Quantum cohomology as a deformation of symplectic cohomology

We prove that under certain conditions, the quantum cohomology of a positively monotone compact symplectic manifold is a deformation of the symplectic cohomology of the complement of a simple crossings symplectic divisor. We also prove rigidity results for the skeleton of the divisor complement.

Let D = ∪ N i=1 D i ⊂ M be an SC symplectic divisor (in the sense of [36,Section 2]) and set X = M \D. 1 We assume that there exist positive rational numbers λ 1 , . . . , λ N called weights such that Note that the number of weights in the setup depends on the divisor. If PD(D i ) are linearly independent classes in H 2 (M ; R) (e.g., if D is smooth), the weights are canonically determined. Otherwise, the choice of weights is extra data. Note that in the log Calabi-Yau case d = n + 1, all weights λ i must be equal to 2.

Quantum cohomology is a deformation of symplectic cohomology
We fix, once and for all, a commutative ring k. Let A ⊂ Q be the subgroup generated by the weights λ i , and set Λ = k[A] to be the group algebra of A. 2 We define a Q-grading on Λ by putting e a in degree i(e a ) = a. Let a 0 > 0 be a generator of A, and define q := e a0 . Hence, we have an isomorphism of algebras Λ ∼ = k[q, q −1 ].
Throughout the paper, we will consider various filtrations associated with filtration maps (see Sect. A.1 for a review of this notion). We will abuse notation using the same symbol for the filtration map and the associated filtration. In the first instance of this abuse of notation, we introduce the filtration Q ≥• on Λ associated with the filtration map Q : Λ → Z induced by Q(q a ) = a. Thus, Q ≥p Λ consists of all linear combinations of monomials q a with a ≥ p.
We define the graded Λ-module QH * (M ; Λ) := H * (M ; k)⊗ k Λ, equipped with the tensor product grading. 3 We are concerned with the following idealized and vague conjecture:
We will prove a modified version of Conjecture 1.4 in the setup described in Sect. 1.1. Notably, for the analogue of part (b) we will need Hypothesis A. Remark 1.5. Conjecture 1.4 part (b) is not true in general. For example, if we take M = CP n and D a hyperplane, then X = M \D = C n has vanishing symplectic cohomology. But we cannot have a spectral sequence with vanishing E 1 page, converging to the non-vanishing cohomology of CP n ! Note that Hypothesis A is not satisfied in this case by Example 1.3. More generally, it is not satisfied for D a union of N ≤ n hyperplanes; and X = C n+1−N × (C * ) N −1 still has vanishing symplectic cohomology in these cases.
consisting of Floer cohomology groups and continuation maps, where the monotone sequence of Hamiltonians H 1 ≤ H 2 ≤ · · · converges to 0 on K and +∞ outside of K. We consider the telescope complex tel(C), which is constructed so that We define tel(C) to be the degreewise completion of tel(C) with respect to the action filtration, and SH * M (K; Λ) := H * ( tel(C)).

Outline of proofs
In this section, we give an extended overview of the proofs of our main results, trying to convey the main ideas while avoiding technicalities. We assume that we are in the geometric setup described in Sect. 1.1, with the additional properties and data explained in Sects. 1.2 and 1.3.
We will construct a function ρ : M → R which is a smoothing of ρ 0 (really, a family of smoothings ρ R for R > 0 sufficiently small) with the following properties: • it will be continuous on M , and smooth on the complement of L; • ρ| L = 0 and ρ| D ≈ 1; 8 JFPTA • it will satisfy Z(ρ) = ρ on X\L.

Theorem B.
We choose σ ∈ (σ crit , 1), and construct acceleration data (H τ , J τ ) for K σ ⊂ M as follows. Fix 0 < 1 < 2 < · · · such that the Reeb flow on Y = ∂K σ has no n -periodic orbits for all n, and n → ∞ as n → ∞. We choose an increasing family of smooth functions h n : R → R, approximating the piecewise-linear functions max(0, n (ρ − σ)) with increasing accuracy as n → ∞, and being linear with slope n for ρ ≥ σ (see Fig. 2). We consider acceleration data (H τ , J τ ) for K σ ⊂ M such that J τ is of contact type near ∂K σ and H n is equal to a carefully chosen perturbation of h n • ρ. The 1periodic orbits of Hamiltonians H n then fall into two groups (1) SH-type: contained in K σ and (2) D-type: outside of K σ . We also make sure that the SH-type orbits that are not "Reeb type" are constant. We now consider the Floer 1-ray C(H τ , J τ ) := CF * (M, H 1 ; Λ) → CF * (M, H 2 ; Λ) → · · · associated with our choice of acceleration data. We decompose the associated telescope complex as a direct sum of the SH-type generators and the D-type generators: This is a direct sum as Λ-modules, not as cochain complexes: the differential, which we denote by ∂, mixes up the factors. By restricting the acceleration data with K σ , we also obtain a Floer 1-ray of k-cochain complexes C SH (H τ , J τ ) := CF * (K σ , H 1 | Kσ ; k) → CF * (K σ , H 2 | Kσ ; k) → · · · and we set SC * (X; k) := tel(C SH ). We denote the differential by d. Strictly speaking, this is the cochain complex defining the symplectic cohomology of the Liouville domain K σà la Viterbo [45]. Our notation is justified by the fact that in [23,Section 4], McLean shows that H * (SC * (X; k)) only depends on X.
We associate a canonical fractional cap u in to each SH-type orbit γ, by setting u in := u − u · λ for an arbitrary cap u (one easily checks that u in is independent of u). There is then an isomorphism of Λ-modules (recall Equation (1.1)) ( 1.2) However, this is not a chain map: indeed, the matrix component ∂ SH,SH need not even be a differential.
One could think of Proposition 1.12 as a manifestation of positivity of intersection of Floer trajectories with the components of the divisor D (c.f. [37,Lemma 4.2]), although we actually prove it using an argument related to Abouzaid-Seidel's 'integrated maximum principle' [2,Lemma 7.2].
The consequence of Proposition 1.12 is that d ⊗ id Λ − ∂ SH,SH strictly increases the Q-filtration. Using PSS isomorphisms, we also see that the homology of tel(C) is isomorphic to QH * (M ; Λ). Thus, we are some way towards proving Theorem B, but we are troubled by the existence of D-type orbits. The following proposition is the most important ingredient in the proof of Theorem B, as it allows us to 'throw out' the D-type orbits. Sketch of proof when D is smooth. The Hamiltonian H n is approximately equal to n (ρ − σ) near D. When D is smooth we have ρ = r/κλ, where r is the moment map for a Hamiltonian circle action rotating a neighbourhood of D about D with unit speed. In particular, the Hamiltonian flow of H n approximately rotates around D at speed n /κλ, and the D-type orbits are approximately constant. (This is in contrast to the Hamiltonians used, for example, in [37], which are approximately constant near D, and which have non-constant D-type orbits linking D. ) We compute the mixed index with respect to the approximately constant cap, which is called u out in the body of the paper. As the Hamiltonian flow of H n rotates around D at speed n /κλ, we have i(γ, u out ) ≈ 2 n /κλ. On the other hand, we have H n ≈ h n (1) ≈ n (1 − σ) along D, and ω(u out ) ≈ 0, so A(γ, u out ) ≈ n (1 − σ). Combining we have ≥ κ −1 n (σ − σ crit ), which gives the desired result, as we chose σ > σ crit .
Our first thought, in trying to 'throw out' the D-type orbits, might be to consider the submodule of tel(C) spanned by orbits satisfying i mix (γ) < κ −1 δ n , as that is contained in tel(C) SH by Proposition 1.13. However this does not behave well with respect to the differential: it is neither subcomplex, quotient complex, nor subquotient. Instead, we consider a family of subquotient complexes (SC (p) Λ , ∂ p ) of tel(C), indexed by p ∈ R, spanned by generators (γ, u) satisfying i(γ, u) < p ≤ A(γ, u) + δ n κ .
(Note that these are contained in tel(C) SH by Proposition 1.13, which is identified with SC Λ by (1.2).)

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To see that this is a subquotient of tel(C), we first observe that the differential clearly increases the quantity F(γ, u) = A(γ,u)+δ n κ : it increases action, and increases n and hence n by the definition of the telescope complex. Therefore, it defines a filtration map, so F ≥p tel(C) is a subcomplex. On the other hand, the degree truncation σ <p C • := i<p C i is always a quotient complex of any cochain complex. Thus (SC is a subquotient of SC Λ , whose generators are all of SH-type by Proposition 1.13. Proposition 1.14. For any p ∈ R, both F ≥p tel(C) ⊂ tel(C) and F ≥p tel(C SH ) ⊂ tel(C SH ) are quasi-isomorphic subcomplexes.
Sketch of proof. We may identify F ≥p tel(C) as the telescope complex of the 1-ray of Floer groups A ≥κp−δ n CF * (M, H n , Λ). The key point is that κp − δ n → −∞ as n → ∞, and the action filtration is exhaustive, so the direct limit 'eventually catches everything' (see Appendix A.2). The argument for If we were willing to weaken the statement in Theorem B, and only achieve the isomorphism of item (3) up to degree p − 1, we would now be done: we could simply take SC Λ = SC (p) Λ , with Q equal to the filtration induced by Q. However, to get the corresponding statement in all degrees, we observe that there are natural maps SC Λ for all p ≥ q, induced by the inclusion F ≥p ⊂ F ≥q and the projection σ <p σ <q . We define ( SC Λ , ∂) to be the homotopy inverse limit of the inverse system of chain complexes (SC (p) Λ , ∂ p ), and Q the filtration induced by the Q-filtration on SC Λ . The result is that as desired. (We remark that this step requires us to check that lim Λ , ∂ p ) = 0; indeed the inverse system is easily seen to satisfy the Mittag-Leffler property.) This completes the sketch proof of Theorem B.

Theorem C.
To prove Theorem C, it suffices to prove that the Qfiltration is bounded below and exhaustive, by the 'Classical Convergence Theorem' [46,Theorem 5.5.1]. The Q-filtration on each SC (p) Λ is exhaustive by definition, but the Q-filtration on SC Λ is not exhaustive, due to the direct product taken in the construction. Nevertheless one can show that the inclusion ∪ q Q ≥q SC Λ ⊂ SC Λ is a quasi-isomorphism, and the Q-filtration on this quasi-isomorphic subcomplex is exhaustive by construction.
Thus the main thing to prove, to apply the Classical Convergence Theorem, is that the Q-filtration is bounded below. The key ingredient is the following: Vol. 24 (2022) Quantum cohomology as a deformation of symplectic cohomology Page 11 of 77 48 Proposition 1.15. Suppose that Hypothesis A is satisfied. Then for any SHtype orbit γ, we have i(γ, u in ) ≥ 0.
Sketch of proof when D is smooth. Note that the result is trivial for constant SH-type orbits, as i(γ, u in ) is equal to a Morse index which is non-negative. For a Reeb-type orbit γ, we define u out be the small cap passing through D. Then the orbit γ winds ν = u out · D times around D, so i(γ, u out ) ≈ 2ν. Thus we have We now show that the Q-filtration is bounded below. To be precise, we need to show that for any i there exists q(i) such that Q ≥q(i) SC i Λ = 0. 9 Indeed, we observe that for i(γ ⊗ e a ) = i fixed, we have The following result is an immediate consequence of Theorem D and the Mayer-Vietoris property of relative symplectic cohomology [40]. However it also admits a simple direct proof using Proposition 1.15, which we feel is illuminating, so we give it here.

Theorem D.
To prove Theorem D, we need to consider the dependence of our constructions on the 'smoothing parameter' R > 0, so we include it in the notation. The proof starts with the same strategy that was used in the proof of [38,Theorem 1.24]. For R sufficiently small and σ sufficiently close to 1, M \K R σ is stably displaceable (this follows from an h-principle as popularized by McLean in [25]). Therefore, SH M M \K R σ ; Λ = 0 for such R, σ. We then prove that there exists a continuous function σ D crit (R), with σ D crit (0) = σ crit , such that the following holds: Proposition 1.17 (Proposition 5.14). Let σ crit (R) < σ 1 < σ 2 < 1. Then, there exists an isomorphism The proof of Proposition 1.17 uses the 'contact Fukaya trick' of [38]. This allows us to set up acceleration data (H τ , J τ ) for M \K σ2 and (H τ ,J τ ) for M \K σ1 , so that there is an isomorphism of Floer 1-rays C(H τ , J τ ) ∼ = C(H τ ,J τ ), which however need not respect action filtrations. The key to proving the Proposition, then, is to show that the action filtrations on the corresponding telescope complexes are topologically equivalent. The reason why this last step worked in [38] was the index-boundedness property (also popularized in [25]). In our setting we need estimates on the mixed index, which have a different nature.

When Hypothesis A holds, this inclusion is an equality.
We first observe that the Conjecture is consistent with the fact that Vol. 24 (2022) Quantum cohomology as a deformation of symplectic cohomology Page 13 of 77 48 It is motivated by this together with the natural expectation that the isomorphism of Theorem B (3) sends PD(C) → e i∈I λi · PSS log (C) + (higher-order terms) , where C is a cycle contained in D I , and PSS log is the log PSS map of [14]. Thus we expect Q(PD(C)) ≥ i∈I λ i /a 0 . Remark 1. 19. The filtration in Conjecture 1.18 exhibits intriguing parallels with the weight filtration in Hodge theory, c.f. [9,19].
1.6.2. Analogue of Theorem C in the absence of Hypothesis A. Let us consider the spectral sequence associated with the filtered complex ( SC Λ , ∂, Q ≥• ) of Theorem B. If Hypothesis A holds, then it converges to QH * (M ; Λ) by Theorem C; but it is also interesting to study the spectral sequence when this Hypothesis does not hold.
As we saw in Sect. 1.5.2, the reason Hypothesis A is necessary for Theorem C to hold is that it guarantees the Q-filtration on SC Λ is bounded below, and in particular complete. Let us denote by (SC Λ , ∂) the completion of ( SC Λ , ∂) with respect to the Q-filtration. Note that taking the completion does not change the spectral sequence.
We give a conjectural description of H * (SC Λ , ∂), based on suggestions made to us independently by Pomerleano and Seidel. For each i ∈ I, define QH * (M ; Λ) i to be the 0-generalized eigenspace of the operator PD(D i ) (−) on QH * (M ; Λ), where denotes the quantum cup product. I.e., it is the subspace of α ∈ QH * (M ; Λ) such that PD(D i ) k α = 0 for some k. We then define As evidence for the conjecture, we use Conjecture 1.18 to argue that whenever λ i > 2, the degree-0 class c − e −2 PD(D i ) is invertible in the Qcompleted quantum cohomology, for any c = 0. Indeed its inverse is by multiplying on the left by the inverse.
Assuming that the k-linear endomorphisms e −2 PD(D i ) (−) admit Jordan normal forms, the above argument suggests that only the 0-generalized eigenspaces can 'survive'. This gives some evidence for Conjecture 1.20 in the case that k is an algebraically closed field. It is reasonable to believe that one

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can bootstrap from there to the case of a general commutative ring k. For the rest of this section we will assume that k is an algebraically closed field. Remark 1.21. We strongly expect that H * (SC Λ , ∂) is nothing but the relative symplectic cohomology of the skeleton of X. There is an intriguing contrast between Conjecture 1.20 and Ritter's work [29]: precisely, let us consider the case that D is smooth and λ > 2, and let N be the total space of the inverse of the normal bundle to D. Then Conjecture 1.20 (together with the above expectation) says that QH * (M ) crit , which is the 0-generalized eigenspace of QH * (M ), 'lives on the skeleton of X'; whereas Ritter shows that SH * (N ) is the quotient of QH * (N ) by its 0-generalized eigenspace. Note that we can obtain N from the Liouville completion X of X by replacing a neighbourhood of the skeleton with a copy of D (more precisely, the symplectic cut of X along the hypersurface {ρ = 1} is M N ).  Proof. We first observe that for any even element α in a supercommutative Frobenius algebra, the decomposition into generalized eigenspaces of α (−) is orthogonal (with respect to the pairing and the algebra structure), and hence the generalized eigenspaces are ideals generated by idempotents. It follows for each i, the subspace QH * (M ; Λ) i is an ideal generated by an idempotent; so the intersection is an ideal generated by the product of these idempotents.  Conjecture 1.24 implies, for example, that L must intersect every a-Floer-theoretically essential (over k) monotone Lagrangian, where the latter condition means that CO(a ⊗ Λ k) ∈ HF 0 (L; k) is non-zero. (Here we have used the algebra homomorphism Λ → k, which sends q → 1, to define an idempotent a ⊗ Λ k ∈ QH 0 (M ; k)).

Maurer-Cartan element.
For the purpose of this section, we assume that k is a field of characteristic zero, and we assume that Hypothesis A holds.
Recall that the symplectic cochain complex SC * (X; k) carries an L ∞ structure [13]. This consists of a sequence of operations k : SC * (X; k) ⊗k → SC * (X; k) of degree 3 − 2k, satisfying the L ∞ relations; and 1 = d is the standard differential. We extend these linearly to make SC * Λ into an L ∞ algebra. We recall that a Maurer-Cartan element for the L ∞ algebra (SC * Λ , k ) is an element β ∈ Q ≥1 SC 2 Λ , satisfying the Maurer-Cartan equation: We remark that this is in fact a finite sum, because the terms live in successively higher levels of the Q-filtration, which Hypothesis A ensures is bounded below (see Sect. 1.5.3). A Maurer-Cartan element β can be used to deform the L ∞ structure to get a new one k β on SC Λ (see, e.g. [16,Section 4]). In particular, the resulting operation 1 β defines a new differential on SC Λ .

Conjecture 1.25.
There exists a Maurer-Cartan element β ∈ SC 2 Λ such that in the statement of Theorem B, we may take SC Λ = SC Λ and ∂ = 1 β . Remark 1.26. Cieliebak and Latschev have outlined ideas closely related to Conjecture 1.25 (but in a more general context) in talks as far back as 2014.
Remark 1.27. Moreover, one expects that Floer-theoretic operations on quantum cohomology of M (such as the quantum cup product) are deformations of the corresponding operations on symplectic cohomology of X by β, c.f. [12].
Remark 1.28. In the proof of Theorem B presented in this paper, we need to replace SC Λ with SC Λ . Conjecture 1.25 suggests an alternative proof, in which no such replacement is necessary. The cost is that the construction is significantly more elaborate, relying on the L ∞ structure and a version of the homotopy transfer theorem, which makes it harder to see the key geometric ideas, which are the same in both proofs. Remark 1.29. It is natural to envision generalizations of our results, as well as of Conjecture 1.25, where M is allowed to be only a partial compactification of X; and furthermore, where some of the weights λ i are allowed to be equal to 0. We present several examples in Sect. 1.7 below which illustrate such a generalization. For example, Remark 1.41 gives evidence for this generalized conjecture in the case M = T * RP 2 , with D ⊂ M a smooth divisor equipped with weight λ = 0; the generalized conjecture in this case says that SC * (M ; k) is a 'deformation' of SC * (X; k) (note that there is no need for a Novikov ring in the definition of symplectic cohomology of M , as it is exact). We put scare quotes around 'deformation' because when the weights are 0, the extra terms in the deformed differential may simply preserve the Q-filtration, rather than strictly increasing it; so there is no sense in which they are 'small'. To make a useful version of the conjecture one would need an alternative to the Qfiltration, which is strictly increased by the extra terms; it would probably be defined in terms of the grading.
Note that the projection of β to Gr 1 SC 2 Λ is d-closed, and hence defines a class [β 1 ] ∈ Gr 1 SH 2 (X; Λ). It is immediate from Conjecture 1.25 that the differential on the E 1 page of the spectral sequence is given by We now explain how our conjectures connect with work of Tonkonog [37]. Tonkonog considers the following setup:M is a compact Fano variety equipped with its monotone Kähler form,D ⊂M a simple normal crossings anticanonical divisor, X =M \D, and M =M \ ∪ J i=1D i is a partial compactification of X, with compactifying divisor D = M ∩D. Tonkonog defines a class BS ∈ SH 0 (X; k) by counting pseudoholomorphic 'caps' in M , such that the following holds: This fits into the generalized geometric setup alluded to in Remark 1.29 (we are in the log Calabi-Yau setting, and we equip each component of D with its canonical weight 2). It connects with our conjectures as follows: In many settings, we can tightly constrain the class β using grading considerations. For each i we can define a cocycle B i ∈ SC 2−λj (X; k) by 'counting caps passing through D i ', following [37] or [15]. We define As evidence for the Conjecture, we first observe that Gr Q 1 SC 2 Λ is generated by the classes qB i , together with the unit q · 1; and argue that the coefficient of the unit in β must count certain Chern-number-1 spheres. We further observe that Q ≥2 SC 2 Λ = 0. This follows as we have a 0 = 2, so any generator γ ⊗ q j of SC 2 Λ with j ≥ 2 must have i(γ) ≤ −2; however, i(γ) ≥ 0 by Proposition 1.15. In settings where Conjecture 1.32 holds, the Maurer-Cartan element β is determined up to gauge equivalence by the cohomology classes [B i ]. Furthermore, the components of β get 'turned on' one by one as the corresponding divisors get added compactifying X.

Mirror symmetry in the log Calabi-Yau case. Let us consider the log
Calabi-Yau case, where X = M \D and X is equipped with its preferred Liouville structure and trivialization of canonical bundle. In this case we Assume that Y is a mirror scheme to X over k, which is smooth. Even though we choose to leave what this means vague, we will assume that it implies and in particular Therefore, the classes B i ∈ SH 0 (X; k) are mirror to functions We set W := i w i . This sum includes the constant term w 0 , which may be non-zero in the case that the minimal Chern number of M is 1. Now let Y Λ denote the base change of Y to Λ, and W Λ = qW be a function on Y Λ . 36. In fact, Conjecture 1.35 should extend beyond the log Calabi-Yau case we consider here. However, it becomes difficult (and confusing) to interpret the mirror in terms of the language of classical algebraic geometry: the polyvector fields on Y Λ are given a non-standard grading, and in general W Λ may be a polyvector field rather than a function. In contrast, in the log Calabi-Yau case one can give a transparent interpretation of Conjecture 1.35 in terms of the classical algebraic geometry of the Landau-Ginzburg model (Y, W ) defined over k, which we now do. (We discuss the non-log-Calabi-Yau case in Remark 1.39 at the end of this section.)

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We consider the Koszul complex associated with the section dW of T * Y : This is a complex of vector bundles over Y . When the critical locus Z := Crit(W ) is isolated, K(dW ) is a resolution of O Z , and therefore, its hypercohomology gives the algebra of functions on the critical locus: , because this hypercohomology is, essentially by definition, the graded algebra of functions on the 'derived critical locus of W ' (see e.g. [42] We expect that the mirror to the spectral sequence of Theorem C on the RHS, is the hypercohomology spectral sequence on the LHS, in a sense we now make clear. We recall the construction of the hypercohomology spectral sequence [46,Section 5.7]. We take a Cartan-Eilenberg resolution C p,q of K(dW ), and consider the resulting bicomplex C p,q = Γ(C p,q ). We define a filtration map on this complex by Q(c) = p for c ∈ C p,q (i.e., we have Q(c) = −p for c a section of Λ p T Y ). The resulting Q-filtration induces the spectral sequence with E 1 page as above. The differential on the E 1 page is given by contracting with dW .
We now consider the bicomplex C p,q ⊗ k Λ, and equip it with the filtration map Q(c ⊗ r) = Q(c) + Q(r). We conjecture that the resulting filtered complex is filtered quasi-isomorphic to ( SC Λ , ∂, Q ≥• ), and in particular the corresponding spectral sequence is isomorphic to the one from Theorem C. As evidence, we compute that the spectral sequence has which is clearly isomorphic to the E 1 page of the spectral sequence from Theorem C.
Remark 1.37. The attentive reader may notice the presence of an extra 'p' in the exponent of q, compared with the E 1 page from Theorem C. This is because the isomorphism of E 1 pages This reflects the fact that Vol. 24 (2022) Quantum cohomology as a deformation of symplectic cohomology Page 19 of 77 48 We now explain how this fits with the picture from the previous section. The isomorphism (1.3) is expected to respect the natural graded Lie algebra structures on both sides (among other things), where the Lie bracket on the polyvector field cohomology is given by the Schouten-Nijenhuis bracket. The differential on the E 1 page of the symplectic spectral sequence is given by [B, −]. The differential on the E 1 page of the hypercohomology spectral sequence is given by contraction with qdW , which coincides with [qW, −] (as one can see from the definition of the Schouten-Nijenhuis bracket); thus the two differentials match.
More precisely, we expect that the isomorphism of Lie algebras (1.3) can be refined to a quasi-isomorphism of L ∞ algebras, and the Maurer-Cartan element β matches with the Maurer-Cartan element qW up to gauge equivalence. This would yield a chain-level quasi-isomorphism underlying (1.4), which would imply the isomorphism of spectral sequences discussed above.
Note that when Y is affine, there is no need to take a Cartan-Eilenberg resolution: we may take C p,0 = Γ(Λ −p T Y ) and C p,q = 0 for q = 0, with differential given by contracting with dW , and the bicomplex is simply a complex. In particular, the hypercohomology spectral sequence degenerates at E 2 . This leads us to make the following: Conjecture 1.38. If X in addition (to the conditions from the first paragraph of this section) admits a homological Lagrangian section and SH 0 (X; k) is a smooth algebra, then the spectral sequence of Theorem C degenerates at E 2 page.
Under these assumptions on X one can take Y to be the smooth affine scheme Spec(SH 0 (X; k)) (see [27]), which would satisfy (1.3), which is our reason to make this conjecture.
For example, the conjecture holds in the toric Fano examples (see Sect. 1.7.1), essentially by the argument given above. This degeneration also follows from the fact that one can construct SC * (X; k) with zero differential in this case! Remark 1. 39. We now discuss the non-log-Calabi-Yau case of Conjecture 1.35, which will appear in several examples in Sect. 1.7 below. There are three complicating factors: (1) The mirror to X will in general be a Landau-Ginzburg model (Y, w), rather than simply a variety Y ; (2) The algebra of polyvector fields on Y must be equipped with a nonstandard grading; (3) a priori, β will be mirror to a gauge equivalence class of Maurer-Cartan elements for the differential graded Lie algebra of polyvector fields on (Y, w), rather than simply a function W on Y .
Issue (2) is already present if one wants to talk about the mirror of T * S 1 with a non-standard trivialization of its canonical bundle and then consider the correspondence between compactifications and deformations. In this case one cannot use a traditional SYZ approach as the zero section of T * S 1 does not even have vanishing Maslov class with respect to such a trivialization.

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It seems that to develop some general geometric intuition in the non-log Calabi-Yau cases, it would be helpful to use the language of derived algebraic geometry but we do not feel comfortable enough to do this at this point.
Concerning issue (3), we actually expect that β should be mirror to a function in broad generality, although it is not clear how to prove this. In some cases, it follows from grading considerations, as in Conjecture 1.32 and the ensuing remarks.
Suppose that X = M \D is mirror to Y as at the start of this section. This means that we could choose all weights λ i = 2; we assume, however, that there exists a valid choice of weights with λ i > 0 for all 1 ≤ i ≤ N , We equip X with the trivialization of its canonical bundle corresponding to these weights, and equip the algebra SH 0 (X ; k) with its induced grading. We posit that this is the graded algebra of functions on the mirror of X (with the alternative trivialization), which we regard as a 'graded scheme'. We set X = M \D, and posit that the mirror to We furthermore posit that the Maurer-Cartan element β corresponding to X ⊂ M is mirror to W Λ = N i=1 e λi w i , and therefore that the mirror to M is (Y Λ , w + W Λ ).

Examples 1.7.1. Fano toric varieties.
Let Δ ⊂ R n be a Fano Delzant polytope. This means that it is a Delzant polytope equal to the intersection of half-spaces (with no redundancy) for κ > 0 and ν i ∈ (Z n ) ∨ primitive. Using the symplectic boundary reduction construction (one of the many options), we construct a symplectic manifold (M Δ , ω) with a Hamiltonian T n action and moment map The image of the moment map is by construction Δ. Finally, note that M Δ satisfies the monotonicity condition 2κc We define the toric SC divisor D Δ as the preimage of the boundary of Δ under the moment map. Note that Denoting the coordinates on R n by x 1 , . . . , x n and the circle valued coordinates on We note the short exact sequence A choice of weights is (as always) equivalent to the choice of a rational class which is sent to 2c 1 (T M Δ ) by g and which has positive coordinates. We have a preferred lift given by Let us also use the natural isomorphism Hence, the set of all possible positive weights is the image of the rational points in the interior of 1 κ Δ under the map R n → R m given by We see that the only weight that satisfies Hypothesis A is the canonical weight, which corresponds to 0 ∈ 1 κ Δ. Now let us outline how Theorems B and C work in this context, assuming the conjectural results of Sect. 1.6.3. We can arrange that 1 , . . . , z ±1 n , ∂/∂z 1 , . . . , ∂/∂z n ] where the variables z i are commuting and have degree 0, and the variables ∂/∂z i are anticommuting and have degree 1 (where the degrees are induced by λ can ). We can also arrange that the L ∞ structure is trivial, with the exception of the Lie bracket 2 , which coincides with the Schouten-Nijenhuis bracket. We can compute, for instance via Theorem 1.30 and Cho-Oh's computation of the disc potential of toric Fano varieties [5], that β = qW , where Now Conjecture 1.25 says that in the statement of Theorem B, we can take As explained in Sect. 1.6.4, this is the Koszul complex for dW , tensored with Λ. One can show that W has isolated singularities, so the cohomology of the Koszul complex is

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which is the familiar statement of closed-string mirror symmetry for toric Fano varieties, c.f. [4]. Note that the spectral sequence of Theorem C has E 0 = E 1 = SC Λ , E 2 = Jac(W )⊗ k Λ, and degenerates at E 2 because the differential on SC Λ vanishes (or alternatively, because Jac(W ) is concentrated in even degree). Now let us outline how Theorem D works in this context. For each i = 1, . . . , m, we have a Hamiltonian circle action with moment map ν i • π, which rotates around D i , and these actions commute on the overlaps. It follows that they define a system of commuting Hamiltonians for D Δ , in the sense of Sect. 2. For any p ∈ int(Δ) we define the corresponding weights The relative de Rham class of (ω, θ p ) is easily seen to be f (p) + κλ can = κλ p . The Liouville vector field corresponding to θ p is Z p = (x i − p i )∂/∂x i . It follows that θ p is adapted to the system of commuting Hamiltonians in the sense of Sect. 2. The skeleton L p for θ p is nothing but the Lagrangian torus above p. The corresponding subset K crit,p is easily computed to be π −1 (K crit,p ), whereK crit,p ⊂ Δ is the smallest rescaling of Δ, centred at p, which contains the origin. In particular, K crit,p coincides with L p if and only if λ p = λ can , if and only if Hypothesis A is satisfied.
Our Theorem D says that the monotone torus fiber L 0 is SH-full. It follows that it is not stably displaceable. This result can also be obtained using Lagrangian Floer theory, using the fact that the disc potential always has a critical point in this case. Our result says nothing about the skeleta L p for p = 0. Indeed it is known that for n ≤ 3 all of these non-monotone fibers are displaceable by probes [22, Corollary 3.9 and Proposition 4.7].
The fact that L 0 is SH-full also implies that it intersects every Floer theoretically essential (over some commutative ring) monotone Lagrangian. This result also follows from the fact that L 0 , equipped with appropriate local systems, split-generates each component of the monotone Fukaya category over an arbitrary field [11, Corollary 1.3.1].

Skeleta in S 2 .
Let us move on to a non-toric example. Consider S 2 with a symplectic structure ω such that [ω] = 4κPD(pt). Let D be the union of N distinct points p 1 , . . . , p N ∈ S 2 . Consider weights λ 1 , . . . , λ N > 0, which needs to satisfy Let θ be a primitive of ω on S 2 \D compatible with the weights and with some choice of local moment maps for the circle actions rotating about the p i . Let L be the induced skeleton. The complement S 2 − L is a disjoint union of open disks U i , i = 1, . . . N, one for each point p 1 , . . . p N . L itself is the union of all critical points, homoclinical and heteroclinical orbits, and periodic orbits of the Liouville vector field by the Poincaré-Bendixson theorem. It is elementary to compute (using the compatibility with weights) that the symplectic area of U i is equal to κλ i . If we restrict the function ρ : M → R to the disc U i , then it extends continuously to 0 along the boundary of the closed disk, it is equal to 1 at p i , and it generates a Hamiltonian circle action rotating U i about p i .
Hypothesis A is satisfied if and only if no weight is bigger than 2, which means no disc U i has area more than half the area of S 2 . In this case the subset K crit coincides with the skeleton L. Otherwise, we have λ i > 2 for some i, and K crit is the union of L with a collar around the boundary of U i , so that the rest of U i has area equal to half the area of S 2 . Theorem D says that K crit is SH-full. This implies that it is not stably displaceable, and furthermore that no two such subsets can be disjoint from each other. It is easy to see explicitly that it is necessary to add the collar to K crit in order for these results to hold. We start by sketching how Theorem B works in this case. It is possible to take simpler models for SC * (X; k) and SC Λ than those which appear in the actual proof of the Theorem.
We take a model for SC * (X; k) which is isomorphic to k[z, zθ] where z is a commutative variable of degree −2, and θ is anticommutative of degree 1. The generator 1 corresponds to the unique constant orbit, z j to the fundamental cycle of the Reeb orbit going j times around D, and z j θ to the point class of the same Reeb orbit. The differential d sends z j → 0 and z j θ → z j−1 . In particular the cohomology vanishes: symplectic cohomology of the disc is zero.
We have Λ = C[q] with i(q) = 4. We take SC Λ = SC Λ , and consider the deformed differential ∂, where ∂ − d sends z j → 0 and z j θ → qz j+1 . The cohomology of this differential is free of rank 2 over Λ, with a basis given by 1 and qz. In particular, it is isomorphic to QH * (M ; Λ), in accordance with Theorem B: the class 1 corresponds to 1 ∈ QH 0 (M ; Λ), and the class qz corresponds to PD(pt) ∈ QH 2 (M ; Λ).
Theorem C does not apply in this case, because Hypothesis A is not satisfied: we have λ = 4 > 2. And indeed the conclusion of the Theorem fails, because we cannot have a spectral sequence with E 1 page vanishing, converging to QH * (M ; Λ) = 0. The reason the proof of Theorem C does not run is that the Q-filtration on SC Λ is not degreewise complete. For example, the classes q k z 2k all have degree 0, but their Q-values go to +∞. The convergence theorems for spectral sequences all require completeness, and indeed it could not be otherwise: taking the completion does not change the spectral sequence associated with a filtered complex, by inspection of the construction. It is easy to verify that the degreewise completion of (SC Λ , ∂) is acyclic: for example, This confirms Conjecture 1.20 in this case, as QH * (S 2 ; Λ) crit = 0. We now offer another perspective on this computation following Remark 1.40, which will be useful in the next two sections. First, we take M = S 2 and D = D 1 ∪ D 2 to be an anticanonical divisor on S 2 , where D 1 and D 2 are distinct points. If we equip each point with weight λ i = 2, then this is a special case of Sect. 1.7.1: we see QH * (M ; Λ ) as a deformation of where ∂ 2 x = 0, the generator q of the Novikov ring is in degree 2, x is in degree 0, and ∂ x is in degree 1; the differential ∂ is Λ [x, x −1 ]-linear and sends As expected, this chain complex is degree-wise complete with respect to the Q-filtration and we obtain Now we consider the case that λ 1 = 0, λ 2 = 4. Following the recipe of Remark 1. 40 As expected, this chain complex is acyclic. The chain complex ( , with x and ∂ x graded as before, and the generator q of Λ in degree 4; the differential ∂ sends Note that as expected, we have an isomorphism of chain complexes We learned nothing new so far but we believe that this exercise might help unraveling the more complicated examples in the next two sections below. Consider the graded ring where |x| = −1, |y| = 1, |z| = 0, and consider elements w 1 = y 2 z and w 2 = x of R. We set Y = Spec(R). Then X should be mirror to the Landau-Ginzburg model (Y, w 1 ) while M should be mirror to (Y Λ , w 1 + qw 2 ), where |q| = 3. We expect (SC * (X; k), d) to be quasi-isomorphic to with the filtration map given by Q(c ⊗ q a ) = −p + a for c ∈ Λ p T Y . We can compute the cohomology of this complex: it comes out as the Jacobian ring of w 1 + qw 2 , which is This agrees with Theorem B in this case. Now we turn to Conjecture 1.20. We consider two cases: Case 1: 2 is invertible in k. We easily deduce that 1 − q(x/2) 3 is nullhomologous; it is also clearly invertible in the Q-completion. This implies that the cohomology vanishes after Q-completion. This leads us to conjecture that RP 2 ⊂ CP 2 is SH-full if the characteristic of k is 2, but not otherwise (Entov's result that RP 2 is [CP 2 ]-superheavy over Z/2 can be considered as further evidence for this conjecture). This would imply that RP 2 is non-stably displaceable (which is known), and intersects any monotone Lagrangian which is Floer-theoretically essential over a field of characteristic 2 (note that this does not include the Chekanov torus, as can easily be seen from the superpotential computed in [3]). Remark 1.41. We sketch some evidence for the mirror symmetry statement (1.5), in the case that char(k) = 2. Note that the completion of X is symplectomorphic to T * RP 2 , so SH * (X; k) ∼ = H * (LRP 2 ; k) by Viterbo's theorem JFPTA [1,43,44]. We can compute by [48], where the first factor comes from the manifold RP 2 of constant loops, and the subsequent factors come from the manifolds S(T RP 2 ) of 'length-k' geodesics. Of course H * (RP 2 ; k) ∼ = k[y]/y 3 with |y| = 1, while H * (S(T RP 2 ); k) has rank 1 in degrees 0, 1, 2, 3. On the other hand, one may compute that where v = x∂ x − y∂ y is an anticommuting variable. We identify k[y]/y 3 as corresponding to the constant loops, and the subsequent factors as corresponding to the length-k geodesics. The degrees match up (we observe that |v| = 1). We remark that x = w 2 is the basic loop around D, which corresponds to the family of length-1 geodesics, so it makes sense that multiplying by x takes us to the next k-value.

Fano hypersurfaces.
We consider some examples motivated by [33]. They follow a similar philosophy to Remark 1.40, but are a bit different as they are obtained by partially compactifying an affine variety which is of log general type, rather than being log Calabi-Yau.
Let M = M n,a be a smooth hypersurface of degree a ≤ n + 1 in CP n+1 , and D = D n,a,i a union of i ≤ n + 2 generic hyperplanes. This fits into the setup of Sect. 1.1, and we may take the weights all to be equal to λ = 2(n+2−a) i . In particular, Hypothesis A is satisfied if and only if n + 2 − a ≤ i. This corresponds to the variety X n,a,i = M n,a \D n,a,i being log Calabi-Yau (in the case of equality) or log general type (otherwise). Hypothesis A is not satisfied precisely when X n,a,i is log Fano.
We conjecture that the mirror to X n,a,i is the Landau-Ginzburg model (Y n,a,i , W n,a, Here we assume that k contains all ath roots of unity. The group G n,a,i acts torically, preserving W n,a,i . The variables z j have degree (2 − λ)/a for j ≤ i and degree 2/a for j > i, and q has degree λ. Now let us drop the n, a, i from the notation. By taking M to be a Fermat hypersurface, we obtain a natural action of the dual group G * on M , respecting D. Restricting to the invariant pieces of the relevant group actions, mirror symmetry predicts that and in fact that there is an underlying quasi-isomorphism of L ∞ algebras. In accordance with Conjecture 1.25, this gives us and hence The Jacobian ring has relations Multiplying them together we get that This allows us to compute that The class H corresponds to the hyperplane class (except for the case n + 2 − a = 1, when it corresponds to the hyperplane class plus a!q i ). One can check that this is the correct answer for QH * (M ; Λ) G * , see [17,20]. This is in agreement with Theorem B.
We can also check Conjecture 1.20 in this case. We can factor the defining relation in the Jacobian ring as: Note that we have H = z 1 . . . z n+2 = qz a 1 , from the first relation in the Jacobian ring. Thus Q(H) = 1. On the other hand, Q(q i/(n+2−a) ) = i/(n+2− a). Therefore, precisely when Hypothesis A is not satisfied, the factors (H − ζq i/(n+2−a) ) become invertible in the Q-completion, as argued in Sect. 1.6.2.
The result is that the Q-completion gives Λ[H]/H a−1 , which corresponds to the zero generalized eigenspace (note that Hypothesis A is satisfied for all i ≥ 1 in the anomalous case n + 2 − a = 1, when this corresponds to the −a!q i generalized eigenspace.) JFPTA

Outline
In Sect. 2, we examine the structure of our symplectic manifold in a neighbourhood of the divisor D. In particular, we introduce the notion of a 'system of commuting Hamiltonians near D', and say what it means for a Liouville one-form to be 'adapted' to such a system. This completes the statement of the results in Sect. 1.3, where these notions were used without being defined.
In Sect. 3, we establish our conventions for Hamiltonian Floer theory and relative symplectic cohomology in M , and explain how they are related to symplectic cohomology of the exact symplectic manifold X. In particular, we establish that the map (1.2) respects index and action; and we prove the 'positivity of intersection'-type result which is used to prove Proposition 1.12.
In Sect. 4, we construct the functions ρ R which are smoothings of ρ 0 . We consider degenerate Hamiltonians of the form h • ρ R , explain how to perturb them to obtain non-degenerate time-dependent Hamiltonians H, and give estimates for the index and action of their orbits.
In Sect. 5, we prove our main results.

Remark 2.2. In [23, Section 5] McLean proved that any SC divisor D ⊂ (M, ω)
can be smoothly isotoped in the space of SC divisors to an orthogonal SC divisor D ⊂ (M, ω); and that X = M \D is convex deformation equivalent to X = M \D. This implies that SH * (X; k) ∼ = SH * (X ; k), by [23,Lemma 4.11]. These results mean that it suffices to prove Theorems B and C under the assumption that D is orthogonal.
Setting X = M \D, by Lefschetz duality (e.g. Proposition 7.2 of [7]) we have 2) The inverse is given by mapping the ith basis vector to a disk Assume that Remark 2.3. In the setup from Sect. 1.1, κ will be κλ. Now consider a de Rham representative (ω, θ) for κ consisting of the symplectic form ω together with a one-form θ ∈ Ω 1 (X) satisfying dθ = ω| X , and Following McLean [23,24] we call κ i the wrapping numbers for D with respect to θ, though we use the opposite sign convention than in [23]. We will denote a scH near D of radius R with the notation {r i :

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Note that for any scH of radius R, we can 'shrink' it to an scH of radius R < R by replacing UD i with {r i < R } for each i.

Proposition 2.5. Let D be an SC divisor in a closed symplectic manifold (M, ω). If D admits a scH, then it is orthogonal.
Proof. Assume that {r i : UD i → [0, R)} is a scH near D. We need to show that for all i = j and x ∈ D i ∩ D j the symplectic orthogonal (T We consider the action of S := R/Z on T x M induced by r i . The action on T x D i ⊂ T x M is trivial, since D i is fixed pointwise under the action of S. The action on UD i ∩ UD j leaves {r j = 0} ∩ UD i ∩ UD j invariant by the Poisson commutativity property. Therefore, T x D j is an invariant subspace of T x M under the S action. Finally, since the action of S on T x M preserves the symplectic pairing, (T x D i ) ω ⊂ T x M is also an invariant subspace.
Note that the action of S on (T x D i ) ω cannot be trivial by the Bochner linearization theorem, as x does not have a neighborhood on which S acts trivially. Now we finish the proof with the following claim: • Assume that V is a finite dimensional symplectic representation of S, which is the direct sum of two representations W ⊕ E, where W is the trivial representation on a symplectic codimension 2 subspace, E is not the trivial representation, and E and W are symplectically orthogonal. Let W be another codimension 2 symplectic subspace of V which is invariant under the action of S. Then if W is transverse to W , it has to contain E.
The proof of this statement is as follows. There exists w + e ∈ W with e = 0, as W is transverse to W . For any θ ∈ S we have θ · (w + e) ∈ W ; hence, θ · (w + e) − (w + e) = θ · e − e ∈ W . We may choose θ so that θ · e = e, so W ∩ E = {0}. This implies that E ⊂ W as required. Proof. This is an immediate consequence of [23, Lemma 5.14], where for each i, we use the well-defined radial coordinate of the symplectic disk bundle over D i in the statement as our r i (the domain is the symplectic disk bundle of course). It is trivial to see that this gives a scH near D. Proof. We use a scH as in the proof of Proposition 2.7. Then, a one-form θ on M \D produced by [23,Lemma 5.17] is adapted in the sense of Definition 2.9, as we show below. Note that by the relative de Rham isomorphism, there is a primitive θ defined on M \D such that the relative cohomology class in H 2 (M, M \D) defined by (ω, θ ) is equal to κ i · PD(D i ), which is why we can use McLean's lemma. Using McLean's notation for the moment, on the fibers of the projections π I : UD I → D I we have where we use the action of (R/Z) I on C I × C n−|I| given by θ · ((z i ) i∈I , w) = ((e 2πiθi z i ) i∈I , w) for all θ ∈ (R/Z) I and ((z i ) i∈I , w) ∈ C I × C n−|I| . Proof. This immediately follows from the equivariant Darboux theorem [18,Theorem 22.1].
We now choose an arbitrary Riemannian metric on M , and let inj(M ) be the injectivity radius with respect to this metric. We call a standard chart (U, φ) in UD I admissible if U is contractible and has diameter < inj(M )/2. The significance of admissibility for us is that it guarantees uniqueness of caps: Lemma 2.14. If γ : S 1 → M is a loop contained in some admissible standard chart, then there exists a disc bounding γ, whose image is contained inside an admissible chart. Moreover, such a disc is independent of the choice of admissible chart containing γ, up to homotopy rel. boundary in M .
Proof. The existence is clear, as admissible standard charts are contractible. The uniqueness follows as the union of two admissible standard charts containing γ has diameter < inj(M ), hence is contained in a ball of radius < inj(M ). As the ball is contractible, the caps in the two charts are homotopic rel. boundary in M . Proof. First note that any standard chart around x ∈ D I can be shrunk so that it is admissible. Therefore, we have a neighbourhood of D I given by the union of all admissible standard charts. By shrinking the scH sufficiently, we may ensure that UD I is contained in the neighbourhood, for all I. Remark 2.17. In Sect. 3.1, we will define a cap for a loop γ : S 1 → M to be an equivalence class of discs u bounding γ under the equivalence relation Therefore, we could get away with the following weaker notion of admissibility for the purposes of the present paper. We call a standard chart weakly admissible if it is simply connected. Assume that we have a loop γ inside UD I that is the orbit of a point under the action of a one dimensional subgroup S of (R/Z) I . We claim that the symplectic area of a cap of γ that is contained inside a weakly admissible standard chart U (assuming such charts exist) only depends on γ, i.e. it is independent of U and the cap chosen inside of U . The reason is because we can then compute the symplectic area by transporting everything into C I × C n−|I| and see that it is equal to l(0) − l(p), where l : R I → R is a function whose pre-composition with r I generates the action of S and p is the point of R I ≥0 above which γ lives. Hence, for such γ existence of a weakly admissible standard chart determines uniquely an equivalence class of caps. This would be enough for our purposes.
Let where Given a non-degenerate Hamiltonian F : S 1 × M → R, let P F denote the set of contractible one-periodic orbits of F , and letP F be the set of orbits equipped with a cap. Elementsγ = (γ, u) ∈P F have a Z-grading and an action and these are compatible with the action of A in that i(a · (γ, u)) = i(γ, u) + a and A(a · (γ, u)) = A(γ, u) + κa .
Note that the 'mixed index' is independent of the cap u.
Define CF * (M, F ) to be the free Z-graded k-module generated byP F . It is naturally a graded Λ -module, via e a · (γ, u) := a · (γ, u). It also admits a Floer differential after the choice of a generic S 1 -family of ω-compatible almost complex structures (which we suppress from the notation). The differential is Λ -linear, increases the grading by 1, does not decrease action, and squares to zero.
One can also define continuation maps CF (M, F 0 ) → CF (M, F 1 ) in the standard way by choosing a smooth function F : R s × S 1 × M → R, which is equal to F 0 for s 0 and to F 1 for s 0, as well as an R × S 1 dependent family of ω-compatible almost complex structures, which together satisfy a regularity condition. Continuation maps are Λ -linear chain maps. If the continuation maps are defined using monotone Floer data, which means ∂F ∂s ≥ 0, then the continuation map CF (M, F 0 ) → CF (M, F 1 ) does not decrease action.
Remark 3.1. We would like to stress that the discussion of Hamiltonian Floer theory that we gave here is slightly simpler than the general theory due to our positive monotonicity assumption. In particular, we did not need to complete

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Λ or our Hamiltonian Floer groups, which is necessary in general for the potential infinite sums to make sense. For details, we refer the reader to [31]. Apart from the ones that we have explicitly stated above, our conventions for Hamiltonian Floer theory agree with (1), (2), (3) and (5) of Section 3.1 in [40].
, with the same grading convention i(e a ) = a ∈ Q; then we have an inclusion Λ ⊂ Λ. (Eventually we will take A and Λ to be as defined in the beginning of Sect. 1.2 but we choose to be more general for a while.) Let us define the Λ-cochain complex We denote the cohomology of this cochain complex by HF * (M, F ; Λ) := H * (CF * (M, F ; Λ), ∂). There exists a natural PSS chain map: which is known to be a quasi-isomorphism [28]. The PSS map is well defined up to chain homotopy and compatible with chain level continuation maps up to chain homotopy. We now introduce the notion of 'fractional caps' of orbits. A fractional cap for γ is a formal expression u + a, where u is a cap for γ and a ∈ R, and we declare u + a ∼ u + a if and only if a − a ∈ A and u = (a − a ) · u. There is a well-defined index and action associated with a fractional cap: There is a natural bijection between the k-basis (γ, u) ⊗ e a of CF * (M, F ; Λ), and the set of fractionally capped orbits (γ, u + a) with a ∈ A.

Relative symplectic cohomology
Let M, ω, κ, Λ be as in Sect. 3.1. We now define relative symplectic cohomology for compact subsets of M over Λ, referring to [40] for the details. As briefly mentioned in the introduction (see Sect. 1.3, especially the footnote on pages 4-5), the construction below is slightly different than the one in [40]. Namely, here we use capped orbits (in particular we only consider contractible orbits) and keep track of the caps rather than weighting Floer solutions using a formal variable. Let K ⊂ M be compact. We call the following data a choice of acceleration data for K: We denote the acceleration data as a single family of time-dependent Hamiltonians and almost complex structures (H τ , J τ ), τ ∈ R ≥1 . We also fix an non-decreasing surjective smooth map (−∞, ∞) → [0, 1]. Given a [i, i+1]dependent family of Hamiltonians and almost complex structures, we use this map to write down a Floer equation for maps from R × S 1 to M . Let us call the resulting R × S 1 -family of Hamiltonians and almost complex structures the associated Floer data.
We require the acceleration data (H τ , J τ ) to satisfy the following two assumptions: Given acceleration data (H τ , J τ ), Hamiltonian Floer theory provides a 1-ray of Floer Λ-cochain complexes, called a Floer 1-ray: The horizontal arrows are Floer continuation maps defined using the monotone homotopies appearing in the acceleration data. Recall that a cylinder u contributing to a Floer differential or a continuation map has nonnegative topological energy

Remark 3.2.
We also note that the inequality in (3.1) comes from the more general inequality where u is a solution of the Floer equation for an arbitrary H : R×S 1 ×M → R which is s-independent at the ends.
From now on, we will use the terminology introduced in Sect. A.3 freely. We apologetically ask the reader to take a look at it before moving further. Using the grading and action considerations from Sect. 3.1, C(H τ , J τ ) becomes a 1-ray in F iltCh Λ . We define the Λ-cochain complexes tel(C(H τ , J τ )) and tel(C(H τ , J τ )) as in Sect. A.3. We can now repeat Section 3.3.2 of [40] in this set-up.   H s , J)) .

Proposition 3.4.
There are canonical restriction maps of Q-graded Λ-modules for K ⊂ K : We finally list the three properties we will need of relative symplectic cohomology. Here is the first one. 2) and the fact that a direct limit of quasi-isomorphisms is a quasi-isomorphism.
Before we state the second property, we note the following important statement from Hamiltonian Floer theory.
Let H : S 1 × M → R a non-degenerate Hamiltonian and J an S 1dependent almost complex structure compatible with ω. Assume that (H, J) is regular and fix Δ ≥ 0. is the naive map which sends each capped orbit to itself. Yet, note that the action of the capped orbit for H + Δ is Δ more than its action for H.
Let us fix a non-decreasing Ψ for the proof below. Let us denote the continuation map above for any H and Δ by c Ψ by abuse of notation. time dependent Hamiltonian vector field X t , t ∈ I of h : I × M → R and the time-T flow of X t , t ∈ I of (h • (ψ × id)) · dψ dt : I × M → R are the same map M → M . 11 Let us fix a non-decreasing function ψ : [0, 1/2] → [0, 1], which is locally constant in a neighborhood of the endpoints of [0, 1/2].
Using the reparametrization construction with ψ, starting with H L , H R : M × [0, 1] → R we can cook up a new Hamiltonian H L φH R : M × R/Z → R, such that the H L and H R parts are supported in (1/2, 1) and (0, 1/2) respectively. The Hamiltonian flow of H L φH R is tangent to X HR first. After not moving for a short period, it arrives at φ 1 HR in less than 1/2-time, and stops for a while. At some point after time 1/2, it starts moving again, this time being tangent to X HL , and reaches to φ 1 HL • φ 1 HR before time-1. It then stops for a little until time 1, after which it repeats this flow.
We define SH * M (K, H; Λ) via the family HφH s in the same way we defined SH * M (K; Λ). Note that this construction does not use that H displaces K. In particular, we can define SH * M (K, 0; Λ), and it follows from Lemma 4.2.1 of [39] that SH * M (K, 0; Λ) is isomorphic (as a graded Λ−module) to SH * M (K; Λ). Here and in the future, by abuse of notation, we denote the constant function M × [0, 1] → R, sending everything to Δ ∈ R by Δ.
The next step is to show that SH * M (K, H; Λ) is isomorphic to SH * M (K, 0; Λ), which is true for arbitrary H. We can find a Δ ≥ 0 such that where c > 0 is a constant that depends on our choice of ψ.
The composition of the first two maps is filtered chain homotopic to the map obtained from c Ψ s as explained right before the theorem using a filling in 3-slits argument. The same result is true for the composition of last two maps. Using Lemma A.2's last statement and the second bullet point of Lemma A.3, we obtain that there is a chain of maps The main point of the proof is to show that SH * M (K, H; Λ) = 0 for the displacing Hamiltonian H from the beginning of the argument. This uses Lemma A.5. The more detailed claim is that a slightly modified version of the family HφH s gives rise to a 1-ray that satisfies the conditions of Lemma JFPTA A.5. The actual proof of this is too long to include here (see Section 4.2.3 of [39]). Let us instead explain the intuition behind the proof. Let γ : S 1 → M be a 1-periodic orbit of HφH s for some s. Because φ displaces D, either γ(0) or γ(1/2) = φ −1 (γ(0)) needs to lie outside of D. Then conservation of energy and that ∂D is a level set of H s for all times shows that in fact we have γ([0, 1/2]) ⊂ M \D. Now if we could use parametrized moduli spaces and cascades instead of continuation maps, we would have our proof. This relies in the fact that γ([0, 1/2]) ⊂ M \D holds for all 1-periodic orbits of all HφH s and that ∂Hs ∂s = 1 in M \D: the actions increase with a constant rate as we follow the orbits and accidental solutions can only further increase the action. There are technical difficulties in making this work, so we refer the reader to [39] for the actual proof.
We move on to the case when K is only stably displaceable. Let T 2 be a symplectic torus such that where γ is a meridian in T 2 . Note that M × T 2 also satisfies the conditions of our construction of relative symplectic cohomology over Λ as T 2 is aspherical.
We will prove that SH * M (K; Λ) naturally injects into SH * M ×T 2 (K; Λ), which finishes the proof. It is easy to see that acceleration data can be chosen for γ ⊂ T 2 where each Hamiltonian in the cofinal family has exactly 4 contractible orbits, and the differentials on each of the corresponding Hamiltonian Floer groups vanish. Using the the chain level Künneth isomorphism for Hamiltonian Floer theory and that completion commutes with tensor product with a finite dimensional Λ-module, we easily prove the desired claim.
We come to the third and final property of relative symplectic cohomology that we will discuss in this section. Recall from the introduction that a compact set K ⊂ M is called SH-invisible if SH * M (K; Λ) = 0. Theorem 3.7. If a compact subset K ⊂ M is SH-invisible, then any compact subset K ⊂ K is also SH-invisible.
Proof. The proof is identical to that of Theorem 1.2 (4) in [38]. The key point (Proposition 2.5 of [38]) is that there is a distinguished element 1 K ∈ SH M (K, Λ), called the unit, with the following properties.
• Restriction maps send units to units.
The element 1 K is constructed so that it is the unit of a pair-of-pants type product structure on SH M (K, Λ). The details are in Section 5 of [38].

Towards the symplectic cohomology of the divisor complement
We return to the geometric setup of Sect. 1.1: (M, ω) will be a closed symplectic manifold that is monotone will be a simple crossings divisor and λ 1 , . . . , λ n ∈ Q >0 will be the weights. We will denote X = M \D, λ ∈ H 2 (M, X; R) will be the Vol. 24 (2022) Quantum cohomology as a deformation of symplectic cohomology Page 39 of 77 48 associated lift of 2c M 1 , and θ ∈ Ω 1 (X) will be a primitive of ω| X such that the relative de Rham cohomology class of (ω, θ) is κλ.
First we recall the action and index of orbits in the exact symplectic manifold (X, θ). Let F : S 1 × X → R be a Hamiltonian, and γ : S 1 → X a non-degenerate orbit of F . Its action is defined to be To associate an index to orbits, we require an additional piece of data: a homotopy class of trivializations η of Λ top C (T X) ⊗2N , for some integer N > 0. To define the index i η (γ) of an orbit γ, we first choose a trivialization Φ of γ * T X; we denote the Conley-Zehnder index with respect to this trivialization by CZ(γ, Φ). The trivialization Φ induces a trivialization of Λ top C (γ * T X) ⊗2N , and we define w(Φ, η) ∈ Z to be the winding number of One easily checks that the index is independent of the trivialization Φ. Note that it is fractional: i η (γ) ∈ 1 N Z. In our setting, the relevant choice of trivialization η is determined by λ. Let N be an integer such that Nλ i ∈ Z for all i. ⊗2N ) by definition, so we may choose a section of Λ top C (T M) ⊗2N which is non-vanishing over X, and vanishes with multiplicity Nλ i along D i . Restricting this section to X defines a homotopy class of trivializations of Λ top C (T X) ⊗2N , which we denote by η λ . We will write i(γ) for i η λ (γ). Now let F : S 1 × M → R be a Hamiltonian, and γ : S 1 → X a nondegenerate orbit of F which is contractible in M , and contained inside X. We define a canonical fractional cap u in for γ, by setting u in := u − u · λ for an arbitrary cap u; the result is clearly independent of u. One should think of u in as a 'cap inside X': indeed, if u were a cap contained inside X, we would have u in = u. and Proof. Let us choose an arbitrary u : D → M capping γ. We start with the action. Directly from the definitions: Therefore, the result follows from the assumption that the relative de Rham cohomology class of (ω, θ) is κλ.

JFPTA
Recalling definitions for indices: where we choose Φ to be the trivialization of γ * T X induced by the cap u, and Therefore, we need to show that This follows because η λ actually induces a section of Λ top C (u * T M) ⊗2N . Using any trivialization of Λ top C (u * T M) ⊗2N , we can think of this section as a map D → C, which does not vanish along the boundary. The degree of this map at 0 ∈ C is easily computed to be Nu·λ using that η λ vanishes with multiplicity Nλ i along D i . It is an elementary fact that the same degree is also equal to the winding number that we are interested in, so the result follows.

Positivity of intersection
In this section, we prove a result based on Abouzaid-Seidel's 'integrated maximum principle'. We will later use it to prove Proposition 1.12, although the result is more broadly applicable.
Let (W, ω) be a symplectic manifold with a concave boundary modelled on the contact manifold (Y, θ). This means that ∂W = Y , and there is a symplectic embedding of the symplectization (Y × [c, c + ), d(ρ · θ)) onto a neighbourhood of the boundary, where ρ ∈ [c, c + ) is the Liouville coordinate. Note that as ω| Y = cdθ, we have a relative de Rham cohomology class [ω; cθ] ∈ H 2 (W, Y ). We will consider u : (Σ, ∂Σ) → (W, Y ) satisfying the pseudoholomorphic curve equation for a certain class of almost-complex structures and Hamiltonian perturbations, and give a criterion guaranteeing that [ω; cθ](u) ≥ 0, with equality if and only if u ⊂ Y .
To define our pseudoholomorphic curve equation, we choose a complex structure j on Σ, a family J of ω-compatible almost-complex structures J z parametrized by z ∈ Σ, and a Hamiltonian-valued one-form K ∈ Ω 1 (Σ; C ∞ (W )). Note that differential forms on Σ×W decompose into types: so we may interpret K as a one-form on Σ × W . The de Rham differential decomposes as and The isomorphism C ∞ (T W ) → Ω 1 (W ) sending v → ω(v, −) allows us to turn d W K into a Hamiltonian-vector-field-valued one-form X K ∈ Ω 1 (Σ; C ∞ (T W )). We will consider the pseudoholomorphic curve equation Note that the (0, 1)-projection of v ∈ Ω 1 (Σ; C ∞ (T W )) is given by 1 We introduce the geometric energy of a pseudoholomorphic curve u: It is manifestly non-negative. Letũ : Σ → Σ × W denote the graph of u. We have the standard computation (e.g. Equation (8.12) of [32]): where the final term lives in Ω 2 (Σ, C ∞ (W )) and is defined by We also introduce the topological energy Note that (2) There exist one-forms α, β ∈ Ω 1 (Σ) such that K = α · ρ + β in a neighbourhood of Y Proof. We have By hypothesis (2), the first and second terms combine to give We can analyse the remaining term using the argument in [2,Lemma 7.2]. Let v ∈ T z ∂Σ be a positively-oriented boundary vector. Using the Floer equation We analyse each term on the RHS. For the first, we note that j(v) points into Σ. Therefore u * (j(v)) points into W . Such vectors can be written as the sum of a non-negative multiple of the Liouville vector and a vector that is tangent to Y . Because J is of contact type, this implies that For the second, we note that hypothesis (2) where R is the Reeb vector field on Y . Thus, θ(X K (v)) = −α(v). For the third, hypothesis (2) again ensures that X K (j(v)) is a multiple of the Reeb vector field; because J is of contact type, θ(JX K j(v)) = 0. Putting it all together, we have Combining, we finally obtain If equality holds then we have E geom (u) = 0, which implies that du = X K . Hypothesis (2) then implies that u * (v) = X K (v) is a multiple of the Reeb vector field R in a neighbourhood of Y , for all v; as R is tangent to Y , this implies that u is contained in Y .

Special Hamiltonian
Our goal in this section is to construct the special functions ρ R : M → R, defined for R > 0 sufficiently small, as mentioned Sect. 1.5. Recall their key properties: • ρ R is continuous on M , and smooth on the complement of the skeleton L; • ρ R | L = 0 and ρ R | D ≈ 1; Having constructed the functions ρ R , we use them to construct the Hamiltonians on M which we use in our main arguments; and we compute the action and index of the orbits of these Hamiltonians. The results are expressed in Lemmas 4.21 and 4.24.
We use the geometric setup of Sect. 1.1 with slight modifications in light of Sect. 2. Let us spell this out fully. We have a closed symplectic manifold (M, ω) which is monotone is an orthogonal simple crossings divisor and λ 1 , . . . , λ N ∈ Q >0 is a choice of weights. We denote X = M \D and λ ∈ R N ∼ = H 2 (M, X; R) is the associated lift of 2c M 1 . We also choose an admissible system of commuting Hamiltonians {r i : UD i → [0, R 0 )} near D and a primitive θ ∈ Ω 1 (X) of ω| X such that the relative de Rham cohomology class of (ω, θ) is κλ. We assume that θ is adapted to {r i : UD i → [0, R 0 )} and that R 0 < κλ i , for all i. (4.1) The last condition can be achieved by shrinking the ascH (as explained in Sect. 2.2). In fact, we will consider the (0, R 0 )-family of such data obtained by shrinking the ascH to radius R ∈ (0, R 0 ), while keeping all else fixed. The parameter R will also 13 be used as the 'smoothing parameter' for ρ R . In Sect. 5, we will want R to be sufficiently small for certain arguments to work. The approximations in this section (such as ρ R | D ≈ 1) will be more and more accurate as R tends to 0. The dependence on R of our constructions below should be understood in this light.
carefully chosen so that the definition agrees on the overlaps and ρ R satisfies the desired key properties. In fact,ρ R I will be well defined on the larger region Let us briefly discuss how we will ensure that ρ R thus defined satisfies Z(ρ R ) = ρ R . We translate this into a property of the functionsρ R I . We denote the standard projection by pr I : R N → R I , and set λ I := pr I (λ). We consider the (Euler-type) vector fieldZ I on R I defined by

Lemma 4.1.
For all x ∈ UD I \D, .
Proof. Follows from the fact that Z(r i ) = r i − κλ i , as θ is adapted to the scH.
In fact, UD max  The functionsρ R I will be constructed roughly as follows. We will choose a hypersurfaceỸ R I ⊂ V I ∩R I ≥0 which is a smoothing ofỸ 0 I := ∂R I ≥0 , satisfying certain properties (see Lemma 4.8). Then, we will defineρ R I as the function that is linear along the rays emanating from κλ I , converging to zero at that point, and takes the value 1 onỸ R I . For the other key properties of ρ R let us mention the following slightly sketchy point to orient the reader. Recall that ρ R is supposed to be a smoothing of the continuous function ρ 0 : M → R introduced in Sect. 1.3, which has all of the properties we need (e.g., it satisfies ρ 0 | L = 0, ρ 0 | D = 1 and Z(ρ 0 ) = ρ 0 ), except it is not smooth. We now give an alternative description of the function ρ 0 , which is parallel with the construction of ρ R . We extend the function κλi−r max i κλi : UD max i → R to M by defining it to be 0 everywhere outside of its original domain of definition. Let us momentarily denote this extension with the same notation. Then we have  Proof. For any i, I, we define the following function on V I :

The Hamiltonian and its orbits
Note that it is non-negative by property (3) ofỸ I . We claim that for any i, and any J ⊂ I, we havẽ If i ∈ I, this follows by Lemma 4.10 (there are two cases: i ∈ J and i ∈ I\J). If i / ∈ I, it is obvious as both functions are 0. This allows us to mimic the construction of ρ: we set ν i =ν I,i • r max I overŮD max I . We finally observe that dρ I = − iν I,i dr i , which completes the proof.
For any m ∈ M − L, we define I(m) := {i : ν i (m) = 0}. We have m ∈ UD max I(m) . We define ν : M \L → R N to be the smooth function with coordinates (ν 1 , . . . , ν N ). We note that the function h (ρ) · ν : M \L → R N extends smoothly to M , and we denote this extension by ν h : M → R N . Note that ν h is constant along orbits of h • ρ, so we have a well-defined ν h (γ) ∈ R N associated with such an orbit γ. We can interpret ν h i (γ) as 'the number of times γ wraps around D i ' (it is an integer unless γ is contained in D i , see Lemma 4.16 below). We define where P I consists of families of orbits γ with I(γ) = I. The two cases I = ∅, I = ∅ must be treated differently. To describe P ∅ , let us suppose that is maximal so that h is constant on (−∞, ]. Then Associated with p ∈ P ∅ is a set of constant orbits C p , which can be identified with a subset of M : On the other hand, for I = ∅ we have For each p ∈ P , C p is a manifold-with-corners on which the flow of h • ρ is 1-periodic, yielding a manifold-with-corners of orbits which is diffeomorphic to C p . We now perturb h•ρ, in such a way as to make the orbits nondegenerate.
Proof. This is an elementary computation.
as required.
Note that the assumption on the boundary of u is automatic if π is involutive; even more specifically, when π is a moment map for a Hamiltonian torus action. Also note that if f is an affine function, then f (B) − f (A) is equal to the linear part of f evaluated at the vector − − → AB considered as an element of R k . If π is a moment map for a Hamiltonian (R/Z) k -action, and f is integral affine, then u as in the statement of the lemma satisfies u(0, s) = u(1, s), for all s ∈ [0, 1].
We will only use this special case of the lemma below, where u can also be thought of as a map R/Z×[0, 1] → M . As a final remark that will be relevant, note that the blow down map is orientation reversing, where we use the standard orientation of C.
Let us now get back to the action computation that we wanted to undertake, continuing the notation used in the previous section.
There is a canonical cap u out associated with any orbit γ of h • ρ, which we now describe. If I(γ) = ∅, then γ is a constant orbit. We define u out to be the constant cap in this case. Otherwise, γ is contained in UD max I(γ) . If γ is contained in UD I(γ) then it is contained in an admissible standard chart, Vol. 24 (2022) Quantum cohomology as a deformation of symplectic cohomology Page 53 of 77 48 and we define u out to be the cap contained in that chart. Note that u out is well-defined by Lemma 2.14.
Note that if γ is an orbit on D, it is contained in UD I(γ) . For an orbit γ not contained in D, we define u out to be the union of the cylinder swept by γ along the Liouville flow taking it into UD I(γ) , with the canonical cap in an admissible chart.
At this point the reader might also benefit from looking at Remark 2.17, which gives a simpler version of admissibility and suffices for the purposes of this paper. It works because of the following Lemma.
We claim that the second term is Consider the map Notice that γ is a one periodic orbit of the Hamiltonian vector field off := f • r max I (see Lemma 4.14). We break u out into two pieces: the piece u out,1 lying in an admissible chart, and the piece u out,2 = ∪ t∈[0,T ] ϕ t (γ) swept out by the Liouville flow. Assume that the boundary of u out,1 is contained in r −1 I ((a i ) i∈I ). Using the symplectic embedding of the admissible chart into C I ×C n−|I| , we see that uout,1 ω is equal to the symplectic area of an arbitrary cap of a 1-periodic orbit of Xf contained inside the fiber above (a i ) i∈I of the moment map C I ×C n−|I| → R I . Choosing the cap obtained by radially scaling the loop to the origin inside the slice C I × {c} that it is contained in, we immediately obtain (e.g. using Lemma 4.19): For the area of the second piece, we use Lemma 4.19 for the map r max

JFPTA
Note that here we used the r max I -relatedness of the Liouville vector field and the Euler vector field (i.e., Lemma 4.4).
Putting the computations together, we get the desired result.
We define the fractional inner cap u in := u out − ν h (γ) · λ as in Sect. 3.3. Strictly speaking we do not need the following result for our argument, but we thought it was informative. Note that it is a slight generalization of the well-known formula in [45,Section 1.2], which gives the result for SH-type orbits.
Proof. By Lemma 4.20, setting ρ = ρ(γ), we have where the last step follows asZ I (ρ I ) =ρ I andρ I • r max I = ρ. Proof. For constant orbits the result is easy, so we assume that γ is nonconstant. We may assume that γ and u out lie in an admissible chart C I(γ) × C n−|I(γ)| , as the index does not change as we flow along the Liouville flow. The flow of h • ρ in the admissible chart decomposes as a product of the flow ϕ t (r, θ) = (r, θ + 2πtν h (r)) on C I(γ) (written in action-angle coordinates) with the trivial flow on C n−|I(γ)| . Thus CZ(γ, u out ) = CZ(Dϕ t ). We have CZ(Dϕ t ) = CZ diag e 2πit·ν h (z) · 1 + 2πit · ∂ν h j (z) ∂z i = CZ diag e 2πit·ν h (z) Vol. 24 (2022) Quantum cohomology as a deformation of symplectic cohomology Page 55 of 77 48 + CZ diag e 2πi·ν h (z) · 1 + 2πit · ∂ν h j (z) ∂z i by a standard argument (c.f. [26,Section 3.3]). The first term is equal to 2 i ν h i (γ) (see [26,Section 3.2]). For the second, we decompose C I(γ) = C J ⊕C I(γ)\J . Note that ∂ν h j /∂z i = 0 for i / ∈ J, because r i has vanishing derivative along {z i = 0}, where our orbit is contained. Also note that e 2πi·ν h i (z) = 1 for i ∈ J. Putting these together, one finds that the second term is equal to the Conley-Zehnder index of the path 1 J + 2πit · [∂ν h j /∂z i ] i,j∈J . Writing this in the basis given by action-angle coordinates (i.e., (r i ∂/∂r i , ∂/∂θ i ) i∈J ), we see that it takes the form of a symplectic shear, whose Conley-Zehnder index is equal to where 0 ≤ δ(γ) ≤ 2n.
Combining the stated Lemmas, we have Recalling that i(γ, u out ) := n + CZ(γ, u out ), the result is immediate. In particular, when Hypothesis A is satisfied, we have i(γ, u in ) ≥ 0.

Proof. We have maps of inverse systems
both of which induce a filtered quasi-isomorphism on the corresponding inverse telescope complex. For the upper map, this follows from Lemma 5.7. The lower map requires a little more argument. We first observe that H j (Gr k σ <p F ≥p SC Λ ) ∼ = H j (Gr k SC Λ ) for j < p−1. It follows easily that for each j, the inverse system H j (Gr k σ <p F ≥p SC Λ ) satisfies the Mittag-Leffler condition, so its lim ← − 1 vanishes. Therefore, the cohomology of the kth-associated graded of the inverse telescope of the bottom inverse system is by Lemma A.7. This completes the argument. Finally, we observe that there is a filtered quasi-isomorphism from the inverse telescope of the top inverse system to SC Λ . Indeed, we take the composition where the first map is the natural one (i.e., the one appearing in the proof of Lemma A.7), and the second map is given by projecting to any of the identical factors. Because this inverse system clearly satisfies the Mittag-Leffler condition, the proof of Lemma A.7 shows that the induced map on cohomology is the obvious isomorphism lim ← − p H * (SC Λ ) ∼ = H * (SC Λ ). Therefore, the chain map is a quasi-isomorphism, and applying the same argument to the associated graded pieces shows that it is a filtered quasiisomorphism. This completes the necessary zig-zag of filtered quasi-isomorphisms.
Proposition 5.10. (= Proposition 1.12) For any Floer solution u that contributes to C(H τ , J τ ) with both ends asymptotic to SH-type orbits, we have u · λ ≥ 0. In case of equality, u is contained in K R σ .
Proof. Let u : R × S 1 → M be a pseudoholomorphic curve contributing to C(H τ , J τ ), with both ends asymptotic to SH-type orbits. We choose > 0 so that u is transverse to ∂K R σ+ , and in a neighbourhood of ∂K R σ+ we have that H τ = h τ • ρ R and J τ is of contact type. We will apply Proposition 3.9 to the part of u that lies in {ρ R ≥ σ + }, to show that u is contained in