cid:102) M , E ) with (
X, E + ), where X := P ( O ( − ⊕ C ) isthe projectivized normal bundle to E = P n − and E + = P ( O ( − ⊕ { } ) ∼ = P n − isthe section with positive normal bundle. Using this decomposition, we show that theexistence of the nonzero absolute invariant (3.5) implies the existence of a nontrivialrelative invariant (3.7) for the pair ( (cid:102) M , E ). Next we identify the blow down M as thefiber sum of ( (cid:102) M , E ) with the pair ( P n , P n − ) and deduce from the nontriviality of (3.7)the nontriviality of a suitable absolute invariant for M .In the following we denote by ε ∈ H ( (cid:102) M ) the class of the line in the exceptionaldivisor E , and write the class (cid:101) β ∈ H ( (cid:102) M ) as β − dε , where d := (cid:101) β · E and β ∈ H ( (cid:102) M (cid:114) E ) = H ( M ). The homology of the exceptional divisor E ∼ = P n − is generatedby the hyperplane class in H n − ( E ). As a homology class we identify E with the classin H n − ( M ) it represents. Hence the generator of H n − ( E ) is E , and ε = E n − . Therelative constraints for ( (cid:102) M , E ) have the form E j , j = 1 , . . . , n . With our conventions(which are different from [15]) the constraint in E dual to E j is − E n − j +1 , i.e.(3.6) E j · E ( − E n − j +1 ) = pt. Lemma 3.6.
If there is a nonzero absolute invariant of the form (3.5) for a given p ≥ and q ≥ then there is a nonzero (connected) relative invariant of the form (3.7) (cid:10) a (cid:48) , . . . , a (cid:48) m | E i , . . . , E i r (cid:11) f M,Eβ (cid:48) − (cid:96)ε, ( (cid:96) ...,(cid:96) r ) , i j ≥ , (cid:88) (cid:96) i = (cid:96) > , where ≤ m ≤ p , β (cid:48) ∈ H ( M ) and a i ∈ H ∗ ( M ) .Proof. Decompose (cid:102) M as the fiber sum of (cid:102) M with the ruled manifold ( X, E + ) as above.Apply the decomposition formula to evaluate the nonzero invariant (cid:10) a , . . . , a p , E j , . . . , E j q (cid:11) f M e β putting all the a , . . . , a p insertions into (cid:102) M (cid:114) E and the insertions E j , . . . , E j q into X (cid:114) E + . Because (cid:101) β / ∈ H ( E ), each term in the decomposition formula must correspond AMILTONIAN S -MANIFOLDS ARE UNIRULED 27 to a splitting (cid:101) β = (cid:101) β (cid:48) + α where 0 (cid:54) = (cid:101) β (cid:48) ∈ H ( (cid:102) M ). Hence there is a nonzero relativeinvariant for ( (cid:102) M , E ) in some class (cid:101) β (cid:48) (cid:54) = 0 that goes through all the constraints a , . . . , a p .It counts curves modelled on the possibly disconnected graph Γ and hence is a productof connected invariants, each of which has the form (3.7) for some subset of a , . . . , a p .Note that each such connected invariant has nonempty intersection with E because theinitial (cid:101) β -curve in (cid:102) M is connected and q ≥ (cid:3) Lemma 3.7.
If there is a nonzero relative invariant for ( (cid:102) M , E ) of the form (3.7) forsome m ≥ and r ≥ then there is a nonzero absolute invariant on M (cid:10) a , . . . , a t , τ k pt, . . . , τ k s pt (cid:11) Mβ for some β (cid:54) = 0 , t ≤ m , ≤ s ≤ r , k j ≥ and a i ∈ H ∗ ( M ) . Proof.
Choose a class β of minimal energy ω ( β ) such that there is a nonzero connectedrelative invariant in some class (cid:101) β := β − (cid:96)ε, (cid:96) > , of the form (3.7) with t ≤ m absoluteconstraints. Denote by s the smallest r > m = t , this class β , and r relative constraints. Denote the correspondingrelative constraints by E n − j +1 , . . . , E n − j s +1 (cf. equation (3.6)) and the multiplicitiesby (cid:96) = ( (cid:96) , . . . , (cid:96) s ). Note that (cid:80) (cid:96) j = (cid:96) >
0. In particular s > M into the fiber sum of (cid:102) M with Y = P n by identifying E ⊂ (cid:102) M with thehyperplane E + = P n − in Y . Apply the decomposition formula to evaluate(3.8) (cid:10) a , . . . , a t , τ k pt, . . . , τ k s pt (cid:11) Mβ , k i := n(cid:96) i − j i , putting all the point insertions into Y and the others into (cid:102) M (cid:114) E . We claim that (3.8)is nonzero.Note first that by Lemma 3.2(iii) there is a nonzero term T in this decompositionformula given by taking the product of the nonzero relative invariant (3.7) with s termsof the form (cid:10) τ k i pt | E j i (cid:11) P n , P n − (cid:96) i λ, ( (cid:96) i ) . We need to check that all other terms in this formulavanish.Consider an arbitrary nonzero term T (cid:48) in this formula. It is a product of a relativeinvariant for ( (cid:102) M , E ) modelled on a graph Γ and in some class β (cid:48) − dε with a relativeinvariant for ( P n , P n − ) in class dλ . Since the classes β (cid:48) − dε and dλ combine to give β (cid:48) , we must have β (cid:48) = β . The minimality of ω ( β ) implies that Γ is connected, sinceotherwise each of its components is in a class β i − d i ε with 0 < ω ( β i ) < ω ( β ).Now look at the other side. By Lemma 3.2 (i) each connected relative invariant in( P n , P n − ) must go through some absolute constraint. Hence Γ has at most s compo-nents. But it cannot have fewer than s components, because if it did Γ would havefewer than s relative constraints (since the genus zero requirement means that Γ meetseach component of Γ at most once), which contradicts the minimality of s . Thereforethere are s components of Γ and each goes through precisely one absolute constraint.Thus each component is a nonzero 2-point invariant of the form (cid:10) τ k i pt | E j (cid:11) P n , P n − df, ( d ) forsome i, d and 1 ≤ j ≤ n . But k i has a unique decomposition of the form n(cid:96) i − j i where ≤ j i ≤ n . Hence, by Lemma 3.2 (iii) the relative constraints in Γ are precisely the E j i , i = 1 . . . , s, with intersection multiplicities (cid:96) . Thus T (cid:48) = T . (cid:3) Proof of Proposition 3.5.
By Lemma 3.7 it suffices to show that M is uniruled iff aninvariant of the form (cid:10) τ i a , . . . , τ i k − a k − , τ i k pt (cid:11) Mβ is nonzero. Hu–Li–Ruan [15, The-orem 4.9] prove this by an inductive argument (similar to the proof of Proposition 4.4below) that is based on the identity (3.9). (cid:50) Blowing down section invariants.
It remains to prove Proposition 2.19. Webegin with a general remark about the invariants of pairs of spaces (
X, D ) that arefibered over P with fiber ( F, D F ). It concerns invariants in the class β ∈ H ( F ) of afiber. Denote by ι : H ∗ ( F ) → H ∗ ( X ) the map induced by inclusion. Lemma 3.8.
Let π : X → P be a Hamiltonian fibration with fiber F that induces afibration F D → D → P on the divisor D . If a := ι ( a ) is a fiber class and β ∈ H ( F ) then (cid:10) ι ( a ) , . . . | b , . . . (cid:11) X,Dβ,d = (cid:10) a , a ∩ F, . . . | b ∩ D F , . . . (cid:11) F,D F β,d . Similarly, if b := ι ( b ) is a fiber class then (cid:10) a , . . . | ι ( b ) , . . . (cid:11) X,Dβ,d = (cid:10) a ∩ F, . . . | b , b ∩ D F , . . . (cid:11) F,D F β,d . In particular, any invariant with two fiber insertions must vanish. Corresponding re-sults hold for absolute invariants.Proof.
This holds because one can define M [ vir ] β in such a way that each of its elementsrepresents a curve lying in some fiber of π . Therefore, if we represent the class a bya cycle lying in the fiber F z := π − ( z ) all the curves that contribute to the invariantlie entirely in this fiber and hence must intersect the other cycles in representativesfor a i ∩ F, b j ∩ F . The details of the proof in the absolute case are spelled out in [27,Prop 1.2(ii)]. The relative case is similar. (cid:3) We saw earlier that (cid:101) P is the blow up of P along a section s max in F max × P . Similarly, (cid:101) P (cid:48) is the blow up of P (cid:48) along the corresponding section s (cid:48) min in F (cid:48) min × P . Note that s (cid:48) min has normal bundle O (1) ⊕ C n − , where O ( m ) → P denotes the holomorphic linebundle of Chern class m . We denote by D the exceptional divisor in (cid:101) P (cid:48) . Thus it is a P n − -bundle π D : D := P ( O (1) ⊕ C n − ) → P . Consider the section s D := P ( { } ⊕ · · · ⊕ { } ⊕ C ) of D → P . It lies with trivial normalbundle in the product divisor V := P ( { } ⊕ C n − ) ∼ = s D × P n − and so c ( s D ) = 3. Theexceptional divisor E of (cid:102) M can be identified with π − D ( pt ). The line ε in E intersects V once. Therefore classes s D − mε with m > D since they have negative intersection with V and cannot be represented in V itself. On the other hand the section class s Z := s D − ε has the unique representative P ( O (1) ⊕ { } ⊕ · · · ⊕ { } ). AMILTONIAN S -MANIFOLDS ARE UNIRULED 29 The proof of Proposition 2.19 has two steps: we need to show that the absoluteinvariant in (cid:101) P (cid:48) equals a suitable relative invariant for ( (cid:101) P (cid:48) , D ), and then look at the blowdown correspondence. The first of these steps turns out to be the hardest, requiringa detailed study of the the invariants of the ruled manifold ( Y, D + ). Here Y is theprojectivization P ( L ⊕ C ) of the normal bundle to D ⊂ (cid:101) P (cid:48) and D + := P ( L ⊕ { } )as above; see Fig. 3.1. Thus ( Y, D + ) fibers over P with fiber ( X, E + ) equal to thepair consisting of the 1-point blow up of P n and hyperplane class E + , and there is acommutative diagram P → ( X, E + ) ρ X → E ↓ ↓ ↓ P → ( Y, D + ) ρ Y → Dπ Y ↓ π D ↓ P = P . We denote by D the divisor P ( { } ⊕ C ) in Y that is disjoint from D + , and by s D thesection in D corresponding to s D . Since D has the same normal bundle as D ⊂ (cid:101) P (cid:48) , c ( s D ) = 3. Also D ∩ X can be identified with the exceptional divisor E in (cid:102) M . Thus c ( λ X ) = n − λ X is the class of a line in X ∩ D . Note that H ( Y ) is generatedby s D , λ X and f . Moreover ρ Y ( s D − λ X ) = s Z . Thus we write s Z := s D − λ X . Figure 3.1.
Diagram (I) is a 3 D picture of the moment polytope ofthe toric manifold D , while (II) is a 3 D picture of that for Y with D reduced to 2-dimensionsThe next lemma contains some preliminary calculations. Here, and subsequently, wechoose the relative constraints b ∗ i from the elements of the following self-dual basis for H ∗ ( D ): D i ∈ H n +1 − i ) ( D ) , ≤ i ≤ n, E j ∈ H n − j ) ( D ) , ≤ j ≤ n. As always E j denotes the image in D of the class in (cid:102) M represented by the j -foldintersection of E with itself in (cid:102) M . Thus the E j are fiber constraints for the projection π D : D → P , while the D i are nonfiber constraints. Also D i · E j = 0 unless i + j = n +1. Lemma 3.9. (i) If m ≤ − , all invariants of ( Y, D + ) in the classes s D + mλ X + df vanish. They also vanish in the classes mλ X + df with m ≤ − .(ii) Let j = 1 or . Then (cid:10) E j | b ∗ , . . . , b ∗ r (cid:11) Y,D + s D + df + mλ X ,d = 0 for all d such that ≤ d ≤ m + 1 .(iii) Let d > and ≤ j ≤ . If (cid:10) E j | b ∗ , . . . , b ∗ r (cid:11) Y,D + df + mλ X ,d (cid:54) = 0 then b ∗ i is a nonfiberconstraint for all i . Moreover if n ≥ and r = 1 then m = 0 and d = 1 . The sameholds if n = 2 and j = r = 1 .(iv) If (cid:10) | b ∗ , . . . , b ∗ r (cid:11) Y,D + df + mλ X , ( d ) (cid:54) = 0 for some m ≥ and ≤ r ≤ then m < d .Proof. (i) Since ρ Y : Y → D is holomorphic every class with nonzero GW invariantsin Y must map to a class with a holomorphic representative in D . Hence their imagemust have nonnegative intersection with the divisor V containing s D . Since s D · V = 1and ε · V = − m +1 ≥ d the invariant(ii) can be nonzero only if m = − j = 2 and d = r = 0. Hence it remains to considerthe invariant (cid:10) E | (cid:11) Y,D + s Z , where s Z := s D − λ X . Although this could be nonzero asfar as dimensions are concerned, the geometry shows that it must be zero. To see this,consider the holomorphic fibration π Y : Y → P . Since the normal bundle to s Z in D is a sum of copies of O ( − D in class s Z . Since anyholomorphic curve in Y in class s Z projects to an s Z curve in D , the s Z curves in Y lie in the surface ρ − Y ( s Z ); cf. Figure 3.1 where this surface is crosshatched. Thissurface has 2-dimensions, while the constraint E has codimension 3 in Y . (Rememberthat E is the exceptional divisor in (cid:102) M and E denotes its intersection in (cid:102) M .) Hencethere is a cycle in Y representing E that does not meet any s Z curves. This proves(ii).In case (iii) we are counting curves in a fiber class of the holomorphic projection π Y : Y → P . Since E j is a fiber constraint, the first claim is an immediate consequenceof Lemma 3.8. To prove the second, note that by Lemma 3.8 the invariant reduces toone in ( X, E + ). A count of dimensions shows that m = 0. But then f is a fiber classfor the projection ρ X so that the invariant is nonzero only if ρ X ( b ) ∩ ρ X ( E j ) (cid:54) = 0. Afurther dimension count now implies that d = 1 (or one can use the fact that one iscounting d -fold covered spheres in P with 2 homological constraints). The proof of thethird claim is similar.Consider (iv). Since the fibers X of the bundle Y → P can all be identified (asK¨ahler manifolds) with the 1-point blow up of P n (with X ∩ D + the hyperplane class),this invariant can be nonzero only if one of the relative constraints, say b ∗ is a fiberconstraint while the others are not. Therefore δ b ≥ r −
1. Now count dimensions. (cid:3)
AMILTONIAN S -MANIFOLDS ARE UNIRULED 31 Lemma 3.10.
Suppose that ( M, ω ) is not uniruled and that n ≥ . Then (cid:10) | E (cid:11) e P (cid:48) ,Dσ − ε = (cid:10) E (cid:11) e P (cid:48) σ − ε . Proof.
As in the proof of Lemma 3.6 decompose (cid:101) P (cid:48) as the connected sum of ( (cid:101) P (cid:48) , D )with the ruled manifold ( Y, D + ) considered above. Consider a typical term in thedecomposition formula for (cid:10) E (cid:11) e P (cid:48) σ − ε where we put the constraint E in D ⊂ Y : (cid:10) | b , . . . , b r (cid:11) Γ , e P (cid:48) ,Dα − (cid:96)ε,(cid:96) (cid:10) E | b ∗ , . . . , b ∗ r (cid:11) Γ ,Y,D + α + (cid:96)f,(cid:96) where α ∈ H ( P (cid:48) ) and α ∈ H ( D ). (Note that the intersections on each side of D match because α · D = 0 for all α i ∈ H ( P (cid:48) ).) There is one term in this formula with α = σ, (cid:96) = r = 1 and b = E . The first factor is then (cid:10) | E (cid:11) e P (cid:48) ,Dσ − ε while the second is (cid:10) E | D n − (cid:11) Y,D + f = 1 . Hence we must see that all other terms vanish.The argument has several steps. Throughout the following discussion the words fiber/section class and fiber/nonfiber constraint apply to the fibration π over P . More-over a component of Γ or Γ is called a fiber component if it represents a class in thehomology H ( (cid:102) M ) of the fiber.(a) No component of Γ can lie in a fiber class (cid:101) β and go through a fiber constraint b . For if there were such a component, Lemma 3.8 would imply that the correspondinginvariant (cid:10) | b, b , . . . (cid:11) e P (cid:48) ,D e β,k equals the fiber invariant (cid:10) | b, b ∩ (cid:102) M , . . . (cid:11) f M,E e β,k , and so ( M, ω )would be uniruled by Lemma 3.7.(b)
Every fiber component of Γ has r ≤ ; moreover a fiber component through E j has r = 1 . If a component of Γ has two nonfiber relative constraints, say b ∗ , b ∗ , then thegenus zero restriction implies that at least one of the dual fiber constraints b , b mustlie in a fiber component of Γ (since Γ has at most one section component.) But thiscontradicts Step (a). Thus the second claim holds by the first part of Lemma 3.9(iii),while the first claim holds by Lemma 3.8 which says that only one of the relativeconstraints for each component can be a fiber class.(c) If Γ is a section class, its section component must go through the absolute con-straint. For otherwise Γ contains an invariant of the type considered in Lemma 3.9(iii) whose relative constraint is dual to a fiber constraint. Hence Γ would have tocontain a fiber component with a fiber constraint, contradicting (a).(d) Γ cannot be a section class . If Γ is a section class then by (c) the fiber componentsin Γ have no absolute constraints. Hence by Lemma 3.9(i) and (iv) (which applies by(b)) the fiber classes have the form d i f + m i λ X with 0 ≤ m i < d i . Suppose the sectionclass is s D + d f + m λ X . Then these classes in Γ combine with α − dε to make σ − ε where σ ∈ H ( P (cid:48) ). But λ X projects to the class ε in D . Hence − d + (cid:88) i ≥ m i = (cid:88) i ≥ ( m i − d i ) = − . We saw above that m i − d i ≤ − i >
0. Therefore either Γ is connected and m − d = − m − d ≥
0. Neither of these cases occur by Lemma 3.9(ii).(e)
Completion of the proof.
As in step (d) d = (cid:80) d i and (cid:80) i ≥ ( m i − d i ) = −
1. Thecomponent of Γ through E has r = 1 by (b) and hence m = 0 and d = 1 byLemma 3.9(iii). Moreover all other components of Γ have m i ≥ m i < d i by Lemma 3.9(iv). Therefore Γ is connected and lies in class f .Hence Γ is also connected and lies in class σ − ε . Therefore this is the term alreadyconsidered. (cid:3) We now relate the relative invariant in (cid:101) P (cid:48) to an absolute invariant in P (cid:48) . To thisend, consider the P n -bundle W := P ( O (1) ⊕ C n − ⊕ C ) → P . It contains a copy D + := P ( O (1) ⊕ C n − ⊕ { } ) of D with normal bundle L ∗ D (where as usual L D is thenormal bundle of D ), and a disjoint section s W := P ( { } ⊕ { } ⊕ C ). This can beidentified with s D and so c ( s W ) = 3. We also denote by λ the line in the fiber of W → P . Lemma 3.11.
Let n ≥ . Then:(i) (cid:10) τ pt | b ∗ , . . . , b ∗ r (cid:11) W,D + s W + dλ,d = 0 for all b ∗ i ∈ H ∗ ( D + ) .(ii) (cid:10) τ pt | b ∗ , . . . , b ∗ r (cid:11) W,D + dλ,d (cid:54) = 0 only if d = r = 1 and b ∗ is a multiple of ( D + ) n − .(iii) If d > , (cid:10) | b ∗ , . . . , b ∗ r (cid:11) W,D + dλ,d = 0 for all b ∗ i .Proof. (i) Since c ( s W ) = 3 and c ( λ ) = n + 1 this invariant is nonzero only if n + 1 + 3 + d ( n + 1) + r + 1 − δ b − − ( d − r ) = ( r + 1)( n + 1) , which reduces to ( d − r ) n + δ b + r = 0. Hence it cannot hold when d ≥ r ≥ c ( s W ). If d > r then we need n = 2 , r = 1 , d = 2 and δ b = 0. But then theinvariant has two point constraints and hence vanishes by Lemma 3.8: it counts curvesin a fiber class through points that could lie in different fibers. If d = r , we need δ b + r = 3. If r > b ∗ i is a point constraint, and the invariantvanishes as before. Hence we must have r = 1, δ b = 2. Since b cannot be a fiber class, itreduces to (cid:10) τ pt | b ∩ F (cid:11) P n , P n − which is nonzero when b = ( D + ) n − by Lemma 3.2(iii).Claim (iii) holds by a dimension count (or by Lemma 3.2(i)). (cid:3) Corollary 3.12.
Suppose that ( M, ω ) is not uniruled and that n ≥ . Then there is c (cid:54) = 0 such that (cid:10) τ pt (cid:11) P (cid:48) σ = c (cid:10) | E (cid:11) e P (cid:48) ,Dσ − ε .Proof. Identify P (cid:48) with the fiber sum of ( (cid:101) P (cid:48) , D ) with ( W, D + ) and calculate (cid:10) τ pt (cid:11) P (cid:48) σ using the decomposition formula, putting the point into W (cid:114) D (cid:48) . A typical term in thedecomposition formula (3.4) has the form (cid:10) | b , . . . , b r (cid:11) Γ , e P (cid:48) ,Dα − dε,d (cid:10) τ pt | b ∗ , . . . , b ∗ r (cid:11) Γ ,W,D + α + dλ,d where α = 0 or s W , and λ is the class of a line in the fiber of W . AMILTONIAN S -MANIFOLDS ARE UNIRULED 33 We saw in Lemma 3.11 that in a nonzero term α = 0. Further each nonzerocomponent of Γ must have an absolute constraint and d = r = 1. Hence there is onlyone such component and it has b ∗ = ( D + ) n − . Hence b = E and Γ must consist ofthe single component (cid:10) | E (cid:11) e P (cid:48) ,Dσ − ε, (1) . Hence there is only one term in the decompositionformula. The result follows. (cid:3) Proof of Proposition 2.19(i).
This follows immediately from Corollary 3.12 and Lemma 3.10. (cid:50)
We now turn to the proof Proposition 2.19(ii) which concerns the case n = 2. Lemma 3.13.
Let n = 2 . If (cid:10) E (cid:11) e P (cid:48) σ − ε (cid:54) = 0 then (cid:10) | E, D (cid:11) e P (cid:48) ,Dσ − ε, (1 , (cid:54) = 0 . Proof.
As before, we calculate the nonzero invariant (cid:10) E (cid:11) e P (cid:48) σ − ε by considering (cid:101) P (cid:48) as aconnected sum of ( (cid:101) P (cid:48) , D ) with ( Y, D + ), putting E into Y . Steps (a), (b) and (c) of theproof of Lemma 3.10 go through as before. So consider (d) and suppose that Γ is asection class. The homology class count is now (cid:88) i ≥ ( m i − d i ) = − . By Lemma 3.9 (i), m ≥ − m i ≥
0. Therefore by Lemma 3.9 (ii) m − d < − m i − d i ≤ − i >
0. Therefore Γ must be connected and m − d = −
2. Since d ≥ m ≤ −
2, which is impossible.Since m i − d i ≤ − has at mosttwo fiber components. One of these goes through the E constraint and has m = 0 , d = 1by Lemma 3.9 (iii). Hence there is exactly one other that has 0 ≤ m (cid:48) = d (cid:48) − r ≤
2. Since this is a fiber invariant it reduces to an invariant in (
X, E + ) of the form (cid:10) | a , . . . , a r (cid:11) X,E + d (cid:48) f +( d (cid:48) − λ X ,d (cid:48) . Since n = 2, this is possible only if d (cid:48) = 1. Hence both components of Γ lie in theclass f , and the Γ factor must be (cid:10) E | D (cid:11) Y,D + f (cid:10) | pt (cid:11) Y,D + f . Hence (cid:10) | E, D (cid:11) e P (cid:48) ,Dσ − ε, (1 , (cid:54) = 0 . (cid:3) Proof of Proposition 2.19(ii).
Blow down D by summing with ( W, D + ) as in Corollary 3.12. Consider the resultingdecomposition formula for (cid:10) pt, s W (cid:11) P (cid:48) σ where the absolute constraints are put into W .(a) There is no term in this formula with Γ = ∅ . In such a term σ would be a sectionclass in W with c W ( σ ) = 3. Since the only section classes are s W + mλ and c W ( λ ) = 3,we must have σ = s W . But the only holomorphic representatives of s W that meet theproduct divisor Z = P ( { } ⊕ C ) ∼ = s W × P are parallel copies of s W lying entirely in Z . Hence one can place the constraints pt, s W in Z in such a way that no holomorphic s W -curve meets them both.(b) Γ cannot be a section class. For if so, by (a), there would have to be a nonzeroinvariant (cid:10) a , . . . , a m | b ∗ , . . . , b ∗ r (cid:11) W,D + s W + dλ,d , d > , where 0 ≤ m ≤ a i form a subset of { pt, s W } . This is impossible for dimen-sional reasons.(c) Γ has at most components. Each has d = r = 1. By (b) each component in Γ liesin some class df . Even if one puts all the constraints into one component, a dimensioncount shows that the invariant (cid:10) pt, s W | b , . . . , b r (cid:11) W,D + dλ,d is nonzero only if d = 1. Itfollows that d = 1 in all cases. Thus we are counting lines, and so each componentmust have two constraints and hence since r = 1 each must have an absolute constraint.Therefore the only possibilities for Γ are: one component of type (cid:10) pt, s W | D (cid:11) W,D + λ, (1) = (cid:10) pt, pt | E + (cid:11) F,E + λ, (1) , or the two terms in the product (cid:10) pt | s D (cid:11) W,D + λ, (1) (cid:10) s W | pt (cid:11) W,D + λ, (1) . The corresponding Γ terms are (cid:10) | pt (cid:11) e P (cid:48) ,Dσ − ε , and (cid:10) | D, E (cid:11) e P (cid:48) ,Dσ − ε . We saw in Lemma 3.13 that the second of these terms is nonzero. Therefore, if thefirst term vanishes, (cid:10) pt, s W (cid:11) P (cid:48) σ (cid:54) = 0. However, if the first term does not vanish, we mayapply Corollary 3.12 to conclude that (cid:10) τ pt (cid:11) P (cid:48) σ (cid:54) = 0. This completes the proof. (cid:50) Identities for descendent classes.
To complete the proof of Theorem 1.1 wemust establish Lemma 2.21. Its proof is based on some identities for genus zeroGromov–Witten invariants that we now explain. We shall denote by ψ i the first Chernclass of the cotangent bundle to the domain of a stable map at the i th marked point,and by W β the configuration space of all (not necessarily holomorphic) stable mapsin class β . If i (cid:54) = j consider the subset D i,β | j,β of W β consisting of all stable mapswith at least two components, one in class β containing z i and the other in class β := β − β containing z j . Further, if i, j, k are all distinct consider the subset D i | jk of W β consisting of all stable maps with at least two components, one in some class β containing z i and the other in class β − β containing z j , z k . The virtual moduli cycle M [ vir ] β has a natural map to W β that may be chosen to be transverse to the abovesubsets. These therefore pullback to real codimension 2 sub(branched) manifolds in M [ vir ] β which we denote by the same names. AMILTONIAN S -MANIFOLDS ARE UNIRULED 35 In [22, Thm 1], Lee–Pandharipande prove the following identities in the algebraiccase: ψ i = D i | jk (3.9) ev ∗ i ( L ) = ev ∗ j ( L ) + ( β · L ) ψ j − (cid:88) β + β = β ( β · L ) D i,β | j,β , (3.10)where the two sides are considered as elements of an appropriate Picard group and L ∈ H n − ( M ; Z ) ∼ = H ( M ) is any divisor. We might try to think of these equationsas identities in the second cohomology group of M [ vir ] β . However, M [ vir ] β is not reallya space in its own right since it is only well defined up to certain kinds of cobordism.Hence it does not have a well defined H . Therefore we shall interpret these equationsas identities for Gromov–Witten invariants. For example, (3.9) states that for all k ≥ i > i j ≥ β ∈ H ( M ) , a i ∈ H ∗ ( M ), (cid:10) τ i a , . . . , τ i k a k (cid:11) Mβ = (cid:88) j,S,β + β = β (cid:10) τ i − a , . . . , ξ j (cid:11) Mβ (cid:10) ξ ∗ j , τ i a , τ i a , . . . (cid:11) Mβ . Here the sum is over the elements of a basis ξ j of H ∗ ( M ) (with dual basis ξ ∗ j ), all de-compositions β + β = β of β , and all distributions S of the constraints τ i a , . . . , τ i k a k over the two factors, subject only to the restriction that if β = 0 then the first factormust include at least one of the τ i (cid:96) a (cid:96) , (cid:96) ≥ , so that it is stable. The second identityhas a similar interpretation; cf. Lemma 3.14.Identity (3.9) is proved for the space of genus zero stable curves M ,k by Getzlerat the beginning of § ψ i differs from the pullback of the corresponding class on M ,k precisely by the boundaryclass consisting of the elements in M [ vir ] β such that the component containing the i th marked point is unstable. Note that the proof of Proposition 3.5 and hence ofTheorem 1.1 requires the full strength of this identity.Lee–Pandharipande’s proof of (3.10) in [22, § M [ vir ] β in which the elements in H ( M [ vir ] β ) have “nice” representatives.One could work with ad hoc methods as in [27] and interpret the idea of transversalityusing the normal cones to the strata of the moduli space provided by the gluing data,but it is cleaner to work in the category of polyfolds newly introduced by Hofer–Wysocki–Zehnder since this provides the moduli spaces with a smooth structure. Notethat because one needs to use multivalued perturbations the moduli spaces are ingeneral branched. (Here one can work with various essentially equivalent definitions,see [28, Def 3.2] and [13, Def 1.3].)To prove Theorem 1.1 we only need the very special case of this identity relevant toLemma 2.21. We will sketch the proof in this case to give the general idea. Note thathere i = 1 , j = 2 and the term ev ∗ ( L ) does not contribute since L ∩ pt = 0. When weprove Lemma 2.21 we will explain how to construct the moduli space B in this context; since we are working in a fibered 6 dimensional manifold, it is not hard to obtain it byad hoc methods. Lemma 3.14.
Given a class σ ∈ H ( M ) and a divisor H ∈ H n − ( M ) , denote by B := M [ vir ] σ, ( pt ) the virtual moduli space of stable maps in class σ with two markedpoints, one through a fixed point x ∈ M and the other through a cycle representing H .Suppose that B can be constructed as a smooth branched manifold of real dimension that is transverse to the strata in W . Then, for any divisor L ∈ H n − ( M ) (cid:10) LH, pt (cid:11) Mσ = ( σ · L ) (cid:10) H, τ pt (cid:11) Mσ − (cid:88) j, α + α = σ ( α · L ) (cid:10) H, ξ j (cid:11) Mα (cid:10) ξ ∗ j , pt (cid:11) Mα . Proof.
For present purposes we may think of a branched 2-dimensional manifold asthe realization of a rational singular 2-cycle formed by taking the union of a finitenumber of (positively) rationally weighted oriented 2-simplexes ( λ i , ∆ i ), where λ i ∈ Q > , appropriately identified along their boundaries. (This might have singularitiesat the vertices, but since these are codimension 2 this does not matter. See [28, 13]for a more complete description.) Since B is transverse to the strata in W β , there isa finite set Sing B of points in B corresponding to stable maps whose domain has twocomponents, and the other points have domain S . We assume that each b ∈ Sing B lies in the interior of a 2-simplex and therefore has a weight λ b .Let π : C → B be the universal curve formed by the domains with evaluation map f : C → M . The marked points define two disjoint sections s , s of C → B , numberedso that f ◦ s ( b ) ∈ H, f ◦ s ( b ) = x . We may assume that these sections are transverseto the pullback divisor (codimension 2 cycle) f ∗ L . Note that C is the blow up of anoriented S -bundle P → B at a finite number of points, one in each fiber over Sing B .For each such b we choose the exceptional sphere E b to be the component that doesnot contain s ( b ). Since the marked points never lie at nodal points of the domain, thesections s , s blow down to disjoint sections s (cid:48) , s (cid:48) of the S bundle P → B . Hence P can be considered as the projectivization P ( V ⊕ C ) where V → B is a line bundle and s (cid:48) = P ( V ⊕ s (cid:48) = P (0 ⊕ C ). Note also that s = s (cid:48) while s is the blow up of s (cid:48) overthose points b ∈ Sing B for which s ∩ E b (cid:54) = ∅ . For such b ∈ Sing B set δ ( b ) := λ b , andotherwise set δ ( b ) := 0.If B were a manifold then, as in the proof of Theorem 1 in [22, § (cid:10) H, τ pt (cid:11) Mσ = − s · s = − (cid:90) B c ( V ) . In our situation we must take the weights on B into account: each point y in the inter-section s · s should be given the (positive) weight λ ( π ( y )) ∈ Q of the correspondingpoint π ( y ) ∈ B as well as the sign o ( y ) ∈ {± } of the intersection. Thus we find (cid:10) H, τ pt (cid:11) Mσ = − (cid:88) y ∈ s · s o ( y )Λ( π ( y )) =: − (cid:90) B c ( V ) . AMILTONIAN S -MANIFOLDS ARE UNIRULED 37 We want to calculate (cid:10)
LH, pt (cid:11) Mσ = (cid:10) LH, pt (cid:11) Mσ − (cid:10) H, Lpt (cid:11) Mσ = (cid:90) B ev ∗ ( L ) − ev ∗ ( L )= (cid:90) s f ∗ ( L ) − (cid:90) s f ∗ ( L )= (cid:88) y ∈ s · f ∗ ( L ) o ( y )Λ( π ( y )) − (cid:88) y ∈ s · f ∗ ( L ) o ( y )Λ( π ( y )) . This can be done just as in [22]. The divisor f ∗ ( L ) intersects a generic fiber F of C → B with multiplicity σ · L , and intersects the exceptional divisor E b with multiplicity α ( b ) · L , where α ( b ) = [ f ∗ ( E b )]. Since H ( C ; Q ) splits as the sum H ( P ; Q ) ⊕ (cid:80) b ∈ Sing B [ E b ] Q ,we may consider the difference [ s ] − [ s ] ∈ H ( C ; Q ) as the sum of [ s (cid:48) ] − [ s (cid:48) ] with − (cid:80) b ∈ Sing B δ ( b )[ E b ]. But [ s (cid:48) ] − [ s (cid:48) ] = k [ F ] where k := − (cid:82) B c ( V ) F and [ F ] denotesthe fiber class of P → B . It follows that (cid:90) s f ∗ ( L ) − (cid:90) s f ∗ ( L ) = − ( σ · L ) (cid:90) B c ( V ) − (cid:88) b ∈ Sing B δ ( b ) α ( b ) · L = ( σ · L ) (cid:10) H, τ pt (cid:11) Mσ − (cid:88) α + α = σ ( α · L ) N α ,α where N α ,α = (cid:80) j (cid:10) H, ξ j (cid:11) Mα (cid:10) ξ ∗ j , pt (cid:11) Mα is the number of two-component curves, one inclass α through H and the other in class α through the point x . This completes theproof. (cid:3) Proof of Lemma 2.21
If a symplectic 4-manifold (
M, ω ) is not the blow up of a ra-tional or ruled manifold then it has a unique maximal collection { ε , . . . , ε k } of disjointexceptional classes (i.e. classes that may be represented by symplectically embeddedspheres of self-intersection − M , ω )(called the minimal reduction of (
M, ω )) has trivial genus zero Gromov–Witten invari-ants. We show that for these M it is impossible for an invariant of the type (cid:10) pt, s (cid:11) Pσ to be nonzero.Since ( M, ω ) is not strongly uniruled, it follows from Remark 2.10 that (cid:10) pt, a (cid:11) Pσ = 0for all a ∈ H ( M ) . Therefore (cid:10) pt, s (cid:11) Pσ is the same for all section classes s . Chooseclasses h , h ∈ H ( M ; Z ) with h h (cid:54) = 0. Pull them back to M and then extend themto H ( P ; Q ) (which is possible by [27, Thm 1.1] for example.) Multiplying them by asuitable constant we get integral classes h , h ∈ H ( P ; Z ) with Poincar´e duals H , H .By construction s := H H is a section class and H j · ε i = 0 for all exceptional divisors ε i in M .We now claim that we can apply Lemma 3.14 to evaluate the nonzero invariant (cid:10) H H , pt (cid:11) Pσ . For this, it suffices to show that the space of (regularized) stable mapsin class σ and through the point x can be constructed as a branched 2-manifold B (cid:48) . To this end, consider an Ω-tame almost complex structure J P on ( P, Ω) for whichthe projection π : ( P, J P ) → ( S , j ) is holomorphic. Then J P restricts on each fiber P z := π − ( z ) to an ω -tame almost complex structure on M . Every J P -holomorphicstable map in class σ consists of a holomorphic section plus some fiberwise bubbles.Since the family of such stable maps through some fixed point x has real dimension2, we can assume that each such bubble is a k -fold cover of an embedded regular curvein some class β with c ( β ) ≥ ω ( kβ ) ≤ κ = Ω( σ ), and that the sections through x with energy ≤ κ are regular and so lie in classes with 2 ≤ c ( σ (cid:48) ) ≤
3. For any J P ,each exceptional class ε i is represented by a unique embedded sphere. Since ( M, ω ) isnot rational or ruled, it follows from Liu [24] that these are the only J P -holomorphicspheres in classes β with c ( β ) >
0. (For details of this argument see [30, Cor 1.5].)Since regular sections through x in classes with c ( σ (cid:48) ) = 2 are isolated, there can bea finite number of two component stable maps whose bubble is an exceptional sphereand there are no stable maps involving multiply covered exceptional classes. Howeverthere may be some with multiply covered bubbles in classes β with c ( β ) = 0. To dealwith these, choose a suitable very small multivalued perturbation ν over the modulispaces of fiberwise curves with class kβ for ω ( kβ ) ≤ κ so that there are only isolatedsolutions of the corresponding perturbed equation. Since the sections through x areregular, they can meet one of these isolated bubbles only if they lie in a moduli spaceof real dimension 2 and for generic choices of J P and ν they will meet only one suchbubble. Therefore for this choice of ν the perturbed moduli space contains isolated two-component stable maps. It remains to extend the perturbation over a neighborhood ofthis stratum in W σ (tapering it off to zero outside this neighborhood), and to define B (cid:48) as the solution space of the resulting perturbed Cauchy–Riemann equation.Lemma 3.14 now implies that (cid:10) H H , pt (cid:11) Pσ = ( σ · H ) (cid:10) H , τ pt (cid:11) Pσ − (cid:88) j,α + α = σ ( α · H ) (cid:10) H , ξ j (cid:11) Pα (cid:10) ξ ∗ j , pt (cid:11) Pα , where ξ j runs over a basis for H ∗ ( P ) with dual basis ξ ∗ j . Note that we may choose thisbasis so that exactly one of each pair ξ j , ξ ∗ j is a fiber class.The first term must vanish, since otherwise ( M, ω ) is strongly uniruled by Lemma 2.20.Therefore there must be some nonzero product. If α is a fiber class then by Lemma 3.8 (cid:10) ξ ∗ j , pt (cid:11) Pα = (cid:10) ξ ∗ j ∩ M, pt (cid:11) Mα so that ( M, ω ) is strongly uniruled by definition. Hence theseproduct terms vanish. Further if α is a section class and ξ ∗ j (cid:54) = [ M ] is a fiber class then( M, ω ) is strongly uniruled by Remark 2.10. On the other hand if ξ ∗ j = [ M ] then thereis a nonzero invariant (cid:10) H , s (cid:11) Pα for α ∈ H ( M ) and some section class s which impliesthat c M ( α ) = 1. Also because α · H (cid:54) = 0, the choice of H implies that α is not one ofthe exceptional classes ε i . But this is impossible since we saw above that the only bub-bles with Chern class 1 are the exceptional spheres. Therefore there must be a nonzeroterm in which both α and ξ j are fiber classes. In this case, (cid:10) H , ξ j (cid:11) Pα = (cid:10) H , ξ j ∩ M (cid:11) Mα is an invariant in M . But the only nonzero classes α ∈ H ( M ) with nontrivial 2-point AMILTONIAN S -MANIFOLDS ARE UNIRULED 39 invariants are the exceptional divisors ε i and ε i · H = 0 by construction. Thereforethese terms must vanish as well. This completes the proof. (cid:50) Special cases
We now discuss some special S -actions for which it is possible to prove directly that( M, ω ) is strongly uniruled.
Proposition 4.1.
Suppose that ( M, ω ) is a semifree Hamiltonian S -manifold. Then ( M, ω ) is strongly uniruled.Proof. Denote by γ the element in π (Ham M ) represented by the circle action and by S ( γ ) ∈ QH ∗ ( M ) × its Seidel element. Since the action is semifree, [33, Thm 1.15] showsthat S ( γ ) ∗ pt = a ⊗ q − d t κ + l.o.t.where a is a nonzero element of H d ( M ) with d >
0. But if (
M, ω ) is not stronglyuniruled, S ( γ ) ∗ pt = (1l ⊗ λ + x ) ∗ pt = pt ⊗ λ by Lemma 2.1. An alternative proof isgiven in Proposition 4.3. (cid:3) The ideas of [33] also work when the isotropy weights have absolute value ≤
2. (Inthis case, we say that the action has at most 2-fold isotropy.) This property is stableunder blow up along the maximal or minimal fixed point sets: cf. Lemma 2.18.
Proposition 4.2.
Suppose that ( M, ω ) is an effective Hamiltonian S -manifold withat most -fold isotopy and isotropy weights along F max . Then ( M, ω ) is stronglyuniruled.Proof. By Proposition 2.11 S ( γ ) = a ⊗ q d t K max + (cid:88) β ∈ H ( M ; Z ) , ω ( β ) > a β ⊗ q m − c ( β ) t K max − ω ( β ) , where a β ∈ H ∗ ( M ) and a is in the image of H ∗ ( F max ) in H ∗ ( M ). Suppose that ( M, ω )is not strongly uniruled. By Lemma 2.1 (iii), there is at least one term in S ( γ ) with a β = r
1l where r (cid:54) = 0. Consider the term of this form with minimal ω ( β ). Let J be ageneric ω -tame and S -invariant almost complex structure on M , with correspondingmetric g J . Proposition 3.4 of [33] shows that in order for r (cid:54) = 0 there must be, forevery point y ∈ F min , an S -invariant J -holomorphic genus zero stable map in class β that intersects F max and y . Such an invariant element consists of a connected stringof 2-spheres from F max to y , possibly with added bubbles. Components of the string(called beads in [33]) either lie in the fixed point set M S or are formed by the orbitsof the g J -gradient trajectories of K . The energy ω ( β (cid:48) ) of an invariant sphere in class β (cid:48) that joins the two fixed point components F , F is at least | K ( F ) − K ( F ) | /q ,where q is the order of the isotropy at a generic point of the sphere. Therefore, if theisotropy has order at most 2 the energy needed to get from F max to y ∈ F min is at least( K max − K min ) /
2. In the case at hand, it is strictly larger than ( K max − K min ) / the first element of the string has trivial isotropy. Therefore there is r (cid:54) = 0 , x ∈ Q − such that S ( γ ) = 1l ⊗ (cid:0) rt K max − κ + l.o.t (cid:1) + x, κ > ( K max − K min ) / . Similarly, there is r (cid:48) (cid:54) = 0 , x (cid:48) ∈ Q − such that S ( γ − ) = 1l ⊗ (cid:0) r (cid:48) t − K min − κ (cid:48) + l.o.t (cid:1) + x (cid:48) , κ (cid:48) > ( K max − K min ) / . Since S is a homomorphism, we know that S ( γ ) ∗ S ( γ − ) = 1l. Now assume that M isnot strongly uniruled. Then by Lemma 2.1(ii) the above expressions imply that S ( γ ) ∗ S ( γ − ) = rr (cid:48) ⊗ (cid:0) t δ + l.o.t (cid:1) , δ < , a contradiction. (cid:3) The previous results give conditions under which (
M, ω ) is strongly uniruled, butthey do not claim that the specific invariant (cid:10) pt (cid:11) Mα is nonzero, where α is the orbit ofa generic gradient flow line from F max to F min . There are two cases when we can provethis. Note that condition (ii) is not very general since F max is often obtained by blowup and any such manifold is uniruled. Proposition 4.3.
Suppose that ( M, ω ) is a Hamiltonian S -manifold whose maximaland minimal fixed point sets are divisors. Suppose further that at least one of thefollowing conditions holds: (i) the action is semifree, or (ii) there is an ω -tame almost complex structure J on F max such that the nonconstant J -holomorphic spheres in F max do not go through every point.Then (cid:10) pt (cid:11) Mα (cid:54) = 0 . Proof.
Suppose first that the action is semifree and let J be a generic S -invariantalmost complex structure on M . Then by [33, Lemma 4.5] the gradient flow of themoment map with respect to the associated metric g J ( · , · ) = ω ( · , J · ) is Morse–Smale.Hence for a generic point x of F max all the gradient flow lines that start at x end on F min . The union of these flow lines is an invariant J -holomorphic α -sphere through x .Moreover there is no other invariant J -holomorphic stable map in class α through x .For as in the previous proof this would have to consist of a sphere C through x in F max together with a string of 2-spheres from a point x in F max to a point y in F min , possiblywith added bubbles. (The string has to reach F min since α · F min = 1.) But becausethe action is semifree the energy of such a string is at least ω ( α ). Since ω ( C ) > J -holomorphic stable map in class α through x . Since this is regular, (cid:10) pt (cid:11) Mα = 1.A similar argument works in case (ii). Choose J to be a generic S -invariant exten-sion of the given almost complex structure on F max . The arguments of [33, § X of points in F max that flow down to some intermediate fixed point setof K is closed and of codimension at least 2. Moreover by perturbing J in M (cid:114) F maxAMILTONIAN S -MANIFOLDS ARE UNIRULED 41 we may jiggle X so that there is a point x ∈ F max (cid:114) X that does not lie on any J -holomorphic sphere in F max . Hence again there is only one invariant stable map inclass α through x . The result follows as before. (cid:3) We end by discussing the case when H ∗ ( M ; Q ) is generated by H ( M ). Our mainresult here is the following. Proposition 4.4.
Assume that H ∗ ( M ; Q ) is generated by H ( M ) . Then ( M, ω ) isstrongly uniruled iff it is uniruled. The proof is given below. By Theorem 1.1 we immediately obtain:
Corollary 4.5.
Suppose that ( M, ω ) is a Hamiltonian S manifold such that H ∗ ( M ; Q ) is generated by H ( M ) . Then ( M, ω ) is strongly uniruled. Remark 4.6. (i) Observe that if M is a Hamiltonian S -manifold such that H ∗ ( M ; Q )is generated by H ( M ) then the same holds for the blow up of M along any of its fixedpoint submanifolds F . For because the moment map K is a perfect Morse functionthe inclusion F → M induces an injection on homology. (Any class c in H ∗ ( F ) can bewritten as c − ∩ F + , where c − , F + are the canonical downward and upward extensions of c, [ F ] defined for example in [33, § H ∗ ( F ) is generated by the restrictionsto F of the classes in H ( M ). Since the exceptional divisor E is a P k bundle over F , H ∗ ( E ) is also generated by H ( E ): in fact the generators are the pullbacks of theclasses in H ( F ) plus the first Chern class c ∈ H ( E ) of the canonical line bundle over E . But c is the restriction to E of the class (cid:101) c in (cid:102) M that is Poincar´e dual to E . It followseasily (using the Mayer-Vietoris sequence) that H ∗ ( (cid:102) M ) is generated by the pullback ofthe classes in H ( M ) together with (cid:101) c ; see the discussion of the cohomology of a blowup given in [15, § S manifolds have the property that H ∗ ( M ) isgenerated by H ( M ). It is not enough that the fixed points are isolated. For example,Sue Tolman pointed out that the complex Grassmannian Gr (2 ,
4) of 2-planes in C has H of dimension 1 and H of dimension 2 so that H (cid:54) = ( H ) . It also has an S action with precisely 6 fixed points. However, it is enough to have isolated fixedpoints plus semifree action, since in this case Tolman–Weitsman [38] show that H ∗ ( M )is isomorphic as a ring to the cohomology of a product of 2-spheres. Also H ∗ ( M ) isgenerated by H ( M ) in the toric case. However, these cases are uninteresting in thepresent context since we already know that these manifolds are strongly uniruled, inthe former case by Proposition 4.3 and in the latter by the fact that toric manifoldsare projectively uniruled.The proof of Proposition 4.4 uses the identities (3.9) and (3.10). Note that eachtime we apply one of these formulas we must take special care with the zero class. Thefollowing lemma is well known: cf. [15, Lemma 4.7]. Private communication.
Lemma 4.7. (cid:10) τ k a , . . . , τ k p a p (cid:11) M (cid:54) = 0 for some p ≥ only if the intersection productof the classes a i is nonzero (i.e. the (real) codimensions of the a i sum to n ) and (cid:80) k i = p − .Proof. This is immediate from the definition if k i = 0 for all i . To prove the general case,one can either argue directly or can construct an inductive proof based on the identity(3.9). To understand why the invariant vanishes when dim( a ∩ · · · ∩ a m ) >
0, observethat in this case the moduli space M of constant maps with fixed marked points andthrough the constraints can be identified with a ∩ · · · ∩ a m and so has dimension > ψ i are trivial on M and hence the integral of any product of the ψ i over the full moduli space (with varying marked points) must vanish. (cid:3) Proof of Proposition 4.4.
Since any strongly uniruled manifold is uniruled, it suffices to prove the converse.This in turn is an immediate consequence of the next lemma.
Lemma 4.8.
Suppose that H ∗ ( M ; Q ) is generated by H ( M ) and that there is anonzero invariant of the form (4.1) (cid:10) τ k pt, τ k a , . . . , τ k m a m (cid:11) Mβ , a i ∈ H ∗ ( M ) , k i ≥ , β (cid:54) = 0 . Then ( M, ω ) is strongly uniruled.Proof. The first two steps in this argument apply to all M and are contained in theproof of [15, Thm 4.9]. We include them for completeness. Without loss of generalitywe consider a nonzero invariant (4.1) such that m is minimal and ω ( β ) is minimalamong all nonzero invariants (4.1) of length m with β (cid:54) = 0. We then order the indices k i so that k ≤ · · · ≤ k m and suppose that the k i for i > m, ω ( β ). Finally we choose a minimal k for thegiven m, ω ( β ) , and k i , i > Step 1:
We may assume that k i = 0 for i > . If not, let r be the minimal integer greater than one such that k r (cid:54) = 0. Suppose firstthat m ≥ r > i = r, j = 1 and k = 2. Then the invariant(4.1) is a sum of products (cid:10) τ k r − a r , ξ, . . . (cid:11) Mβ (cid:10) ξ ∗ , τ k pt, a , . . . (cid:11) Mβ − β , where ξ runs over a basis for H ∗ ( M ) with dual basis { ξ ∗ } , and the dots representthe other constraints τ k (cid:96) a (cid:96) (which may be distributed in any way.) There must be anonzero product of this form.We now show that this is impossible. Suppose first that there is such a productwith ω ( β ) >
0. Then the second factor is an invariant of type (4.1) in a class β (cid:48) with ω ( β (cid:48) ) < ω ( β ) and at most m constraints. Since our assumptions imply that all suchterms vanish, this is impossible. Hence any nonzero product must have ω ( β ) = 0 andhence β = 0. But then the second factor has at least one fewer nonzero k i , sinceit has the homological constraint ξ ∗ instead of τ k r a r . This contradicts the assumedminimality of k , . . . , k m . AMILTONIAN S -MANIFOLDS ARE UNIRULED 43 This completes the proof when r >
2. If m ≥ r = 2 use the same argumentbut take k = 3 instead of k = 2. If m < m in any essential way. Step 2: k = 0 . If k > i = 1 , j = 2 and k = 3. Again, there must be a nonzeroproduct of the form (cid:10) τ k − pt, ξ, . . . (cid:11) Mβ (cid:10) ξ ∗ , a , a , . . . (cid:11) Mβ − β . Since the first factor can have at most ( m −
1) constraints, the minimality of m impliesthat β = 0. But then pt ∩ ξ (cid:54) = 0, so that ξ = [ M ]. Hence ξ ∗ = pt and the second factoris an invariant of the required form with k = 0. Step 3:
Completion of the proof.
By hypothesis on M and Step 2, there is a nonzero invariant (cid:10) pt, H i , . . . , H i m m (cid:11) Mβ with m constraints, where H j ∈ H n − ( M ). Moreover, we may assume that all in-variants (4.1) in a class β (cid:48) with ω ( β (cid:48) ) < ω ( β ) or m (cid:48) < m vanish and that the set i ≤ · · · ≤ i m is minimal in the lexicographic ordering. Note that i > m by using the divisor equation. We must show that m ≤ i = 2 and j = 1. Since H ∩ pt = 0 the first term onthe RHS vanishes. The second is a multiple of (cid:10) τ pt, H i − , H i , . . . , H i m m (cid:11) Mβ . Supposeit is nonzero and apply (3.9) to it, with i = 1 and j, k, = 2 ,
3. This gives a sum of terms (cid:10) pt, ξ, . . . (cid:11) Mβ (cid:10) H i − , H i , ξ ∗ , . . . (cid:11) Mβ − β . Since the first factor has < m constraints, wemust have β = 0. But then ξ = [ M ] so that ξ ∗ = pt . Hence all the other constraintsmust lie in the second factor (since it must have at least m constraints). Therefore thefirst factor is a constant map with only two constraints, i.e. it is unstable. But this isnot allowed. Therefore, the second term in (3.10) must vanish.It remains to consider the product terms in (3.10), namely( β · H ) (cid:10) H i − , ξ, . . . (cid:11) Mβ (cid:10) pt, ξ ∗ , . . . (cid:11) Mβ , where β + β = β . In any nonzero product of this form, β (cid:54) = 0. Also the stabilitycondition on the second factor implies that if β = 0 this term must have anotherconstraint. Since this must have the form H i j j with i j >
0, this is not possible byLemma 4.7. Therefore 0 < ω ( β ) < ω ( β ). But then the second factor vanishes by theminimality of ω ( β ). (cid:3) Remark 4.9.
Suppose that (
M, ω ) is uniruled with even constraints, i.e. there is anonzero invariant of the form (4.1) in which all the a i have even degree. Then theproof of Lemma 4.8 goes through if we assume only that the even part H ev ( M ) of thecohomology ring is generated by H . For if the a i have even degree, odd dimensionalhomology classes appear in the above proof only as elements ξ, ξ ∗ . Since these alwaysappear as part of invariants where all the other insertions have even dimension, all terms involving odd dimensional ξ, ξ ∗ must vanish. The appendix contains other results aboutsuch manifolds; cf. Propositions A.2 and A.4. Appendix A. The structure of QH ∗ ( M ) for uniruled M In this appendix we explore the extent to which the uniruled property can be seenin quantum homology, completing the discussion begun in Lemma 2.1. To simplify weshall ignore contributions to the quantum product from the odd dimensional homologyclasses. Hence our results are not as general as they might be.Let F := Λ be the field Λ univ of generalized Laurent series. Observe that the evenquantum homology QH ev ∗ ( M ) := n (cid:77) i =0 H i ( M ; R ) ⊗ Λ[ q, q − ]is a subring of QH ∗ ( M ) because when a, b ∈ H ∗ ( M ) have even degree (cid:10) a, b, c (cid:11) Mβ = 0unless c also has even degree. We shall denote by A its subring QH n ( M ) = QH ev n ( M )of elements of degree 2 n , regarded as a commutative algebra over F . (Equivalently wecan think of A as the algebra obtained from QH ev ( M ) by setting q = 1.)Choose a basis ξ i for H ev ( M ; Q ) with ξ = pt , ξ N = 1l and so that 0 < deg( ξ i ) < n for the other i . These elements form a finite basis for A considered as a vector spaceover F . Hence there is a well defined linear map f : A → F given by f ( N (cid:88) i =0 λ i ξ i ) = λ . Since the corresponding pairing ( a, b ) := f ( ab ) on A is nondegenerate, ( A , f ) is acommutative Frobenius algebra. It is easy to see that it satisfies the other conditionsof the next lemma, with p = pt and M = span { ξ i : i (cid:54) = 0 , N } . In particular f ( pa ) = 0when a ∈ Q − := p F ⊕ M because all Gromov–Witten invariants of the form (cid:10) a, b, [ M ] (cid:11) Mβ , β (cid:54) = 0 , a, b ∈ H < n ( M )vanish. Lemma A.1.
Let ( A , f ) be a finite dimensional commutative Frobenius algebra overa field F that decomposes additively as A = p F ⊕ M ⊕ F where f ( p ) = 1 and ker f = M ⊕ F . Suppose further that f ( pa ) = 0 for all a ∈ Q − := p F ⊕ M . Then pa = 0 forall a ∈ Q − iff there are no units in Q − .Proof. One implication here is obvious: if a ∈ Q − is a unit then pa (cid:54) = 0. Conversely,suppose that pa (cid:54) = 0 for some a ∈ Q − , but that there are no units in Q − . We shallshow by a sequence of steps that there are no Frobenius algebras ( A , f ) that have thisproperty as well as satisfying the other conditions in the statement of the lemma. A pair ( A , f ) consisting of a commutative finite dimensional unital algebra together with a linearfunctional f : A → F satisfies the Frobenius nondegeneracy condition iff ker f contains no nontrivialideals. AMILTONIAN S -MANIFOLDS ARE UNIRULED 45 Decompose A as a sum A ⊕ · · · ⊕ A k of indecomposables and let e , . . . , e k be thecorresponding unitpotents. Thus for all i, je i e j = δ ij e i , and A i = A e i . Each e i may be written uniquely in the form λ i
1l + x i where x i ∈ Q − and λ i ∈ F . Orderthe e i so that the nonzero λ i are λ , . . . , λ (cid:96) . Since (cid:80) e i = 1l, we must have (cid:80) (cid:96)i =1 λ i = 1.In particular, (cid:96) ≥ Step 1:
All units in A i , i > , lie in Q − . Hence (cid:96) = 1 . Suppose there is a unit in A i of the form µ i
1l + x , where µ i (cid:54) = 0 and i >
1. Choosenonzero ν j ∈ Λ for j (cid:54) = i, j ≤ (cid:96) , so that (cid:80) ν j λ j = µ i and set ν j := 1 when j > (cid:96) . Then u := (cid:80) j (cid:54) = i ν j e j − u i is a unit of A since it is a sum of units, one from each factor. Byconstruction, the coefficient 1l in u vanishes. Hence u is a unit of A lying in Q − , which,by hypothesis, is impossible. Step 2:
All nilpotent elements in A lie in Q − . If n = 1l − x is nilpotent for some x ∈ Q − , then x = 1l − n is a unit of A lying in Q − . Step 3:
For all i > , each A i ⊂ Q − and pe i = 0 . The standard theory of indecomposable finite dimensional algebras over a field im-plies that every nonzero element in A i either is a unit or is nilpotent; see Curtis–Reiner[5, pp.370-2]. The units in A i , i >
1, lie in Q − by Step 1, and the nilpotent elementsdo too by Step 2. This proves the first statement. Thus e i x ∈ Q − for all x ∈ A and i >
1, so that by our initial assumptions f ( pe i x ) = 0 for all x . But the restriction of f to each summand A i is nondegenerate. Hence this is possible only if pe i = 0. Step 4:
Completion of the argument.
Let e = (cid:80) i> e i . By the above we may assume that p = p (1l − e ) ∈ A . Decompose A additively as the direct sum N ⊕U where N is the subspace formed by the nilpotentelements and U is a complementary subspace. Step 2 implies that N ⊂ A ∩ Q − . Onthe other hand, as in Step 3, any nonzero element in A ∩ Q − that is not nilpotent is aunit u in A . If any such existed then u + e would be a unit of A lying in Q − . Hence byhypothesis we must have N = A ∩ Q − . Thus p is nilpotent and dim U = 1, spannedby e . Further because Q − is spanned by N and the A i , i > p does not annihilateall elements in N .We now need to use further information about the structure of A . Recall that thesocle S of an algebra A is the annihilator of N . Therefore, our assumption on p impliesthat p / ∈ S . But there always is w ∈ N such that pw is a nonzero element of S . To seethis, choose a set x = p, x , . . . , x k of multiplicative generators for N that includes p .There is N such that all products of the x i of length > N must vanish. (Take N tobe the sum of the orders of the x i .) Therefore there is a nonzero product of maximallength that contains p and so can be written as pw . Since pwx i = 0 for all i , pw ∈ S .(A more precise version of this argument is given in Abrams [1, Prop.3.3].)The argument is now quickly completed. For, by the Frobenius nondegeneracy con-dition there must be z ∈ A such that ( pw, z ) = f ( pwz ) (cid:54) = 0. By Step 3 we may assumethat z ∈ A so that z = λe + n for some n ∈ N . But then pwz = pw ( λe ) = λpw sothat f ( λpw ) = λf ( pw ) (cid:54) = 0. But w ∈ N ⊂ Q − by construction. Hence f ( pw ) = 0 by our initial assumptions. This contradiction shows that our assumption that Q − has nounits must be wrong. (cid:3) We shall say that (
M, ω ) is uniruled with even constraints if there is a nonzeroGromov–Witten invariant of the form (cid:10) pt, a , . . . , a k (cid:11) Mβ with β (cid:54) = 0 and all a i of evendegree. Similarly, ( M, ω ) is strongly uniruled with even constraints if there is anonzero invariant of this kind with k = 3. For example, [15, Cor. 4.3] shows that anyprojective manifold that is uniruled is in fact strongly uniruled with even constraints. Proposition A.2. ( M, ω ) is strongly uniruled with even constraints iff the even quan-tum homology ring QH ev ∗ ( M ) has a unit in Q − .Proof. Note that QH ev ∗ ( M ) has a unit iff its degree 2 n part A has. Also the subring Q − ( A ) of A considered above is just the intersection Q − ∩ QH ev n ( M ). Therefore thisis an immediate consequence of Lemma A.1. (cid:3) If (
M, ω ) is uniruled rather than strongly uniruled we can still see some effect onquantum homology if instead of considering the small quantum product ∗ we considerthe whole family of products ∗ a , a ∈ H . Here we shall take H := H ev ( M ; C ) andcorrespondingly allow the coefficients r i of the elements (cid:80) r i t κ i in Λ to be in C . Let ξ i , i = 0 , . . . , N, be a basis for H with ξ = pt as before, and identify H with C N +1 bythinking of this as the standard basis in C N +1 . Denote by T , . . . , T N the correspondingcoordinate functions on H , thought of as formal variables. If α = ( α , . . . , α p ) is amulti-index with 0 ≤ α i ≤ N define ξ α := ( ξ α , . . . , ξ α p ) ∈ H p , T α := p (cid:89) i =1 T α i . In this language, the (even) Gromov-Witten potential Φ( t, T ) is the formal power seriesin the variables t and T given byΦ( t, T ) := (cid:88) α (cid:88) β | α | ! (cid:10) ξ α (cid:11) β t − ω ( β ) T α . Let us assume that the following condition holds: Condition (*): there is δ > such that this series converges if we consider the T i tobe complex numbers such that | T i | ≤ δ . Then we may think of Φ as a function defined near 0 ∈ H with values in the fieldΛ = Λ univ C . Further, as explained for example in [32, Ch 11.5], the structure constants ofthe associative product x ∗ a y are given by evaluating the third derivatives of Φ( t, T ) with We make this assumption to simplify our subsequent discussion. It is satisfied in the case ofmanifolds such as C P n (cf. [32, Ch 7.5]), but there is at present little understanding of when it issatisfied in general. Even if it were not satisfied, one could presumably adapt the results below in anyparticular case of interest. AMILTONIAN S -MANIFOLDS ARE UNIRULED 47 respect to the variables T i at the point T = a . In other words, ξ i ∗ a ξ j = (cid:80) k c kij ( a ) ξ ∗ k , where c kij ( a ) = ∂ Φ( t, T ) ∂T i ∂T j ∂T k (cid:12)(cid:12)(cid:12) T = a = (cid:88) m m ! (cid:10) ξ i , ξ j , ξ k , a, . . . , a (cid:11) Mm +3 ,β t − β . (A.1)We denote the corresponding (ungraded) rings by QH aev ( M ). As before, they areFrobenius algebras over the field F := Λ.The main point for us is the following lemma: Lemma A.3.
Assume that condition (*) holds and that there is a nonzero invariant (cid:10) pt, a , . . . , a k (cid:11) Mβ where β (cid:54) = 0 and the a i have even degree. Then there are a, b ∈ H with deg b < n such that pt ∗ a b (cid:54) = 0 .Proof. Let m + 3 be the minimal k for which some (cid:10) pt, a , . . . , a k (cid:11) Mk,β does not vanish. If m ≤ a = 1l so that, by Lemma 4.7, ∗ a is the usual product. If m > (cid:10) pt, ξ j , ξ k , a, . . . , a (cid:11) Mm +3 ,β isnonzero. (This holds because GW invariants are symmetric and multilinear functionsof their arguments.) But by equation (A.1) the coefficients c kij ( a ) are power series inthe coordinates of a ∈ H . Hence the fact that one coefficient of the power series for c k j ( a ) does not vanish implies that by perturbing a slightly if necessary we can arrangethat c k j ( a ) itself is nonzero. It follows that pt ∗ a ξ j (cid:54) = 0. (cid:3) A similar argument proves the analog of the other statements in Lemma 2.1. More-over, if ( A a , f ) denotes the Frobenius algebra QH a n ( M ), then one can check as beforethat ( A a , f ) satisfies the conditions of Lemma A.1. Thus we deduce: Proposition A.4.
Assume that condition (*) holds. Then ( M, ω ) is uniruled witheven constraints iff there is a ∈ H such that the even quantum ring QH aev ( M ) has aunit in Q − . Remark A.5. (i) Suppose that ( A , f ) is a Frobenius algebra over a field F with theproperty that f (1l) = 0. Suppose further that there is p ∈ A such that f ( p ) (cid:54) = 0while f ( p ) = 0. Since the functional ker f → F given by x (cid:55)→ f ( px ) does not vanishwhen x = 1l, its kernel M is a complement to 1l F in ker f . Moreover the assumption f ( p ) = 0 implies that the subspace of A orthogonal to p is p F ⊕ M =: Q − . Hence A decomposes additively as p F ⊕ M ⊕ F as in Lemma A.1. Therefore the conditions inthis lemma are satisfied for some M provided only that f (1l) = 0 and there is p with f ( p ) (cid:54) = 0 , f ( p ) = 0.(ii) The fact that there is such a nice characterization of the uniruled property in termsof the structure of quantum homology leads immediately to speculations about rational One needs to interpret these formulas with some care. When we set T = a we are thinkingof T as the set of numbers ( T , . . . , T k ) that are the coordinates of a = P T (cid:96) ξ (cid:96) . On the otherhand, the arguments of a Gromov–Witten invariant are homology classes. Thus ˙ a, . . . , a ¸ Mp,β = P α ˙ ξ α ¸ Mp,β t − ω ( β ) T α | T = a . connectedness. A projective manifold is said to be rationally connected if there isa holomorphic P through every generic pair of points: see Kollar [18]. This impliesthat there is a holomorphic P through generic sets of k points, for any k , but it isnot yet known whether these spheres are visible in quantum homology, e.g. it is notknown whether there must be a nontrivial Gromov–Witten invariant with more thanone point constraint. This would correspond to the point class p := pt in QH aev ( M )having nonzero square p . This raises many questions. Are there symplectic manifoldswith p nilpotent but with p (cid:54) = 0? If p is not nilpotent is the quantum homology semi-simple? Abrams’ condition for semi-simplicity in [1] involves the quantum Euler class.What is its relation to the class p ? There are many possible choices for the coefficientring Λ; how do these affect the situation?(iii) The purpose of Proposition A.4 is to show that there is not much conceptual dif-ference between the usual quantum product and its deformations ∗ a . All the usualapplications of quantum homology (such as the Seidel representation and spectral in-variants) should have analogs for ∗ a . For example, given a ∈ H d ( M ; Q ) let G a bethe extension of π (Ham( M )) whose elements are pairs ( γ, (cid:101) a ) consisting of an element γ ∈ π (Ham( M )) with a class (cid:101) a ∈ H d +2 ( P γ ) such that (cid:101) a ∩ [ M ] = a ; cf. the discus-sion of the group (cid:98) G in [32, Ch 12.5]. Then (assuming that the appropriate version ofcondition (*) holds) one can define a homomorphism S a : G a → (cid:0) QH aev ( M ) (cid:1) × by setting S a (cid:0) ( γ, (cid:101) a ) (cid:1) = (cid:88) σ,m,i m ! (cid:10) ξ i , (cid:101) a, . . . , (cid:101) a (cid:11) m +1 ,σ ξ ∗ i ⊗ t − u γ ( σ ) , as in equation (2.2). It is not hard to check that this satisfies the analog of (2.4),namely S a (cid:0) ( γ, (cid:101) a ) (cid:1) ∗ a b = (cid:88) σ,m,i m ! (cid:10) b, ξ i , (cid:101) a, . . . , (cid:101) a (cid:11) m +2 ,σ ξ ∗ i ⊗ t − u γ ( σ ) . References [1] L. Abrams, The quantum Euler class and the quantum cohomology of the Grassmannians,q-alg/9712025[2] M. Audin, Hamiltoniens periodiques sur les vari´et´es symplectiques compactes de dimension4. In
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