MMONODROMY IN HAMILTONIAN FLOER THEORY
DUSA MCDUFF
Abstract.
Schwarz showed that when a closed symplectic manifold (
M, ω ) is sym-plectically aspherical (i.e. the symplectic form and the first Chern class vanish on π ( M )) then the spectral invariants, which are initially defined on the universal coverof the Hamiltonian group, descend to the Hamiltonian group Ham( M, ω ). In this notewe describe less stringent conditions on the Chern class and quantum homology of M under which the (asymptotic) spectral invariants descend to Ham( M, ω ). For ex-ample, they descend if the quantum multiplication of M is undeformed and H ( M )has rank >
1, or if the minimal Chern number is at least n + 1 (where dim M = 2 n )and the even cohomology of M is generated by divisors. The proofs are based oncertain calculations of genus zero Gromov–Witten invariants. As an application, weshow that the Hamiltonian group of the one point blow up of T admits a Calabiquasimorphism. Moreover, whenever the (asymptotic) spectral invariants descend itis easy to see that Ham( M, ω ) has infinite diameter in the Hofer norm. Hence ourresults establish the infinite diameter of Ham in many new cases. We also show thatthe area pseudonorm — a geometric version of the Hofer norm — is nontrivial onthe (compactly supported) Hamiltonian group for all noncompact manifolds as wellas for a large class of closed manifolds.
Contents
1. Introduction 22. Spectral invariants 92.1. Quantum homology 92.2. Spectral invariants and norms 103. The main argument 183.1. The Seidel representation 183.2. Calculating the coupling class. 224. Calculations of Gromov–Witten invariants 244.1. Preliminaries. 244.2. Technical lemmas. 264.3. Proof of Proposition 3.7. 275. Examples 32
Date : 26 December 2007, last revised February 15, 2010.2000
Mathematics Subject Classification.
Key words and phrases.
Hamiltonian group, Floer theory, Seidel representation, spectral invariant,Hofer norm, Calabi quasimorphism.partially supported by NSF grant DMS 0604769. a r X i v : . [ m a t h . S G ] F e b DUSA MCDUFF
References 341.
Introduction
Let (
M, ω ) be a closed symplectic manifold. Denote by Ham := Ham(
M, ω ) itsgroup of Hamiltonian symplectomorphisms and by (cid:93)
Ham the universal cover of Ham.Each path { φ t } ≤ t ≤ in Ham is the flow of some time dependent Hamiltonian H t and,following Hofer [8], we define its length L ( { φ t } ) to be: L ( { φ t } ) = (cid:90) (cid:16) max x ∈ M H t ( x ) − min x ∈ M H t ( x ) (cid:17) dt. The Hofer (pseudo)norm (cid:107) (cid:101) φ (cid:107) of an element (cid:101) φ = ( φ, { φ t } ) in the universal cover (cid:93) Hamof Ham is then defined to be the infimum of the lengths of the paths from the identityelement id to φ that are homotopic to { φ t } . Similarly we define the norm (cid:107) φ (cid:107) of anelement in Ham to be the infimum of the lengths of all paths from id to φ . It is easyto see that (cid:107) φ (cid:107) is conjugation invariant and satisfies (cid:107) φψ (cid:107) ≤ (cid:107) φ (cid:107) + (cid:107) ψ (cid:107) , but harder tosee that it is nondegenerate, i.e. (cid:107) φ (cid:107) = 0 iff φ = id . (This was proved for compactlysupported symplectomorphisms of R n by Hofer [8] and for general M by Lalonde–McDuff [14].) It is unknown whether (cid:107) · (cid:107) is always nondegenerate (and hence a norm)on (cid:93) Ham since it may vanish on some elements of the subgroup π (Ham). On the otherhand, there is no known counterexample; for some results in the positive direction seeRemark 2.9 below.The question of whether (cid:107) · (cid:107) is uniformly bounded makes sense even if (cid:107) · (cid:107) is justa pseudonorm. If is it unbounded on (cid:93) Ham or Ham we shall say that this group has infinite (Hofer) diameter.
Ostrover [24] showed that (cid:93)
Ham always has infinite Hoferdiameter (we sketch the proof below), while the corresponding result is unknown forHam in many cases. For example, it is shown in [20] that Ham has infinite diameterwhen M is a “small” blow up of C P but it is unknown whether this remains the casewhen M is monotone (i.e. the exceptional divisor is precisely one third the size of theline) or is a still bigger blow up.However, if both [ ω ] and c ( M ) vanish on π ( M ) then Schwarz [27] showed thatHam does have infinite diameter. To prove this he established that for such M eachelement (cid:101) φ ∈ (cid:93) Ham has a set of so-called spectral invariants { c ( a, (cid:101) φ ) | a ∈ QH ∗ ( M ) , a (cid:54) = 0 } ⊂ R . Later, Oh [22, 23] and Usher [29] showed that the numbers c ( a, (cid:101) φ ) are well defined on (cid:93) Ham for all symplectic manifolds. It follows easily from their properties (explained in § (cid:93) Ham has infinite Hofer diameter. However, they do not in generaldescend to well defined functions on Ham; in other words it may not be true that c ( a, (cid:101) φ ) = c ( a, (cid:101) ψ ) whenever (cid:101) φ, (cid:101) ψ project to the same element of Ham. Schwarz showedthat when both [ ω ] and c ( M ) vanish on π ( M ) the invariants do descend to Ham. As ONODROMY IN HAMILTONIAN FLOER THEORY 3 we explain below, it is then an easy consequence of Ostrover’s construction that Hamhas infinite diameter.Another case in which Ham was known to have infinite diameter is S . The originalproof in Polterovich [26] initially appears somewhat different in spirit from the approachpresented here, but the arguments in Entov–Polterovich [3] using quasimorphisms bringthis result also within the current framework; cf. case (ii) of Theorem 1.3.In fact there are two questions one can ask here. Do the spectral invariants them-selves descend, or is it only their asymptotic versions that descend? Here, followingEntov–Polterovich [3], we define the asymptotic spectral invariants c ( a, (cid:101) φ ) for nonzero a ∈ QH ∗ ( M ) by setting c ( a, (cid:101) φ ) := lim inf k →∞ c ( a, (cid:101) φ k ) k , for all (cid:101) φ ∈ (cid:93) Ham . Ostrover’s construction again implies that Ham has infinite diameter whenever an as-ymptotic spectral number descends to Ham; see Lemma 2.7.We shall extend Schwarz’s result in two directions, imposing conditions either on ω via the Gromov–Witten invariants or on c . Recall that the quantum product on H ∗ ( M ) is defined by the 3-point genus zero Gromov–Witten invariants (cid:10) a , a , a (cid:11) Mβ , β ∈ H ( M ) , a i ∈ H ∗ ( M ) , and reduces to the usual intersection product if these invariants vanish whenever β (cid:54) = 0.In the latter case we shall say that the quantum product (or simply QH ∗ ( M )) is un-deformed . The condition [ ω ] | π ( M ) = 0 is much stronger; in this case there are no J -holomorphic spheres at all, and so the quantum product on M is of necessity unde-formed. If [ ω ] | π ( M ) = 0, Schwarz’s argument easily extends to show that the spectralinvariants descend; see Proposition 3.1(i). However, its generalization in Theorem 1.1below concerns the asymptotic invariants.We shall denote by N the minimal Chern number of ( M, ω ), i.e. the smallest positivevalue of c ( M ) on π ( M ). If c | π ( M ) = 0 then we set N := ∞ . We always denotedim M = 2 n . We shall say that ( M, ω ) is spherically monotone if there is κ > c | π ( M ) = κ ω | π ( M ) and is negatively monotone if c = κ [ ω ] on π ( M ) for some κ <
0. Recall also that (
M, ω ) is said to be (symplecically) uniruled if some genus zeroGromov–Witten invariant of the form (cid:10) pt, a , . . . , a m (cid:11) β , β (cid:54) = 0, does not vanish. It iscalled strongly uniruled if this happens for m = 3. Theorem 1.1.
Suppose that ( M, ω ) is a closed symplectic manifold such that QH ∗ ( M ) is undeformed. Then the asymptotic spectral invariants descend to Ham except possiblyif the following three additional conditions all hold: rank H ( M ) = 1 , N ≤ n, and ( M, ω ) is spherically monotone. Remark 1.2. (i) The exceptional case does not occur if M has dimension 4. For if QH ∗ ( M ) is undeformed then ( M, ω ) is minimal. (The class of an exceptional spherealways has nontrivial Gromov–Witten invariant; see [19].) Moreover it follows fromthe results of Taubes–Li–Liu that it cannot be spherically monotone, for if it were it
DUSA MCDUFF would be uniruled, in particular the quantum product would be deformed. Similarly,the exceptional case does not occur for smooth projective varieties of any dimension.For N (cid:54) = ∞ implies that at least some nonzero elements in H ( M ) are represented byspheres. Hence they must all be (since rank H ( M ) = 1). Therefore the conditionsimply that M is Fano, and hence, by an argument of Kollar–Ruan that is explained in[10], also symplectically uniruled. It is unknown whether every spherically monotonesymplectic manifold is uniruled. If this were true then there would be no exceptions atall.(ii) Every minimal 4-manifold that is not rational or ruled (such as a K N . We denote the even degree homology of M by H ev ( M ). Theorem 1.3.
Let ( M, ω ) be a closed symplectic n -dimensional manifold with min-imal Chern number N . Suppose further that either H ev ( M ) is generated as a ring bythe divisors H n − ( M ) or that QH ∗ ( M ) is undeformed. Then: (i) If N ≥ n + 1 , the spectral invariants are well defined on Ham except possibly if N ≤ n and ( M, ω ) is strongly uniruled. (ii) If n + 1 ≤ N ≤ n the asymptotic spectral invariants are well defined on Ham . (iii) The conclusion in (ii) still holds if N = n except possibly if ( M, ω ) is stronglyuniruled or if rank H ( M ) = 1 . (iv) The conclusion in (ii) also holds when ( M, ω ) is negatively monotone, independentlyof the values of n, N . For example, if M is a 6-dimensional K¨ahler manifold then H ev ( M ) is generatedas a ring by H ( M ) because ∧ [ ω ] : H ( M ) → H ( M ) is an isomorphism. Hence allCalabi–Yau 3-folds satisfy the conditions of this theorem. Remark 1.4. (i) In Theorem 1.3 the conditions in the second sentence may be replacedby the weaker but somewhat technical condition (D); cf. Definition 3.5.(ii) Theorem 1.3 is sharp. To see that the non-uniruled hypothesis in (i) is necessary,observe that by Entov–Polterovich [3] the spectral invariants do not descend for M = C P n although the asymptotic ones do. This condition is also needed in (iii). Forexample, consider M = S × S which has N = n = 2 and is strongly uniruled.Ostrover [25] showed that the asymptotic spectral invariants descend if and only if( M, ω ) is monotone, i.e. the two 2-spheres have equal area. Further, the results donot extend to smaller N . Proposition 1.8 below gives many examples of manifoldswith N = n − N = n but rank H ( M ) = 1. If ( M, ω ) isnegatively monotone, then by Proposition 3.1 the asymptotic spectral invariants alwaysdescend (without any condition on QH ∗ ( M )), but it is not clear what happens in the ONODROMY IN HAMILTONIAN FLOER THEORY 5 positive case. (Cf. the similar missing case in Theorem 1.1.) If (
M, ω ) were alsoprojective then (
M, ω ) would be uniruled and one would not expect the invariants todescend but in the general case considered here all we can say is that our methods fail.The relevant part of the proof of Proposition 3.1 fails for N = n and κ >
0, while theargument in Lemmas 4.3 and 4.4 definitely needs rank H ( M ) > S × S in Remark 1.4(ii) above suggests that perhaps the as-ymptotic spectral invariants descend for all monotone manifolds. But this is not true.Consider, for example the monotone one point blow up of C P with its obvious T ac-tion. It is easy to see that there are circles in T that represent elements γ ∈ π (Ham)for which c (1l , γ ) (cid:54) = 0; see [20, 25]. Corollary 1.5. If ( M, ω ) satisfies any of the conditions in Theorems 1.1 and 1.3, then Ham has infinite Hofer diameter.Proof.
This holds by Lemma 2.7. (cid:3)
Of course, one expects Ham always to have infinite Hofer diameter, but this questionseems out of reach with current techniques. However there are other ways to tackle thisquestion. For example, in [20] we show that a small blow up of C P has infinite Hoferdiameter even though the spectral invariants do not descend by using an argumentbased on the asymmetry of the spectral invariants, i.e. the fact that the function V ofRemark 1.11 does not vanish on π (Ham). Also if π ( M ) is infinite, one can sometimesuse the energy–capacity inequality as in Lalonde–McDuff [15].Another related problem is the question of when the area pseudonorm ρ + + ρ − :Ham → R defined in [17] is nonzero. Here ρ + ( φ ) := inf (cid:90) max x ∈ M H t dt, where the infimum is taken over all mean normalized Hamiltonians with time 1 map φ . Further ρ − ( φ ) := ρ + ( φ − ). It is easy to see that ρ + + ρ − is a conjugation invariantpseudonorm on Ham. Therefore, because Ham is simple, ρ + + ρ − is either identicallyzero or is nondegenerate and hence a norm. The difficulty in dealing with it is that onemay need to use different lifts (cid:101) φ of φ to (cid:93) Ham to calculate ρ + and ρ − . However it hasa very natural geometric interpretation. For by [17, Prop. 1.12] ρ + ( φ ) + ρ − ( φ ) = inf (cid:0) Vol( P, Ω) / Vol(
M, ω ) (cid:1) where the infimum is taken over all Hamiltonian fibrations ( M, ω ) → ( P, Ω) → S withmonodromy φ around some embedded loop in the base. This ratio is called the area ofthe fibration P → S since for product fibrations it would be the area of the base.Since the area pseudonorm ρ + + ρ − is never larger than the Hofer norm the nextresult also implies Corollary 1.5. i.e (cid:82) M H t ω n = 0 for all t . Also, this discussion of one sided norms is the one place in this paperwhere the choice of signs is crucial. In order to be consistent with [17] we shall define the flow of H t to be generated by the vector X H t satisfying ω ( X H , · ) = − dH t . DUSA MCDUFF
Corollary 1.6. If ( M, ω ) satisfies any of the conditions in Theorems 1.1 and 1.3, thenthe area pseudonorm ρ + + ρ − is an unbounded norm on Ham .Proof.
See Lemma 2.8. (cid:3)
This result improves [17, Thm 1.2] which established the nontriviality of ρ + + ρ − only for the cases ω | π ( M ) = 0 and M = C P n .The real problem in understanding the onesided pseudonorms ρ ± is caused by thepossible existence of short loops , i.e. loops in Ham that are generated by Hamiltoniansfor which ρ + (or ρ − ) is small; see [17, § M is noncompact.Let ( M, ω ) be a noncompact manifold without boundary, and U ⊂ M be open withcompact closure. Denote by Ham c U the group of symplectomorphisms generated byfunctions H t with support in U and by (cid:93) Ham c U its universal cover. Denote by (cid:101) ρ + U thepositive part of the Hofer norm on (cid:93) Ham c U and by ρ + U the induced function on Ham c U .Notice that in principle (cid:101) ρ + U might depend on U . But clearly U (cid:48) ⊂ U implies (cid:101) ρ + U (cid:48) ≥ (cid:101) ρ + U .A similar remark applies to ρ + U .The following result was suggested by a remark in an early version of [7]. Notethat ( M, ω ) can be arbitrary here; in particular it need satisfy no special conditions atinfinity.
Proposition 1.7.
Suppose that ( M, ω ) is noncompact and that U is an open subsetof M with compact closure. Then for all H : M → R with support in U there is δ = δ ( H, U ) > such that ρ + U ( φ Ht ) = t max H for all ≤ t ≤ δ . In particular ρ + U + ρ − U is a nondegenerate norm on Ham c U . The proof is given at the end of §
2. Note that (cid:93)
Ham c U always has infinite Hoferdiameter because of the existence of the Calabi homomorphism. However, even though (cid:93) Ham c U is not a simple group, the kernel of the Calabi homomorphism is simple. Hencethe second statement in the above proposition follows from the first because the element φ Ht belongs to this kernel when (cid:82) U Hω n = 0. For a brief discussion of other issues thatarise in the noncompact case, see [20, Remark 3.11].Finally we describe another class of examples, the one point blow ups. These have N | ( n −
1) but may be chosen to satisfy the other conditions of Theorem 1.3.
Proposition 1.8. (i)
Let M be a sufficiently small one point blow up of any closedsymplectic manifold ( X, ω X ) such that at least one of [ ω X ] , c ( X ) does not vanish on π ( X ) . Then the asymptotic spectral invariants do not descend to Ham . (ii) If M is the one point blow up of a closed symplectic -manifold X such that both [ ω X ] and c ( X ) vanish on π ( X ) then the asymptotic spectral invariants do descend to Ham . The proof is given in § Corollary 1.9. If M is as in Proposition 1.8(ii) then Ham supports a nontrivial Calabiquasimorphism.
ONODROMY IN HAMILTONIAN FLOER THEORY 7
The proof is contained in the following remark.
Remark 1.10.
Other potential applications of these results arise from the work ofEntov–Polterovich [3, 4, 5, 6]. They denote the asymptotic spectral invariant given byan idempotent e in A M := QH n ( M ) by µ e : (cid:93) Ham → R , µ e ( · ) := c ( e, · ) . It is immediate that µ e descends to Ham iff µ e | π (Ham) vanishes; see Proposition 2.3.The Entov–Polterovich results about nondisplaceablity (see [5] for example) applywhether or not µ e descends to Ham. But if one is interested in questions about thestructure of the Hamiltonian group itself, for example what quasimorphisms or normsit might have or what discrete subgroups it might contain, then our results are relevant.If e is an idempotent such that e A M is a field, then Entov–Polterovich show that µ e is a homogeneous quasimorphism, i.e. there is C > k ∈ Z and (cid:101) φ, (cid:101) ψ ∈ (cid:93) Ham µ e ( (cid:101) φ k ) = kµ e ( (cid:101) φ ) , | µ e ( (cid:101) φ (cid:101) ψ ) − µ e ( (cid:101) φ ) − µ e ( (cid:101) ψ ) | ≤ C. In particular, the restriction of µ e to the abelian subgroup π (Ham) ⊂ (cid:93) Ham is ahomomorphism. Therefore, µ e descends to a quasimorphism on Ham exactly if itvanishes on π (Ham). It has the Calabi property of [3] by construction.There are rather few known manifolds (besides C P n ) for which A M contains anidempotent e such that both e A M is a field and µ e descends. For these conditions workin opposite directions; we need many nontrivial Gromov–Witten invariants for e A M to be a field, but not too many (or at least the Seidel representation of § µ e is to descend. In this paper the one set of new examples with asuitable idempotent e are those of Proposition 1.8 (ii). Indeed, the calculations in [19, §
2] (see also [6]) show that in this case there is an idempotent e ∈ A M such that e A M is a field, while we show here that µ e descends.However, even if e A M is not a field, Entov–Polterovich show in [4, §
7] that µ e interacts in an interesting way with the geometry of M . For example, it provides alower bound for the so-called fragmentation norm (cid:107) · (cid:107) U on (cid:93) Ham of the form | µ e ( (cid:101) φ (cid:101) ψ ) − µ e ( (cid:101) φ ) − µ e ( (cid:101) ψ ) | ≤ K min {(cid:107) (cid:101) φ (cid:107) U , (cid:107) (cid:101) ψ (cid:107) U } , for all (cid:101) φ, (cid:101) ψ ∈ (cid:93) Ham , When A M is semisimple, it is tempting to think that µ e is a quasimorphism for every idempotent,and in particular for e = 1l. However, this is not true, as is shown by the example of the small one pointblow up of C P . The calculations in [17] (see also [25]) show that there is an element Q ∈ A M such that ν ( Q k )+ ν ( Q − k ) → ∞ . (See § c and an element α ∈ π (Ham)such that µ ( α k ) := ν ( Q k ) + kc for all k ∈ Z . But if µ were a quasimorphism it would restrict toa homomorpism on the abelian subgroup π (Ham) and we would have ν ( Q k ) + ν ( Q − k ) = 0. Henceit cannot be a quasimorphism. Rather it is related to the maximum of two different quasimorphisms:since 1l = e + e where each e i is minimal, µ ≤ max( µ e , µ e ) with equality at all elements where the µ e i take different values; cf. equation (20) in [5]. If U is an open subset of M , (cid:107) (cid:101) φ (cid:107) U is defined to be the minimal number k such that (cid:101) φ can bewritten as a product of k symplectomorphisms each conjugate to an element in (cid:93) Ham c ( U ), the universalcover of the group of compactly supported Hamiltonian symplectomorphisms of U . DUSA MCDUFF where K is a constant that depends only on the open set U and U is assumed dis-placeable, i.e. there is ψ ∈ Ham M such that U ∩ ψ ( U ) = ∅ . If e A M is a field thenthe quantity on LHS is bounded and µ e is a quasimorphism as in (ii) above. On theother hand, if this quantity is unbounded then the fragmentation norm (cid:107) · (cid:107) U is alsounbounded on (cid:93) Ham. Theorems 1.1 and 1.3 allow one to transfer these results to Hamin many cases; cf. Burago–Ivanov–Polterovich [2, Ex. 1.24].
Remark 1.11. (i) (Properties of the spectral norm and its variants.) Consider thefunction V on (cid:93) Ham given by V ( (cid:101) φ ) := c (1l , (cid:101) φ ) + c (1l , (cid:101) φ − ) , and the corresponding function v induced on Ham: v ( φ ) := inf (cid:8) V ( (cid:101) φ ) | (cid:101) φ lifts φ (cid:9) . Schwarz and Oh showed that v is a conjugation invariant norm on Ham, called the spectral norm . As noticed by Entov–Polterovich [3], this norm is bounded when M = C P n or, more generally, when QH ∗ ( M ) is a field with respect to suitable coefficients. For these hypotheses imply that c (1l , · ) is an (inhomogeneous) quasimorphism on (cid:93) Ham,so that V ( (cid:101) φ ) = | c (1l , id ) − c (1l , (cid:101) φ ) − c (1l , (cid:101) φ − ) | ≤ const . Note that Ostrover’s argument does not apply here since V ( (cid:101) ψ s ) remains bounded onthe path (cid:101) ψ s , s ≥ , defined in equation (2.9).The supremum of v ( φ ) for φ ∈ Ham is called the spectral capacity of M , cf. Albers[1, equation (2.47)]. Since V ( (cid:101) φ ) ≥ c (1l , (cid:101) φ ) − c ( pt, (cid:101) φ ) > (cid:101) φ (cid:54) = id , this capacity canbe thought of as a measure of spectral spread. When M is the standard torus T n it iswell known that there are (normalized) functions H : M → R whose flow (cid:101) φ Ht has theproperty that V ( (cid:101) φ Hk ) = kV ( (cid:101) φ H ) (cid:54) = 0; see the discussion after Question 8.7 in [4]. SinceTheorem 1.3 implies that V = v in this case, the spectral capacity of T n is infinite.In general the spectral capacity is poorly understood. For example, it is not knownwhether it is always infinite when QH ∗ ( M ) is very far from being a field, for exampleif ( M, ω ) is aspherical.As we explain in the proof of Lemma 2.8, c (1l , (cid:101) φ ) ≤ ρ + ( (cid:101) φ − ). Hence one mightthink of minimizing c (1l , (cid:101) φ ) and c (1l , (cid:101) φ − ) separately over the lifts of φ as we did for theHofer norm. However, in general this procedure gives nothing interesting since c (1l , (cid:101) φ )certainly can be negative and may well have no lower bound over a set of lifts. Instead,one can consider the functions V : (cid:93) Ham → R , (cid:101) φ (cid:55)→ c (1l , (cid:101) φ ) + c (1l , (cid:101) φ − ) , and v : Ham → R , φ (cid:55)→ inf (cid:8) V ( (cid:101) φ ) | (cid:101) φ lifts φ (cid:9) . Cf. Albers [1, Lemma 5.11]. In this context it matters which coefficients are used for quantumhomology; compare the approaches of Ostrover [25] and Entov–Polterovich [6].
ONODROMY IN HAMILTONIAN FLOER THEORY 9 (Thus V is a symmetrized version of the function µ considered in the previous remark.)If c (1l , · ) is a quasimorphism then c (1l , · ) is a homogeneous quasimorphism and hencesatisfies c (1l , (cid:101) φ ) = − c (1l , (cid:101) φ − ). Therefore, in this case, V ≡
0. On the other hand, aswe pointed out in Remark 1.10, V does not vanish on π (Ham) ⊂ (cid:93) Ham when M is asmall blow up of C P . Our remarks above imply that T n has infinite V -diameter, butagain very little is known about V for general M .Although these variants of v have some uses, they are unlikely to be (pseudo)norms,since, as we explain in Remark 2.2 (iii), they probably never have the property m ( f g ) ≤ m ( f ) + m ( g ). One might also think of replacing the class 1l by some other idempotent e . But it is easy to see that ν ( e ) > e . Hence the resulting function wouldnot take the value 0 at the identity id ∈ (cid:93) Ham.(ii) The proof of Corollary 1.6 compares ρ + + ρ − with the Schwarz–Oh norm v . Sincethere are (cid:101) φ (cid:54) = 1l such that c (1l , (cid:101) φ ) ≤ ρ + evervanishes on some φ (cid:54) = 1l. Acknowledgements
Many thanks to Leonid Polterovich for very helpful commentson an earlier draft of this paper, and also to Peter Albers, Alvaro Pelayo and the refereefor detailed comments that have helped improve the clarity of the exposition.2.
Spectral invariants
In this section we discuss the basic properties of spectral invariants and prove Lem-mas 2.7 and 2.8 concerning the Hofer diameter of Ham, as well as Proposition 1.7. Weassume throughout that M is closed unless explicit mention is made to the contrary.2.1. Quantum homology.
To fix notation we list some facts about the small quantumhomology QH ∗ ( M ) := H ∗ ( M ) ⊗ Λ. We shall take coefficients Λ := Λ univ [ q, q − ], where q is a variable of degree 2 and Λ univ is the field of generalized Laurent series in t − with elements λ = (cid:88) i ≥ r i t ε i , r i , ε i ∈ R , ε i > ε i +1 , ε i → −∞ . Thus Λ univ has a valuation ν : Λ univ → [ −∞ , ∞ ) given by ν ( λ ) := max { ε i : r i (cid:54) = 0 } .Observe that ν (0) = −∞ , and for all λ, µ ∈ Λ ν ( λ + µ ) ≤ max (cid:0) ν ( λ ) , ν ( µ ) (cid:1) , ν ( λµ ) = ν ( λ ) + ν ( µ ) , ν ( λ − ) = − ν ( λ ) . This valuation extends to QH ∗ ( M ) in the obvious way: namely, for any a i ∈ H ∗ ( M )we set ν (cid:16)(cid:88) a i ⊗ q d i t ε i (cid:17) = max { ε i : a i (cid:54) = 0 } . We use the ground field R here since later on we use homology with R coefficients, but couldequally well take r i ∈ C as in Entov–Polterovich [5]. The quantum product a ∗ b of the elements a, b ∈ H ∗ ( M ) ⊂ QH ∗ ( M ) is defined asfollows. Let ξ i , i = 0 , . . . , m, be a basis for H ∗ ( M ) with dual basis { ξ ∗ M i } . Thus ξ ∗ M i · M ξ j = δ ij . We use this slightly awkward notation to reserve ξ ∗ i for later use; cf. equation (4.3).Also · M (which is often simplified to · ) denotes the intersection product H d ( M ) ⊗ H n − d ( M ) → H ( M ) ≡ R . Further, denote by H S ( M ) the spherical homology group, i.e. the image of the Hurewiczmap π ( M ) → H ( M ). Then(2.1) a ∗ b := (cid:88) i,β ∈ H S ( M ) (cid:10) a, b, ξ i (cid:11) Mβ ξ ∗ M i ⊗ q − c ( β ) t − ω ( β ) , where (cid:10) a, b, ξ i (cid:11) Mβ denotes the Gromov–Witten invariant that counts curves in M of class β through the homological constraints a, b, ξ i . Note that deg( a ∗ b ) = deg a + deg b − n ,and the identity element is 1l := [ M ]. The product is extended to H ∗ ( M ) ⊗ Λ bylinearity over Λ.Later it will be useful to consider the Λ-submodule Q − := (cid:77) i< n H i ( M ) ⊗ Λ . (Here as usual 2 n = dim M .) The following (easy) result was proved in [19]. Lemma 2.1. (i) If M is not strongly uniruled then Q − is an ideal in QH ∗ ( M ) . (ii) If Q − is an ideal and if u ∈ QH n ( M ) is invertible, then u = 1l ⊗ λ + x for somenonzero λ ∈ Λ and some x ∈ Q − . Note that if quantum multiplication is undeformed then the elements of Q − arenilpotent so that all elements 1l ⊗ λ + x with λ (cid:54) = 0 and x ∈ Q − are invertible. However,if we assume only that Q − is an ideal, then an element of the form 1l ⊗ λ + x mightnot be invertible. For example it could be a nontrivial idempotent. See [19] for furtherdetails. Note: whenever we write a unit as 1l ⊗ λ + x we assume, unless explicit mention is madeto the contrary, that λ (cid:54) = 0 and x ∈ Q − .2.2. Spectral invariants and norms.
One way to estimate the length of a Hamil-tonian path is to use the Schwarz–Oh spectral invariants ; see [27, 22] and Usher [29].(They are also explained in [21, Ch 12.4].) For each element (cid:101) φ ∈ (cid:93) Ham and each nonzeroelement a ∈ QH ∗ ( M ) the number c ( a, (cid:101) φ ) ∈ R has the following properties: −(cid:107) (cid:101) φ (cid:107) ≤ c ( a, (cid:101) φ ) = c ( a, (cid:101) ψ (cid:101) φ (cid:101) ψ − ) ≤ (cid:107) (cid:101) φ (cid:107) for a ∈ H ∗ ( M ) , (cid:101) ψ ∈ (cid:93) Ham(2.2) c ( λa, (cid:101) φ ) = c ( a, (cid:101) φ ) + ν ( λ ) for all λ ∈ Λ , (2.3) c ( a, (cid:101) φ ◦ γ ) = c ( S ( γ ) ∗ a, (cid:101) φ ) for all γ ∈ π (Ham) , (2.4) c ( a ∗ b, (cid:101) φ ◦ (cid:101) ψ ) ≤ c ( a, (cid:101) φ ) + c ( b, (cid:101) ψ ) for all a, b ∈ QH ∗ ( M ) , (cid:101) φ, (cid:101) ψ ∈ (cid:93) Ham . (2.5) ONODROMY IN HAMILTONIAN FLOER THEORY 11
The third property explains how these numbers depend on the path (cid:101) φ . Here, theelement S ( γ ) ∈ QH ∗ ( M ) is called the Seidel element of the loop γ (see [28, 16]). It isan invertible element of degree 2 n = dim M in QH ∗ ( M ); we give a brief definition in § c ( a, id ) = 0 for all a ∈ H ∗ ( M ), where id denotesthe constant loop at the identity. Hence, for all γ ∈ π (Ham) c (1l , γ ) = c ( S ( γ ) , id ) = ν ( S ( γ )) , c (1l , γ ) = lim k →∞ ν ( S ( γ k )) k . Remark 2.2. (i) Equation (2.2) implies that the lim inf defining the asymptotic in-variants c ( a, · ) always exists. When a = a standard arguments based on (2.5) showthat one can replace the lim inf by an ordinary limit; cf. [3, § π (Ham) lies in the center of (cid:93) Ham; cf. the proof of Proposition 2.3 below. Moreover, if (cid:101) φ, (cid:101) ψ commute then the fourthproperty is also inherited by c . (This is the basis for the discussion in [4] of partialsymplectic quasi-states .)(iii) If e is an idempotent it is easy to see that c ( e, (cid:101) φ (cid:101) ψ ) ≤ c ( e, (cid:101) φ ) + c ( e, (cid:101) ψ ) for all (cid:101) φ, (cid:101) ψ .But in general the asymptotic invariants only have good algebraic properties when e is an idempotent such that e A M is a field; cf. Remark 1.10. However even in this caseit is impossible to have c ( e, (cid:101) φ (cid:101) ψ ) ≤ c ( e, (cid:101) φ ) + c ( e, (cid:101) ψ ) for all (cid:101) φ, (cid:101) ψ . Indeed, as Polterovichpoints out, because c ( e, (cid:101) φ − ) = − c ( e, (cid:101) φ ) for such e , if this inequality did always holdone would have − c ( e, (cid:101) φ (cid:101) ψ ) = c ( e, ( (cid:101) φ (cid:101) ψ ) − ) ≤ c ( e, (cid:101) φ − ) + c ( e, (cid:101) ψ − ) = − c ( e, (cid:101) φ ) − c ( e, (cid:101) ψ ) . But then we would have equality, i.e. c ( e, · ) would be a surjective homomorphism.It would then descend to a nontrivial homomorphism Ham → R / Γ, where Γ is theimage of the countable group π (Ham). Since Ham is perfect, this is impossible.(Incidentally, this argument also shows that when e A M is a field it is impossible that c ( e, (cid:101) φ ) = c ( e, (cid:101) φ ) for all (cid:101) φ . This is obvious from the explicit calculation in formula (2.9)below. On the other hand, the current argument is structural and hence might applyin other situations as well.)Much of following proposition is implicit in Entov–Polterovich [3]. Proposition 2.3. (i)
The following conditions are equivalent: (a)
All spectral invariants c ( a, · ) , a ∈ QH ∗ ( M ) , a (cid:54) = 0 , descend to Ham ; (b) The spectral invariant c (1l , · ) descends to Ham ; Private communication. A proof of this classical result is sketched in [21, Remark 9.5.6]. In the published version, it is claimed that (i:a) below is equivalent to the statement that just oneinvariant c ( a, · ) descends, and the same claim is made for (ii:a). However, this could be false, though Iam not aware of any explicit counterexample at this time. Corrected versions of (i:b) and (ii:b) appearbelow; they are not used elsewhere in the paper. (c) c (1l , γ ) = ν ( S ( γ )) = 0 for all γ ∈ π (Ham) ; (d) For all γ ∈ π (Ham) , S ( γ ) = 1l ⊗ λ + x where ν ( λ ) = 0 and ν ( x ) ≤ . (ii) The following conditions are equivalent: (a)
All asymptotic spectral invariants c ( a, · ) , a ∈ QH ∗ ( M ) , a (cid:54) = 0 , descend to Ham ; (b) The asymptotic spectral invariant c (1l , · ) descends to Ham ; (c) For all γ ∈ π (Ham) , c (1l , γ ) ≥ , and c (1l , γ ) = lim k →∞ ν ( S ( γ k )) k = 0 . (iii) If the spectral invariants descend then so do the asymptotic invariants.Proof.
Consider (i). The equivalence of the first three conditions is immediate from(2.3), (2.4) and (2.5). Part (ii) of the next lemma shows that (c) implies (d). Conversely,suppose that (d) holds, i.e. all the S ( γ ) may be written as 1l ⊗ λ + x where ν ( λ ) = 0 ≥ ν ( x ). Then ν ( S ( γ )) ≤ γ . Moreover, if ν ( S ( γ )) < γ then0 = ν (1l) = ν (cid:0) S ( γ ) S ( γ − ) (cid:1) ≤ ν (cid:0) S ( γ ) (cid:1) + ν (cid:0) S ( γ − ) (cid:1) < c descend then in particular we must have c (1l , γ ) = c (1l , id ) = 0 for all γ . Further, if c (1l , γ ) = − ε < c (1l , γ k ) ≤ − kε which implies that c (1l , γ ) <
0. Therefore (a) implies (b) implies (c). To prove that (c)implies (a), observe that because γ ∈ π (Ham) is in the center of (cid:93) Ham, the subadditivityrelation (2.5) for c implies that c ( a, (cid:101) φ ◦ γ ) = lim inf k (cid:0) c ( a, (cid:101) φ k γ k ) (cid:1) ≤ c ( a, (cid:101) φ ) + c (1l , γ ) . Hence if c (1l , γ ) = 0 always, c ( a, (cid:101) φ ◦ γ ) ≤ c ( a, (cid:101) φ ) = c ( a, ( (cid:101) φ ◦ γ ) ◦ γ − ) ≤ c ( a, (cid:101) φ ◦ γ ) . Thus we must always have c ( a, (cid:101) φ ◦ γ ) = c ( a, (cid:101) φ ). Therefore (a), (b) and (c) are equivalent.This proves (ii); (iii) is immediate. (cid:3) Lemma 2.4. (i)
Every element of the form u = 1l ⊗ λ + y with ν ( λ ) ≥ ν ( y ) is invertiblein QH ∗ ( M ) . (ii) Let u ∈ QH ∗ ( M ) be an invertible element of the form u = 1l ⊗ µ + y where µ mightbe zero, and suppose that ν ( u ) = 0 . Then either ν ( µ ) = 0 = ν ( u − ) or ν ( u − ) > .Proof. First note that by the definition of the quantum product(2.6) ν ( x ∗ z − x ∩ z ) ≤ ν ( x ) + ν ( z ) − δ for all x, z ∈ QH ∗ ( M ) , where δ > ω ( β ) of a class β (cid:54) = 0 with nonzero Gromov–Witteninvariant (cid:10) a , a , a (cid:11) β . Hence, because all elements in the undeformed ring H < n ( M )are nilpotent of order ≤ n + 1, ν ( a n +1 ) ≤ − δ for all a ∈ H < n ( M ) . ONODROMY IN HAMILTONIAN FLOER THEORY 13 (Here and subsequently a k denotes the k -fold quantum product a ∗ · · · ∗ a .) Hence if z ∈ Q − has ν ( z ) ≤ ν ( z ( n +1) k ) ≤ − kδ → −∞ as k → ∞ . Because ν ( z i ) is anonincreasing sequence when ν ( z ) ≤
0, it also diverges to −∞ . Thus (cid:80) i ≥ z i is a welldefined element of QH ∗ ( M ). Hence 1l ⊗ λ + y is invertible with inverse u − := λ − (1l + λ − y ) − = λ (cid:88) ( − i ( λ − y ) i ∈ QH ∗ ( M ) . This proves (i).To prove (ii) suppose that u = 1l ⊗ µ + y is a unit with ν ( u ) = 0. Because 1l andthe components of y are linearly independent, ν ( u ) = max (cid:0) ν ( µ ) , ν ( y ) (cid:1) . Similarly, if wewrite u − = 1l ⊗ λ + x , we have ν ( u − ) = max (cid:0) ν ( λ ) , ν ( x ) (cid:1) .If ν ( µ ) = 0 then the above formula for u − shows that ν ( u − ) = 0 . So suppose that ν ( µ ) <
0. Because u u − = 1l, at least one of the terms 1 ⊗ λµ and xy must contain1l with a nonzero rational coefficient. If it is the former then ν ( λ ) = − ν ( µ ) >
0, whileif it is the latter then we must have ν ( x ) + ν ( y ) ≥ δ so that ν ( x ) ≥ δ . In either case ν ( u − ) > (cid:3) Corollary 2.5.
The asymptotic spectral invariants descend to
Ham if for all γ thereis m such that S ( γ m ) = 1l ⊗ λ + x where ν ( λ ) = 0 and x is nilpotent.Proof. Suppose that x N = 0. Let K = max { ν ( x k ) | ≤ k ≤ N } . Then because S is ahomomorphism and ν is subadditive ν (cid:0) S ( γ mk ) (cid:1) = ν (cid:0) ( S ( γ m )) k (cid:1) = ν ( λ k + kλx + . . . ) ≤ max { kν ( λ ) , ν ( λx ) , . . . } ≤ K, for all k ∈ Z and all γ . Hence c (1l , γ m ) = 0, and hence c (1l , γ ) = 0. (cid:3) Remark 2.6.
It is not hard to find conditions under which every invertible element in QH ∗ ( M ) has the form 1l ⊗ λ + x where x is nilpotent. For example, as we noted above,this is always true if the quantum multiplication is undeformed. For other cases seeLemma 3.2. Therefore the main difficulty in showing that the (asymptotic) spectralinvariants descend lies in ensuring that the condition ν ( λ ) = 0 holds.The numbers c ( a, (cid:101) φ ) are defined by looking at the filtered Floer complex of thegenerating mean normalized Hamiltonian H , and turn out to be particular criticalvalues of the corresponding action functional A H ; cf. Oh [23] and Usher [29]. Thuseach c ( a, (cid:101) φ ) corresponds to a particular fixed point of the endpoint φ ∈ Ham of (cid:101) φ .It is usually very hard to calculate them. However, if (cid:101) φ := (cid:101) φ H is generated by a C -small mean normalized Morse function H : M → R , then(2.7) c ( a, (cid:101) φ H ) = c M ( a, − H ) , a ∈ QH ∗ ( M )where c M ( a, − H ) are the corresponding invariants obtained from the Morse complexof − H . These are defined as follows. Denote by CM ∗ ( M, K ) the usual Morse complex In fact this property is essential to the existence of the spectral invariants as functions on (cid:93)
Ham;the spectral invariants are first defined as functions on the space of Hamiltonians H and one needs thespectrality property to conclude that they actually depend only on the element in (cid:93) Ham defined by theflow of H . for the Morse function K . For each κ ∈ R , it has a subcomplex CM κ ∗ ( M, K ) generatedby the critical points p ∈ Crit( K ) with critical values K ( p ) ≤ κ . Denote by ι κ theinclusion of the homology H κ ∗ of this subcomplex into H ∞∗ ∼ = H ∗ ( M ). Then for each a ∈ H ∗ ( M )(2.8) c M ( a, K ) := inf { κ : a ∈ Im ι κ } . Ostrover’s construction.
As pointed out by Ostrover [24], one can use the continuityproperties of the c ( a, (cid:101) φ ) to find a path s (cid:55)→ (cid:101) ψ s in (cid:93) Ham whose spectral invariants tendto ∞ as s → ∞ .Normalize ω so that (cid:82) M ω n = 1. Let H be a small mean normalized Morse function,choose an open set U that is displaced by φ H (i.e. φ H ( U ) ∩ U = ∅ ) and let F : M → R be a function with support in U and with nonzero integral I := (cid:82) M F ω n , so that F − I ismean normalized. Denote the flow of F by f t and consider the path (cid:101) ψ s := { f ts φ Ht } t ∈ [0 , in (cid:93) Ham as s → ∞ . For each s , (cid:101) ψ s is generated by the Hamiltonian F st H := sF + H ◦ f st . The corresponding mean normalized Hamiltonian is K s := sF + H ◦ f st − sI. By construction, f s φ H has the same fixed points as φ H , namely the critical points of H . Hence the continuity and spectrality properties of c ( a, (cid:101) ψ s ) imply that for each a thefixed point p a whose critical value is c ( a, (cid:101) ψ s ) remains unchanged as s increases. But thespectral value does change. In fact, if a ∈ H ∗ ( M ), then when s = 0 there is a criticalpoint p a of H such that c ( a, (cid:101) ψ ) = c M ( a, − H ) = − H ( p a ) . Hence(2.9) c ( a, (cid:101) ψ s ) = − K s ( p a ) = − H ( p a ) + sI, for all s ∈ R , a ∈ H ∗ ( M ) . By (2.2) it follows that (cid:93)
Ham has infinite Hofer diameter.The next result is well known.
Lemma 2.7.
Ham has infinite diameter if either a spectral number or an asymptoticspectral number descends to
Ham .Proof.
Let π : (cid:93) Ham → Ham denote the projection. We will show that, for the abovepath (cid:101) ψ s , (cid:107) π ( (cid:101) ψ s ) (cid:107) = (cid:107) ψ s (cid:107) → ∞ as s → ∞ . By definition, for each s (cid:107) ψ s (cid:107) = inf {(cid:107) (cid:101) ψ s ◦ γ (cid:107) : γ ∈ π (Ham) } ≥ inf { c ( a, (cid:101) ψ s ◦ γ ) : γ ∈ π (Ham) } where the inequality follows from equation (2.2). If the spectral number c ( a, · ) descendsto Ham then c ( a, (cid:101) ψ s ◦ γ ) = c ( a, (cid:101) ψ s ) for all γ so that the result follows from equation(2.9).Now suppose that the asymptotic spectral number c ( e, · ) descends to Ham for someidempotent e . We first claim that there are elements (cid:101) g i , i = 1 , . . . , k − , in (cid:93) Ham
ONODROMY IN HAMILTONIAN FLOER THEORY 15 that are conjugate to (cid:101) φ H and such that (cid:101) ψ sk (cid:101) g · · · (cid:101) g k − = ( (cid:101) ψ s ) k . To see this, denote a := (cid:101) f s , b := (cid:101) φ H so that (cid:101) ψ sk = a k b and ( (cid:101) ψ s ) k = ( ab ) k . Then use the fact that( ab ) k = a k b · b − a − k ab . . . ab = a k b · b − (cid:16) ( a − k +1 ba k − )( a − k +2 ba k − ) . . . ( a − ba ) (cid:17) b. Next, observe that by (2.5) we have c ( e, ( (cid:101) ψ ) k ) ≤ kc ( e, (cid:101) ψ ) for all k >
1. Hence c ( e, (cid:101) ψ s ◦ γ ) = lim k c ( e, ( (cid:101) ψ s ◦ γ ) k ) ≤ c ( e, (cid:101) ψ s ◦ γ ) ≤ (cid:107) (cid:101) ψ s ◦ γ (cid:107) . On the other hand, c ( e, (cid:101) ψ s ◦ γ ) = c ( e, (cid:101) ψ s ) = lim k c ( e, ( (cid:101) ψ sk ) (cid:101) g − k − · · · (cid:101) g − ) ≥ lim k (cid:16) c ( e, (cid:101) ψ sk ) − k − (cid:88) i =1 c ( e, (cid:101) g i ) (cid:17) , ≥ sI − (cid:107) (cid:101) φ H (cid:107) , where the first inequality uses the identity c ( e, f g ) ≥ c ( e, f ) − c ( e, g − ) which followsfrom (2.5), and the second uses (2.9) and the fact that the (cid:101) g i are conjugate to (cid:101) φ H . (cid:3) Lemma 2.8.
The function ρ + + ρ − is unbounded on Ham whenever the asymptoticspectral numbers descend to
Ham .Proof.
The proof of the inequalities in (2.2) actually shows that (cid:90) min( − H t ) dt ≤ c (1l , (cid:101) φ ) ≤ (cid:90) max( − H t ) dt for every mean normalized H t that generates (cid:101) φ ; see for example [21, Thm.12.4.4]. Nowsuppose that c (1l , · ) descends. Equation (2.5) implies that c (1l , (cid:101) φ k ) ≤ kc (1l , (cid:101) φ ). Hence if (cid:101) φ is any lift of φ , c (1l , φ ) = c (1l , (cid:101) φ ) ≤ (cid:90) max( − H t ) dt. Therefore, because − H − t generates (cid:101) φ − , we have c (1l , φ ) ≤ ρ + ( φ − ). Applying this tothe image ψ s ∈ Ham of the element (cid:101) ψ s of (2.9) we find that c (1l , ψ s ) = s ≤ ρ + ( ψ s − ) . Thus, because ρ − ≥
0, the function ρ + + ρ − is not identically zero and hence is a norm.Moreover it is clearly unbounded. (cid:3) Remark 2.9.
As remarked in the introduction, the Hofer pseudonorm (cid:107) · (cid:107) is a normon (cid:93)
Ham if and only if it restricts to a norm on π (Ham). The only way that I knowto estimate (cid:107) γ (cid:107) for γ ∈ π (Ham) is via the spectral invariant c (1l , γ ) = ν ( S ( γ )) whichis a lower bound for (cid:107) γ (cid:107) by (2.2). If the spectral invariants descend to Ham then c (1l , γ ) = 0 for all γ and one gets no information. For example, there are no currentmethods to detect the Hofer lengths of the elements of π (Ham) when ( M, ω ) is thestandard 2 n -torus. (Of course, in this case if n > π (Ham) itself is nonzero.) On the other hand, there are manifolds such as certain blow ups of C P and S -bundles over S for which π (Ham) is known and ν ( S ) does restrict to aninjective homomorphism; see [20, 17] and the references cited therein.We end this section by proving Proposition 1.7 about the behavior of ρ + in thenoncompact case. We need to prove: Lemma 2.10.
For each H : M → R with support in U (cid:54) = M there is δ = δ ( H, U ) > such that ρ + U ( φ tH ) = t max H, ≤ t ≤ δ. Proof.
We will prove this for open sets U with smooth boundary. Since any precompactopen set U (cid:48) is contained in such U the result for U (cid:48) follows easily, since ρ + U (cid:48) is definedby taking an infimum over a smaller set than ρ + U .Put a collar neighborhood Y × [ − , ⊂ M round the boundary of U so that U ∩ ( Y × [ − , Y × [ − , . Let U λ := U ∪ ( Y × [ − , λ ]). Choose any ω -compatible almost complex structure J on M and choose r > y ∈ Y × { / } there is a symplectic embedding f y of the standard ball of radius r into Y × [1 / , /
4] with center at f y (0) = y . Then bymonotonicity there is δ > J -holomorphic sphere u : S → U whoseimage meets both Y × { / } and Y × { / } has energy ≥ δ .One condition on the constant δ is that δ (cid:107) H (cid:107) < δ , where (cid:107) H (cid:107) = max H − min H .The other is that when 0 < t ≤ δ the function tH should be sufficiently small in the C norm for it to be possible to embed a ball of capacity (cid:107) tH (cid:107) in R − tH ( ε ), the region“under the graph” of tH . This region is described in [17, § δ . Thus if F := tH for some t ∈ [0 , δ ] we suppose that there is another Hamiltonian path φ Kt inHam c U with φ K = φ F that is generated by a Hamiltonian K t with support in U andsuch that (cid:90) (cid:90) (max x ∈ M K t ) ω n dt < (cid:90) (max x ∈ M F ) ω n . Under these conditions we show in [17, §
2] that there is a Hamiltonian bundle( U , ω ) → ( P , Ω) → S with the following properties:(i) P is trivial outside U . More precisely, ( P , Ω) → S contains a subbundle ( P, Ω) → S with fiber ( U, ω ) such that ( P (cid:114) P, Ω) → S may be identified with the productbundle (( U (cid:114) U ) × S , ω × α ) where α is an area form on S .(ii) (cid:82) S α < (cid:107) F (cid:107) .(iii) ( P, Ω) contains a symplectically embedded ball B of capacity (cid:107) F (cid:107) . P is constructed as the union R K,F (2 ε ) of two bundles over D , one a smoothing ofthe region in M × [0 , × R between the graphs of K t and of the function t (cid:55)→ max K t ONODROMY IN HAMILTONIAN FLOER THEORY 17 and with anticlockwise boundary monodromy { φ Kt } t ∈ [0 , , and the other a smoothingof the region between the graphs of F and of the constant function t (cid:55)→ min F and withclockwise boundary monodromy { φ Ft } t ∈ [0 , . The bundle is trivial outside U because F, K have support in U . Property (ii) is an immediate consequence of the construc-tion. The region below the graph of F contains the ball, which is embedded near themaximum of F .Now consider the Gromov–Witten invariant that counts holomorphic spheres in P in the class σ = [ p × S ] ∈ H ( P ) for p ∈ U (cid:114) U and through one point p . If we restrictthe class of allowable almost complex structures to those that are Ω-tame and equalto J × j outside P and choose p ∈ U / , then this invariant is well defined becausethe energy Ω( σ ) = (cid:82) S α of each curve is less than δ , the amount of energy neededfor a curve to enter the boundary region P (cid:114) P / ∼ = ( Y × (1 / , × S . (Note thatbecause ( P (cid:114) P, J × j ) is a product, the energy of any curve that enters this region isat least as much as the energy of its projection to a fiber.) We shall call this invariant (cid:10) pt (cid:11) P σ , though in principle it might depend on the product structure imposed near theboundary of P .Since there is a unique σ -sphere though each point in P / (cid:114) P , (cid:10) pt (cid:11) P σ = 1. Butthen, by a standard argument, the nonsqueezing theorem of [17, § r embedded in Int P has capacity πr ≤ Ω( σ ). Butthis contradicts properties (ii) and (iii) of P .We conclude with a sketch proof of the nonsqueezing theorem. Let f : B r → Int P be a symplectic embedding of the standard r -ball with center f (0) and image B . Choosean Ω-tame almost complex structure J on P that is normalized on P (cid:114) P and equalto f ∗ ( j n ) on B , where j n is the standard structure on C n . By hypothesis there is a J -holomorphic sphere C in class σ through f (0), and soΩ( σ ) = (cid:90) C Ω > (cid:90) C ∩ B Ω ≥ (cid:90) f − ( C ∩ B ) ω ≥ πr , where the last inequality holds because f − ( C ∩ B ) is a properly embedded holomorphiccurve through the center of B r ⊂ C n . For more details, see [21, Ch. 9]. (cid:3) Remark 2.11.
Because the homology of M has no fundamental class when M isnoncompact, it is not clear how to understand quantum homology and the Seidelrepresentation in this case. Nevertheless, as we will see below, for the problems underconsideration here we do not need to know everything about the Seidel element S ( γ ),but just some facts about the coefficient of the fundamental class 1l in S ( γ ). Thiscoefficient is given by counting the number of sections of the corresponding fibration P γ → S that go through a point, i.e. by a Gromov–Witten invariant of the form (cid:10) pt (cid:11) P γ σ . Therefore the above argument fits naturally into the framework developedbelow for the closed case. The main argument
This section explains the main ideas and proves Theorems 1.1 and 1.3 modulo somecalculations of Gromov–Witten invariants that are carried out in § The Seidel representation.
Our aim in this section is to prove the followingresult.
Proposition 3.1. (i) If ω vanishes on π ( M ) then the spectral invariants descend to Ham . (ii) If rank H ( M ) = 1 , n + 1 ≤ N < ∞ and ( M, ω ) is not strongly uniruled then thespectral invariants descend. The same conclusion holds if N ≥ n − and ( M, ω ) isnegatively monotone. (iii) If rank H ( M ) = 1 and n + 1 ≤ N < ∞ then the asymptotic spectral invariantsdescend. The main tool is the Seidel representation S . We will see that the stringent hypothe-ses above guarantee that the relevant moduli space of spheres has no bubbling so thatit is compact. The argument goes back to Seidel though it was first published bySchwarz, who showed in [27, § ω and c vanish on π ( M ). We generalize this result in (i) above. Part (iii) is a mild gen-eralization of a result of Entov–Polterovich [3] who proved that the asymptotic spectralinvariants descend when M = C P n .After defining S , we shall calculate it under the conditions of the above proposition(see Lemma 3.2). Finally we prove the various cases of the proposition.We shall think of the Seidel representation as a homomorphism S : π (Ham( M, ω )) → QH n ( M ) × , to the degree 2 n multiplicative units of the small quantum homology ring, where 2 n :=dim M . To define it, observe that each loop γ = { φ t } in Ham M gives rise to an M -bundle P γ → S defined by the clutching function γ : P γ := ( M × D + ) ∪ ( M × D − ) / ∼ , ( φ t ( x ) , e πit ) + ∼ ( x, e πit ) − . (Here D ± are two copies of the unit disc with union S .) Because the loop γ isHamiltonian, the fiberwise symplectic form ω extends to a closed form Ω on P γ , thatwe can arrange to be symplectic by adding the pullback of a suitable form on the base S .The bundle P γ → S carries two canonical cohomology classes, the first Chern class c Vert1 of the vertical tangent bundle and the coupling class u γ , which is the unique classthat extends the fiberwise symplectic class [ ω ] and is such that u n +1 γ = 0. Then, withnotation as in (2.1), we define(3.1) S ( γ ) := (cid:88) σ,i (cid:10) ξ i (cid:11) P γ σ ξ ∗ M i ⊗ q − c Vert1 ( σ ) t − u γ ( σ ) ∈ H ∗ ( M ) ⊗ Λ , Private communication.
ONODROMY IN HAMILTONIAN FLOER THEORY 19 (cf. [21, Def. 11.4.1].) Thus S ( γ ) is obtained by “counting” all section classes in P γ through one fiberwise constraint ξ i . As in Seidel’s original paper [28], one can alsothink of it as the Floer continuation map around the loop γ ; cf. [21, § b ∈ H ∗ ( M )(3.2) S ( γ ) ∗ b = (cid:88) σ,i (cid:10) b, ξ i (cid:11) P γ σ ξ ∗ M i ⊗ q − c Vert1 ( σ ) t − u γ ( σ ) , i.e. it is given by counting section classes with two fiberwise constraints.We will also often use the fact that if σ is the homology class of a section of P γ → S , then every other section in H ( P γ ) may be written as σ + β for a unique β ∈ H ( M ) ⊂ H ( P γ ). Note that(3.3) c Vert1 ( σ + β ) = c Vert1 ( σ ) + c ( β )where as usual c denotes c ( T M ).The first parts of the following lemma are due to Seidel [28].
Lemma 3.2.
Let N be the minimal Chern number of M and γ ∈ π (Ham M ) . (i) If N ≥ n + 1 then S ( γ ) = 1l ⊗ λ ; (ii) If n + 1 ≤ N ≤ n then S ( γ N ) = 1l ⊗ λ . Moreover S ( γ ) = 1l ⊗ λ provided that ( M, ω ) is not strongly uniruled; (iii) If N = n and if ( M, ω ) is not strongly uniruled then S ( γ ) = 1l ⊗ λ + pt ⊗ q n µ forsome λ (cid:54) = 0 ; (iv) If ( M, ω ) is negatively monotone then S ( γ ) = 1l ⊗ r t ε + x where x ∈ Q − , ν ( x ) < ε . (v) If ω = 0 on π ( M ) then S ( γ ) = (1l + x ) ⊗ r t ε where x ∈ H < n ( M )[ q ] .Proof. Suppose that N ≥ n + 1 and that (cid:10) a, b, c (cid:11) Mβ (cid:54) = 0, where a, b, c ∈ H ∗ ( M ) havedegrees 2 d a , d b , d c respectively. Then n + c ( β ) + d a + d b + d c = 3 n . This equationhas no solution if | c ( β ) | ≥ n + 1. Hence c ( β ) = 0, and d a + d b + d c = 2 n . Moreover,if a = pt then d b = d c = n which is impossible because (cid:10) pt, M, M (cid:11) Mβ = 0 when β (cid:54) = 0.Hence M is not strongly uniruled, and so by Lemma 2.1 all units in QH ∗ ( M ) have theform 1l ⊗ λ + x where λ (cid:54) = 0 , x ∈ Q − . Therefore to prove (i) it suffices to show that x = 0.Next observe that the Seidel element is given by invariants of the form (cid:10) ξ (cid:11) P γ σ where ξ ∈ H ∗ ( M ). If σ is a section class, then c ( T P )( σ ) = c Vert1 ( σ ) + 2, where the 2 is theChern class of the tangent bundle to the section. Hence the dimension over C of themoduli space of parametrized σ -curves in P γ is n + 1 + c Vert1 ( σ ) + 2. Therefore the aboveinvariant can be nonzero only if(3.4) 2 c Vert1 ( σ ) + deg ξ = 0 . In particular, we must have − n ≤ c Vert1 ( σ ) ≤ S ( γ ) is nonzero. Then there is a nonzeroinvariant of the form (cid:10) pt (cid:11) P γ σ , which implies that the corresponding section σ has c Vert1 ( σ ) = 0. Equation (3.3) then implies that every other section σ + β either has c Vert1 ( σ + β ) = 0 so that this section also contributes to the coefficient of 1l, or issuch that c Vert1 ( σ + β ) = c ( β ) has absolute value at least N . Hence if N > n equation(3.4) shows that the class σ + β cannot contribute to S ( γ ). Thus S ( γ ) = 1l ⊗ λ . Thisproves (i).In case (ii) the argument in the previous paragraph shows that all sections of P γ contributing to S ( γ ) have the same value of c Vert1 , say − d γ . For short let us say that γ is at level d γ . If d γ > S ( γ ) ∈ Q − . Since S ( γ ) is invertible, it follows fromLemma 2.1 that this is possible only if ( M, ω ) is strongly uniruled. This proves thesecond statement in (ii). To complete the proof of (ii) we must show that γ N alwayslies at level 0 even if ( M, ω ) is strongly uniruled.To see this, observe that for any loops γ, γ (cid:48) one can form the fibration correspondingto their product γ γ (cid:48) by taking the fiber connect sum P γ P γ (cid:48) . Thus if σ γ is a section of P γ and σ γ (cid:48) is a section of P γ (cid:48) one can form a section of P γγ (cid:48) by taking their connectedsum σ γ σ γ (cid:48) . Under this operation the vertical Chern class adds. Thus the sectionsthat contribute to S ( γ γ (cid:48) ) either have level d γ + d γ (cid:48) or have level d γ + d γ (cid:48) − N , dependingon which of these numbers lies in the allowed range [0 , n ]. Hence if γ lies at level d , forall k ≥ m ≥ kd − mN ∈ [0 , n ]. Since N > n the only possiblesolution of this equation when k = N is m = d . Therefore S ( γ N ) lies at level 0. Thisproves (ii).Almost the same argument proves (iii). Note that if there is a section at level d > S ( γ ) lie at this level. On the other hand if there isa section at level 0 there might also be a section at level n . It follows as before that S ( γ ) must have a section at level 0. Therefore S ( γ ) = 1l ⊗ λ + pt ⊗ q n µ. Moreover, ourassumption on QH ∗ ( M ) implies that λ (cid:54) = 0.To prove (iv) note that ( M, ω ) is not uniruled, so that by Lemma 2.1 S ( γ ) = 1l ⊗ λ + x where λ (cid:54) = 0. Moreover, if σ is a section class of P of minimal energy with (cid:10) pt (cid:11) σ (cid:54) = 0then every other section class σ with (cid:10) a (cid:11) σ (cid:54) = 0 for some a ∈ H ∗ ( M ) has the form σ = σ + β where c ( β ) ≤
0. If c ( β ) = 0 then ω ( β ) = 0 and a = pt . These invariantscontribute to the coefficient of t ε in λ , where ε = − u γ ( σ ). On the other hand if c ( β ) < ω ( β ) > x withvaluation < ε − ω ( β ).Finally, if ω vanishes on π ( M ), then, because all Gromov–Witten invariants vanish,the quantum multiplication is undeformed. Moreover, all sections of P γ have the sameenergy. Hence S ( γ ) = (1l + x ) ⊗ r t ε where 0 (cid:54) = r ∈ Q and x ∈ Q − . This proves(v). (cid:3) Definition 3.3. If S ( γ ) = 1l ⊗ λ + x we define σ to be the section class of P γ ofminimal energy that contributes nontrivially to the coefficient λ . Moreover we write λ = (cid:80) i ≥ r i t ε i where r (cid:54) = 0 and ε i > ε i +1 for all i . Thus(3.5) r = (cid:10) pt (cid:11) P γ σ , ν ( λ ) = ε = − u γ ( σ ) . ONODROMY IN HAMILTONIAN FLOER THEORY 21
Note that c Vert1 ( σ ) = 0 by equation (3.4). The conditions in Proposition 3.1 are chosenso that the moduli space of sections in class σ is compact. In what follows we shalloften simplify notation by writing P instead of P γ . Proof of Proposition 3.1.
Suppose that ω | π = 0. Then S ( γ ) = (1l + x ) ⊗ r t ε by Lemma 3.2 (v). We will show that ε = 0. The claimed result then follows fromProposition 2.3 (i:d).Since ω | π ( M ) = 0 the spaces of sections that give the coefficients of S ( γ ) are compact(when parametrized as sections) since there is no bubbling. To be more precise, notethat it suffices to consider almost complex structures J on P := P γ that are compatiblewith the fibration π : P → S in the sense that π is holomorphic and the restriction J z of J to the fiber over z ∈ S is ω -compatible for each z ∈ S . Then all stable mapsin the compactification of a space of sections consist of a section together with somebubbles in the fibers. But if ω | π ( M ) = 0 there are no J z -holomorphic spheres for any z . Hence in this case the sections in class σ form a compact n -dimensional manifold M := M ( σ ). (Here we assume that the elements of M are parametrized as sections.)Therefore there is a commutative diagram S × M ev −→ Ppr ↓ π ↓ S = S where the top arrow is the evaluation map. Moreover, σ = ev ∗ [ S × { pt } ].Now observe that the Gromov–Witten invariant r = (cid:10) pt (cid:11) Pσ of equation (3.5) isjust the degree of the map { z } × M → M = π − ( z ) that is induced by ev . Thusour hypotheses imply that ev has nonzero degree. Since the coupling class u γ satisfies u γ | M = [ ω ] and u n +1 γ = 0, it easily follows that there is a class a ∈ H ( M ) such that ev ∗ ( u γ ) = pr ∗ ( a ). Hence ε = ν ( λ ) = − u γ ( σ ) = − (cid:90) S ×{ pt } ev ∗ u γ = 0as required. This proves (i).Next consider case (ii). Lemma 3.2 implies that in all cases considered here S ( γ ) =1l ⊗ λ + x where ν ( x ) < ν ( λ ). Hence the spectral invariants descend provided that ν ( λ ) = 0. But for generic π -compatible J on P we claim that the moduli space M isagain compact. Hence the previous argument applies to show that ν ( λ ) = 0.To prove the claim, note that the only possible stable maps in the compactification M of M consist of the union of a section in the class σ − kβ where k > k i β . Suppose first that c ( M ) = κ [ ω ] where κ <
0. Thenbecause ω ( β ) > c ( β ) ≤ − N . But if J is generic it induces a generic 2-parameterfamily J z , z ∈ S , of almost complex structures on M . Therefore such a class β couldbe represented only if − N ≥ c ( β ) ≥ − n , which is possible only if N ≤ n − In fact, since r must be an integer it must be ±
1. For its product with the corresponding integerfor γ − must be 1. Since we assume N ≥ n − c ( M ) = κ [ ω ] on π ( M ) where κ > c ( β ) ≥ N . Hence c ( σ − kβ ) ≤ − N . Butthe section is embedded and hence is regular for generic J . Thus for such a section toexist we must have n + 1 + (2 − N ) ≥
3, i.e. n ≥ N . (This is precisely the argumentin [3].) This proves (ii).If N ≥ n + 1 but ( M, ω ) is strongly uniruled, then we can apply the above argumenttogether with Corollary 2.5 to γ N to conclude that the asymptotic spectral invariantsdescend. (cid:50) Calculating the coupling class.
An essential ingredient of Schwarz’s argumentis that the vanishing of ω on π implies that there is no bubbling so that the modulispace M is a compact manifold. One cannot replace S × M here by the universalcurve M , ( σ ) over some compactified virtual moduli cycle M ( σ ) since the “bundle” f : M , ( σ ) → M ( σ ) is singular over the higher strata of M ( σ ). The argument wegive below shows that if the relevant Gromov–Witten invariants of M and P := P γ vanish then this potential twisting does not effect ev ∗ ( u γ ) too much, so that this classstill has zero integral over the fiber of f . Observe that it is not enough here that theinvariants of M vanish; we need some control on the invariants of P in either sectionor fiber classes.Our reasoning is very similar to that in [16, § H ∗ ( P ) splits as a product, i.e. it is isomorphicas a ring to H ∗ ( M ) ⊗ H ∗ ( S ). As the next lemma makes clear, what we need here isa partial splitting of this ring. Lemma 3.4.
Suppose S ( γ ) = 1l ⊗ λ + x and that there is an element H ∈ H n ( P γ ; R ) such that (a) H ∩ [ M ] is Poincar´e dual in M to [ ω ] ; (b) H · σ = 0 ; (c) H n +1 = 0 .Then ν ( λ ) = 0 .Proof. Let u := PD P ( H ), the Poincar´e dual in P := P γ to the divisor class H . Then(a) implies that u | M = [ ω ], while (c) implies that u n +1 = 0. Hence u is the couplingclass u γ . Therefore ν ( λ ) := − u γ ( σ ) = − u ( σ ) = − H · σ = 0 , as required. (cid:3) It is possible that an element H ∈ H n ( P ) with the above properties exists whenever N ≥ n + 1 and S ( γ ) = 1l ⊗ λ . However, we can only prove this under an additionalassumption on the structure of QH ∗ ( M ) that we now explain. Roughly speaking we This condition, though essential to the proof, is very technical and there seems no intrinsic reasonwhy it should be necessary.
ONODROMY IN HAMILTONIAN FLOER THEORY 23 require that the divisor classes in H ∗ ( M ) carry all the nontrivial quantum products.To be more precise, we make the following definition. Definition 3.5.
Let D be the subring of H ev ( M ) generated by the divisors, i.e. theelements in H n − ( M ) . We say that QH ∗ ( M ) satisfies condition (D) if D hasan additive complement V in H ev ( M ) such that the following conditions hold for all d ∈ D , v ∈ V and β ∈ H ( M )( a ) d · v = 0; and ( b ) (cid:10) d, v (cid:11) Mβ = 0 . This condition is used in Lemma 4.8 in order to allow certain computations to bedone recursively.
Remark 3.6. (i) This condition is trivially satisfied if either the quantum product isundeformed (since then (cid:10) d, v (cid:11) Mβ = 0 always) or if D = H ev ( M ).(ii) When N ≥ n + 1 the only classes β for which (cid:10) d, v (cid:11) Mβ could be nonzero have c ( β ) = 0. (If N = n + 1 there is one potentially nonzero invariant with two insertionsand c ( β ) (cid:54) = 0, namely (cid:10) pt, pt (cid:11) Mβ . But pt ∈ D and so this does not affect condition (b)in Definition 3.5.)By condition (a) in Definition 3.5 we may choose a basis ξ i , ≤ i ≤ m, for H ev ( M )so that the first m elements span the subring D while the others span V . Hence thefirst m elements of the dual basis { ξ ∗ M i } will also span D . In other words, for all d ∈ D and β ∈ H ( M )(3.6) (cid:10) d, ξ i (cid:11) Mβ (cid:54) = 0 = ⇒ ξ ∗ M i ∈ D for all i. Notice also that because there is an open subset of H n − ( M ) consisting of elements D such that D n (cid:54) = 0 we may assume that ξ = 1l and that for 1 ≤ i ≤ m each ξ i = D k for some such D . Then each of the corresponding dual elements ξ ∗ i is also a sum ofelements of the form D k .Now consider the map s : H ∗ ( M ) → H ∗ +2 ( P ) defined by the identity(3.7) s ( a ) · P v := r (cid:10) a, v (cid:11) Pσ , v ∈ H ∗ ( P ) , where r is as in equation (3.5). Proposition 3.7.
Suppose that S ( γ ) = 1l ⊗ λ + x , let σ be as in Definition 3.3and define s as above. Then the element H := s (PD M ( ω )) satisfies the conditions ofLemma 3.4 in each of the following situations: (i) N ≥ n + 1 , S ( γ ) = 1l ⊗ λ and QH ∗ ( M ) satisifies condition (D) ; (ii) N = n , ( M, ω ) is not strongly uniruled, rank H ( M ) > , and condition (D) holds; (iii) ( M, ω ) is negatively monotone and condition (D) holds; (iv) QH ∗ ( M ) is undeformed and ( M, ω ) is not spherically monotone with rank H ( M ) =1 . We defer the proof to § Corollary 3.8.
Theorems 1.1 and 1.3 hold.Proof.
Theorem 1.1 concerns the case when QH ∗ ( M ) is undeformed. Then S ( γ ) alwayshas the form 1l ⊗ λ (cid:48) + x where x is nilpotent, since these are the only invertible elementsin QH ∗ ( M ). Moreover, if N ≥ n + 1, S ( γ ) = 1l ⊗ λ by Lemma 3.2 (ii). Hence wemay suppose that the conditions of either part (i) or part (iv) of Proposition 3.7 hold.Therefore this result, together with Lemma 3.4 and Corollary 2.5, prove the theorem.Note that if n ≤ N we cannot conclude that the spectral invariants descend, since it ispossible that ν ( x ) > γ .Similarly, part (iv) of Theorem 1.3 follows from part (iii) of Proposition 3.7. Nextsuppose that N ≥ n + 1 and, if N ≤ n , that ( M, ω ) is not strongly uniruled. Then S ( γ ) = 1l ⊗ λ by Lemma 3.2. Moreover ν ( λ ) = 0 by Lemma 3.4. Hence the spectralinvariants descend by part (i) of Proposition 2.3. This proves part (i) of Theorem 1.3.Parts (ii) and (iii) of this theorem follow similarly. In case (ii) we have S ( γ N ) = 1l ⊗ λ while in case (iii) S ( γ ) = 1l ⊗ λ + pt ⊗ q n µ . In either case ν ( λ ) = 0. Therefore theasymptotic spectral invariants descend by Corollary 2.5. (cid:3) Calculations of Gromov–Witten invariants
This section contains the proof of Proposition 3.7.4.1.
Preliminaries.
We shall use the following identity of Lee–Pandharipande [12]: ev ∗ i ( H ) = ev ∗ j ( H ) + ( α · H ) ψ j − (cid:88) α + α = α ( α · H ) D i,α | j,α , (4.1)where ψ i is the first Chern class of the cotangent bundle to the domain at the i thmarked point and H ∈ H n ( P ) ∼ = H ( P ) is any divisor. Lee–Pandharipande wereworking in the algebraic context and hence interpreted both sides as elements of anappropriate Picard group. Thus provided that we are working with stable maps thathave at least three marked points, D i,α | j,α is the divisor consisting of all stable mapswith two parts, one in class α and containing z i and the other in class α = α − α containing z j .We shall interpret (4.1) as an identity for Gromov–Witten invariants. Thus taking i, j = 1 , u, v, w ∈ H ∗ ( P ) and all α ∈ H ( P ), (cid:10) Hu, v, w (cid:11) α = (cid:10) u, Hv, w (cid:11) α + ( α · H ) (cid:10) u, τ v, w (cid:11) α (4.2) − (cid:88) j,α + α = α ( α · H ) (cid:10) u, η j , . . . (cid:11) α (cid:10) η ∗ j , v, . . . (cid:11) α , where the sum is over the elements of a basis { η j } of H ∗ ( P ) (with dual basis { η ∗ j } ) andall decompositions α + α = α of α , and where the dots indicate that the constraint w Here α · H denotes the intersection number of two homology classes α, H that lie in complementarydegrees. If u ∈ H i ( P ) where i is arbitrary, we shall denote by Hu the cap product H ∩ u ∈ H i − ( P ).Further, when H = M is the class of the fiber, we shall write u ∩ M for the cap product when consideredas an element in H i − ( M ). This last distinction is not very important since H ∗ ( M ) injects into H ∗ ( P )by the result of [16]. ONODROMY IN HAMILTONIAN FLOER THEORY 25 may be in either factor except if α = 0 in which case it must be in the second factor forreasons of stability. Note that τ here denotes a descendent invariant. In fact we shallonly use (4.1) in cases when H · α = 0 so that this term vanishes. If it does not, oneshould get rid of the τ insertion by using the identity ψ i = D i | jk , where D i | jk denotesthe divisor consisting of all stable maps with two parts, one containing the point z i andthe other containing the points z j , z k . (But often one then gets no information sincethe term on LHS appears in the expansion for τ .) For further discussion of this pointas well as a brief proof of (4.1) in the symplectic context see [19].Notice that if we apply an identity such as (4.2) to even dimensional classes a i weonly need to consider even dimensional η j , η ∗ j . We will make this restriction now inorder to avoid irrelevant considerations of sign. Also, we will simplify the argumentsthat follow by choosing a basis { η j } for H ev ( P ) of special form. We start with a basis ξ i , i ∈ I, for H ev ( M ; R ) that satisfies the condition in (3.6) and extend this to a basisfor H ev ( P ; R ) by adding elements ξ ∗ j , j ∈ I, so that for all i, j (4.3) ξ i · P ξ ∗ j = δ ij , ξ ∗ i · P ξ ∗ j = 0 . Thus ξ i is a fiber class but ξ ∗ j is not. Note that this basis { ξ i , ξ ∗ j } for H ev ( P ) is self-dual.Further, ξ ∗ Mi = ξ ∗ i ∩ M .With this choice of basis the sum in (4.2) breaks into two, depending on which of η j , η ∗ j is a fiber class. To analyze the resulting product terms, we frequently use thefollowing fact about invariants of P in a fiber class β . Lemma 4.1.
Suppose that a, b ∈ H ∗ ( M ) , v, w ∈ H ∗ ( P ) and β ∈ H ( M ) . Then: (i) (cid:10) a, b, v (cid:11) Pβ = 0 . (ii) (cid:10) a, v, w (cid:11) Pβ = (cid:10) a, v ∩ M, w ∩ M (cid:11) Mβ . Proof.
This is part of [16, Prop 1.6]. These statements hold because, as is shown in [16],one can calculate these invariants using an almost complex structure and perturbing1-form that are compatible with the fibration π : P → S . Hence every element inthe virtual moduli cycle is represented by a curve that lies in a single fiber. (i) is thenimmediate, since a, b can be represented in different fibers. Similarly, (ii) holds becauseevery β -curve through a must lie in the fiber containing a . The fact that v ∩ M, w ∩ M do not vanish means that these cycles take care of the needed transversality normal tothis fiber. (cid:3) To make our argument work we also need information about certain section invariantsof P . When N ≥ n + 1 the following lemma suffices; the proof is easy since it is basedon a dimension count. Lemma 4.2.
Suppose that N ≥ n and that if N = n then ( M, ω ) is not stronglyuniruled. Suppose further that S ( γ ) = 1l ⊗ λ + pt ⊗ q n µ , and let σ = σ − β where ω ( β ) > . Then for all a ∈ H < n ( M ) : (i) For all w ∈ H ∗ ( P ) , (cid:10) a, w, M (cid:11) σ = 0 unless c Vert1 ( σ ) = 0 and deg w + deg a = 2 n ; (ii) (cid:10) a, b, M (cid:11) σ = 0 for all b ∈ H ∗ ( M ) ; (iii) For all w ∈ H ∗ ( P ) , (cid:10) a, w, M (cid:11) σ depends only on w ∩ M .Proof. Statement (i) holds by a dimension count as in the proof of Lemma 3.2. (ii)holds because an invariant of this form with only two nontrivial fiber insertions isdetermined by S ; namely by (3.2) (cid:10) a, ξ i (cid:11) σ is the coefficient of ξ ∗ M i ⊗ q − c Vert1 ( σ ) t − u γ ( σ ) in S ( γ ) ∗ a = (1l ⊗ λ + pt ⊗ q n µ ) ∗ a = a ⊗ λ. Since − u γ ( σ ) > ν ( λ ) this vanishes. Note that for this argument to apply we need thateither µ = 0 (which happens when N ≥ n + 1) or pt ∗ a = 0, i.e. M not stronglyuniruled.Statement (iii) is an immediate consequence of (ii) because any two classes w, w (cid:48) with w ∩ M = w (cid:48) ∩ M differ by w − w (cid:48) ∈ H ∗ ( M ). (cid:3) Technical lemmas.
This section contains two rather technical results about thevanishing of certain Gromov–Witten invariants needed for some of the cases consideredby Proposition 3.7. However, they are not needed when N ≥ n + 1, and the readermight do well to read the next section first, coming back to this section later.If N = n we also need the following lemma about the fiber invariants of P . Lemma 4.3.
Suppose that N = n , that ( M, ω ) is not strongly uniruled, and that rank H ( M ) > . Suppose further that S ( γ ) = 1l ⊗ λ + pt ⊗ q n µ where λ, µ ∈ Λ univ .Then for any classes s , s ∈ H ( P ) and any β ∈ H ( M ) , (cid:10) s , s (cid:11) β = 0 . Proof.
A dimension count shows that the invariant is zero unless c ( β ) = n . If thereare any nonzero invariants of this form, choose β with minimal energy ω ( β ) so that (cid:10) s , s (cid:11) β (cid:54) = 0.If s , s are both fiber classes then the invariant vanishes by Lemma 4.1. If just one(say s ) is a fiber class then the same lemma implies that the invariant equals (cid:10) s , pt (cid:11) Mβ which vanishes because M is not strongly uniruled. Hence (cid:10) s , s (cid:11) β does not dependon the choice of section classes s i .We now make a specific choice of s . Since rank H ( M ) > a ∈ H ( M )such that a ( β ) = 0. Let F ∈ H n ( P ) be any extension of PD M ( a ) ∈ H n − ( M ). (Since H ∗ ( M ) injects into H ∗ ( P ) by [16] such a class exists.) Choose b ∈ H ( M ) such that F · P b = pt , and let v ∈ H ( P ) be any extension of b . Then vF := v ∩ F ∈ H ( P ) is asection class of P . Therefore if σ is any section class, it suffices to show that (cid:10) vF, σ, H (cid:11) β = 0 , where H ∈ H n ( P ) is chosen so that H · β = 1.To prove this, apply (4.1) with i = 1 , j = 2. We obtain (cid:10) vF, σ, H (cid:11) β = (cid:10) v, F σ, H (cid:11) β + ( β · F ) (cid:10) v, τ σ, H (cid:11) β − (cid:88) i,β + β = β ( β · F ) (cid:16)(cid:10) v, ξ i , . . . (cid:11) β (cid:10) ξ ∗ i , σ, . . . (cid:11) β + (cid:10) v, ξ ∗ i , . . . (cid:11) β (cid:10) ξ i , σ, . . . (cid:11) β (cid:17) , ONODROMY IN HAMILTONIAN FLOER THEORY 27 where the dots indicate that the H insertion could be in either factor. The first termin RHS vanishes because F σ is a multiple of a point. Hence Lemma 4.1 implies that (cid:10) v, F σ, H (cid:11) β = (cid:10) v ∩ M, pt, H ∩ M (cid:11) Mβ which vanishes because ( M, ω ) is not uniruled.The second term on RHS vanishes since β · F = 0. Further in the sum neither β nor β is zero because of the factor β · F . Since c ( β ) = n = N one of the β i has c = 0and the other has c = n . (Other possibilities such as c ( β i ) = − n can be ruled out bydimensional considerations as in the proof of Proposition 3.1.)Consider the first sum. Since ξ i is a fiber class and dim( v ∩ M ) = 2 there are twopossibilities; either c ( β ) = 0 and dim ξ i = 2 n or c ( β ) = n and dim ξ i = 0. In the firstcase, ξ ∗ i is a section class so that (cid:10) ξ ∗ i , σ, H (cid:11) β = 0 by the minimality of ω ( β ). Thereforesuch a term does not contribute. But the second case also does not contribute because( M, ω ) is not strongly uniruled. Therefore the first sum vanishes.Now consider the second sum. Applying Lemma 4.1 again, we find that because ξ i is a fiber class the second factor here equals (cid:10) ξ i , pt, H ∩ M (cid:11) Mβ . Hence it vanishes byhypothesis on M . (cid:3) When all 3-point Gromov–Witten invariants in M vanish the same argument givesthe following conclusion. Lemma 4.4.
Suppose that the quantum multiplication in M is undeformed and that rank H ( M ) > . Then for all nonzero β ∈ H ( M ) , v ∈ H ∗ ( P ) and F ∈ H n ( P ) wehave: (i) (cid:10) F k , v (cid:11) β = 0 ; (ii) (cid:10) σ, v (cid:11) β = 0 for all section classes σ .Proof. Consider (i). If any of F k , v are fiber classes then the invariant reduces to aninvariant in M and hence vanishes by assumption. Therefore, as above we may assumethat F · β = 0. Now suppose that there is some nonzero invariant as in (i) and choosea minimal k such that (cid:10) F k , v (cid:11) β (cid:54) = 0. Then k > H so that β · H = 1 and expand (cid:10) F k , v, H (cid:11) β as before. The first term vanishes by theminimality of k and the second since β · F = 0. Moreover, in the sum neither of β , β vanish. Therefore, because each term in the sum has a fiber constraint in at least oneof its factors the sum vanishes. This proves (i).To prove (ii) let F be any extension of the Poincar´e dual to [ ω ]. Then F n ∩ M = pt .Hence F n = σ is a section class. Therefore (cid:10) σ, v (cid:11) β = (cid:10) F n , v (cid:11) β = 0 by (i). (cid:3) Proof of Proposition 3.7.
For simplicity we will now assume that [ ω ] is nor-malized so that (cid:82) M ω n = 1. Further, we set h := PD M ( ω ) so that h n = pt . Recallthat we define s : H ∗ ( M ) → H ∗ +2 ( P ) by: s ( a ) · P v := r (cid:10) a, v, M (cid:11) Pσ , v ∈ H ∗ ( P ) . Since [ ω ] ∈ H ( M ) need not be a rational class, h ∈ H n − ( M ; R ). Therefore, in this section oneshould assume that homology groups have coefficients R unless otherwise indicated. In particular, s ( M ) · P pt = r (cid:10) M, pt, M (cid:11) Pσ = 1 so that s ( M ) = P . Lemma 4.5.
Suppose that S ( γ ) = 1l ⊗ λ + x where for all b ∈ H < n ( M ; R ) either ν ( x ∗ b ) < ν ( λ ) or x ∗ b = qy where y ∈ QH ∗ ( M ) involves only nonnegative powers of q . Then: (i) s ( pt ) = σ , (ii) s ( a ) ∩ M = a for all a ∈ H ∗ ( M ) .Proof. σ was chosen so that (cid:10) pt, M (cid:11) Pσ = (cid:10) pt (cid:11) Pσ = r (cid:54) = 0. A count of dimensionsshows that (cid:10) pt, v (cid:11) Pσ (cid:54) = 0 only if v is a divisor class. Thus for all v ∈ H n ( P ) the divisoraxiom implies that s ( pt ) · v = r (cid:10) pt, v (cid:11) Pσ = r ( σ · v ) (cid:10) pt (cid:11) Pσ = σ · v. This proves (i).Since we have already checked the case a = [ M ] in (ii), it suffices to take a ∈ H < n ( M ). Observe that by equation (3.2) (cid:10) a, ξ i (cid:11) σ is the coefficient of ξ ∗ M i ⊗ t ε in S ( γ ) ∗ a = (1l ⊗ λ + x ) ∗ a = a ⊗ λ + x ∗ a, The hypothesis on x implies that x ∗ a does not contribute to this coefficient. Hence (cid:10) a, ξ i (cid:11) σ = r a · M ξ i , and so r ( s ( a ) ∩ M ) · M ξ i = r s ( a ) · P ξ i def = (cid:10) a, ξ i (cid:11) σ = r a · M ξ i where the second equality holds by the definition (3.7) of s . Hence s ( a ) ∩ M = a asrequired. (cid:3) There are many different cases in Proposition 3.7. In an attempt to avoid confusionwe will first prove parts (i) and (ii). Thus we assume that N ≥ n , with some furtherconditions when N = n . Lemma 4.6.
Suppose that the hypotheses of Lemma 4.2 hold, and that if N = n thoseof Lemma 4.3 hold as well. Then (cid:10) h, σ , M (cid:11) σ = 0 .Proof. Given any divisor class in P extending h we may add a suitable multiple of M to obtain a class K ∈ H n ( P ) such that K ∩ M = h, K · σ = 0 . We then find by formula (4.1) that (cid:10) h, σ , M (cid:11) σ = (cid:10) KM, σ , M (cid:11) σ = (cid:10) M, Kσ , M (cid:11) σ + ( σ · K ) (cid:10) M, τ σ , M (cid:11) σ − (cid:88)(cid:0) ( σ − α ) · K (cid:1) (cid:16)(cid:10) M, ξ i , . . . (cid:11) σ − α (cid:10) ξ ∗ i , σ , . . . (cid:11) α + (cid:10) M, ξ ∗ i , . . . (cid:11) σ − α (cid:10) ξ i , σ , . . . (cid:11) α (cid:17) , ONODROMY IN HAMILTONIAN FLOER THEORY 29 where the dots indicate that the M insertion could be in either factor. Note that thefirst two terms vanish because K · σ = 0. Suppose there is a nonzero contribution fromthe first sum. Because the first factor has two fiber constraints, Lemma 4.1 impliesthat σ − α must be a section class. Moreover0 (cid:54) = ( σ − α ) · K = − α · M h = ω ( α ) . Thus ω ( α ) > α -invariant. But then (cid:10) M, ξ i (cid:11) σ − α = (cid:10) M, ξ i , M (cid:11) σ − α vanishes by Lemma 4.2(ii) except possibly if dim ξ i = 2 n . But this case can occuronly if N = n and c ( α ) = n . But then ξ ∗ i is a section class, so that the secondfactor (cid:10) ξ ∗ i , σ , . . . (cid:11) α vanishes by Lemma 4.3. Therefore in all cases this sum makes nocontribution.Now consider the second sum. Suppose first that β := σ − α were a fiber class.Note that β (cid:54) = 0 because the sum is multiplied by β · K . Then, because M is a fiberconstraint, Lemma 4.1 implies that the β invariant is either (cid:10) M, ξ ∗ i ∩ M, M ∩ M (cid:11) β ,or is (cid:10) M, ξ ∗ i ∩ M (cid:11) β , which both vanish when β (cid:54) = 0 because the first marked point isnot constrained. Thus the only nonzero terms have σ − α a section class, and hencehave the form (cid:10) M, ξ ∗ i , M (cid:11) Pσ − α (cid:10) ξ i , pt (cid:11) Mα where ω ( α ) (cid:54) = 0 since K · σ = 0. Again thereare two cases. If N > n then Lemma 4.2(i) implies that this can be nonzero only ifdeg ξ ∗ i = 0. But ξ ∗ i (cid:54) = pt since it is not a fiber class. Therefore this is impossible. Onthe other hand if N = n then the first factor might be nonzero, but the second hasto vanish since M is not uniruled. Therefore in all cases the second sum vanishes aswell. (cid:3) Corollary 4.7.
Under the conditions of the above lemma, if H := s ( h ) , we have H · σ = 0 . Proof.
By the definition of s in (3.7), s ( h ) · σ = (cid:10) h, σ , M (cid:11) σ = 0. (cid:3) Lemma 4.8.
Suppose that ( M, ω ) , S ( γ ) and N satisfy the hypotheses of Lemma 4.6and that QH ∗ ( M ) satisfies condition (D) . Then: (i) For all section classes σ = σ − β where ω ( β ) > and all k we have (cid:10) F k , a, M (cid:11) σ = 0 whenever F ∈ H n ( P ) and a ∈ H < n ( M ) . (ii) (cid:10) H n +1 − k , h k , M (cid:11) σ = 0 for all k .Proof. Consider (i). Choose σ of minimal energy (i.e. ω ( β ) is maximal) and thenthe minimal k so that (cid:10) F k , a, M (cid:11) σ (cid:54) = 0 for some a and F . Note that k > (cid:10) F, a, M (cid:11) σ = ( F · σ ) (cid:10) a, M (cid:11) σ vanishes by Lemma 4.2(ii).Again by Lemma 4.2(ii) we may add an arbitrary fiber class to F k without changing (cid:10) F k , a, M (cid:11) σ . Hence, by replacing F by F − cM for suitable c , we may arrange that F · σ = 0. Now use (4.1) as in the previous lemma, moving F from the first constraint to the second. Because F · σ = 0 we find as before that (cid:10) F k , a, M (cid:11) σ = (cid:10) F k − , F a, M (cid:11) σ − (cid:88) ( α · F ) (cid:16)(cid:10) F k − , ξ i , . . . (cid:11) α (cid:10) ξ ∗ i , a, . . . (cid:11) σ − α + (cid:10) F k − , ξ ∗ i , . . . (cid:11) α (cid:10) ξ i , a, . . . (cid:11) σ − α (cid:17) . The first term is zero by our choice of k . Consider the first sum. If α is a section class,then because of the factor α · F we may assume that α (cid:54) = σ . But then α has less energythan σ since ω ( σ − α ) > α invariant vanishes by the minimality of theenergy of σ . So we may suppose that α is a fiber class, in which case the other factoris an invariant (cid:10) ξ ∗ i , a, M (cid:11) σ − α in a section class with smaller energy than σ . We claimthat condition (D) implies that the product of these factors must still vanish.For suppose not. Then by Lemma 4.1 (cid:10) F k − , ξ i (cid:11) Pα = (cid:10) f k − , ξ i (cid:11) Mα (cid:54) = 0where f := F ∩ M . Therefore (3.6) implies that ξ ∗ M i = ξ ∗ i ∩ M ∈ D . Thus ξ ∗ i ∩ M is alinear combination of elements of the form g k , g ∈ H n − ( M ). But if we write g = G ∩ M for some G ∈ H n ( P ) all invariants of the form (cid:10) G k , a, M (cid:11) σ − α vanish by the minimalitycondition on the energy of σ . But ξ ∗ i ∩ M = G k ∩ M . Hence (cid:10) ξ ∗ i , a, M (cid:11) σ − α = 0 byLemma 4.2(iii).Thus the first sum vanishes. Now consider the second sum. The second invarianthas at least two fiber constraints. Hence σ − α must be a section class, so that theinvariant vanishes by Lemma 4.2(ii). This proves (i).Now consider (ii). When k = n the invariant is (cid:10) H, pt, M (cid:11) σ which vanishes because H · σ = 0. Suppose that it does not vanish for all k and choose the maximal k forwhich it is nonzero. Expand (cid:10) H n +1 − k , h k , M (cid:11) σ by transferring one H to the secondconstraint. As usual the first two terms in the expansion vanish and we obtain (cid:10) H n +1 − k , h k , M (cid:11) σ = − (cid:88) ( α · H ) (cid:16)(cid:10) H n − k , ξ i , . . . (cid:11) α (cid:10) ξ ∗ i , h k , . . . (cid:11) σ − α + (cid:10) H n − k , ξ ∗ i , . . . (cid:11) α (cid:10) ξ i , h k , . . . (cid:11) σ − α (cid:17) . Consider the first sum. If α is a section class then we may suppose α (cid:54) = σ becauseof the factor α · H . Therefore it has smaller energy than σ so that the α -invariantvanishes by part (i) of this lemma. On the other hand if α is a fiber class then as abovecondition (D) implies that ξ i ∈ D and the second factor is a sum of terms of the form (cid:10) K n − k +1 , h k , M (cid:11) Pσ where σ has less energy than σ . Therefore this factor vanishes bypart (i). Therefore the first sum vanishes. But the second factor in the second sum hasat least two fiber constraints. Therefore σ − α is a section class, and its energy is lessthan that of σ since ω ( α ) (cid:54) = 0. Hence this factor must vanish by Lemma 4.2(ii). Thusthe RHS of the above expression vanishes. Therefore the LHS is zero also, contrary tohypothesis. The result follows. (cid:3) Corollary 4.9.
If the hypotheses of Lemma 4.8 hold then H n +1 = 0 . ONODROMY IN HAMILTONIAN FLOER THEORY 31
Proof.
Putting k = 1 into claim (ii) of Lemma 4.8 we find that (cid:10) h, H n , M (cid:11) Pσ = 0. Butby equation (3.7) this is a multiple of the intersection s ( h ) · H n . Since s ( h ) = H , weobtain H n +1 = 0. (cid:3) Proof of Proposition 3.7. If N ≥ n + 1, condition (D) holds and S ( γ ) = 1l ⊗ λ ,then the hypotheses of Lemma 4.2 hold. Hence we may apply Lemmas 4.6 and 4.8.Therefore, the conditions of Lemma 3.4 hold by Corollaries 4.7 and 4.9 and Lemma 4.5.This proves (i). To prove (ii) note that the extra conditions here precisely match theconditions of Lemma 4.3. Hence the proof goes through as before.Now consider part (iii) of the proposition. The assumption here is that ( M, ω ) isnegatively monotone and that condition (D) holds. We saw in Lemma 3.2 that in thiscase S ( γ ) = 1l ⊗ rt ε + x where ν ( x ) < ε . Hence the conclusions of Lemma 4.5 hold.Further Lemma 4.2(ii) holds (though part (i) may not). To see this, recall that (cid:10) a, ξ ∗ Mi , M (cid:11) σ is the coefficient of ξ i ⊗ q − c Vert( σ ) t − u γ ( σ ) in S ( γ ) ∗ a . Because ν ( x ) < ν ( λ )in S ( γ ), this must vanish when − u γ ( σ ) > − u γ ( σ ). Therefore if we write σ = σ − β ,the invariant vanishes when ω ( β ) >
0. Therefore part (iii) of this lemma also holds.It is now easy to check that the proof of Lemma 4.6 goes through. The argumentfor the vanishing of the first sum needs no change (note that c ( α ) < M, ω )is not uniruled instead of Lemma 4.2(i). It remains to check the proof of Lemma 4.8.But this holds as before, provided that condition (D) hold. This completes the proofof part (iii).Finally consider part (iv). The assumption here is that QH ∗ ( M ) is undeformed, andif rank H ( M ) = 1 that ( M, ω ) is not positively monotone. Since condition (D) holdswhen QH ∗ ( M ) is undeformed, the latter case follows from (i) if c = 0 and from (iii)otherwise. Hence we may assume that rank H ( M ) >
1. Therefore the conditions ofLemma 4.4 hold. Further, because S ( γ ) = 1l ⊗ λ + x has degree 2 n , all terms in x havea positive coefficient of q . When x ∗ b = x ∩ b this remains true for x ∗ b . Hence theconclusions of Lemma 4.5 hold.Next we claim that Lemmas 4.6 and 4.8 hold. To see this, we go though the proofs ofthese lemmas using Lemma 4.4 instead of Lemma 4.2 to show that the requisite termsvanish. Note for example that the fiber invariants in the expansion in Lemma 4.6contain factors of the form (cid:10) ξ ∗ , σ (cid:11) α which vanish by Lemma 4.4(ii). Similarly, inLemma 4.8 we may use Lemma 4.4(i). We do not need condition (D) because in aproduct such as (cid:10) H n − k , ξ i , M (cid:11) α (cid:10) ξ ∗ i , h k (cid:11) σ − α with α ∈ H ( M ) the first factor vanishes, and so we do not need to worry about thesecond factor. Hence the proof goes through as before. (cid:50) Examples
We now prove Proposition 1.8. We shall calculate in the subring QH n ( M ) ∼ = H ∗ ( M ) ⊗ Λ univ , i.e. by fixing the degree of the elements considered we can forgetthe coefficients q i .Recall the following construction from K¸edra [11] and [18]. Let ( X, ω X ) be a sym-plectic manifold. For each map α : S → X , let gr α be the graph ( z, α ( z )) ∈ S × X of α , and let τ be an area form on S of area 1. Choose a constant µ so that(5.1) Ω := µ pr ∗ ( τ ) + pr ∗ ( ω )is nondegenerate on gr α . Denote by ( (cid:101) P , (cid:101) Ω δ ) the δ -blow up of ( S × X, Ω) along gr α .(The parameter δ refers to the symplectic area of a line in the exceptional divisor.)Then the symplectic bundle π : (cid:101) P → S has fiber M := ( (cid:101) X, (cid:101) ω δ ) and corresponds to aHamiltonian loop γ α ∈ π (Ham( M )). Lemma 5.1.
Let ( X, ω X ) be a symplectic manifold of dimension n ≥ . Given a map α : S → X define µ α := (cid:82) S α ∗ ω X and (cid:96) α := c X ( α ) . Suppose that (cid:96) α ≥ and that atleast one of µ α , (cid:96) α is nonzero. Then the Seidel element S ( γ α ) ∈ QH n ( M ) of the loop γ α defined above has the form ⊗ λ + x where ν ( λ ) (cid:54) = 0 for small nonzero δ .Proof. Let D be the exceptional divisor in (cid:101) P . Denote the trivial section of S × X by σ := S × { p } . If p / ∈ gr α this lifts to a section (cid:101) σ of (cid:101) P → S . If ε denotes the class ofthe line in the fibers of the exceptional divisor, then every section class may be writtenas (cid:101) σ − mε + β where β ∈ H S ( X ). If m = 0 then this class is pulled back from S × X ,and because (cid:101) P is obtained from S × X by blowing up along a (complex) curve withnonnegative Chern class, we may apply the results of Hu [9]. Thus for all a ∈ H ∗ ( X )(5.2) (cid:10) a (cid:11) (cid:101) P (cid:101) σ + β = (cid:10) a (cid:11) S × Xσ + β . Hence (cid:10) a (cid:11) (cid:101) P (cid:101) σ + β = 1 if β = 0 and vanishes otherwise.Define κ := − u γ ( (cid:101) σ ). The above argument shows that the coefficient λ of 1l in S ( γ )contains the term t κ . There might be other classes (cid:101) σ − mε + β that contribute tothe coefficient of 1l in S ( γ ) but these all appear with the coefficient t κ + mδ − ω ( β ) where m (cid:54) = 0. Thus ν ( λ ) either equals κ or equals κ + mδ − ω ( β ) for some m (cid:54) = 0 and β ∈ H ( X ).We now calculate κ . The coupling class u γ has the form (cid:101) Ω δ + c δ pr ∗ τ where c δ is chosen so that (cid:82) (cid:101) P u n +1 γ = 0. Further, because (cid:101) Ω δ | (cid:101) σ X = Ω | σ X by construction and (cid:82) σ Ω = µ by equation (5.1), − κ := u γ ( (cid:101) σ ) = c δ + µ . We showed in [18] (see also [20]) that if V := n ! (cid:82) X ω n then n +1)! (cid:82) (cid:101) P ( (cid:101) Ω δ ) n = µ ( V − v δ ) − v δ (cid:16) µ α − (cid:96) α n +1 δ (cid:17) , ONODROMY IN HAMILTONIAN FLOER THEORY 33 where v δ := δ n n ! is the volume of the ball cut out of X . But0 = n +1)! (cid:82) (cid:101) P ( (cid:101) Ω δ + c δ pr ∗ τ ) n +1 = (cid:16) n +1)! (cid:82) (cid:101) P ( (cid:101) Ω δ ) n +1 (cid:17) + c δ ( V − v δ ) , since V − v δ is the volume of ( M, (cid:101) ω δ ). Therefore − κ := c δ + µ = v δ V − v δ (cid:16) µ α − (cid:96) α n +1 δ (cid:17) . Thus, provided that at least one of µ α , (cid:96) α are nonzero, κ is a rational function of δ with isolated zeros. Therefore neither κ nor κ + mδ − ω ( β ) vanish for sufficiently small δ (cid:54) = 0. The result follows. (cid:3) Proof of Proposition 1.8.
The previous lemma proves (i), and so it remains toconsider the case when X has dimension 4 and [ ω ] and c vanish on π ( X ). It isshown in [18, Prop 6.4] that under the given hypotheses on X every Hamiltonianbundle ( M, (cid:101) ω δ ) → ( (cid:101) P , (cid:101) Ω) → S is constructed by blowing up some section σ X of someHamiltonian bundle ( X, ω ) → ( P, Ω) → S .Let us denote by (cid:101) γ X the loop in π (Ham M ) corresponding to the M -bundle (cid:101) P → S and by γ X that corresponding to its blow down P → S . As in the proof ofProposition 3.1, S ( γ X ) = r
1l for all γ X ∈ π (Ham X ). Therefore there is at leastone section σ X := σ X + β of P such that (cid:10) pt (cid:11) Pσ X (cid:54) = 0. Moreover both c Vert1 and thecoupling class u X of X vanish on the section σ X , and hence on all other sections of P , in particular on σ X .The section classes in (cid:101) P have the form (cid:101) σ X − mε + β , where (cid:101) σ X is the lift of σ X and β ∈ H S ( M ). For such a class to contribute to the corresponding Seidel element S ( (cid:101) γ ) we need − ≤ c Vert1 ( (cid:101) σ X − mε + β ) = − m ≤
0. Moreover, since we may choose Ωso that the section σ X of ( P, Ω) is symplectic, Hu’s results imply that (cid:10) pt (cid:11) (cid:101) P (cid:101) σ X = (cid:10) pt (cid:11) Pσ X (cid:54) = 0 . Therefore if κ := − u (cid:101) γ X ( (cid:101) σ X ) S ( (cid:101) γ X ) = r ⊗ t κ + ξ ⊗ t κ + δ + r pt ⊗ t κ +2 δ , where ξ ∈ H ( M ) . We can now repeat the calculation of κ in Lemma 5.1. All we need change is the inter-pretation of the constant µ , which we now define to be the area Vol( P, Ω) / Vol(
X, ω X )of the fibration P → S . Hence κ = 0.Now observe that as in [19, § ε ∗ ε = − pt + ε ⊗ t − δ , There may be several such since each coefficient of S ( (cid:101) γ ) is a sum of contributions from all sectionswith given values of c Vert1 and µ (cid:101) γ . We do not need to use Hu [9] here. All that matters is that the coefficient of 1l in S ( (cid:101) γ X ) isnonzero, which follows from the fact that M is not uniruled. and that quantum multiplication (in M ) by the elements of H ≤ ( X ) ⊂ H ≤ ( M ) isundeformed. In particular, pt ∗ ε = 0 so that ν (cid:0) ( ε ⊗ t δ ) k (cid:1) = δ, k ≥ . Hence, if we decompose the class ξ appearing in S ( (cid:101) γ X ) as ξ = sε + ξ (cid:48) for some s ∈ Q , ξ (cid:48) ∈ H ( X ), we have ξ (cid:48) ∗ ε = ξ (cid:48) ∗ pt = 0, and we easily find that0 ≤ ν ( S ( (cid:101) γ k )) ≤ δ, k ≥ . Hence the asymptotic invariants descend by Proposition 2.3. (cid:50)
Remark 5.2. (i) It is perhaps worth pointing out that the equality in (5.2) does notalways hold if you blow up along the graph of a class with c negative. For example,suppose that you take α := L to be the line in X := C P . Then it is not hard to checkthat S ( γ α ) = 1l ⊗ t κ + ( L − E ) ⊗ t κ + δ , where E denotes the exceptional divisor in M .(The second term comes from counting sections in class (cid:101) σ − E .) Let us normalize thesymplectic form on X so that ω X ( L ) = 1. Since γ − α = γ − α , we must then have S ( γ − α ) = S ( γ α ) − = ( − E + pt ⊗ t δ ) t − δ − κ . Thus γ α , which is formed by blowing up along the graph of − L , has a Seidel elementin which the coefficient of 1l vanishes. These calculations are carried out in detail in[17, § γ − α is three times the generator of π (Ham M ) that is called α in [17]. Thus the element of QH ∗ ( M ) called Q − in [17] has the form S ( γ α ) t κ (cid:48) forappropriate κ (cid:48) . See also [18, Remark 1.8].(ii) One should be able to use the methods of [19] and Lai [13] to show that in thesituation of Lemma 5.1 classes with m (cid:54) = 0 do not contribute to S ( γ ). This calculationwill be carried out elsewhere. References [1] P. Albers, On the Extrinsic Topology of Lagrangian submanifolds, arXiv:math/0506016.[2] D. Burago, S. Ivanov and L. Polterovich, Conjugation invariant norms on groups of geo-metric origin, arXiv:math/0710.1412.[3] M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, SG/0205247.
International Mathematics Research Notes, (2003), 1635–76.[4] M. Entov and L. Polterovich, Quasi-states and symplectic intersections,arXiv:math/0410338, Comment. Math. Helv. (2006), 75–99.[5] M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds, arXiv:math/0704.0105[6] M. Entov and L. Polterovich, Symplectic quasi-states and semi-simplicity of quantumhomology, arXiv:math/0705.3735[7] M. Entov and L. Polterovich, C rigidity of Poisson brackets, preprint (2007)[8] H. Hofer, On the topological properties of symplectic maps. Proceedings of the RoyalSociety of Edinburgh , (1990), 25–38.[9] J. Hu, Gromov–Witten invariants of blow ups along points and curves, Math. Z. (2000), 709–39.[10] J. Hu, T.-J. Li and Yongbin Ruan, Birational cobordism invariance of uniruled symplecticmanifolds, arXiv:math/0611592 to appear in
Invent. Math. .[11] J. K¸edra, Evaluation fibrations and topology of symplectomorphisms,
Proc. Amer. Math.Soc (2005), 305-312.
ONODROMY IN HAMILTONIAN FLOER THEORY 35 [12] Y.-P. Lee and R. Pandharipande, A reconstruction theorem in Quantum Cohomology andQuantum k -theory, AG/0104084, Amer. Journ. Math. (2004), 1367–1379.[13] Hsin-Hong Lai, Gromov–Witten invariants of blow-ups along submanifolds with convexnormal bundles, arXiv:math/0710.3968.[14] F. Lalonde and D. McDuff, The geometry of symplectic energy,
Annals of Mathematics , (1995), 349–371.[15] F. Lalonde and D. McDuff, Hofer’s L ∞ geometry: geodesics and stability, I, II. Invent.Math. (1995), 1–33, 35–69.[16] D. McDuff, Quantum homology of Fibrations over S , International Journal of Mathe-matics , , (2000), 665–721.[17] D. McDuff, Geometric variants of the Hofer norm, arXiv:math/0103089, Journal of Sym-plectic Geometry , (2002), 197–252.[18] D. McDuff, The symplectomorphism group of a blow up, arXiv:math/06103089, to appearin Geom. Dedicata .[19] D. McDuff, Hamiltonian S -manifolds are uniruled, arXiv:math/0706.0675.[20] D. McDuff, Loops in the Hamiltonian group – a survey, arXiv:math/0711.4086[21] D. McDuff and D. Salamon, J -holomorphic Curves and Symplectic Topology , ColloquiumPublications , American Mathematical Society, Providence, RI (2004).[22] Yong-Geun Oh, Spectral invariants, analysis of the Floer moduli spaces and geometry ofthe Hamiltonian diffeomorphism group, Duke Math J. (2005), 199-295.[23] Yong-Geun Oh, Floer mini-max theory, the Cerf diagram and the spectral invariants,arXiv:math/0406449[24] Y. Ostrover, A comparison of Hofer’s metrics on Hamiltonian diffeomorphisms and La-grangian submanifolds, arXiv:math/0207070
Commun. Contemp. Math (2003) 2123–2141.[25] Y. Ostrover, Calabi Quasimorphisms for some nonmonotone symplectic manifolds,arXiv:math/0508090[26] L. Polterovich, Hofer’s diameter and Lagrangian intersections, Intern. Math. Res. Notices (1998), 217–223.[27] M. Schwarz, On the action spectrum for closed symplectially aspherical manifolds, PacificJourn. Math (2000), 419–461.[28] P. Seidel, π of symplectic automorphism groups and invertibles in quantum cohomologyrings, Geom. and Funct. Anal. (1997), 1046 -1095.[29] M. Usher, Spectral numbers in Floer theories, arXiv:math/0709.1127 Department of Mathematics, Barnard College, Columbia University, New York, NY10027-6598, USA.
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