A characterization of heaviness in terms of relative symplectic cohomology

For a compact subset K$K$ of a closed symplectic manifold (M,ω)$(M, \omega)$ , we prove that K$K$ is heavy if and only if its relative symplectic cohomology over the Novikov field is nonzero. As an application, we show that if two compact sets are not heavy and Poisson commuting, then their union is also not heavy. A discussion on superheaviness together with some partial results is also included.


Introduction
Let (M, ω) be a closed symplectic manifold.A subset K ⊂ M is called non-displaceable from another subset K ′ if for any Hamiltonian diffeomorphism φ we have φ(K) ∩ K ′ = ∅.If K is non-displaceable from itself, then we say it is non-displaceable.Otherwise we say it is displaceable.A subset K of M is called stably non-displaceable if K times the zero section Z S 1 of (T * S 1 , ω st ) is non-displaceable in M × T * S 1 .We refer the reader to [9] for a detailed discussion of these notions.
As an example, note that for any K ⊂ M , the subset K × Z S 1 ⊂ M × T * S 1 is displaceable from itself by a symplectic isotopy.On the other hand there are many interesting examples of stably non-displaceable subsets, e.g.K = M .
To detect obstructions in displaceability questions that involve some interesting rigidity, Floer theory provides powerful tools.For example, Lagrangian Floer cohomology is very effective in finding obstructions to displacing a Lagrangian submanifold from another one.For general compact subsets, Entov-Polterovich [8,9] introduced the symplectomorphism invariant notions of heaviness and superheaviness, which have the following properties.
Theorem 1.1 (Theorem 1.4 [9]).For a compact subset K of M : (1) If K is heavy, then it is stably non-displaceable from itself.
(2) If K is superheavy, then K intersects any heavy set.
(3) Superheavy sets are heavy, but, in general, not vice versa.
The definition of heavy and superheavy subsets is based on spectral invariants [18] in Hamiltonian Floer theory (see Section 2.2 for the definition in our conventions).
Remark 1.2.For experts we note that throughout the introduction we only use the unit in quantum cohomology as our idempotent for simplicity, but in the main body of the paper our results are stated (and hold) for all idempotents uniformly.
Relative symplectic cohomology [27,28] of compact subsets arose as another alternative to deal with displaceability questions.In particular, the third author and Tonkonog suggested the following definitions [24,3], which are also symplectomorphism invariant.
An important example is when K 1 and K 2 are disjoint.As far as we know, results of this form were only proved for very special M and K i , for example, when M is aspherical and K i are incompressible Liouville domains [13,23].
This corollary in turn implies a positive answer to [6,Question 3.4] about dispersion-freeness of symplectic quasi-states constructed using the homogenized spectral invariant, see [6,Theorem 3.1].For a more in depth discussion of this question see [20,Section 6.2].This implication was communicated to us by Adi Dickstein, Yaniv Ganor, Leonid Polterovich and Frol Zapolsky after the first preprint version of the paper was released.It would take us too far to try to introduce the relevant definitions or notions regarding symplectic quasi-states here, so we refer the reader to [6] for those.As noted in Remark 1.2, the results stated in the introduction are in fact proved for all idempotents (not just for the unit) and we use this below.
Corollary 1.10.Every symplectic quasi-state constructed in [6,Theorem 3.1] corresponding to the unit in a field factor of QH(M ; Λ) is dispersion-free.
Proof.A quasi-state z is dispersion-free if and only if it satisfies z(f 2 ) = z(f ) 2 for each continuous function f .
Let z be the symplectic quasi-state corresponding to the unit in a field factor of QH(M ; Λ).By the C 0 -Lipschitz property of z (cf.Theorem 2.5(4)), it is enough to show that z(f 2 ) = z(f ) 2 for each smooth function f .Consider f as an involutive map f : M → R. Corollary 1.9 implies that this map has a heavy fiber, say at the point t ∈ R. Since z is a genuine quasi-state, this fiber is also superheavy by [6,Theorem 4.1.e.].It then follows that the pushforward functional f * z is in fact the evaluation at t directly from the definition of heavy and superheavy (see e.g.Definition 2.8).In particular z(f 2 ) is the evaluation of the function x → x 2 at t, that is just Another immediate corollary is a solution to Conjecture 1.3 in [24].
Corollary 1.11.If L is a Lagrangian submanifold which admits a bounding cochain with non-zero Floer cohomology, then L is not contained in an SH-invisible set K.
Proof.By Theorem 1.6 in [10], L is heavy so Theorem 1.7 part (A) implies that SH M (L; Λ) = 0. Remark 1.12.If K is a union of Lagrangian submanifolds {K i } one can also observe certain parallels between the notion of (split)-generating the Fukaya category and the notion of superheaviness/SHfullness. Yash Deshmukh presented to us a convincing (and not too difficult) argument showing that if {K i } satisfies the Abouzaid generation criterion then K = ∪K i is SH-full.
We note the following partial converse to part (C) of Theorem 1.7.
Theorem 1.13.Let K be a compact subset of M , and let K 0 ⊃ K 1 ⊃ . . .be compact subsets that all contain K in their interior such that K i = K.If c 1 (T M ) vanishes on π 2 (M ) and the Z-graded relative symplectic cochain complexes SC * M (K n ; Λ) have finite boundary depth in all degrees for all n ≥ 1, then K being SH-full implies that it is superheavy.Remark 1.14.Our methods would extend to show a similar result in certain negative and positive monotone cases as well (see Proposition 3.13 and the discussion afterwards) but we are not confident of their usefulness and optimality.Hence we refrain from a detailed discussion.
For example, as a corollary, the skeleta from Theorem 1.24 of [24] can now be proved superheavy.We omit a full proof of this in order not to introduce notation that is not used elsewhere in the paper.
We obtain Theorem 1.13 from a more general result Theorem 3.18 that involves a condition involving reduced symplectic cohomology SH red M (M ; Λ).This can be defined as the quotient of SH M (M ; Λ) by its valuation ∞ subspace (see the discussion surrounding (3.6)).
Finally, let us note the following result, which is a by-product of our discussion: Proposition 1.15.SH red M (K; Λ) = 0 implies SH M (K; Λ) = 0.The proof is given in Section 3.3 after all the necessary notions are properly introduced.
Acknowledgements.We thank Leonid Polterovich and Sobhan Seyfaddini for helpful discussions and their interest.We also thank the anonymous referee for helpful suggestions.
C.M. was supported by the Simons Collaboration on Homological Mirror Symmetry #652236 and the Royal Society University Research Fellowship.U.V. was supported by the T ÜB İTAK 2236 (Co-Circ2) programme with a grant numbered 121C034.

Preliminary
First we specify the ring and field that will be used.The Novikov field Λ is defined by where T is a formal variable.There is a valuation val : Λ → R ∪ {+∞} given by val( ∞ i=0 a i T λ i ) := min i {λ i |a i = 0} and val(0) := +∞.Then for any real number r, we define Λ >r be the subset of elements with valuation greater than r.Similarly we define Λ ≥r .In particular, Λ ≥0 is called the Novikov ring.It is a commutative ring with a unit 1 ∈ Q.Both Λ >r and Λ ≥r are modules over Λ ≥0 .When r ≥ 0, they are ideals of Λ ≥0 .
Another related Z-graded algebra Λ ω = n∈Z Λ n ω is defined by Note that we have an algebra map Λ ω → Λ by e A → T ω(A) .
(3) make a completion to come to the completed group-algebra of the abelian group ω(π 2 (M )).
Remark 2.2.We defined our Novikov rings with ground field being Q.But all our theorems work for any ground field containing Q.Moreover, if the full Hamiltonian Floer theory package on M can be defined over some other commutative ring, then our results work for those as well.
2.1.Spectral invariants.From the outset let us note that we need to use virtual techniques to do Hamiltonian Floer theory on a general closed symplectic manifold.Here we follow the same convention in [27], where Pardon's approach [19] was used.On the other hand, the analysis in this article is independent of this choice and we will not include all the necessary technical details and language regarding Pardon's implicit atlases etc. in our discussion.We will mostly omit discussing almost complex structures as well.
With this in mind, we briefly review the definition of Hamiltonian Floer groups, referring to [12] for details. 1 Let (M, ω) be a closed symplectic manifold and H be a non-degenerate Hamiltonian function M × S 1 → R. Let γ be a contractible one-periodic orbit of H and u be a disk capping of γ.The action of (γ, u) is and we can associate to it an integer degree using the Conley-Zehnder index CZ(γ, u).We define an equivalence relation on capped orbits by The equivalence classes are denoted by [γ, u].The classical Hamiltonian Floer complex in degree n ∈ Z, i.e.CF n (H) consists of all formal sums Here the coefficient x i 's are in Q, and [γ i , u i ]'s are equivalence classes of capped orbits of degree n.The valuation of such a Q-linear combination of capped orbits is defined as the minimum of the actions of summands with nonzero coefficients.Novikov ring Λ ω acts on CF * (H) by e A • [γ, u] := [γ, u + A], making CF * (H) a free graded Λ ω -module.Choosing a reference cap for each 1-periodic orbit will give a particular basis for CF * (H).
The Floer differential d : CF (H) → CF (H) is defined by counting Floer cylinders.We remark that our conventions about Floer equations and Conley-Zehnder indices follow [27,3].In particular, the differential increases the action and index of a capped orbit.The resulting homology is written as HF * (H).For any p ∈ R we define Since the differential d increases the action, we also have a homology HF * ≥p (H) := H * (CF ≥p (H), d).And there is a natural map i * p : HF * ≥p (H) → HF * (H) induced by the inclusion.Next let P SS H : QH * (M ; Λ ω ) → HF * (H) be the PSS isomorphism [21] between the quantum cohomology of M and HF (H).We usually omit the superscript H when it is clear.For any non-zero pure degree class a ∈ QH n (M ; Λ ω ) the spectral invariant c(a; H) is defined as Remark 2.3.For writing max in the definition of the spectral invariant, we are relying on the existence of best representatives result of Usher, Theorems 1.3 and 1.4 from [25].This is the setup generally used in the spectral invariant literature.We will simplify it a bit (using a simpler Novikov cover) and justify why nothing is lost by doing this.Consider the Floer complex CF (H; Λ) which is the Λ-vector space generated by the 1-periodic orbits (without cappings).We equip it with the Z/2 grading coming from the Lefschetz signs of the corresponding fixed points.The cylinders contributing to the Floer differential are weighted by the topological energy.More precisely, 1 In this article we always work on closed symplectic manifolds and only use contractible orbits to define Hamiltonian Floer groups.
let u be a Floer cylinder connecting two orbits γ − , γ + .Then its contribution in the differential is weighted by T Etop(u) , where Note that we have E top (u) ≥ 0. We write the resulting homology as HF (H; Λ).For x = x i γ i with γ i 's being orbits and x i ∈ Λ, we define val(x) := min i {val(x i )}.
We have the map It is an isomorphism of chain complexes, which also preserves the valuations.
We have a P SS-map over Λ given as the composition: }.And a handy comparison result is the following.Proposition 2.4.Let H be a non-degenerate function and a ∈ QH(M ; Λ ω ) be a non-zero pure integral degree class.Let ã ∈ QH(M ; Λ) be the image of a under the map where f is a Morse function on M and the map counts spiked disks connecting critical points of f and orbits of H.By base changing and using the action scaling map above we also obtain cP SS H Λ : CM (f, Λ) → CF (H; Λ), so that we have a commutative square of chain maps For the opposite direction note that for all n ∈ Z, the maps CF n (H) → CF (H; Λ) are valuation preserving (and hence injective) and HF n (H) → HF (H; Λ) are injective.
We consider a simple case to illustrate the proof.Suppose that . By the commutativity of the above square, we have that for some γ 2 .Then we must have rigid Floer cylinders u 2 from γ 2 to γ 1 , and u 3 from γ 2 to γ.We have We can directly check that [γ, u 3 − u 2 + u 1 ] is a capped orbit of the same degree as [γ 1 , u 1 ] which represents P SS H (a) and with action E.
We now give the proof.Since H is non-degenerate, we have finitely many one-periodic orbits {γ j } j=1,••• ,N in the mod 2 degree of a, that is deg(ã), and x l j T E l j γ j , x l j ∈ Q be an element representing P SS H Λ (ã) with valuation c Λ (ã; H), and be a pure degree element representing P SS H (a).After the base change we get such that x and y T are homologous.That is, we have another element z with dz = y T − x.We write Let n be the degree of a.
for some index n cap u of γ j , we call it good; otherwise we call it bad.Consider the Q-linear expression of d(T E β k ).The argument in the simple case above shows that either all of its terms are good or they are all bad.Let us call rT E β k with r ∈ Q * good or bad accordingly as well.Define z ′ by removing from the Q-linear expression of z all the bad terms.Therefore, dz ′ is a Q-linear combination of (possibly infinitely many) good terms.We define Clearly x ′ is in the image of CF n (H) → CF (H; Λ) (say of ỹ) and it represents P SS Λ (ã).It follows that the valuation of ỹ is at least as large as that of x because the terms in the Q-linear expansion of x ′ is simply a subset of the terms of x.Therefore the valuation of ỹ and hence c(a; H) is at least as large as c Λ (ã; H).
In the rest of this article we will use the second definition of spectral invariant, but with the first notation c(a; H).It has lots of good properties, which we list here and refer to [18,10,9] for proofs and more discussions.
Theorem 2.5.Let a be a non-zero class in QH(M ; Λ), and let H be a non-degenerate Hamiltonian function.The number c(a; H) has the following properties.
By the Lipschitz property, we can extend the definition of spectral numbers to all continuous functions M × S 1 → R by C 0 approximation.Remark 2.6.Let us also note that by monotone approximation one can define spectral numbers for lower (or upper) semi-continuous functions M → R ∪ {±∞}.An important example of a lower semi-continuous function is Next we recall a Poincaré duality property.For two classes a, b ∈ QH(M ; Λ) we define Here * denotes the quantum product, •, • : H * (M ; Λ) × H * (M ; Λ) → Λ is the pairing between cohomology and homology, and π : Λ → Q is the projection to the constant term.where the Hamiltonian H(x, t) := −H(φ t H (x), t) generates (φ t H ) −1 .The identity (2.3) has the same mathematical content as [7, Lemma 2.2] but is in our sign conventions.
Given a time-independent smooth function H on M and a non-zero idempotent a ∈ QH(M ; Λ), the homogenized spectral invariant is defined as This limit is well-defined and has several properties.See [8].We note which is immediate from the triangle inequality.
When a is the unit, we often omit it from the notation.
Remark 2.9.The original definition of (super)heavy sets in [9] uses a homological version of spectral invariants.And there is a duality formula (Section 4.2 in [15]) relating them with our cohomological spectral invariants.Sticking to the unit case for simplicity, we have The definition in [9] of heaviness is Remark 2.11.We cannot use c(a; H) instead of µ(a; H) to define superheaviness.Consider the unit sphere S 2 ⊂ R 3 with its standard area form and let z : S 2 → [−1, 1] be the height function.Let K ⊂ S 2 be the upper hemisphere defined by z ≥ 0. It is well-known that K is superheavy.Consider the non-degenerate autonomous Hamiltonian H := ǫz with 0 < ǫ ≪ 1.By superheaviness, we have Now we recall two criteria for heaviness and superheaviness.
Lemma 2.12.Let K be a compact set in M .
(1) K is a-heavy if and only if for any non-negative function H on M with H| K = 0, we have that µ(a; H) = 0; (2) K is a-superheavy if and only if for any non-positive function H on M with H| K = 0, we have that µ(a; H) = 0.
Proof.This is Proposition 4.1 in [9].Note that we have a different sign convention than the original proof.So the positivity and negativity are swapped.
Remark 2.13.Recall Remark 2.6.Using monotonicity and the triangle inequality it is easy to show that for K compact, c(1; χ K ) is equal to either 0 or ∞.K is heavy precisely when the former possibility is true.
One might be tempted to characterize superheavy sets by c(1; −χ K ) = 0, but this equality in fact only holds for K = M .Choosing a non-positive C 2 -small function that is zero on K and negative somewhere proves this claim.The value of c(1; −χ K ) could also be a negative real number.

2.3.
Relative symplectic cohomology.Now we review the construction of the relative symplectic cohomology, which was introduced in [27,28].The construction could be seen as a categorification of the definition of c(a; χ K ) from Remark 2.6 using monotone approximation.
Let M be a closed symplectic manifold and K be a compact subset of M .Consider (2) for x ∈ K, H n (x, t) converges to 0 as n → ∞; (3) for x / ∈ K, H n (x, t) converges to +∞ as n → ∞.Then we choose a monotone homotopy of Hamiltonian functions connecting H n and H n+1 for each n.The sequence of Hamiltonian functions together with chosen monotone homotopies will be called an acceleration datum for K.
Now given an acceleration datum for K, we denote the differential of CF (H n ; Λ) by d n : CF (H n ; Λ) → CF (H n ; Λ) and the continuation maps by h n : CF (H n ; Λ) → CF (H n+1, ; Λ).
Both Floer differentials and continuation maps are defined by counting suitable cylinders u : R × S 1 → M satisfying the Floer equation.We emphasize that both d n and h n are weighted by powers of the Novikov parameter T Etop(u) .Since our homotopies are chosen to be monotone, this topological energy E top (u) is always non-negative for any u in the definitions of d i and h i .
We have a one-ray of Floer complexes and we can form a new Z/2-graded complex tel(C) = ⊕ n (CF (H n ; Λ) ⊕ CF (H n ; Λ) [1]) by using the telescope construction, see Section 3.7 [2] and Section 2.1 [28].The Λ-linear differential δ is defined as follows, if This telescope comes with a min valution induced by the valuations on CF (H n ; Λ).Then we have an induced non-Archimedean norm and an induced metric Finally we complete tel(C) using this norm.The resulting complete normed vector space is written as tel(C).More concretely, we are adding infinite sums +∞ i=1 (x n + x ′ n ) to tel(C), where the valuations of x n and x ′ n go to positive infinity.Since the differential on tel(C) is continuous (in fact contracting), it induces an differential on tel(C).The relative symplectic cohomology of K in M is defined as the δ-homology SH * M (K; Λ) := H * ( tel(C), δ).Remark 2.14.The above construction also works over the Novikov ring Λ ≥0 .Let CF (H n ; Λ ≥0 ) be the free Λ ≥0 -module generated by one-periodic orbits of H n .By our choice of monotone homotopies, all the Floer differentials and continuation maps are weighted by non-negative powers of T .Hence the resulting T -adically completed telescope tel(C; Λ ≥0 ) is a Λ ≥0 -module with a differential δ.We call the δ-homology SH M (K; Λ ≥0 ) := H( tel(C; Λ ≥0 ), δ) the relative symplectic cohomology over Λ ≥0 .
Note that the T -adic completion of a free Λ ≥0 -module A tensored with Λ can be canonically identified with the completion of A ⊗ Λ ≥0 Λ with respect to the induced non-Archimedean norm (see e.g.Equation (3.1)).Since Λ is flat over Λ ≥0 , we have that On the other hand, SH M (K; Λ ≥0 ) does have torsion and carries more quantitative information compared with SH M (K; Λ).We will often use the Λ ≥0 version in the arguments below.
In [28], it is proved that SH M (K; Λ) and SH M (K; Λ ≥0 ) are independent of choices.Moreover, they enjoy several good properties, notably a Mayer-Vietoris sequence.
These satisfy the presheaf property.
Theorem 2.16.Global section and the displaceability: (1) (Theorem 1.3.1,[27]) Definition 2.17.(Definition 1.22, [5]) Let K 1 , K 2 be two compact sets in M .We say they are Poisson commuting if there are two Poisson commuting functions 0). Theorem 2.18.Let K 1 , K 2 be two compact sets such that they are Poisson commuting.Then there is a long exact sequence We refer to Theorem 1.3.4[28] for the most general form of this theorem.

Theorem 2.19 ([24]
). Relative symplectic cohomology SH M (K; Λ) has a canonical product structure and unit e K which makes it a graded commutative unital Λ-algebra.Restriction maps respect the unit and the product structure.
The techniques of [24] can also be used to show the following result.
The following is also standard.
Theorem 2.21.For every compact K ⊂ M and H : M × S 1 → R a smooth function which is negative on K, there are canonical Λ ≥0 -module maps κ : HF (H; Λ ≥0 ) → SH * M (K; Λ ≥0 ).These maps satisfy Definition 2.22.Let K be a compact set in M .Letting a ∈ QH(M ; Λ) denote a non-zero element, we make the following definition.
(1) K is called SH-a-visible if P SS K (a) = 0, otherwise it is called SH-a-invisible.
(2) K is called SH-a-full if every compact set contained in M − K is SH-a-invisible.
(3) K is called nearly SH-a-visible if any compact domain containing K in its interior is SH-avisible.When a is the unit, we omit it from the notation.
By unitality, SH-a-visiblity is of course equivalent to P SS K (a) • SH M (K; Λ) = 0, and in particular SH-visibility is equivalent to SH M (K; Λ) = 0.When a is an idempotent, so is P SS K (a).Therefore, P SS K (a) • SH M (K; Λ) ⊂ SH M (K; Λ) is an ideal, which is also on its own an algebra with unit P SS K (a).Our results in this paper will only concern the case that a is an idempotent.
Let us finally note that for an idempotent A ∈ SH M (K; Λ), we have the following finer Mayer-Vietoris sequence (under the same assumptions): [24] for a detailed discussion.
2.4.Symplectic ideal-valued quasi-measures.Let K be a compact set of a closed symplectic manifold M .Dickstein-Ganor-Polterovich-Zapolsky [5] defined the quantum cohomology ideal-valued quasi-measure of K as where U runs over all open sets containing K.

2.5.
Chain level product structure.Let k be a commutative ring.Let us call a Z/2-graded kalgebra, not necessarily associative or commutative, equipped with a differential satisfying the graded Leibniz rule a k-chain complex with product structure.
Using the results of [1], the following theorem is immediate.
Theorem 2.24.For every compact K ⊂ M , we can construct a Λ ≥0 -chain complex C K with product structure whose underlying Λ ≥0 -module is torsion free and T -adically complete such that (1) There are preferred isomorphisms of Z/2-graded Λ ≥0 -algebras (2) Let H : M × S 1 → R be a non-degenerate smooth function which is negative on K. Then there is a Λ ≥0 -chain map Proof.We define as in Theorem 1.4 of [1].The product comes from an arbitrary closed element of [1].Then Theorem 1.8 and 1.9 therein prove the remaining statements.
Remark 2.25.Actually, we do not even need ι K to respect the product structures for our application.All we need is that the composition respects the product structures.Theorem 2.24 implies this composition respects the product structures because it equals to Remark 2.26.It should be possible to construct a product directly on the telescope model over Λ ≥0 (and hence the completed telescope).We know this a posteriori because using homotopy transfer results as in Corollary 1.13 of [1] one can equip it with an A ∞ -structure (and under a characteristic 0 assumption with a BV ∞ -structure).We stress that this also involves choosing extra data: for example, a one sided homotopy inverse to the quasi-isomorphism from the telescope to the big model.The existence of this data is proven using model theoretic arguments in [1] without an explicit construction.
To show the difficulty let us try to multiply an element from CF (H n ; Λ ≥0 ) with an element from CF (H m ; Λ ≥0 ) in the telescope (the degrees unshifted copies).The product could plausibly live in any CF (H k ; Λ ≥0 ) with H k sufficiently larger than H n and H m (see Definition 2.18 of [1] for details).Yet, which one?Or should it have more than one component perhaps?We do not know how to proceed with a direct approach like this.We end with a question: is there a product on the telescope model over Λ ≥0 which satisfies the Leibniz rule and induces the correct homology level product; and moreover, only involves moduli problems with at most two inputs?

3.
1.An algebraic lemma.We first state an algebraic lemma, which will play a key role in the proof of our main theorem.The construction of an algebraic structure on a suitable chain model of the relative symplectic cohomology satisfying this lemma is in [1], and what we need is Theorem 2.24.
A Λ ≥0 -chain complex C with product structure whose underlying Λ ≥0 -module is T -adically complete gives such a structure on C ⊗ Λ ≥0 Λ.Here the valuation on C ⊗ Λ ≥0 Λ is defined by where ι : C → C ⊗ Λ is the natural map.We define xyz := x • (y • z).And x k should be understood similarly for any positive integer k.Multiply y, from the left, to both sides of the equation we get Multiply y, from the left, to this equation we get Then we can repeat this process to get Formally summing over k we get If val(y) > 0, then the term y k α has valuation going to infinity, as k goes to infinity.Hence +∞ k=0 y k α is convergent.This implies that [y] = 0, a contradiction.
In fact, we can prove the more general result below by the same proof.Lemma 3.2.Let (A, d, val) be as above.Let a, b ∈ H(A) satisfy ba = λa with λ ∈ Λ * and assume that b has a representative with valuation larger than val(λ).Then a = 0.
To recover the previous lemma, set a = b and λ = 1.The reader is invited to compare our discussion with [3, Section 1.6.2].

3.2.
Heaviness.Now we use this algebraic lemma to prove one of our main theorems.Let a ∈ QH(M ; Λ) be a non-zero idempotent throughout this section.
Proof.We can easily find a non-degenerate smooth function H ′ : M × S 1 → R that is negative on K with c(a; H ′ ) > 0. We consider the map g : CF (H ′ ; Λ) → C K ⊗ Λ ≥0 Λ as in Theorem 2.24, which gives us the following commutative diagram.

QH(M
Proof.We only prove a-heaviness.The proof of a-superheaviness is similar.We will use Lemma 2.12 again to certify that K is heavy.Let H be a non-negative smooth function on M which vanishes on K. Define ǫ n as the maximum of H on K n .Clearly we have ǫ n → 0 as n → ∞.Since K n is a-heavy for each n, then µ(a; H) ≤ max Kn H = ǫ n .So µ(a; H) = 0 and K is a-heavy.
On the other hand, it is not clear that how the relative symplectic cohomology would behave under approximations.Particularly, whether the two notions SH-visible and nearly SH-visible are equivalent was not known previously.Now we have an affirmative answer by combining Corollary 3.4, Theorem 3.5 and Lemma 3.7.
Corollary 3.8.A compact subset K of M is SH-a-visible if and only if it is nearly SH-a-visible.
Proof.Note that for any compact K we can construct compact domains K 1 ⊃ K 2 ⊃ ... such that K n = K, for example by finding a non-negative smooth function on M whose vanishing set is precisely K and using Sard's theorem.If K is nearly SH-visible, then K n is SH-visible and hence heavy for all n.Lemma 3.7 shows that K is heavy and therefore SH-visible.
Another application is about the notion of SH-heaviness.Hence for an SH-heavy set K, any open set U containing K has SH M ( Ū ; Λ) = 0. We deduce that K is nearly SH-visible.Then Corollaries 3.8 and 3.4 imply that K is heavy.This answers one direction of Conjecture 1.6 in full generality.We do not know the answer for the opposite direction.
3.3.Superheaviness.Our first result is a corollary of the fact that a superheavy set intersects any heavy set.
Corollary 3.10.Let a ∈ QH(M ; Λ) be a non-zero idempotent.For a compact subset K of M , if K is a-superheavy then K is SH-a-full.
Proof.Suppose that K is not SH-a-full.Then there exists some compact subset K ′ of M which is disjoint from K and SH-a-visible.By Corollary 3.4, K ′ is a-heavy, which contradicts to that K is a-superheavy as a-heavy sets intersect a-superheavy sets.
We do not know how to prove the converse in general, but we will discuss a strategy that gives a partial converse.We restrict to a = 1 for simplicity in the rest of the paper.We start with the key but simple lemma.Next we prove an inequality in the other direction.Pick a class a with Π(a, 1) = 0. Since (M, ω) is symplectic Calabi-Yau on spherical classes, the quantum product respects the natural Z-grading on QH(M, Λ).In particular, we need to have where the degree of a ′ is less than 2n, and a 0 ∈ Λ with val(a 0 ) = 0. Recall for two classes x, y with different degrees, the spectral invariants satisfy that Remark 3.12.The equation (3.4) also holds when M is (1) negatively monotone, or (2) positively monotone and its minimal Chern number is greater than n, where 2n is the real dimension of M .
Recall that in order to prove that K is superheavy, we need to estimate c(1; nH) as n → ∞ for all non-positive H that is zero on K. Using the lemma (when we can), we can instead try to estimate c([vol]; nH) as n → ∞ for all non-negative H that is zero on K.This is the content of Proposition 3.13.Before we explain it, we recall that the boundary depth (cf.[26, Section 2]) of a cochain complex A over Λ is defined to be Proof.This proof shares some similarities with the proof of Theorem 3.5.Instead of showing that a chain level representative of P SS K (a) is not exact, we want to find a primitive of it using the conditions on c(a; nH) and the boundary depth.
Let H n be an acceleration data of K such that Let x ∈ CF (H 1 ) be an element representing P SS G 1 Λ (a), so (x, 0, 0, 0, is a closed element that represents P SS K (a) in SH M (K; Λ).The differential of the telescope is given by (3.2): (3.3) for the equivalent form).The y i and y ′ i will be defined inductively.The first equation of (3.3) requires that y ′ 1 is homologous to x 1 in HF (H 1 ; Λ).In particular, we can take y 1 = 0 and y ′ 1 = x.The third equation of (3.3) shows that y ′ 2 is homologous to h 1 y ′ 1 in HF (H 2 ; Λ).By the compatibility of continuation and PSS maps, h 1 y ′ 1 represents P SS H 2 Λ (a), and hence we can choose y ′ 2 to be a chain level representative of P SS H 2 Λ (a) such that val(y ′ 2 ) = c(a; H 2 ).Since y ′ 2 and h 1 y ′ 1 descend to the same class in HF (H 2 ; Λ), we can find y 2 which solves the equation d 2 y 2 + y ′ 2 + h 1 y ′ 1 = 0.Moreover, by the definition of the boundary depth (3.5), for any ǫ > 0, we can find such y 2 so that val(y ′ 2 + h 1 y ′ 1 ) − val(y 2 ) < β(H 2 ) + ǫ.By induction, using the same argument, we get that for all n ≥ 1, • val(y , the fact that c(a; H n ) goes to infinity as n goes to infinity implies that val(y n ) goes to infinity as n goes to infinity.Therefore (y 1 , y ′ 1 , y 2 , y ′ 2 , • • • ) is an element of the completed telescope, which shows that (x, 0, 0, 0, There is a convenient way to deal with the assumption on boundary depth by replacing relative symplectic cohomology with reduced symplectic cohomology [11], which is defined as follows: take {H n } to be an acceleration data of K and form the completed telescope tel(CF (H n ; Λ)), then the reduced symplectic cohomology SH where im refers to the closure of im with respect to (2.7).The Z/2-graded Λ-vector space SH red M (K; Λ) is well-defined and independent of the choice of acceleration data of K, see [11].There is a natural map SH M (K; Λ) → SH red M (K; Λ), (3.6) which is an isomorphism if and only if im(δ) = im(δ).The kernel is precisely the subspace of elements with infinite valuation.
Let us first give a proof of Proposition 1.15 with all these notions at hand.
Proof of Proposition 1.15.SH red M (K; Λ) = 0 implies that the unit in SH M (K; Λ) has infinite valuation.Using the chain level product structure from Theorem 2.24 and our basic algebraic lemma (Lemma 3.1), we obtain that the unit must be zero.This finishes the proof.Remark 3.17.It is instructive at this point to remember that tel(CF (H n ; Λ)) is naturally Z/2Ngraded with N being the minimal Chern number of M .In this case, we can take the Z/2N -graded completion (which might be different than just completing if N = 0) and both SH M (K; Λ) and SH red M (K; Λ) are still Z/2N -graded.Proposition 1.15 (and its proof ) works in this version as well.
The following theorem is the analogue of Theorem 3.14 with relative symplectic cohomology being replaced by the reduced symplectic cohomology.Theorem 3.18.Let (M, ω) be a closed symplectic manifold such that (3.4) is satisfied.For a compact subset K of M , if the image of P SS ([vol]) ∈ SH M (K; Λ) under (3.6) in SH red M (K; Λ) is not zero, then K is superheavy.
Proof.The proof is almost exactly the same as the proof of Proposition 3.13 and Theorem 3.14.
First of all, if c(a; nH) → ∞ as n → ∞, then we can try to construct a primitive of a chain level representative of P SS K (a) as in Proposition 3.13.Using the notation from Proposition 3.13, it is clear that δ(y 1 , y ′ 1 , y 2 , y ′ 2 , • • • y k , y ′ k , 0, 0, . . . ) is converging to x as k goes to infinity.As a result, x ∈ im(δ) The following lemma gives a sufficient condition for SH M (K; Λ) ≃ SH red M (K; Λ).
The following lemma explains that the boundary depth of tel(CF (H n ; Λ))) does not depend on the choice of acceleration data of K.Because of this, we also call the boundary depth of tel(CF (H n ; Λ))) the boundary depth of K.
Lemma 3.20.The boundary depth β of tel(CF (H n ; Λ))) equals to the maximal torsion exponent τ of SH M (K; Λ ≥0 ), which is independent of the choices of acceleration data.
In the Z-graded context, it is useful to talk about having finite boundary depth in a fixed degree.Note it is possible to have finite boundary depth in all degrees, without having finite boundary depth as defined above.
Corollary 3.21.Let (M, ω) be a closed symplectic manifold with c 1 (T M )| π 2 (M ) = 0. Let K be a compact subset of M .Assume that • K is SH-full and • there is a sequence of compact sets K 1 ⊃ K 2 ⊃ ... with finite boundary depth all containing K in their interior such that K i = K.Then K is superheavy.
Proof.Suppose that K is SH-full.Then by the Mayer-Vietoris property, for any compact set K ′ containing K in its interior, we know that QH(M, Λ) → SH M (K ′ , Λ) is an isomorphism.
As an immediate consequence of Lemma 3.11, Theorem 3.18 and Lemma 3.19., we know that K n is superheavy for all n ≥ 1.We conclude that K is also superheavy using Lemma 3.7.
Recall from Remark 3.17 that we can take Z-graded completion when c 1 (T M ) = 0 and the resulting SH M (K; Λ) is Z-graded.We can then replace the finite boundary depth assumption by the weaker assumption which says that the boundary depth is finite at all degrees.This makes Corollary 3.21 much more useful, see Theorem 1.13.The proof of Theorem 1.13 is complete now with these remarks.

Lemma 3 . 1 .
Let (A, d, val) be as above.Let a be a non-zero idempotent in H(A).Then any degree 0 chain level representative of a has non-positive valuation.Proof.Pick a closed homogeneous element y with [y] = a and deg(y) = 0. Then we have that [y • y] = [y] • [y] = [y].So there exist α ∈ A such that y − y • y = dα.

Proposition 3 . 9 .
If a compact subset K of M is SH-heavy, then it is heavy.Proof.For a compact set K and an open set U containing K, we can find another open set V such that M − U ⊂ V , V ∩ K = ∅ and ∂ V ∩ ∂ Ū = ∅.If SH M ( Ū ; Λ) = 0, then the Mayer-Vietoris sequence for the pair ( Ū , V ) tells us that ker(r : SH M (M ; Λ) → SH M ( V ; Λ)) = 0.
and • there is B > 0 such that the boundary depth β(H n ) := β(CF (H n ; Λ)) ≤ B for all n.
Λ) is defined to be the kernel ker(δ : tel * (CF (H n ; Λ)) → tel * +1 (CF (H n ; Λ))) modulo the closure of the image im(δ : tel * −1 (CF (H n ; Λ)) → tel * (CF (H n ; Λ))) On the other hand, if c(a; H) ≤ max K H for all H, then we have c(a; nH) ≤ max K nH = n max K H for all H.This implies that µ(a; H) ≤ max K H for all H.
and hence our definition above.The translation for superheaviness is similar.Lemma 2.10 (µ-heaviness and c-heaviness).A compact subset K ⊂ (M, ω) is a-heavy if and only if c(a; H) ≤ max K H for all H ∈ C ∞ (M ).Proof.By the inequality (2.4), we have µ(a; H) ≥ c(a; H).Therefore, if µ(a; H) ≤ max K H, then c(a; H) ≤ max K H.
[16,24,5,22]xact.Theorem 3.14.Let K be a compact subset and H ∈ C ∞ (M ) is a non-negative function that is zero on K. Suppose that nH satisfies (3.4) for all n ∈ N and there is a uniform upper bound on the boundary depth of CF (nH; Λ).If P SS K ([vol]) = 0, then K is superheavy.Proof.By P SS K ([vol]) = 0 and Proposition 3.13, we know that c([vol]; nH) is uniformly bounded above.Under condition (3.4) for all nH, we therefore obtain that c(1; −nH) is uniformly bounded below and hence µ(1; −H) = 0.It therefore implies the superheaviness of K.Remark 3.15.One way to achieve a uniform bound on boundary depth is to impose an index bounded condition when K is a Liouville domain.See[16,24,5,22]for more discussions.Remark 3.16.Without the condition (3.4), we would need a uniform bound on all c(a; nH) with Π(a, 1) = 0.This does not follow from P SS K (a) = 0 for all such a and appears more difficult to obtain.On the other hand what we ended up proving is strictly stronger than µ(1; −H) = 0.