CCOMPUTING HIGHER SYMPLECTIC CAPACITIES I
KYLER SIEGEL
Abstract.
We present recursive formulas which compute the recently defined “highersymplectic capacities” for all convex toric domains. In the special case of four-dimensional ellipsoids, we apply homological perturbation theory to the associatedfiltered L ∞ algebras and prove that the resulting structure coefficients count puncturedpseudoholomorphic curves in cobordisms between ellipsoids. As sample applications,we produce new previously inaccessible obstructions for stabilized embeddings ofellipsoids and polydisks, and we give new counts of curves with tangency constraints. Contents
1. Introduction 21.1. Context 21.1.1. Symplectic embeddings 21.1.2. Enumerating punctured curves 41.1.3. Obstructions from curves 71.2. Main results 81.2.1. From geometry to algebra 81.2.2. Applications to symplectic embeddings 91.2.3. Applications to enumerative geometry 10Acknowledgements 112. A family of filtered L ∞ algebras 112.1. L ∞ recollections 112.2. The differential graded Lie algebra V V Ω V Ω and its linear spectral invariants 153. Computing the canonical model of V a,b Φ a,b and Ψ a,b L ∞ homomorphisms 305.2. Well-defined curve counts in cobordisms 325.3. Rounding and partially compactifying 36 Date : February 5, 2020. a r X i v : . [ m a t h . S G ] F e b KYLER SIEGEL
1. Introduction
This paper is about two closely related problems in symplectic geometry:(a) understanding when there is a symplectic embedding of one domain into anotherof the same dimension(b) counting punctured pseudoholomorphic curves in a given domain.Each of these questions has both a qualitative and quantitative version, the former beingmore topological and the latter being more geometric in flavor. The qualitative versionof (a) asks for symplectic embeddings up to a suitable class of symplectic deformations,and the qualitative version of (b) seeks enumerative invariants which are independentof such deformations. The quantitative version of (a) asks for symplectic embeddingson the nose, and the quantitative version of (b) seeks invariants which are potentiallysensitive to the symplectic shape of a given domain.In this paper we will focus on the quantitative theory, and by “domain” we have in mindstar-shaped subdomains of C n for some n ≥ . One can also extend the discussion to awider class of open symplectic manifolds such as Liouville domains or nonexact symplecticmanifolds with sufficiently nice boundary. A class of particular importance in dynamicsis given by the ellipsoids E ( a , . . . , a n ) ⊂ C n with area parameters a , . . . , a n ∈ R > ,defined by E ( a , . . . , a n ) := { ( z , . . . , z n ) ∈ C n : n (cid:88) i =1 π | z i | /a i ≤ } . We will typically assume that the area factors are ordered as a ≤ · · · ≤ a n and arerationally independent, in which case ∂E ( a , . . . , a n ) has precisely n simple Reeb orbits,with actions a , . . . , a n respectively. The motivation for (a) goes back to Gromov’s cele-brated nonsqueezing theorem [Gro], which states that a large ball cannot be squeezedby a symplectomorphism into a narrow infinite cylinder. Gromov proved this resultusing his newly minted theory of pseudoholomorphic curves, thereby giving the firstnon-classical obstructions for symplectic embeddings. This kickstarted the search for anunderstanding of the “fine structure” of symplectic embeddings. The following problemis still largely open for n ≥ and provides a useful metric for progress in this field: Problem 1.1.1 (ellipsoid embedding problem (EEP)) . For which a , . . . a n and a (cid:48) . . . a (cid:48) n is there a symplectic embedding E ( a , . . . , a n ) s (cid:44) → E ( a (cid:48) , . . . , a (cid:48) n ) ? Recall that a Liouville domain is a compact symplectic manifold ( X, ω ) , where ω = dθ for a one-form θ , such that the Liouville vector field X θ characterized by ω ( X θ , − ) = θ is outwardly transverse alongthe boundary of X . OMPUTING HIGHER SYMPLECTIC CAPACITIES I 3
Note that the only classical obstruction is the volume constraint n ! a . . . a n ≤ n ! a (cid:48) . . . a (cid:48) n , while Gromov’s nonsqueezing theorem amounts to the inequality min( a , . . . , a n ) ≤ min( a (cid:48) , . . . , a (cid:48) n ) . There have been a number of important contributions to Problem 1.1.1 and its cousins,and we mention here only a partial list of symplectic rigidity highlights: • the construction by Ekeland–Hofer of an infinite sequence of symplectic capacities c EH1 ( X ) ≤ c EH2 ( X ) ≤ c EH3 ( X ) ≤ . . . associated to a domain X of any dimension, often giving stronger obstructionsthan Gromov’s nonsqueezing theorem • the solution (or at least reduction to combinatorics) by McDuff [McD1] of thefour-dimensional ellipsoid embedding problem E ( a, b ) s (cid:44) → E ( a (cid:48) , b (cid:48) ) • the construction by Hutchings [Hut1] of the embedded contact homology (ECH)capacities c ECH1 ( X ) ≤ c ECH2 ( X ) ≤ c ECH3 ( X ) ≤ . . . associated to a four-dimensional domain X , with many strong applications tofour-dimensional symplectic embedding problems (see e.g. [CCGF + , CG2] andthe references therein).There have also been some important developments on the side of symplectic flexi-bility, including the advent of symplectic folding (see [Sch1]), Guth’s result on polydiskembeddings [Gut], and its subsequent refinement by Hind [Hin1]. For example, the lattertogether with [PVuN] produces a symplectic embedding E (1 , x, ∞ ) s (cid:44) → E ( xx +1 , xx +1 , ∞ ) (1.1.1)for any x ∈ R ≥ . In particular, by [Hin1] there is an embedding E (1 , ∞ , ∞ ) s (cid:44) → E ( c, c, ∞ ) if c ≥ , and in fact this is optimal by [HK].Although the full solution to the higher dimensional ellipsoid embedding problemis still seemingly out of reach, the following special case of Problem 1.1.1 has recentlygained popularity and probes to what extent gauge-theoretic obstructions in dimensionfour persist in higher dimensions. Problem 1.1.2 (stabilized ellipsoid embedding problem) . Fix N ∈ Z ≥ . For which a, b and a (cid:48) , b (cid:48) is there a symplectic embedding E ( a, b ) × C N s (cid:44) → E ( a (cid:48) , b (cid:48) ) × C N ? Note that E ( a, b ) = a · E (1 , x ) for x = b/a . Restricting to the case that the target is astabilized round four-ball, we arrive at: Problem 1.1.3 (restricted stabilized ellipsoid embedding problem) . For N ∈ Z ≥ ,determine the function f N ( x ) := inf { c ∈ R > : E (1 , x ) × C N s (cid:44) → B ( c ) × C N } . Note that we have E ( c, c, ∞ ) = B ( c ) × C . KYLER SIEGEL
The analogous unstabilized function f ( x ) was determined by McDuff–Schlenk [MSch]and dubbed the “Fibonacci staircase”. Based on (1.1.1), McDuff [McD2] has given thefollowing explicit conjecture for the form of f N ( x ) for N ≥ : Conjecture 1.1.4 (restricted stabilized ellipsoid embedding conjecture) . For x ∈ R ≥ and N ≥ , we have f N ( x ) = (cid:40) f ( x ) if x ≤ τ xx +1 if x > τ . Here τ = √ denotes the golden ratio. Combining several results, the following progresshas been made: Theorem 1.1.5 ([HK, CGH, CGHM, McD2]) . Conjecture 1.1.4 holds in the followingcases: • for all x ≤ τ • for x ∈ Z ≥ − • for x in a decreasing sequence of numbers b = 8 , b = , b = , . . . determinedby the even index Fibonacci numbers, with b i → τ . These techniques require a rather strong geometric control on SFT-type degenerationsafter neck-stretching, so they appear difficult to scale as the number of possibilities grows.It has therefore been difficult to say whether Conjecture 1.1.4 holds for other seeminglysimple cases such as x = 7 or x = . One application of the results of this paper is aproof of many new cases of Conjecture 1.1.4 - see Example 1.2.13. We now move to motivation (b). Let X n be a Liouville domain. What does it mean to “count” punctured pseudoholomorphiccurves in X ? Since there is a naturally induced contact structure on ∂X , we can speak ofReeb orbits in ∂X . After passing to the symplectic completion (cid:98) X of X and picking anSFT-admissible almost complex structure J , we can consider pseudoholomorphic curveswhich are asymptotic to Reeb orbits at each of the punctures. If we fix the genus g ,homology class A , and nondegenerate Reeb orbit asymptotes γ , . . . , γ k , this gives rise toa moduli space M of curves in X , of expected dimension ind M = ( n − − g − k ) + k (cid:88) i =1 CZ( γ i ) + 2 c ([ ω ]) · A. Note that the asymptotic condition at a puncture is formulated in such a way that thesecurves are proper. We refer the reader to [Sie1, §3] for more details on the geometricsetup.In principle we can try to count the elements of M , but several basic issues come tomind:(1) Typically, M will have nonzero expected dimension, so it will not contain finitelymany elements. For example, any convex domain X ⊂ C is dynamically convex(see [HWZ]), meaning that M will have strictly positive dimension. Note that all almost complex structures and punctured pseudoholomorphic curves in this paperare in completed symplectic cobordisms, so we will sometimes omit explicit mention of this completionprocess without risk of ambiguity.
OMPUTING HIGHER SYMPLECTIC CAPACITIES I 5 (2) Even if we do get a finite count, it could depend on the choice of J , along with anyother choices we make during the construction (c.f. “wall-crossing” phenomena).Following [Sie1], we can take care of (1) by imposing additional constraints to cutdown the dimension of the moduli space M . The most basic such constraint is to requireour curves to pass through k generic points p , . . . , p k ∈ X , with each p i cutting down thedimension by n − . A different constraint of the same codimension is given by picking ageneric local divisor D at a single point p ∈ X and requiring curves to pass through p withcontact order k to D . This idea goes back to the work of Cieliebak–Mohnke [CM1, CM2],who considered degree one curves in CP n satisfying such a local tangency constraint andused a neck-stretching argument to prove the Audin conjecture. As explained in [Sie1],unlike curves satisfying generic point constraints, curves with a local tangency constraintenjoy nice dimensional stability properties which makes them particularly relevant toProblem 1.1.2.As for (2), recall that in closed symplectic manifolds we can use moduli spaces of closedcurves to define Gromov–Witten invariants, and in favorable cases these do enumeratehonest curves. The key point in Gromov–Witten theory is that the relevant modulispaces admit codimension two compactifications, which means we can hope to avoidboundary phenomena in one-parameter families. In contrast, the SFT compactnesstheorem provides a codimension one compactification of M , and we typically encounternontrivial pseudoholomorphic buildings in one-parameter families. In such a situationwe should generally replace counting invariants with homological invariants, and thisis the purview of symplectic field theory [EGH]. At present we are interested in genuszero curves, so we are in the setting of rational symplectic field theory (see [Sie1, §3,§5]).There is also an alternative approach to these invariants using Floer theory (see e.g. [Sie1,Rmk. 3.9] and the references therein), although the full details have no yet appeared inthe literature.In the special case of an ellipsoid X = E ( a , . . . , a n ) , the differentials and higheroperations involved in the above homological invariants all vanish identically for degreeparity reasons. Moreover, it is proved in [MSie] that a local tangency constraint can bereplaced by removing a small neighborhood symplectomorphic to an infinitely skinnyellipsoid E sk , and then considering punctured curves in the resulting cobordism with anadditional negative puncture. We are thus in the framework of the following problem. Problem 1.1.6 (ellipsoidal cobordism curve counting problem) . What is the count M JE ( a (cid:48) ,...,a (cid:48) n ) \ εE ( a ,...,a n ) (Γ + ; Γ − ) ? Let us explain this notation. Given a , . . . , a n and a (cid:48) , . . . , a (cid:48) n , let ε > be smallenough that we have an inclusion εE ( a , . . . , a n ) ⊂ E ( a (cid:48) , . . . , a (cid:48) n ) . Let E ( a (cid:48) , . . . , a (cid:48) n ) \ εE ( a , . . . , a n ) denote the corresponding complementary cobordism, and let J be ageneric SFT-admissible almost complex structure on E ( a (cid:48) , . . . , a (cid:48) n ) \ εE ( a , . . . , a n ) . Forcollections of Reeb orbits Γ + = ( γ +1 , . . . , γ + k ) in ∂E ( a (cid:48) , . . . , a (cid:48) n ) and Γ − = ( γ − , . . . , γ − l ) in ∂E ( a , . . . , a n ) , let M JE ( a (cid:48) ,...,a (cid:48) n ) \ εE ( a ,...,a n ) (Γ + ; Γ − ) denote the moduli of genus zeropunctured curves in E ( a (cid:48) , . . . , a (cid:48) n ) \ εE ( a , . . . , a n ) with positive punctures asymptoticto Γ + and negative punctures asymptotic to Γ − . Assuming ind M = 0 , we then denoteby M JE ( a (cid:48) ,...,a (cid:48) n ) \ εE ( a ,...,a n ) (Γ + ; Γ − ) the signed count of curves in this moduli space. KYLER SIEGEL
Even after restricting to ellipsoids, Problem 1.1.6 is still not entirely well-defined.For one thing, the presence of index zero branched covers of trivial cylinders in thesymplectizations of ∂E ( a , . . . a n ) and ∂E ( a (cid:48) , . . . , a (cid:48) n ) introduces certain ambiguities andprevents curve counts from being independent of J and ε . More severely, for arbitrary Γ + and Γ − , curves of negative index can appear due to lack of transversality, whichprecludes any hope of naive curve counting.Nevertheless, Problem 1.1.6 is not vacuous, as there are many cases in which we doget well-defined enumerative invariants. Here is a first example: Example 1.1.7.
Consider the slightly perturbed four-ball E (1 , δ ) for δ > sufficientlysmall, and let γ short and γ long denote the short and long simple Reeb orbits respectively of ∂E (1 , δ ) . Put E sk := E (1 , x ) for x (cid:29) sufficiently large, and let η k denote the k -foldcover of the short simple Reeb orbit of E sk . For d ∈ Z ≥ , we take Γ + = ( γ long , . . . , γ long ) (cid:124) (cid:123)(cid:122) (cid:125) d and Γ − = ( η d − ) . Then M JE (1 , δ ) \ εE sk (Γ + ; Γ − ) is finite and independent of δ, ε , andgeneric J . In fact, in the above example, d !(3 d − M JE (1 , δ ) \ εE sk precisely coincides with thecount T d of degree d rational curves in CP satisfying a local tangency constraint ofcontact order d − . These counts were recently computed for all d in [MSie], giving T = 1 , T = 1 , T = 4 , T = 26 , T = 127 , and so on. The computation in [MSie]is based on a recursive formula which reduces these counts to blowup Gromov–Witteninvariants of CP , which can in turn be computed e.g. by [GP].Example 1.1.7 is a special case of the following, which appears as [CGHM, Prop. 3.3.6](see also §5.2 for further discussion and extensions): Proposition 1.1.8.
For x ∈ R > , assume that we have d − k + (cid:98) k/x (cid:99) andthat we cannot find decompositions d = (cid:80) mi =1 d i and k = (cid:80) mi =1 k i for m ∈ Z ≥ and d , . . . , d m ∈ Z ≥ such that d i − k i + (cid:98) k i /x (cid:99) for i = 1 , . . . , m . Put Γ + = ( γ long , . . . , γ long ) (cid:124) (cid:123)(cid:122) (cid:125) d and Γ − = ( γ short; k ) , where γ short; k denotesthe k -fold cover of γ short . Then M JE (1 , \ εE (1 ,x ) (Γ + ; Γ − ) is finite and independent of ε and generic J .Remark . The count in Proposition 1.1.8 is equivalent to the count of degree d rationalcurves in CP \ εE (1 , x ) with one negative puncture asymptotic to γ short; k . Note that thecondition d − k + (cid:98) k/x (cid:99) is equivalent to having ind M JE (1 , \ εE (1 ,x ) (Γ + ; Γ − ) = 0 . Definition 1.1.10.
In the context of Proposition 1.1.8, we put T d ;1 ,x := d ! k M JE (1 , \ εE (1 ,x ) (Γ + ; Γ − ) . As far as we are aware, these counts have not previously been computed except for somespecial cases. Note that we have T d = T d ;1 ,x for x (cid:29) d . We could also take a = a = 1 , after taking into account the necessary Morse–Bott modifications. The extra combinatorial factor d − d ! has to do with our conventions for handling asymptoticmarkers and orderings of punctures - see §5.2. OMPUTING HIGHER SYMPLECTIC CAPACITIES I 7
We now elaborate on the connection between (a)and (b). Given n -dimensional Liouville domains X and X (cid:48) , the basic strategy forobstructing symplectic embeddings X s (cid:44) → X (cid:48) is as follows:(1) use a “homological framework” to argue that if such an embedding existed, therewould have to be a punctured curve u in X (cid:48) \ X with some predetermined positiveReeb orbit asymptotics Γ + = ( γ +1 , . . . γ + k ) and negative Reeb orbit asymptotics Γ − = ( γ − , . . . , γ − l ) (2) apply Stokes’ theorem together with nonnegativity of energy to get an inequalityof the form ≤ E ( u ) = (cid:80) ki =1 A ( γ + i ) − (cid:80) lj =1 A ( γ − j ) We refer the reader to [Sie1] for the precise definition of energy and so on. Roughlyspeaking, “homological framework” will be formalized using the following perspective: • cylinders in X (cid:48) \ X are encoded using action-filtered linearized contact homol-ogy CH lin ( X ) (or alternatively action-filtered positive S -equivariant symplecticcochains SC S , + ( X ) ), and these give the same obstructions as the Ekeland–Hofercapacities • spheres in X (cid:48) \ X with several positive ends and one negative end are encoded usingthe action-filtered L ∞ structure on CH lin ( X ) (or alternatively on SC S , + ( X ) ),and these give the obstructions from [Sie1]. A folklore question asks whether all nonclassical symplectic embedding obstructions aregiven by some pseudoholomorphic curve as in the above strategy. In principle such acurve could have higher genus and/or more than one negative end , necessitating a morerefined homological framework such as higher genus SFT. However, we do not know ofany framework for defining dimensionally stable obstructions which involves such curves(see the discussion in [Sie1, §5.4]).In order to implement the above strategy, we need to compute the filtered L ∞ algebras CH lin ( X ) and CH lin ( X (cid:48) ) . Following [Sie1], we can then try to read off obstructions usingtheir bar complex spectral invariants (this is reviewed in §4). However, we also needa canonical way of referencing homology classes in CH lin ( X ) and CH lin ( X (cid:48) ) . In [Sie1],local tangency constraints accomplish this task, and the capacities g b ( X ) and g b ( X (cid:48) ) give the corresponding “coordinate-free” bar complex spectral invariants of X and X (cid:48) respectively (this is reviewed in §4.1 below). Alternatively, if we can compute the filtered L ∞ homomorphism CH lin ( X (cid:48) ) → CH lin ( X ) induced by the complementary cobordism X (cid:48) \ X , we can read off obstructions directly via Stokes’ theorem, since any homologicallynontrivial structure coefficient must be represented by some curve or building.In the special case of ellipsoids X = E ( a , . . . , a n ) and X (cid:48) = E ( a (cid:48) , . . . , a (cid:48) n ) , the filtered L ∞ algebras CH lin ( X ) and CH lin ( X (cid:48) ) are trivial to compute, since the differentials andall higher L ∞ operations vanish for degree parity reasons. However, computing thecobordism map CH lin ( X (cid:48) ) → CH lin ( X ) essentially amounts to counting all punctured Strictly speaking it is an open conjecture that these obstructions coincide with those defined byEkeland–Hofer. See [GH, Conj. 1.9] for evidence of this conjecture and more details. Note that we willnot make any use of the original definition of the Ekeland–Hofer capacities in this paper. See Remark 1.2.14 for a discussion of our transversality assumptions. Here we are excluding anchors (see e.g. [Sie1, §3]), which behave essentially differently from negativeends.
KYLER SIEGEL spheres in X (cid:48) \ X with several positive ends and one negative end, and this is an intricateenumerative problem, a special case of Problem 1.1.6. For instance, as pointed out byMcDuff, Conjecture 1.1.4 would follow from the existence of some very specific curves: Theorem 1.1.11 ([McD2]) . Let x = p/q for p, q, d ∈ Z ≥ with p + q = 3 d and gcd( p, q ) =1 . If T d ;1 ,x (cid:54) = 0 , then we have f N ( x ) ≥ xx +1 for all N ∈ Z ≥ .Remark . One can also check that numbers of the form p/q as in Lemma 1.1.11are dense in R ≥ , and hence are sufficient to prove Conjecture 1.1.4 for all x . We first recall the class of convex toric domains.Let µ : C n → R n ≥ denote the moment map for the standard T n action on C n , givenexplicitly by µ ( z , . . . , z n ) = ( π | z | , . . . , π | z n | ) . Note that the fiber µ − ( p ) over a point p ∈ R n> is a smooth n -dimensional torus, whilethe fiber over a point p ∈ R ≥ \ R > is a torus of strictly lower dimension. Following[CCGF + , Hut3, GH], we make the following definition: Definition 1.2.1. A convex toric domain is a subdomain of C n of the form X Ω := µ − (Ω) , where Ω ⊂ R ≥ is a subset such that (cid:98) Ω := { ( x , . . . , x n ) ∈ R n : ( | x | , . . . , | x n | ) ∈ Ω } ⊂ R n is compact and convex.This class includes the following examples: • the ellipsoid E ( a , . . . , a n ) is of the form X Ω E ( a ,...,an ) with moment map image Ω E ( a ,...,a n ) := Conv ((0 , . . . , , ( a , , . . . , , (0 , . . . , , a n )) ⊂ R n ≥ • the polydisk P ( a , . . . , a n ) := B ( a ) × · · · × B ( a n ) is of the form X Ω P ( a ,...,an ) with moment map image Ω P ( a ,...,a n ) := [0 , a ] × · · · × [0 , a n ] ⊂ R n ≥ . Compared to arbitrary Liouville domains, the extra torus symmetry present for convextoric domains makes their Reeb dynamics and pseudoholomorphic curve moduli spacesmore amenable to analysis. We can also view convex toric domains as partial compactifi-cations of smooth Lagrangian torus fibrations, making them natural objects of study inmirror symmetry (see §5.5 below).In §2, we define an explicit filtered L ∞ algebra V n Ω associated to any subset Ω ⊂ R n ≥ for which X n Ω is a convex toric domain. More precisely, V n Ω is a certain differential gradedLie algebra V n which is independent of Ω , equipped with an Ω -dependent filtration. If X n Ω (cid:48) is another convex toric domain associated to a subset Ω (cid:48) ⊂ R n ≥ , by slight abuseof notation we take the “identity map” : V Ω (cid:48) → V Ω to be the (possibly unfiltered) L ∞ homomorphism sending each generator of V Ω (cid:48) to the corresponding generator in V Ω .The following theorem provides a complete algebraic model for the filtered L ∞ algebra CH lin ( X Ω ) and for the filtered L ∞ homomorphism Ξ : CH lin ( X Ω (cid:48) ) → CH lin ( X Ω ) inducedby a symplectic embedding X Ω s (cid:44) → X Ω (cid:48) : OMPUTING HIGHER SYMPLECTIC CAPACITIES I 9
Theorem 1.2.2 ([Sie2]) . Let X Ω and X Ω (cid:48) be n -dimensional convex toric domains,and suppose there is a symplectic embedding X Ω × C N s (cid:44) → X Ω (cid:48) × C N for some N ≥ .Then there exist inverse filtered L ∞ homotopy equivalences F Ω : V Ω → CH lin ( X Ω ) and G Ω (cid:48) : CH lin ( X Ω (cid:48) ) → V Ω (cid:48) such that Ξ is unfiltered L ∞ homotopic to F Ω ◦ ◦ G Ω (cid:48) . It will also be convenient to formulate the following “model-independent” version:
Corollary 1.2.3.
Let X Ω and X Ω (cid:48) be n -dimensional convex toric domains, and supposethere is a symplectic embedding X Ω × C N s (cid:44) → X Ω (cid:48) × C N for some N ≥ . Then thereexists a filtered L ∞ homomorphism Q : V Ω (cid:48) → V Ω which is unfiltered L ∞ homotopic tothe identity. Our symplectic embedding obstructions will follow from Corollary 1.2.3 by applying akind of “filtered L ∞ calculus”. In §3, we construct a canonical model for the unfiltered L ∞ algebra V Ω . Namely, we put V canΩ := H ( V Ω ) , and we recursively construct inverse L ∞ homotopy equivalences Φ Ω : V Ω → V canΩ and Ψ Ω : V canΩ → V Ω . In §4, we computethe homology of the bar complex of V Ω and explain how to extract the capacities g b from [Sie1]. In the special case of a four-dimensional ellipsoid, we put V a,b := V Ω E ( a,b ) , Φ a,b := Φ Ω E ( a,b ) , and Ψ a,b := Ψ Ω E ( a,b ) , and we show that V a,b is a canonical model forthe filtered L ∞ algebra V a,b . In §5, we prove that the combinatorially defined maps Φ a,b and Ψ a,b can be used to compute enumerative invariants. It is conjectured in [Sie1] that thecapacities g b give a complete set of obstructions for Problem 1.1.2. For the restricted ver-sion, namely Problem 1.1.3, the following corollary gives a purely combinatorial criterion.We note that there is a natural isomorphism of K -modules V canΩ ∼ = K (cid:104) A , A , A , . . . (cid:105) ,where there at most one generator A i in each degree (see §2.4). The following theorem isproved in §4.3: Theorem 1.2.4.
Fix a, b, a (cid:48) , b (cid:48) ∈ R > , and suppose that we have (cid:104) (Φ a,b ◦ Ψ a (cid:48) ,b (cid:48) ) k ( A i , . . . , A i k ) , A i + ··· + i k + k − (cid:105) (cid:54) = 0 for some k, i , . . . , i k ∈ Z ≥ . Then if there exists a symplectic embedding E ( a, b ) × C N s (cid:44) → E ( a (cid:48) , b (cid:48) ) × C N for some N ∈ Z ≥ , we must have: k (cid:88) j =1 min s + t = i j s,t ∈ Z ≥ max { a (cid:48) s, b (cid:48) t } ≥ min s + t = i + ··· + i k + k − s,t ∈ Z ≥ max { as, bt } . (1.2.1)A geometric interpretation of the above expression is initiated in §2.4, where it isobserved that min i + j = k max { ia, jb } is equal to the k th smallest action of a Reeb orbit in ∂E ( a, b ) . For example, in the case E ( a (cid:48) , b (cid:48) ) = E ( c, c + δ ) for δ > sufficiently smalland i = · · · = i k = 2 , the left hand side of (1.2.1) becomes kc (1 + δ ) . Meanwhile, inthe case E ( a, b ) = E (1 , p/q + δ (cid:48) ) for δ (cid:48) > sufficiently small, with p, q ∈ Z ≥ satisfying p + q = 3 k , the right hand side of (1.2.1) becomes p , with the minimum occurring for ( s, t ) = ( p, q − . With these choices, the inequality (1.2.1) amounts to k ( c + δ ) ≥ p , orequivalently c + δ ≥ xx +1 for x = p/q . This leads to: Corollary 1.2.5.
Suppose we have x = p/q with p, q, d ∈ Z ≥ such that p + q = 3 d and x ≥ τ . Then Conjecture 1.1.4 holds at the value x provided that we have (cid:104) (Φ ,x ◦ Ψ , )( (cid:12) d A ) , A d − (cid:105) (cid:54) = 0 . We do not prove in this paper that the criterion in Corollary 1.2.5 holds in general.However, for any given x the relevant structure coefficients can easily be computed withthe aid of a computer. For example, by the sample computations in §5 we have:
Corollary 1.2.6.
Conjecture 1.1.4 holds for each of the x values appearing in Table 5.1. For Liouville domains which are not necessarily ellipsoids, it turns out that we cansometimes extract stronger obstructions from Corollary 1.2.3 than those visible to thecapacities g b . As observed by Hutchings [Hut3], a phenomenon which is similar in spiritoccurs for ECH capacities. In §4.4, we illustrate this phenomenon with the two followingexamples, which are (are far as we are aware) new for a < : Theorem 1.2.7.
Given a symplectic embedding P (1 , a ) × C N s (cid:44) → P ( c, c ) × C N for a ≥ and N ≥ , we must have c ≥ min( a, . Moreover, this is sharp for N ≥ . Theorem 1.2.8.
Given a symplectic embedding P (1 , a ) × C N s (cid:44) → B ( c ) × C N for a ≥ and N ≥ , we must have c ≥ min( a + 1 , . Moreover, this is sharp for N ≥ . Let E ( a (cid:48) , b (cid:48) ) \ E ( a, b ) be a cobor-dism between two four-dimensional ellipsoids. In principle, we can now read off curvecounts in this cobordism in using the structure coefficients of the induced L ∞ map Ξ : CH lin ( E ( a (cid:48) , b (cid:48) )) → CH lin ( E ( a, b )) . A priori, the filtered L ∞ homomorphisms F Ω and G Ω (cid:48) given by Theorem 1.2.2 are inexplicit, arising from certain auxiliary SFT cobor-dism maps. In §5 we characterize these maps using techniques from embedded contacthomology, proving the following result: Theorem 1.2.9.
For X Ω = E ( a, b ) and X Ω (cid:48) = E ( a (cid:48) , b (cid:48) ) , we can take F Ω = Φ a,b and G Ω (cid:48) = Ψ a (cid:48) ,b (cid:48) in Theorem 1.2.2. Definition 1.2.10.
In the context of Proposition 1.1.8, we put S d ;1 ,x := d ! k (cid:104) Φ ,x ◦ Ψ , ( (cid:12) d A ) , A d − (cid:105) . Corollary 1.2.11.
We have T d ;1 ,x = S d ;1 ,x . Example 1.2.12.
Using Corollary 1.2.11 and the recursive construction of Ψ a,b givenin §3.2, we get a recursive formula for the numbers S d which is completely different from(and much simpler than) the recursive algorithm for T d given in [MSie] . With the aidof a computer, we have independently verified Corollary 1.2.11 for d = 1 , . . . , . Putting S d := S d ;1 ,x for x sufficiently large, we have: S = 1 , S = 1 , S = 4 , S = 26 S = 217 , S = 2110 , S = 22744 , S = 264057 ,S = 3242395 , S = 41596252 , S = 552733376 , S = 7559811021 ,S = 105919629403 , S = 1514674166755 ., S = 22043665219240 , S = 325734154669786 , and so on. A Python implementation is available on the author’s website.
OMPUTING HIGHER SYMPLECTIC CAPACITIES I 11
Example 1.2.13.
The paper [CGHM] proves the restricted stabilized ellipsoid conjecturefor x = 55 / by showing (in our notation) that T , / ≥ . By a computer calculationusing Corollary 1.2.11, we get precisely S , / = 3 . See Table 5.1 for many morecomputations of this nature.Remark . In general, as in [Sie1], we work in a suitablevirtual perturbation framework in order to define the above symplectic field theoreticinvariants, without invoking any specific properties of the particular scheme used (seee.g. [Sie1, Rmk. 3.1]). In fact, for the enumerative invariants discussed in §5, therelevant moduli spaces are regular for any generic choice of admissible almost complexstructure, and hence are counted in the classical sense. Moreover, thanks to the favorableConley–Zehnder index behavior of Reeb orbits in a fully rounded convex toric domain(see §5.3), the moduli spaces involved in the proof of Theorem 1.2.2 are “nearly regular”,in the sense that we can achieve transversality within a classical perturbation framework(see [Sie2] for details). By contrast, in the absence of virtual perturbations, the naivemoduli spaces involved in e.g. the cobordism map
Ξ : CH lin ( E ( a (cid:48) , b (cid:48) )) → CH lin ( E ( a, b )) are often far from regular. Acknowledgements
I am highly grateful to Dan Cristofaro-Gardiner and Dusa McDuff for their input andinterest in this project. I also thank Mohammed Abouzaid for helpful discussions.
2. A family of filtered L ∞ algebras In this section, after recalling some background on L ∞ algebras and setting up notationin §2.1, we define the DGLA V in §2.2, and endow it with its family of filtrations in §2.3.Lastly, as a prelude to §3, in §2.4 we compute the linear spectral invariants of V Ω andshow that they recover the Ekeland–Hofer capacities of X Ω . L ∞ recollections. Here we briefly recall some basic notions about L ∞ algebrasin order to set our conventions for signs, gradings, and filtrations. We refer the reader to[Sie1, §2] and the references therein for more details.Let K be a fixed field containing Q , which we will usually take to be Q itself. Let V be a Z -graded K -module. For k ∈ Z ≥ , let ⊗ k V denote the k -fold tensor product (over K ) of V , and let (cid:12) k V = ⊗ k V / Σ k denote the k -fold symmetric tensor product of V , i.e.we quotient by the signed action of the permutation group. For an elementary tensor v ⊗ · · · ⊗ v k ∈ ⊗ k V , we will denote its image in (cid:12) k V by v (cid:12) · · · (cid:12) v k , and the signs ofthe permutation action are such that permuting adjacent elements v, v (cid:48) “costs” the sign ( − | v || v (cid:48) | , e.g. we have v (cid:12) v (cid:12) v (cid:12) v = ( − | v || v | v (cid:12) v (cid:12) v (cid:12) v . Let sV denote the graded K module given by shifting the gradings of V down by one.Let SV = (cid:80) ∞ i =1 (cid:12) i V denote the (reduced) symmetric tensor coalgebra on V , wherethe coproduct is given by ∆( v (cid:12) ... (cid:12) v k ) := k − (cid:88) i =1 (cid:88) σ ∈ Sh( i,k − i ) ♦ ( σ, V ; v . . . , v k )( v σ (1) (cid:12) · · · (cid:12) v σ ( i ) ) ⊗ ( v σ ( i +1) (cid:12) ... (cid:12) v σ ( k ) ) . Here
Sh( i, k − i ) denotes the subset of permutations σ ∈ Σ k satisfying σ (1) < ... < σ ( i ) and σ ( i + 1) < · · · < σ ( k ) , and the Koszul-type signs are defined by ♦ ( σ, V ; v , . . . , v n ) = ( − {| v i || v j | : 1 ≤ i
Definition 2.1.2. An L ∞ homomorphism Φ : V → W between L ∞ algebras V and W is by definition a degree coalgebra map (cid:98) Φ : SV → SW such that (cid:98) (cid:96) W ◦ (cid:98) Φ = (cid:98) Φ ◦ (cid:98) (cid:96) V .This can alternatively be described by a sequence of degree graded symmetric maps Φ k : ⊗ k V → W for k ∈ Z ≥ satisfying the L ∞ homomorphism equations, and we recover (cid:98) Φ from the maps Φ , Φ , Φ , . . . via the extension formula (cid:98) Φ( v (cid:12)· · ·(cid:12) v n ) := (cid:88) k ≥ i + ··· + i k = n (cid:88) σ ∈ Sh( n ; i ,...,i k ) ♦ ( σ, V ; v , . . . , v n )(Φ i (cid:12)· · ·(cid:12) Φ i k )( v σ (1) (cid:12)· · ·(cid:12) v σ ( n ) ) . Here
Sh( n ; i , . . . , i k ) denotes subset of permutations σ ∈ Σ n satisfying σ (1) < · · · < σ ( i ) , σ ( i + 1) < · · · < σ ( i + i ) , . . . , σ ( i + · · · + i k − + 1) < · · · < σ ( i + · · · + i k ) . Givenan L ∞ homomorphism Φ : V → W , we will switch freely between its representation as asequence of maps Φ , Φ , Φ , . . . and its representation as a chain map (cid:98) Φ : B V → B W .Similarly, a chain homotopy between two L ∞ homomorphisms Φ , Ψ : V → W is definedsuch that there is an induced chain homotopy between the chain maps (cid:98) Φ , (cid:98) Ψ : B V → B W (see [Sie1, §2.1.3]). Note that we will also implicitly identify maps (cid:12) k V → V with multilinear maps with k inputs andone output in V . OMPUTING HIGHER SYMPLECTIC CAPACITIES I 13
Remark . Most of the L ∞ algebras appearing in this paper will be in fact bedifferential graded Lie algebras (DGLAs), i.e. (cid:96) k ≡ for k ≥ . In this case wewill often use ∂ to denote the differential (cid:96) and [ − , − ] to denote the bracket (cid:96) ( − , − ) .However, the corresponding L ∞ homomorphisms will nevertheless tend to have infinitelymany nonzero terms. V . We now introduce our main pro-tagonist, first without any filtration. For each n ∈ Z ≥ , we will define a differentialgraded Lie algebra (DGLA) V n over K . According to Theorem 1.2.2, V n is an L ∞ model for CH lin ( X ) (or alternatively for SC S , + ( X ) ) when X is a n -dimensional convextoric domain in C n . For ease of exposition we will mostly focus on the case n = 2 , and bydefault we put V = V ; the higher dimensional analogues of V and V Ω will be describedin §5.5. Definition 2.2.1.
As a K -module, the DGLA V has generators: • α i,j for each i, j ∈ Z ≥ , of degree | α i,j | = − − i − j • β i,j for each i, j ∈ Z ≥ not both , of degree | β i,j | = − − i − j .The differential is of the form: • ∂α i,j = jβ i − ,j − iβ i,j − • ∂β i,j = 0 .The bracket is given by: • [ α i,j , α k,l ] = ( il − jk ) α i + k,j + l • [ α i,j , β k,l ] = [ β k,l , α i,j ] = ( il − jk ) β i + k,j + l • [ β i,j , β k,l ] = 0 .According to Theorem 1.2.2, V is an L ∞ model for CH lin ( X ) (or alternatively SC S , + ( X ) ) whenever X is a four-dimensional convex toric domain in C . As it turnsout, in the unfiltered setting the bracket carries essentially no information. Indeed, bythe computation in §2.4 below, the homology of V is concentrated in even degrees. Thenby standard homological perturbation theory techniques, V has a canonical L ∞ modelall of whose operations are trivial (see §3 for more details). However, the situation willbe quite different in the presence of filtrations. V Ω . We now the equip L ∞ algebra V n with a family of filtrations which will give rise to rich combinatorial structures.For each convex toric domain X Ω ⊂ C n (see Definition 1.2.1), we define the filteredDGLA V n Ω which after forgetting the filtration is simply V n . We again assume bydefault that X Ω is four-dimensional, corresponding to V Ω = V .By a filtration F on V we mean: • submodules F ≤ r V ⊂ F ≤ r (cid:48) V ⊂ V for all < r ≤ r (cid:48) < ∞• (cid:96) k ( v , . . . , v k ) ∈ F ≤ r + ··· + r k V whenever v i ∈ F ≤ r i V for i = 1 , . . . , k .We will work primarily in the category of filtered L ∞ algebras as in [Sie1, §2.2]. Thismeans that all structure maps must preserve filtrations, which is a rather strict condition. See [Sie1, Rmk. 2.6] for the relationship to typical DGLA grading and sign conventions.
For example, a filtered L ∞ homomorphism Φ : V → W between filtered L ∞ algebrassatisfies Φ k ( v , . . . , v k ) ∈ F ≤ r + ··· + r k V (2.3.1)whenever v i ∈ F ≤ r i V for i = 1 , . . . , k . Similarly, we have a notion of filtered L ∞ homotopy between filtered L ∞ homomorphisms, and a corresponding notion of filtered L ∞ homotopy equivalence between filtered L ∞ algebras.We can succinctly define a filtration on V by endowing each basis element v ∈ V withan “action”, which we denote by A ( v ) ∈ R ≥ . For a nontrivial K -linear combination ofbasis elements of V , we then put A ( c v + · · · + c m v m ) := max {A ( v i ) : c i (cid:54) = 0 } , and we define F ≤ r V to be the span of all basis elements in V with action at most r . In the geometric interpretation of V Ω provided by Theorem 1.2.2, the basis elements of V Ω roughly correspond to Reeb orbits in ∂X (cid:101) Ω , where X (cid:101) Ω is the “fully rounded” versionof X Ω (see §5.3), and A corresponds to the symplectic action functional.Observe that if V is a filtered L ∞ algebra, then its bar complex B V naturally becomes afiltered chain complex. Namely, we define the action of an elementary tensor v (cid:12)· · ·(cid:12) v k ∈B V by A ( v (cid:12) · · · (cid:12) v k ) := k (cid:88) i =1 A ( v i ) . Remark . Recall from [Sie1, §2.2] that there is a close connection between L ∞ algebras over the universal Novikov ring Λ ≥ := (cid:40) ∞ (cid:88) i =1 c i T a i : c i ∈ K , a i ∈ R ≥ , lim i →∞ a i = + ∞ (cid:41) and filtered L ∞ algebras over K in the above sense. In particular, given a filtered L ∞ algebra we can define an L ∞ algebra over Λ ≥ by using the filtration to determinethe T -exponents. Since this procedure forgets the actions of the generators, we willfind it more convenient in this paper to work directly with filtered L ∞ algebras over K . Alternatively, we could adopt the L ∞ augmentation framework of [Sie1] in order torecover the lost information.Now let X Ω ⊂ C n be a convex toric domain with corresponding moment map image Ω ⊂ R n ≥ . Definition 2.3.2 ([GH]) . We define a norm || − || ∗ Ω on R n by || v || ∗ Ω := max {(cid:104) v, w (cid:105) : w ∈ (cid:98) Ω } for v ∈ R n .As pointed out in [GH], if || − || Ω denotes the norm on R n whose unit ball is (cid:98) Ω , then || − || ∗ Ω is the dual norm on R n after identifying ( R n ) ∗ with R n via the Euclidean innerproduct. Note that in terms of action, “filtration preserving” really means “action nondecreasing”.
OMPUTING HIGHER SYMPLECTIC CAPACITIES I 15
Restricting to the case that X Ω is a four-dimensional convex toric domain, we nowdefine V Ω as follows. Definition 2.3.3.
The filtered L ∞ algebra V Ω has underlying unfiltered L ∞ algebra V ,and its filtration F Ω determined by the following action values for its generators: • A Ω ( α i,j ) = || ( i, j ) || ∗ Ω for each i, j ∈ Z ≥ • A Ω ( β i,j ) = || ( i, j ) || ∗ Ω for each i, j ∈ Z ≥ not both . Lemma 2.3.4.
This defines a valid filtered L ∞ algebra.Proof. We take for granted that V satisfies the L ∞ relations, which can be easily checked.To see that the filtration is valid, we need to check that the differential and bracketpreserve the filtration. It suffices to check that we have max {|| ( i − , j ) || ∗ Ω , || ( i, j − || ∗ Ω } ≤ || ( i, j ) || ∗ Ω and || ( i + k, j + l ) || ∗ Ω ≤ || ( i, j ) || ∗ Ω + || ( k, l ) || ∗ Ω . The second inequality follows directly from the triangle inequality. The first inequalityfollows after observing that || − || ∗ Ω satisfies the symmetries || ( x, y ) || ∗ Ω = || ( − x, y ) || ∗ Ω = || ( x, − y ) || ∗ Ω , and hence we have || ( x, yt ) || ∗ Ω ≤ || ( x, y ) || ∗ Ω and || ( tx, y ) || ∗ Ω ≤ || ( x, y ) || ∗ Ω whenever t ∈ [0 , . (cid:3) We will sometimes denote the basis elements of V Ω by α Ω i,j , β Ω i,j if we wish to makeexplicit which filtration is being used. Since four-dimensional ellipsoids play a special rolein this paper, we also introduce the shorthand V a,b := V Ω E ( a,b ) , denoting the correspondinggenerators by α a,bi,j and β a,bi,j . V Ω and its linear spectral invariants. One of ourmain goals is to extract embedding obstructions from Corollary 1.2.3. As a warmup,we consider what happens at the linear level. From the point of view of curves, thiscorresponds to using cylinders rather than spheres with several positive punctures. Wearrive at the following much weaker statement:
Corollary 2.4.1.
In the context of Corollary 1.2.3, the identity map H ( V Ω (cid:48) ) → H ( V Ω ) is filtration preserving. As we now explain, from this statement we naturally recover the capacities from [GH],which conjecturally agree with the Ekeland–Hofer capacities.Observe that that we do not necessarily have A ( α Ω (cid:48) i,j ) ≥ A ( α Ω i,j ) and A ( β Ω (cid:48) i,j ) ≥ A ( β Ω i,j ) .Indeed, what we have is a filtered chain map V Ω (cid:48) → V Ω which is unfiltered chain homotopicto the identity, and this chain map is not necessarily the identity on the nose. What wedo have is the following picture, which is familiar from the study of spectral invariants insymplectic geometry. For a homology class A ∈ H ( V Ω ) , we put A Ω ( A ) := min {A ( v ) : ∂v = 0 , [ v ] = A for some v ∈ V Ω } . (2.4.1)Then in the context of Corollary 2.4.1, we must have A Ω (cid:48) ( A ) ≥ A Ω ( A ) for any homology class A ∈ H ( V ) . By way of terminology, we will say that A Ω ( A ) is the spectral invariant of V Ω in the homology class A . The homology of V can be computed as follows. For convenience, let us make a simplechange of basis by putting, for all i, j , α i,j := ( i − j − α i,j , β i,j := i ! j ! β i,j . In this basis, we have ∂α i,j = β i − ,j − β i,j − , ∂β i,j = 0 . From this we see that H ( V ) is one-dimensional in degrees − , − , − , . . . , and trivialotherwise. This means that, for q ∈ Z ≥ , the cycles β q, , β q − , , . . . , β ,q are all homolo-gous and represent the unique nontrivial class A q (modulo scaling by elements of K ∗ ) in H − − q ( V ) .Now observe that, for q ∈ Z ≥ , we have A Ω ( A q ) = min i,j ∈ Z ≥ i + j = q || ( i, j ) || ∗ Ω . Note that this coincides with the expression for c q ( X Ω ) from [GH, Thm. 1.6]. In thespecial case of E ( a, b ) , we get the expression A a,b ( A q ) = min i,j ∈ Z ≥ i + j = q max { ia, jb } , and one can check that this is precisely the q th Ekeland–Hofer capacity of E ( a, b ) .Alternatively, this is the q th smallest element of the infinite array ( ic : i ∈ Z ≥ , c ∈ { a, b } ) . We summarize this subsection in the following proposition.
Proposition 2.4.2.
For q ∈ Z ≥ , we have H ( V ) = K (cid:104) A , A , A , . . . (cid:105) with | A q | = − − q and A Ω ( A q ) = min i,j ∈ Z ≥ i + j = q || ( i, j ) || ∗ Ω . Given a symplectic embedding X Ω s (cid:44) → X Ω (cid:48) , wemust have A Ω (cid:48) ( A q ) ≥ A Ω ( A q ) for all q .
3. Computing the canonical model of V a,b In order to extract the full power of Corollary 1.2.3, we need to study the bar complexof V Ω , and in particular to understand its homology and corresponding spectral invariants.We first compute in §3.1 the homology H ( B V Ω ) as an unfiltered K -module. To betterunderstand the role of the filtration, we then seek to find a canonical model for V Ω as afiltered L ∞ algebra. This turns out to fail for general V Ω , but we succeed in the ellipsoidcase V a,b , and in §3.2 we recursively construct maps Φ a,b and Ψ a,b which give a filteredcanonical model for V a,b . Note that this is often called the minimal model in the literature, but as pointed in [Fuk, Rmk.2.3.1], this leads to confusion with the notion of minimal model from rational homotopy. In fact, theseare essentially Koszul dual notions.
OMPUTING HIGHER SYMPLECTIC CAPACITIES I 17
We begin with some generalobservations. Firstly, as a K -module up to isomorphism, H ( B V ) does not depend onthe filtration. If we ignore the filtration and view V as an unfiltered L ∞ algebra over K , a standard corollary of the homological perturbation lemma (see e.g. [Fuk, §2.3])states that V is L ∞ homotopy equivalent to an L ∞ algebra V can whose underlying chaincomplex is H ( V ) , with trivial differential. In particular, by basic functoriality propertiesof the bar construction, we get an isomorphism of K -modules H ( B V ) ∼ = H ( B V can ) .Note that in principle V can could have many nontrivial higher L ∞ operations, even if V has only a differential and a bracket (c.f. Massey products). However, our computationin §2.4 shows that H ( V ) is supported in even degrees, whereas the L ∞ operations on V can all have degree +1 . It follows that all of the L ∞ operations on V can are automaticallytrivial for degree reasons. Thus H ( B V can ) = B V can , and we have: Proposition 3.1.1.
For V the L ∞ algebra from Definition 2.2.1, H ( B V ) is abstractlyisomorphic as a K -module to the reduced polynomial algebra S K (cid:104) A , A , A , . . . (cid:105) on formalvariables A q of degree | A q | = − − q for q ∈ Z ≥ . Homological perturbation theory (HPT) in fact produces L ∞ homomorphisms Φ : V → V can and Ψ : V can → V such that Φ ◦ Ψ and Ψ ◦ Φ are both L ∞ homotopic to the identity.These maps are constructed recursively, or can be described more directly as sums overdecorated trees (sometimes interpreted as Feynman diagrams). The construction of Φ and Ψ above is based on the following ground inputs: • a chain map Ψ : V can → V • a chain map Φ : V → V can such that Φ ◦ Ψ = • a chain homotopy h between Ψ ◦ Φ and .See e.g. [Kon, Mar1, Sei, Fuk] for the general strategy and history. We note that whileexplicit (as opposed to obstruction theoretic) recursive and tree-counting formulas for Φ and Ψ in the analogous A ∞ case are given in [Mar2], the explicit formulas for the L ∞ case appear to be somewhat more subtle and we could not find them in the literature.We will take a more direct approach in §3.2 below applied to the L ∞ algebra V .Recall that we want to understand V Ω as a filtered L ∞ algebra. We expect that if allof the ground inputs are filtration preserving, then the resulting L ∞ homomorphisms Φ Ω and Ψ Ω will be filtered L ∞ homotopy equivalences. However, if the homology of V Ω when viewed as a Λ ≥ -module (c.f. Remark 2.3.1) has nontrivial T -torsion, then it willnot be possible to find such ground inputs. Fortunately, in the special case of V a,b we canindeed find ground inputs Ψ a,b , Φ a,b , h a,b which are filtration preserving, e.g. by putting • Φ a,b ( α i,j ) = 0 • Φ a,b ( β i,j ) = A i + j • Ψ a,b ( A q ) = β i ( q ) , j ( q ) • h a,b ( α i,j ) = 0 • h a,b ( β i,j ) = (cid:0) α i ( q )+1 , j ( q ) + · · · + α i ( q )+ j ( q ) , (cid:1) − ( α i +1 ,j + · · · + α i + j, ) for q = i + j .Here the pair ( i ( q ) , j ( q )) ∈ Z ≥ \ { (0 , } is defined as follows. Definition 3.1.2.
Given a four-dimensional convex toric domain X Ω ⊂ C and q ∈ Z ≥ ,we put ( i ( q ) , j ( q )) := argmin ( i,j ) ∈ Z ≥ ,i + j = q || ( i, j ) || ∗ Ω (3.1.1)That is, ( i ( q ) , j ( q )) is the pair ( i, j ) ∈ Z ≥ with i + j = q for which || ( i, j ) || ∗ Ω is minimal.Note that the pair ( i ( q ) , j ( q )) depends quite sensitively on Ω , although we suppress thisdependence from the notation to avoid clutter. In order to avoid borderline cases, forsimplicity we will typically assume that there is a unique minimizer in Definition 3.1.2. Inthe case of the four-dimensional ellipsoid E ( a, b ) , we achieve this by implicitly replacing b with b + δ for δ > sufficiently small.We omit the proof of the following lemma since we will not explicitly need it below: Lemma 3.1.3.
With the above definitions, h a,b is filtration preserving and satisfies h a,b ◦ ∂ + ∂ ◦ h a,b = − Ψ a,b ◦ Φ a,b . Φ a,b and Ψ a,b . Taking the discussion from theprevious subsection as motivation, we now proceed to directly construct maps Φ a,b and Ψ a,b realizing a canonical model for the filtered L ∞ algebra V a,b . Construction 3.2.1.
For any fixed a, b ∈ R > and constants C q ; a,b ∈ K ∗ , q ∈ Z ≥ , werecursively define maps Φ ka,b : (cid:12) k V a,b → V can a,b and Ψ ka,b : (cid:12) k V can a,b → V a,b , k ∈ Z ≥ , by thefollowing properties:(1) for q ∈ Z ≥ we have Ψ a,b ( A q ) = C q ; a,b β i ( q ) , j ( q ) (2) Ψ ka,b ≡ for k ≥ and(1) for ( i, j ) ∈ Z ≥ \ { (0 , } we have Φ a,b ( β i,j ) = i ( q )! j ( q )! i ! j ! C q ; a,b A q with q = i + j (2) Φ ka,b = 0 if any of the inputs is α i,j for some i, j (3) for k ≥ , we have Φ ka,b ( β i ( q ) , j ( q ) , . . . , β i ( q k ) , j ( q k ) ) = 0 for any q , . . . , q k ∈ Z ≥ (4) for k ≥ and ( i , j ) , . . . , ( i k , j k ) ∈ Z ≥ \ { (0 , } , we have j Φ ka,b ( β i − ,j , β i ,j , . . . , β i k ,j k ) − i Φ ka,b ( β i ,j − , β i ,j , . . . , β i k ,j k )+ k (cid:88) m =2 ( i j m − j i m )Φ k − a,b ( β i + i m ,j + j m , β i ,j , . . . , (cid:92) β i m ,j m , . . . , β i k ,j k ) = 0 . Remark . For the time being we leave the arbitrary constants C q ; a,b unspecified.They will not affect the embedding obstructions described in §4. However, they will playa role in the enumerative invariants discussed in §5, and we will nail down a choice in§5.3. Definition 3.2.3.
We will say that a basis element α i,j or β i,j is action minimal if wehave ( i, j ) = ( i ( q ) , j ( q )) for q = i + j .Note that (3) states that Φ ka,b vanishes whenever all of its input basis elements areaction minimal. This is the main place where dependence on a, b enters. Also, (4) is adirect translation of the L ∞ homomorphism relations for Φ a,b . OMPUTING HIGHER SYMPLECTIC CAPACITIES I 19
Using (4), we can iteratively modify the inputs until they are all action minimal.Namely, given inputs β i ,j , . . . , β i k ,j k which are not all action minimal, assume withoutloss of generality that ( i , j ) is not action minimal. In the case i < i ( i + j ) , wecompute Φ ka,b ( β i ,j , . . . , β i k ,j k ) recursively via Φ ka,b ( β i ,j , β i ,j , . . . , β i k ,j k ) = i +1 j Φ ka,b ( β i +1 ,j − , β i ,j , . . . , β i k ,j k ) − j k (cid:88) m =2 ([ i + 1] j m − j i m )Φ k − a,b ( β i + i m +1 ,j + j m , β i ,j , . . . , (cid:92) β i m ,j m , . . . , β i k ,j k ) . (3.2.1)Similarly, if i > i ( i + j ) , we compute Φ ka,b ( β i ,j , . . . , β i k ,j k ) recursively via Φ ka,b ( β i ,j , β i ,j , . . . , β i k ,j k ) = j +1 i Φ ka,b ( β i − ,j +1 , β i ,j , . . . , β i k ,j k )+ i k (cid:88) m =2 ( i j m − [ j + 1] i m )Φ k − a,b ( β i + i m ,j + j m +1 , β i ,j , . . . , (cid:92) β i m ,j m , . . . , β i k ,j k ) . (3.2.2)Here are some example computations: Example 3.2.4.
We compute Φ ,R ( β , , β , ) for R (cid:29) . We have Φ ,R ( β , , β , ) = 2Φ ,R ( β , , β , ) − (2 · − · ,R ( β , )Φ ,R ( β , ) = C ,R A = C ,R A Φ ,R ( β , , β , ) = Φ ,R ( β , , β , ) = 2Φ ,R ( β , , β , ) − (2 · − · ,R ( β , )Φ ,R ( β , ) = C ,R = C ,R A . Using Φ ,R ( β , , β , ) = 0 , we get Φ ,R ( β , , β , ) = C ,R A − C ,R A = C ,R A . Example 3.2.5.
We compute Φ ,R ( β , , β , , β , ) for R (cid:29) . We have Φ ,R ( β , , β , , β , ) = 2Φ ,R ( β , , β , , β , ) − ,R ( β , , β , )Φ ,R ( β , , β , , β , ) = 2Φ ,R ( β , , β , , β , ) + 2Φ ,R ( β , , β , ) − Φ ,R ( β , , β , )Φ ,R ( β , , β , , β , ) = 2Φ ,R ( β , , β , , β , ) + 4Φ ,R ( β , , β , ) = 4Φ ,R ( β , , β , )Φ ,R ( β , , β , ) = 2Φ ,R ( β , , β , ) + 2Φ ,R ( β , )Φ ,R ( β , , β , ) = 5Φ ,R ( β , , β , ) + 2Φ ,R ( β , ) = 2Φ ,R ( β , )Φ ,R ( β , , β , ) = 2Φ ,R ( β , , β , ) − Φ ,R ( β , )2Φ ,R ( β , , β , ) = 4Φ ,R ( β , , β , ) + 4Φ ,R ( β , )Φ ,R ( β , ) = C ,R A = C ,R A Φ ,R ( β , ) = C ,R A = C ,R A Φ ,R ( β , ) = C ,R A = C ,R A . Combining the above, we get Φ ,R ( β , , β , , β , ) = C ,R A . As visible in the above examples, there are choices involved as to what order we applythe recursion. For example, to compute Φ ,R ( β , , β , ) , we can apply (3.2.1) to either β , or β , . It is not a priori obvious that the final answer is independent of these choices.The following lemma alleviates this concern. Lemma 3.2.6.
Let X Ω be any four-dimensional convex toric domain. Each class A ∈ H ( B V Ω ) is uniquely represented by a cycle which is a linear combination of tensorproducts of action minimal basis elements in V Ω .Remark . We warn the reader that the cycle provided by Lemma 3.2.6 is notnecessarily the cycle of minimal action representing A . For example, consider Ω P (1 , ε ) for ε > sufficiently small. The element β , (cid:12) β , + 10 β , ∈ B V Ω P (1 , ε ) which hasaction , and each summand is a tensor product of action minimal basis elements, yet it ishomologous (c.f. Example 3.2.4) to β , (cid:12) β , ∈ B V Ω P (1 , ε ) , which has action ε < . Proof of Lemma 3.2.6.
Suppose that y ∈ B V Ω is a linear combination of tensor productsof action minimal basis elements, and that y is nullhomologous, i.e. y = (cid:98) (cid:96) ( x ) for some x ∈ B V Ω . It suffices to show that y = 0 .For k ∈ Z ≥ , let B ≤ k V Ω denote the subcomplex consisting of linear combinations ofelementary tensors of word length at most k . We claim that the map H ( B ≤ k V Ω ) → H ( B V Ω ) induced by the inclusion of B ≤ k V into B V is injective. In other words, if anelement in B ≤ k V Ω is the differential of an element in B V Ω , then it is also the differentialof an element in B ≤ k V Ω . To justify this claim, let V canΩ denote a canonical model for the L ∞ algebra V Ω , ignoring filtrations. This means that V canΩ is an unfiltered L ∞ algebrawith underlying K -module H ( V Ω ) , with all L ∞ operations necessarily trivial for degreereasons, and we have in particular a commutative diagram H ( B ≤ k V canΩ ) ∼ = (cid:47) (cid:47) (cid:15) (cid:15) H ( B ≤ k V Ω ) (cid:15) (cid:15) H ( B V canΩ ) ∼ = (cid:47) (cid:47) H ( B V Ω ) The left vertical arrow in induced by the inclusion B ≤ k V canΩ → B V canΩ , and the homologylevel map is clearly injective since the differential on B V canΩ is trivial. It follows that theright vertical arrow is also injective, as desired.Now put m := min { k ∈ Z ≥ : y ∈ B ≤ k V Ω } , and consider the quotient complex B ≤ m V Ω / B ≤ m − V Ω . By the earlier claim, we have y = (cid:98) (cid:96) ( x (cid:48) ) for some x (cid:48) ∈ B ≤ m V Ω . Thismeans that the class [ y ] ∈ H ( B ≤ m V Ω / B ≤ m − V Ω ) vanishes. On the other hand, notethat [ y ] is represented by the maximal word length part π m ( y ) ∈ B ≤ m V Ω of y (here π m denotes the projection SV Ω → (cid:12) m V Ω ), and this is a linear combination of tensorproducts of action minimal basis elements.Let S ⊂ Z m ≥ denote the set of integer tuples ( i , . . . , i m ) such that i ≤ · · · ≤ i m , andlet K (cid:104)S(cid:105) denote free K -module spanned by these tuples. We consider the K -linear map Z : B ≤ m V Ω / B ≤ m − V Ω → K (cid:104)S(cid:105) sending v i ,j (cid:12) · · · (cid:12) v i m ,j m to ( i + j , . . . , i m + j m ) , where for each k = 1 , . . . , m we have either v i k ,j k = α i k ,j k or v i k ,j k = β i k ,j k . Recall that for ( i, j ) ∈ Z ≥ we have OMPUTING HIGHER SYMPLECTIC CAPACITIES I 21 ∂α i,j = β i − ,j − β i,j − , and note that the bracket term of B ≤ k V Ω disappears when we passto the quotient complex B ≤ m V Ω / B ≤ m − V Ω . It is then easy to check that all boundariesin B ≤ m V Ω / B ≤ m − V Ω are contained in the kernel of Z , whereas Z ( π m ( y )) (cid:54) = 0 unless wehave π m ( y ) = 0 in B ≤ m V Ω . This contradicts the definition of m unless we have y = 0 in B V Ω , as desired. (cid:3) Remark . It would be interesting to find a more fundamental combinatorial formulafor Φ k , e.g. in terms of lattice point counts in some lattice polytope. Such a descriptioncould give a more conceptual proof of Lemma 3.2.6 and also perhaps shed light on whenthe structure coefficients appearing in Corollary 1.2.5 are nonzero.The rest of this section is occupied with the following two lemmas which verify that Φ a,b and Ψ a,b have the desired properties. Lemma 3.2.9.
Construction 3.2.1 defines a valid L ∞ homomorphism Φ a,b : V a,b → V can a,b and Ψ a,b : V can a,b → V a,b . The induced bar homology maps H ( (cid:98) Φ a,b ) : H ( B V a,b ) → H ( B V can a,b ) and H ( (cid:98) Ψ a,b ) : H ( B V can a,b ) → H ( B V a,b ) satisfy H ( (cid:98) Φ a,b ) ◦ H ( (cid:98) Ψ a,b ) = H ( (cid:98) Ψ a,b ) ◦ H ( (cid:98) Φ a,b ) = . Remark . A more natural formulation would state that Φ a,b ◦ Ψ a,b and Ψ a,b ◦ Φ a,b are homotopic to the identity as filtered L ∞ homomorphisms, but the above formulationin terms of bar complexes suffices for our intended applications. Proof of Lemma 3.2.9.
To see that Ψ a,b as defined is an L ∞ homomorphism, observethat since all the operations in V can a,b are trivial, it suffices to show that (cid:96) kV a,b vanishes oninputs of the form Ψ a,b ( A q ) (cid:12) · · · (cid:12) Ψ a,b ( A q k ) , but this is trivial since the operations on V a,b vanish when all of the inputs are β generators.To see that Φ a,b is an L ∞ homomorphism, we first check that it is a chain map. Sincethe differential of V a,b vanishes on β basis elements, it suffices to check that Φ a,b vanisheson terms of the form ∂α i,j = jβ i − ,j − iβ i,j − for i, j ∈ Z ≥ . Putting q := i + j − , wehave Φ a,b ( jβ i − ,j − iβ i,j − ) = j i ( q )! j ( q )!( i − j ! C q ; a,b A q − i i ( q )! j ( q )! i !( j − C q ; a,b A q = 0 . More generally, since the L ∞ operations on V can a,b are trivial, it suffices to check that Φ ka,b vanishes on inputs of the form (cid:98) (cid:96) V a,b ( v (cid:12) · · · (cid:12) v k ) for v , . . . , v k ∈ V a,b . If amongstthe inputs v , . . . , v k there are either two or more α basis elements, then each term in theabove sum will contain at least one α basis element, and hence automatically vanishes.Similarly, if there are no β inputs then the expression is automatically zero. If there isexactly one α , we get precisely the expression (4) above.Next, we check that H ( (cid:98) Φ a,b ) ◦ H ( (cid:98) Ψ a,b ) = . We have (cid:98) Ψ a,b ( A q (cid:12) · · · (cid:12) A q k ) = C q ; a,b . . . C q k ; a,b β i ( q ) , j ( q ) (cid:12) · · · (cid:12) β i ( q k ) , j ( q k ) , so by property (3) for Φ a,b we have ( (cid:98) Φ a,b ◦ (cid:98) Ψ a,b )( A q (cid:12) · · · (cid:12) A q k ) = C q ; a,b . . . C q k ; a,b Φ a,b ( β i ( q ) , j ( q ) ) (cid:12) · · · (cid:12) Φ a,b ( β i ( q k ) , j ( q k ) )= A q (cid:12) · · · (cid:12) A q k . Finally, we check that H ( (cid:98) Ψ a,b ) ◦ H ( (cid:98) Φ a,b ) = . Since H ( B V a,b ) and H ( B V can a,b ) havethe same finite rank in each degree and H ( (cid:98) Φ a,b ) is a left inverse to H ( (cid:98) Ψ a,b ) it followsimmediately that H ( (cid:98) Φ a,b ) and H ( (cid:98) Ψ a,b ) are both invertible and hence H ( (cid:98) Φ a,b ) is also aright inverse to H ( (cid:98) Ψ a,b ) . (cid:3) Lemma 3.2.11.
The L ∞ homomorphisms Φ a,b and Ψ a,b are filtration preserving.Proof. The fact that Ψ a,b is filtration preserving is manifest. As for Ψ a,b , given β i ,j , . . . , β i k ,j k ∈ V a,b , we need to verify the following action inequality: A a,b (Φ ka,b ( β i ,j , . . . , β i k ,j k )) ≤ k (cid:88) s =1 A a,b ( β i s ,j s ) . (3.2.3)The case k = 1 is clear. For k ≥ , if each of the inputs β i ,j , . . . , β i k ,j k is action minimalthen Φ ka,b ( β i ,j , . . . , β i k ,j k ) = 0 and there is nothing to check, so may assume withoutloss of generality that β i ,j is not action minimal.We will further suppose that i < i ( i + j ) , the case i > i ( i + j ) being closelyanalogous. In order to recursively compute Φ ka,b ( β i ,j , . . . , β i k ,j k ) , the next step is toapply (3.2.1) in order to write it as a linear combination of terms with strictly smaller k or else strictly smaller j . We may assume by induction that we already know A a,b (Φ ka,b ( β i +1 ,j − , β i ,j , . . . , β i k ,j k )) ≤ A a,b ( β i +1 ,j − ) + A a,b ( β i ,j ) + · · · + A a,b ( β i k ,j k ) (3.2.4)and, for m = 2 , . . . , k , A a,b (Φ k − a,b ( β i + i m +1 ,j + j m , β i ,j , . . . , (cid:92) β i m ,j m , . . . , β i k ,j k )) ≤A a,b ( β i + i m +1 ,j + j m ) + A a,b ( β i ,j ) + · · · + (cid:92) A a,b ( β i m ,j m ) + · · · + A a,b ( β i k ,j k ) . (3.2.5)We first observe that the right hand side of (3.2.4) is at most the right hand sideof (3.2.3). Indeed, it suffices to show that A a,b ( β i +1 ,j − ) ≤ A a,b ( β i ,j ) . This followsfrom the assumption i < i ( i + j ) , since the function t (cid:55)→ || ( i ( q ) + t, j ( q ) − t ) || ∗ Ω ismonotonically increasing with t .In order to complete the proof, we need to show the right hand size of (3.2.5) is atmost the right hand side of (3.2.3). It suffices to establish the following inequality: A a,b ( β i +1 ,j ) ≤ A a,b ( β i ,j ) . Indeed, by the triangle inequality we then have A a,b ( β i + i m +1 ,j + j m ) ≤ A a,b ( β i +1 ,j ) + A a,b ( β i m ,j m ) ≤ A a,b ( β i ,j ) + A a,b ( β i m ,j m ) , from which the desired inequality follows. Note that in this case we evidently must have A a,b ( β i +1 ,j ) = A a,b ( β i ,j ) . OMPUTING HIGHER SYMPLECTIC CAPACITIES I 23
Suppose by contradiction that we have A a,b ( β i +1 ,j ) > A a,b ( β i ,j ) , i.e. max { [ i + 1] a, j b } > max { i a, j b } . Then we must have ( i +1) a > j b . On the other hand, from the assumption i < i ( i + j ) we have A ( β i +1 ,j − ) ≤ A ( β i ,j ) , i.e. max { [ i + 1] a, [ j − b } ≤ max { i a, j b } , which necessitates ( i + 1) a ≤ j b , a contradiction. (cid:3) Remark . (1) We note that (3) in Construction 3.2 is often mandated purely for action reasons,but this is not always the case, e.g. in principle a term Φ , ε ( β , , β , ) = A would be permitted by action and index considerations. This is related to theambiguities in punctured curve counts caused by index zero symplectizationcurves, which we further discuss in §5.(2) The analogue of Lemma 3.2.11 does not hold for general Ω . We can, however,iteratively apply analogues of (3.2.1) and (3.2.2) to any cycle in B V Ω in order tofind its representative promised by Lemma 3.2.6. Example 3.2.13.
Consider the filtered L ∞ algebra V Ω P ( a,b ) associated with the polydisk P ( a, b ) (as usual we assume a ≤ b ). Note that basis elements β i,j are action minimal ifand only if we have j = 0 . Now imagine that Φ Ω P ( a,b ) and Ψ Ω P ( a,b ) really were filtered L ∞ homotopy equivalences. Then it would follow that we have a corresponding canonicalfiltered L ∞ model V canΩ P ( a,b ) whose action filtration depends only on a . In particular, in thiscase we would not have any interesting capacities g b ( P ( a, b )) apart from functions of a .As it turns out, whereas the linear spectral invariants of P ( a, b ) indeed are just multiplesof a , the bar complex spectral invariants are much richer than this and do depend on both a and b (c.f. [Sie1, Ex. 1.14] ).
4. Symplectic embedding obstructions
We begin this section by reviewing the capacities g b from [Sie1] in §4.1. In §4.2 wediscuss the role of persistent homology and explain how it can be used to algorithmicallyextract obstructions. We then restrict to the case of ellipsoids in §4.3 and use thecanonical model from the previous section to read off embedding obstructions. Finally,in §4.4 we explore obstructions which lie beyond the bar complex spectral invariants,illustrating this technique in the context of polydisks. For concreteness we mostly stick tothe case that X is four-dimensional, although we expect all of the results in this sectionto have natural extensions to higher dimensions. Let X Ω and X Ω (cid:48) be four-dimensionalconvex toric domains, and suppose we have a symplectic embedding X Ω s (cid:44) → X Ω (cid:48) . ByCorollary 1.2.3, we have a filtered L ∞ homomorphism Q : V Ω (cid:48) → V Ω which is unfiltered L ∞ homotopic to the identity. In particular, this means that the identity map : H ( B V Ω (cid:48) ) → H ( B V Ω ) is filtration preserving, so for any homology class A ∈ H ( B V ) wehave the inequality A Ω ( A ) ≤ A Ω (cid:48) ( A ) . Here we put A Ω ( A ) := min {A Ω ( x ) : ∂x = 0 , [ x ] = A for some x ∈ B V Ω } , with A Ω (cid:48) ( A ) defined similarlyBy Proposition 3.1.1, we have abstract isomorphisms of K -modules H ( B V Ω ) ∼ = H ( B V Ω (cid:48) ) ∼ = S K (cid:104) A , A , A , . . . (cid:105) , (4.1.1)where S ( − ) denotes the reduced symmetric tensor algebra as in §2.1. However, since S K (cid:104) A , A , A , . . . (cid:105) is generally multidimensional in any given degree (unlike H ( V ) , c.f.§2.4), we need to think more carefully about how to reference homology classes. Forexample, in degree − we have K (cid:104) A , A (cid:12) A (cid:105) . Therefore any nonzero linear combination c A + c A (cid:12) A with ( c , c ) ∈ K \ { (0 , } gives a corresponding inequality A Ω ( c A + c A (cid:12) A ) ≤ A Ω (cid:48) ( c A + c A (cid:12) A ) . In fact the quantities A Ω ( A ) and A Ω (cid:48) ( A ) are unaffected if we scale A by an elementof K ∗ , so what we have is a family of spectral invariants indexed by a one-dimensionalprojective space KP .One way to reference homology classes in H ( B V Ω ) and H ( B V Ω (cid:48) ) would be to simplydescribe cycles in terms of their representation in the basis { α i,j , β i,j } . However, this isnot the most natural approach in a general context, since it depends on our precise models V Ω and V Ω (cid:48) and the fact that the cobordism map can be identified with the identity map.Following [Sie1], a more canonical approach is to use the L ∞ homomorphism Ξ sk : CH lin ( X Ω ) → CH lin ( E sk ) induced by the inclusion E sk s (cid:44) → X Ω , where E sk = δE (1 , R ) denotes a “skinny ellipsoid”for δ > sufficiently small and R (cid:29) sufficiently large. This cobordism map makessense for any Liouville domain X and is uniquely determined up to L ∞ homotopy, henceit induces a canonical way to refer to homology classes in B CH lin ( X ) , via the inverse ofthe homology level map H ( (cid:98) Ξ sk ) : H ( B CH lin ( X Ω )) → H ( B CH lin ( E sk )) . By the results in[MSie], Ξ sk can equivalently be defined by counting punctured curves in X with localtangency constraints.As a shorthand, for δ (cid:28) sufficiently small and R (cid:29) sufficiently large we put V sk := V δ,δR , and similarly V cansk := V can δ,δR , Φ sk := Φ δ,δR , Ψ sk := Ψ δ,δR , etc. We haveidentifications of K -modules CH lin ( E sk ) ≈ V cansk ≈ K [ t ] , where A q ∈ V cansk corresponds to t q − for q ∈ Z ≥ . We then also have identifications of K -modules B CH lin ( E sk ) ≈ B V cansk ≈ S K [ t ] . Following [Sie1], for b ∈ S K [ t ] we put g b ( X ) := A B CH lin ( X ) (( H ( (cid:98) Ξ sk )) − ( b )) . Using Ψ sk , note that b also corresponds to an element (cid:98) Ψ sk ( b ) ∈ B V sk , which we canin turn identify with an element of B V Ω whenever X Ω is a four-dimensional convextoric domain. Our algebraic formalism now computes the capacities g b for this class ofdomains: Recall that, with our grading conventions, the degree of the monomial t k in K [ t ] is − − k . OMPUTING HIGHER SYMPLECTIC CAPACITIES I 25
Theorem 4.1.1. If X Ω is a four-dimensional convex toric domain, for any b ∈ S K [ t ] we have g b ( X Ω ) = A Ω ( H ( (cid:98) Ψ sk ( b ))) . Given a symplectic embedding X Ω s (cid:44) → X Ω (cid:48) ,we have the inequality g b ( X Ω ) ≤ g b ( X Ω (cid:48) ) for each choice of b ∈ S K [ t ] , and this gives avery large family of obstructions. However, it turns out there is a much smaller collectionof inequalities which determines all of the others. In fact, it suffices to check just finitelymany choices of b in each degree. This is particularly noteworthy since if we were towork over say K = R there would be a priori uncountably many distinct elements b ∈ K [ t ] ,even after projectivizing.Let p ( q ) denote the dimension of B V Ω in degree − − q , so we have p (1) = 1 , p (2) =1 , p (3) = 2 , p (4) = 2 , p (5) = 4 , p (6) = 4 , etc. According to [ZC], we can find a homoge-neous basis for B V Ω , each element of which is either a left endpoint of a finite barcode, aright endpoint of a finite barcode, or a left endpoint of a semi-infinite barcode. Moreprecisely, for each q ∈ Z ≥ we have nonnegative integers l ( q ) , r ( q ) and a basis ξ Ω q ;1 , . . . , ξ Ω q ; l ( q ) , ζ Ω q ;1 , . . . , ζ Ω q ; r ( q ) , τ Ω q ;1 , . . . , τ Ω q ; p ( q ) for the degree − − q part of B V Ω such that: • for each i ∈ { , . . . , p ( q ) } we have (cid:98) (cid:96) ( τ Ω q ; i ) = 0 • for each i ∈ { , . . . , l ( q ) } we have (cid:98) (cid:96) ( ξ Ω q ; i ) = 0 • for each i ∈ { , . . . , r ( q ) } we have (cid:98) (cid:96) ( ζ Ω q ; i ) = ξ Ω q − j for some unique j ∈ { , . . . , l ( q − } associated to the pair ( q, i ) As explained in [ZC], finding these basis elements essentially reduces to finding the Smithnormal forms of the matrices defining the chain complex B V Ω in each degree, and thereis an efficient algorithm for doing so.In the above basis, each element τ Ω q ; i corresponds to the left endpoint of a semi-infinitebarcode with endpoint at A Ω ( τ Ω q ; i ) ∈ R ≥ . These form a basis for the homology of B V Ω .In particular, the possible values of spectral invariants of B V Ω associated to homologyclasses in degree − − q are given by {A Ω ( τ Ω q ;1 ) , . . . , A Ω ( τ Ω q ; p ( q ) ) } , and we can assume without loss of generality that these actions appear in nondecreasingorder. Meanwhile, a pair of basis elements ( ζ Ω q ; i , ξ Ω q − j ) with (cid:98) (cid:96) ( ζ Ω q ; i ) = ξ Ω q − j corresponds toa finite barcode with left endpoint at A Ω ( ξ Ω q − j ) ∈ R ≥ and right endpoint at A Ω ( ζ Ω q ; i ) ∈ R ≥ . The left endpoints of finite barcodes generate the torsion part of H ( B V Ω ) as amodule over the Novikov ring Λ ≥ , but do not contribute to its homology over K .Fixing q ∈ Z ≥ , we now consider how g b ( X Ω ) changes as we vary b in the degree − − q part of S K [ t ] . For i ∈ { , . . . , p ( q ) } , define b Ω q ; i to be the image of τ Ω q ; i under thecomposition B V Ω −→ B V sk (cid:98) Φ sk −→ B V cansk ≈ S K [ t ] . Note that l ( q ) and r ( q ) also implicitly depend on Ω . Any degree − − q element b ∈ S K [ t ] can be written uniquely as a linear combination c b Ω q ;1 + · · · + c p ( q ) b Ω q ; p ( q ) for some c , . . . , c p ( q ) ∈ K , and we have A Ω ( c τ Ω q ;1 + · · · + c p ( q ) τ Ω q ; p ( q ) ) = max {A Ω ( τ Ω q ; i ) : c i (cid:54) = 0 } . This means that g b ( X Ω ) is constant as b varies along any stratum of the full flag K (cid:104) b Ω q ;1 (cid:105) ⊂ K (cid:104) b Ω q ;1 , b Ω q ;2 (cid:105) ⊂ · · · ⊂ K (cid:104) b Ω q ;1 , . . . , b Ω q ; p ( q ) (cid:105) . We can similarly associate to X Ω (cid:48) the degree − − q elements b Ω (cid:48) q ;1 , . . . , b Ω (cid:48) q ; p ( q ) ∈ S K [ t ] and corresponding actions A Ω (cid:48) ( τ Ω (cid:48) q ;1 ) ≤ · · · ≤ A Ω (cid:48) ( τ Ω (cid:48) q ; p ( q ) ) ∈ R ≥ for each q ∈ Z ≥ . Notethat in general the list of elements b Ω (cid:48) q ;1 , . . . , b Ω (cid:48) q ; p ( q ) might be different from b Ω q ;1 , . . . , b Ω q ; p ( q ) ,and the associated full flags could coincide, intersect transversely, or neither. Thefollowing immediate proposition summarizes the finite number of comparisons which aresufficient in each degree: Proposition 4.2.1.
For each q ∈ Z ≥ and i ∈ { , . . . , p ( q ) } , we have g b Ω (cid:48) q ; i ( X Ω ) ≤ g b Ω (cid:48) q ; i ( X Ω (cid:48) ) . Moreover, these inequalities imply that we have g b ( X Ω ) ≤ g b ( X Ω (cid:48) ) for all degree − − q elements b ∈ S K [ t ] . Note that if we write b Ω (cid:48) q ; i = c b Ω q ;1 + · · · + c p ( q ) b Ω q ; p ( q ) for some c , . . . , c p ( q ) ∈ K , we have g b Ω (cid:48) q ; i ( X Ω ) = max { g b Ω q ; i ( X Ω ) : c i (cid:54) = 0 } . Proof of Proposition 4.2.1.
Assuming that we have g b Ω (cid:48) q ; i ( X Ω ) ≤ g b Ω (cid:48) q ; i ( X Ω (cid:48) ) for all b Ω (cid:48) q ; i ,we need to establish g b ( X Ω ) ≤ g b ( X Ω (cid:48) ) for all b . Fix some arbitrary b ∈ S K [ t ] ofdegree − − q , which we can write as a linear combination b = (cid:80) p ( q ) i =1 c i b Ω (cid:48) q ; i for some c , . . . , c p ( q ) ∈ K . Let m denote the maximal i such that c i (cid:54) = 0 . Observe that we have g b ( X Ω (cid:48) ) = g b Ω (cid:48) q ; m ( X Ω (cid:48) ) and g b ( X Ω ) ≤ max i =1 ,...,m g b Ω (cid:48) q ; i ( X Ω ) , so it suffices to establish theinequality max i =1 ,...,m g b Ω (cid:48) q ; i ( X Ω ) ≤ g b Ω (cid:48) q ; m ( X Ω (cid:48) ) . Now note that for each i = 1 , . . . , m we have g b Ω (cid:48) q ; i ( X Ω ) ≤ g b Ω (cid:48) q ; i ( X Ω (cid:48) ) ≤ g b Ω (cid:48) q ; m ( X Ω (cid:48) ) , from which the desired inequality follows. (cid:3) In the case that X Ω is the four-dimensional ellipsoid E ( a, b ) , we can use the maps Φ a,b and Ψ a,b from §3.2 to replace the filtered L ∞ algebra V a,b with the canonical model V can a,b . This means that in order to compute the spectral invariantof a class A ∈ H ( B V a,b ) we can consider its image under H ( (cid:98) Φ a,b ) : H ( B V a,b ) → H ( B V can a,b ) and then directly read off its action. We arrive at the following refined version ofTheorem 4.1.1 which gives a more direct computation of the capacities g b : OMPUTING HIGHER SYMPLECTIC CAPACITIES I 27
Theorem 4.3.1. If E ( a, b ) is the four-dimensional ellipsoid with area parameters a, b ∈ R > , for any b ∈ S K [ t ] we have g b ( E ( a, b )) = A a,b (( (cid:98) Φ a,b ◦ (cid:98) Ψ sk )( b )) . Remark . The basis { A , A , A , . . . } for V can a,b naturally induces a basis for B V can a,b .Since the differential of B V can a,b is trivial, each element of the latter basis is a left endpointof a semi-infinite barcode as in §4.2. Proof of Theorem 1.2.4.
Given a symplectic embedding E ( a, b ) × C N s (cid:44) → E ( a (cid:48) , b (cid:48) ) × C N for some N ∈ Z ≥ , the induced filtered L ∞ homomorphism Q : V a (cid:48) ,b (cid:48) → V a,b fromCorollary 1.2.3 is well-defined up to unfiltered L ∞ homotopy, and hence so is Φ a,b ◦ Q ◦ Ψ a (cid:48) ,b (cid:48) : V can a (cid:48) ,b (cid:48) → V can a,b . Since the L ∞ operations on V can a,b and V can a (cid:48) ,b (cid:48) are trivial, L ∞ homotopies have no effect, i.e. the compositions Φ a,b ◦ Q ◦ Ψ a (cid:48) ,b (cid:48) and Φ a,b ◦ Ψ a (cid:48) ,b (cid:48) are L ∞ homotopic and therefore must be equal. In particular, each nonzero structure coefficientof the L ∞ homomorphism Φ a,b ◦ Ψ a (cid:48) ,b (cid:48) gives rise to an action inequality, and we canreadily compute these structure coefficients using (3.2.1) and (3.2.2). (cid:3) The basic observation underlying this subsection is asfollows. Suppose that we have a filtered L ∞ homomorphism Q : V Ω (cid:48) → V Ω . Suppose thatfor some basis elements v (cid:48) , . . . , v (cid:48) k ∈ V Ω (cid:48) and v , . . . , v l ∈ V Ω (cid:48) , the induced map on barcomplexes (cid:98) Q : B V Ω (cid:48) → B V Ω has a nonzero structure coefficient (cid:104) (cid:98) Q ( v (cid:48) (cid:12) · · · (cid:12) v (cid:48) k ) , v (cid:12)· · · (cid:12) v l (cid:105) (cid:54) = 0 . Then by filtration considerations we must have the inequality k (cid:88) i =1 A Ω (cid:48) ( v (cid:48) i ) ≥ l (cid:88) j =1 A Ω ( v j ) . (4.4.1)However, (cid:98) Q is not an arbitrary filtered chain map, since it comes from the filtered L ∞ homomorphism Q . In fact, there must be surjective set map M : { , . . . , l } → { , . . . , k } such that for j = 1 , . . . , k we have (cid:88) i ∈ M − ( j ) A Ω (cid:48) ( v (cid:48) i ) ≥ A Ω ( v j ) . (4.4.2)Note that the bar complex spectral invariants only have access to total inequalities ofthe form (4.4.1). In the case of ellipsoids, by triviality of the L ∞ operations for V can a,b these give the same obstructions as the individual inequalities (4.4.2) (c.f. [Sie1, §6.3]).However, for more general domains the latter inequalities could in principle give strongerobstructions.We now illustrate this phenomenon by proving Theorem 1.2.7 and Theorem 1.2.8 fromthe introduction. Before proving these results, we need the following: Proposition 4.4.1.
Put
Ω = Ω P ( a,b ) for a ≤ b .(1) For all d ∈ Z ≥ , the unique representative x of ( (cid:12) d − β , ) (cid:12) β , provided byLemma 3.2.6 satisfies (cid:104) x, β d − (cid:105) (cid:54) = 0 , (2) For all d ∈ Z ≥ , the unique representative x of (cid:12) d β , provided by Lemma 3.2.6satisfies (cid:104) x, β d − (cid:105) (cid:54) = 0 . Proof.
To prove (2), observe that a generator of β i,j is action minimal for V Ω P ( a,b ) if andonly if it is action minimal for V Ω E sk . By Corollary 1.2.11, we have (cid:104) x, β d − (cid:105) = S d = d ! T d .According to [MSie, Cor. 4.1.3], this is positive for all d ∈ Z ≥ .As for (1), we give an example computation, the general being manifest from this. Wehave β , (cid:12) β , (cid:12) β , (cid:12) β , ∼ β , (cid:12) β , (cid:12) β , (cid:12) β , + 3 β , (cid:12) β , (cid:12) β , β , (cid:12) β , (cid:12) β , ∼ β , (cid:12) β , (cid:12) β , + 2 β , (cid:12) β , β , (cid:12) β , ∼ β , (cid:12) β , + β , β , ∼ β , , and hence we have (cid:104) x, β , (cid:105) = (3)(2)(1)(7) (cid:54) = 0 . (cid:3) Remark . One could also try to prove (1) in Proposition 4.4.1 directly from thecombinatorial definition of S d , but this appears to be much less straightforward than theanalogous computation for (2). We have used computer calculations to independentlyverify S d > for d = 1 , . . . , . Proof of Theorem 1.2.7.
Let Q : V Ω (cid:48) → V Ω be a filtered L ∞ homomorphism for Ω (cid:48) :=Ω P ( c,c ) and Ω := Ω P (1 ,a ) as guaranteed by Corollary 1.2.3. Suppose that we have anonzero structure coefficient (cid:104) (cid:98) Q ( (cid:12) d − β , (cid:12) β , ) , β i ,j (cid:12) · · · (cid:12) β i k ,j k (cid:105) (cid:54) = 0 for some d ∈ Z ≥ and β i ,j , . . . , β i k ,j k ∈ V Ω . Then from the definition of (cid:98) Q we canfind d , . . . , d s ∈ Z ≥ such that either (cid:104) Q d s ( (cid:12) d s β , ) , β i s ,j s (cid:105) (cid:54) = 0 or (cid:104) Q d s ( (cid:12) d s − β , (cid:12) β , ) , β i s ,j s (cid:105) (cid:54) = 0 for each s = 1 , . . . , k . Either way, by action and index considerations wemust then have cd s ≥ i s + aj s and i s + j s = 2 d s − for each s = 1 , . . . , k . Eliminating i s , this means we have c ≥ d s − − j s + aj s d s for s = 1 , . . . k .Now suppose by contradiction that we have both c < and c < a . We claim that j s = 0 for s = 1 , . . . , k . Assuming this claim, it follows that (cid:98) Q ( (cid:12) d − β , (cid:12) β , ) isa linear combination of tensor products of action minimal basis elements of V Ω as inLemma 3.2.6. Since (cid:98) Q induces the identity map H ( B V Ω (cid:48) ) → H ( B V Ω ) on homology, wealso have that (cid:98) Q ( (cid:12) d − β , (cid:12) β , ) is homologous to (cid:12) d − β , (cid:12) β , in B V Ω . Then by (1)in Proposition 4.4.1, we have (cid:104) (cid:98) Q ( (cid:12) d − β , (cid:12) β , ) , β d − , (cid:105) (cid:54) = 0 . Action considerations then imply cd ≥ d − , and since d ∈ Z ≥ is arbitrary this gives c ≥ , which is a contradiction. OMPUTING HIGHER SYMPLECTIC CAPACITIES I 29
We now justify the above claim. First suppose that we have a ≥ . Using the inequality d s − − j s + aj s d s < we get j s ( a − ≤ , and hence j s = 0 as claimed.Now suppose that we have a < . Using the inequality d s − − j s + aj s d s < a we get j s < d s ( a −
2) + 1 a − . To conclude that j s = 0 , it suffices to show that we have d s ( a −
2) + 1 a − ≤ , i.e. d s ( a − ≤ a − , which holds since a < and d s ≥ .Finally, to establish the sharpness claim, observe that there is a naive inclusion P (1 , a ) ⊂ P ( a, a ) , so this must be optimal for a ≤ . For a ≥ , we instead use thesymplectic embedding P (1 , a ) × C N s (cid:44) → P (2 , × C N which exists for all N ≥ by [Hin1,Thm. 1.4]. (cid:3) Remark . (1) In the case a ≥ , Proposition 1.2.7 is subsumed by [Irv], which also covers themuch more general case of target P ( c, d ) .(2) In the four-dimensional case, the obstructions in Proposition 1.2.7 also show thatthe naive inclusion P (1 , a ) s (cid:44) → P ( a, a ) is optimal for a ≤ . This is a specialcase of [Hut3, Thm. 1.6], which also covers more general target polydisks. For a ≥ , According to [Sch2, Prop. 4.4.4], for a ≥ symplectic folding gives P (1 , a ) s (cid:44) → P ( c, c ) for any c > a/ , and for a > multiple symplectic foldinggives an even better embedding. We do not know to what extent these areoptimal. Proof of Theorem 1.2.8.
This is similar to the proof of Proposition 1.2.7. By Corol-lary 1.2.3, we have a filtered L ∞ homomorphism Q : V Ω (cid:48) → V Ω , now with Ω (cid:48) := Ω B ( c ) and Ω := P (1 , a ) . Suppose that we have a nonzero structure coefficient (cid:104) (cid:98) Q ( (cid:12) d β , ) , β i ,j (cid:12) · · · (cid:12) β i k ,j k (cid:105) (cid:54) = 0 for some d ∈ Z ≥ and β i ,j , . . . , β i k ,j k ∈ V Ω . Then we can find d , . . . , d s ∈ Z ≥ suchthat (cid:104) Q d s ( (cid:12) d s β , ) , β i s ,j s (cid:105) (cid:54) = 0 for s = 1 , . . . , d . By action and index considerations wemust have cd s ≥ i s + aj s and i s + j s = 3 d s − , and eliminating i s gives c ≥ d s − − j s + aj s d s for s = 1 , . . . , k .Suppose by contradiction that we have both c < and c < a + 1 . We claim that j s = 0 for s = 1 , ..., k . Assuming this claim, it follows as in the proof of Proposition 4.4.1,except using (2) instead of (1) in Proposition 4.4.1, that we have (cid:104) (cid:98) Q ( (cid:12) l β , , β d − (cid:105) (cid:54) = 0 .Action considerations then give cd ≥ d − , and since d ∈ Z ≥ is arbitrary we get c ≥ ,which is a contradiction.To justify the claim, first suppose that a ≥ . Then the inequality d s − − j s + aj s d s < gives j s ( a ) < , and hence j s = 0 as claimed.Now suppose that we have a < . Then the inequality d s − − j s + aj s d s < a gives j s < d s ( a − a − . To conclude that j s = 0 , it suffices to show that d s ( a − a − < .This is equivalent to d s ≥ − a − a , which holds since a < and d s ≥ .Finally, to establish the sharpness claim, note that this is a naive inclusion P (1 , a ) ⊂ B ( a + 1) , so this must be optimal for a ≤ . For a ≥ , we instead use the symplecticembedding P (1 , a ) × C N s (cid:44) → B (3) × C N which exists for all N ≥ by [Hin1, Thm.1.3]. (cid:3) Remark . In the case a ≥ , Proposition 1.2.8 is covered by [Hin2, Thm. 3.2]. Thecombinatorics of our proof is formally similar to and inspired by the works [HL, HO, Hin2].
5. Enumerative implications
In this section we explore to what extent the formulas from §2 be interpreted ascomputations of enumerative invariants. For concreteness we mostly restrict the discussionto four-dimensional ellipsoids. L ∞ homomorphisms. Consider the filtered L ∞ algebra CH lin ( E ( a, b )) . The underlying K -module is freely generated by the Reeb orbits of ∂E ( a, b ) . Assuming that a and b are rationally independent, these Reeb orbits are of theform γ short; k and γ long; k with actions ak and bk respectively, for k ∈ Z ≥ . If we write outthe Reeb orbits in order of increasing action, the Conley–Zehnder index of the k th oneis n − k . For the filtered L ∞ algebra CH lin ( E ( a, b )) , the underlying K -module isfreely generated by the Reeb orbits of ∂E ( a, b ) . With our L ∞ grading conventions as in§2.1, the grading of a Reeb orbit γ is n − CZ( γ ) − , where n = 2 is half the ambientdimension. In the sequel we will sometimes implicitly identify V can a,b and CH lin ( E ( a, b )) byassociating A i with the i th Reeb orbit in the above list. This identification preserves the Here we are computing Conley–Zehnder indices with respect to a global trivialization of the contactvector bundle over ∂E ( a, b ) . This trivialization is denoted by τ ex in [MSie], and it has the property thatthe first Chern class term in the SFT index formula disappears for ellipsoids. OMPUTING HIGHER SYMPLECTIC CAPACITIES I 31 degree and action of generators, while all of the L ∞ operations for both CH lin ( E ( a, b )) and V can a,b vanish for degree parity reasons.Now suppose we have a symplectic embedding E a,b × C N s (cid:44) → E a (cid:48) ,b (cid:48) × C N for some N ∈ Z ≥ . By Theorem 1.2.2 and its proof in [Sie2], we have the following diagram offiltered L ∞ homomorphisms, which commutes up to unfiltered L ∞ homotopy: CH lin ( E ( a (cid:48) , b (cid:48) )) Ξ (cid:15) (cid:15) G a (cid:48) ,b (cid:48) (cid:44) (cid:44) V a (cid:48) ,b (cid:48) (cid:15) (cid:15) F a (cid:48) ,b (cid:48) (cid:111) (cid:111) Φ a (cid:48) ,b (cid:48) (cid:44) (cid:44) V can a (cid:48) ,b (cid:48) Φ a,b ◦ Ψ a (cid:48) ,b (cid:48) (cid:15) (cid:15) Ψ a (cid:48) ,b (cid:48) (cid:108) (cid:108) CH lin ( E ( a, b )) G a,b (cid:44) (cid:44) V a,bF a,b (cid:111) (cid:111) Φ a,b (cid:44) (cid:44) V can a,b Ψ a,b (cid:107) (cid:107) (5.1.1)Here the left vertical map is the SFT cobordism map (c.f. [Sie1, §3.4]), and the Φ and Ψ maps are the ones we constructed in §3.2. The F and G maps also come from certainauxiliary SFT cobordism maps (see §5.3 below).Our aim is to understand the map Ξ , which, as we recall in §5.3 below, enumerates(at least in favorable situations) curves in E ( a (cid:48) , b (cid:48) ) \ E ( a, b ) . The upshot of the abovediagram is that the L ∞ homomorphism Ξ is identified with F a,b ◦ Ψ a,b ◦ (Φ a,b ◦ Ψ a (cid:48) ,b (cid:48) ) ◦ Φ a (cid:48) ,b (cid:48) ◦ G a (cid:48) ,b (cid:48) . That is, we have
Ξ = Φ a,b ◦ Ψ a (cid:48) ,b (cid:48) up to pre-composing and post-composing with filtered L ∞ self homotopy equivalences of CH lin ( E ( a (cid:48) , b (cid:48) )) and CH lin ( E ( a, b )) respectively. In order to better understand the above ambiguity in the filtered L ∞ homomorphism V can a (cid:48) ,b (cid:48) → V can a,b , it is convenient to introduce the following partial order on the basiselements of B V can a,b (and similarly for B V can a (cid:48) ,b (cid:48) ). Note that this also induces a partial orderon the basis elements of B CH lin ( E ( a, b )) via the identification CH lin ( E ( a, b )) ≈ V can a,b . Definition 5.1.1.
We define a partial order on the basis elements of B V can a,b as follows.Firstly, for basis elements v (cid:12) · · · (cid:12) v k ∈ B V can a,b and v ∈ V can a,b = B ≤ V can a,b ⊂ B V can a,b , weput v (cid:22) v (cid:12) · · · (cid:12) v k if the following two conditions hold:(1) (cid:80) ki =1 A a,b ( v i ) ≥ A a,b ( v ) (2) (cid:80) ki =1 | v i | + k − | v | .More generally, for v (cid:48) (cid:12) · · · (cid:12) v (cid:48) l ∈ B V can a,b , we put v (cid:48) (cid:12) · · · (cid:12) v (cid:48) l (cid:22) v (cid:12) · · · (cid:12) v k if thereexists a surjective set map { , . . . , k } → { , . . . , l } such that for each i ∈ { , . . . , l } wehave v i (cid:22) (cid:12) j ∈ f − ( i ) v j . Remark . The conditions (1) and (2) above are precisely the action and indexconditions needed for a filtered L ∞ homomorphism χ : V can a,b → V can a,b to have a nonzerostructure coefficient (cid:104) χ k ( v (cid:12) · · · (cid:12) v k ) , v (cid:105) . We note the similarity to the partial orderdefined in [HT]. Note that each of the arrows in (5.1.1) represents an L ∞ homomorphism as in Definition 2.1.2, i.e.a sequence of k -to- maps for k ∈ Z ≥ , or alternatively a single map on the level of bar complexes. It turns out that we only need to worry about L ∞ self homotopy equivalences whichare the identity at the linear level: Definition 5.1.3. An L ∞ homomorphism χ from an L ∞ algebra to itself is linearlyidentical if the linear term χ is the identity map.We then have: Lemma 5.1.4.
Let Q : V can a (cid:48) ,b (cid:48) → V can a,b be a filtered L ∞ homomorphism. Assume that v (cid:12) · · · (cid:12) v k ∈ B V can a (cid:48) ,b (cid:48) is minimal with respect to the above partial order, and similarlythat v ∈ B V can a,b is maximal. Then the structure coefficients (cid:104) Q k ( v (cid:12) · · · (cid:12) v k ) , v (cid:105) areunchanged if we pre-compose or post-compose Q by linearly identical filtered L ∞ selfhomotopy equivalences of V can a (cid:48) ,b (cid:48) and V can a,b respectively. We now adopt a more geo-metric perspective and consider counts of punctured pseudoholomorphic curves in asymplectic cobordism of the form E ( a (cid:48) , b (cid:48) ) \ E ( a, b ) . Let J be a generic almost com-plex structure on the symplectic completion of E ( a (cid:48) , b (cid:48) ) \ E ( a, b ) . Fix a collection ofReeb orbits Γ + = ( γ +1 , . . . , γ + k ) in ∂E ( a (cid:48) , b (cid:48) ) and Γ − = ( γ − , . . . , γ − l ) in ∂E ( a, b ) , and let M JE ( a (cid:48) ,b (cid:48) ) \ E ( a,b ) (Γ + ; Γ − ) denote the moduli space of genus zero J -holomorphic curvesin E ( a (cid:48) , b (cid:48) ) \ E ( a, b ) with k positive ends asymptotic to γ +1 , . . . , γ + k and l negative endsasymptotic to γ − , . . . , γ − l . More precisely, following [Sie1, §3], M JE ( a (cid:48) ,b (cid:48) ) \ E ( a,b ) (Γ + ; Γ − ) isdefined with the following features: • each puncture of a curve has a freely varying asymptotic marker which is requiredto map to a chosen basepoint on the image of the corresponding Reeb orbit • those punctures asymptotic to the same Reeb orbit are ordered • each curve is unparametrized , i.e. we quotient by the group of biholomorphicreparametrizations.Near any somewhere injective curve u , M JE ( a (cid:48) ,b (cid:48) ) \ E ( a,b ) (Γ + ; Γ − ) is a smooth manifoldof dimension ind M JE ( a (cid:48) ,b (cid:48) ) \ E ( a,b ) (Γ + ; Γ − ) = (2 − − k − l ) + k (cid:88) i =1 CZ( γ + i ) − l (cid:88) j =1 CZ( γ − j ) . (5.2.1)On the other hand, multiply covered curves in E ( a (cid:48) , b (cid:48) ) \ E ( a, b ) tend to appear with higher-than-expected dimension, necessitating abstract perturbations to achieve transversality.In general the SFT compactification M JE ( a (cid:48) ,b (cid:48) ) \ E ( a,b ) (Γ + ; Γ − ) (see [Sie1, §3.3]) will includeboundary strata consisting of pseudoholomorphic buildings with negative expecteddimension. Examples 5.2.8 and 5.2.9 at the end of this subsection illustrate situationswhere a naive counting of regular curves is not available.Recall that the k -to- part Ξ k of the cobordism map Ξ : CH lin ( E ( a (cid:48) , b (cid:48) )) → CH lin ( E ( a, b )) counts index rational curves in E ( a (cid:48) , b (cid:48) ) \ E ( a, b ) with k positive ends and one negativeend. More precisely, for Γ + = ( γ +1 , . . . , γ + k ) a collection of Reeb orbits in ∂E ( a (cid:48) , b (cid:48) ) and Strictly speaking we should count anchored curves, but anchors do not appear for ellipsoids sincethe natural contact forms on their boundaries are dynamically convex.
OMPUTING HIGHER SYMPLECTIC CAPACITIES I 33 Γ − := ( γ − ) a single Reeb in ∂E ( a, b ) , assume that we have ind M JE ( a (cid:48) ,b (cid:48) ) \ E ( a,b ) (Γ + ; Γ − ) =0 . We introduce the following combinatorial factors: • κ γ + i is the covering multiplicity of the Reeb orbit γ + i , and we put κ Γ + := κ γ +1 . . . κ γ + k • µ Γ + is the number of ways of ordering the punctures asymptotic to each Reeborbit (see [Sie1, §3.4.1]).By definition, the structure coefficients are given by (cid:104) Ξ k ( γ +1 (cid:12) · · · (cid:12) γ + k ) , γ − (cid:105) = 1 κ Γ + M JE ( a (cid:48) ,b (cid:48) ) \ E ( a,b ) (Γ + ; Γ − ) . (5.2.2) Remark . If u is a somewhere injective curve in M JE ( a (cid:48) ,b (cid:48) ) \ E ( a,b ) (Γ + ; Γ − ) , then theunderlying curve with unordered punctures and without asymptotic markers makes a totalcontribution of κ γ − µ Γ + to the structure coefficient (cid:104) Ξ k ( γ +1 , . . . , γ + k ) , γ − (cid:105) . This is becausethere are precisely κ γ + i possible placements of the asymptotic marker at the i th positivepuncture, and the number of possible orderings of the positive punctures is precisely µ Γ + . More generally, if u is a regular multiply covered curve in M JE ( a (cid:48) ,b (cid:48) ) \ E ( a,b ) (Γ + ; Γ − ) with covering multiplicity κ u , then each underlying curve after ignoring the asymptoticmarkers and orderings of the punctures contributes κ γ − µ Γ + /κ u . Remark . The structure coefficients of the cobordism map Ξ are only canonicallydefined up to pre-composing and post-composing with linearly identical filtered L ∞ self homotopy equivalences of CH lin ( E ( a (cid:48) , b (cid:48) )) and CH lin ( E ( a, b )) respectively. Indeed,although negative index curves in the symplectizations of ∂E ( a (cid:48) , b (cid:48) ) and ∂E ( a, b ) donot arise, we do have index multiple covers of trivial cylinders. These curves canappear in the SFT compactifications of moduli spaces of curves with two or more positiveends, and hence lead to ambiguities in counting problems (c.f. [MSie, Rmk. 3.2.2] andExample 5.2.9 below). On the other hand, since there is at most one Reeb orbit of ∂E ( a, b ) or ∂E ( a (cid:48) , b (cid:48) ) in any given degree, these ambiguities do not arise at the linearlevel.In favorable situations, one can show that the compactified moduli space M JE ( a (cid:48) ,b (cid:48) ) \ E ( a,b ) (Γ + ; Γ − ) consists only of regular curves for generic J , and hence can be defined without recourse toany virtual perturbation techniques. The idea is to use the fact that somewhere injectivecurves are regular for generic J , and then use action and index considerations to arguethat no multiple covers or nontrivial pseudoholomorphic buildings can appear. One canthen use the SFT compactness theorem to argue that the count is finite and independentof J (provided that it is generic). The following two lemmas illustrate this point. Lemma 5.2.3 ([McD2, MSie2]) . Let Γ + = ( γ +1 , . . . , γ + k ) be a collection of Reeb orbits in ∂E ( a (cid:48) , b (cid:48) ) which is minimal with respect to partial order from Definition 5.1.1. Supposethat γ − = γ short; m is the m -fold iterate of the short simple Reeb orbit of ∂E ( a, b ) forsome m ∈ Z ≥ such that m < b/a . Let J be a generic SFT-admissible almost complex In this paper we find it convenient to use a slightly different convention with respect to the κ γ factorscompared to [Sie1, §3.4.2], which instead puts (cid:104) Ξ k ( γ +1 (cid:12) · · · (cid:12) γ + k ) , γ − (cid:105) = κ Γ − M JE ( a (cid:48) ,b (cid:48) ) \ E ( a,b ) (Γ + ; Γ − ) .These differ by the change of basis γ ↔ κ γ γ . structure on the symplectic completion of E ( a (cid:48) , b (cid:48) ) \ εE ( a, b ) for ε > sufficiently small.Then assuming its index is zero, the compactified moduli space M JE ( a (cid:48) ,b (cid:48) ) \ εE ( a,b ) (Γ + ; ( γ − )) consists entirely of regular curves and coincides with the uncompactified moduli space M JE ( a (cid:48) ,b (cid:48) ) \ εE ( a,b ) (Γ + ; ( γ − )) . Moreover, the count M JE ( a (cid:48) ,b (cid:48) ) \ εE ( a,b ) (Γ + ; ( γ − )) is finiteand independent of the choice of ε and generic J . Lemma 5.2.4 ([MSie2]) . Let Γ + = ( γ long , . . . , γ long ) be a collection of d copies of thelong simple Reeb orbit E (1 , δ ) for δ > sufficiently small. Suppose that γ − is a Reeborbit of ∂E ( a, b ) which is maximal with respect to the partial order from Definition 5.1.1.Let J be a generic SFT-admissible almost complex structure on the symplectic completionof E (1 , δ ) \ εE ( a, b ) for ε > sufficiently small. Then assuming its index is zero,the compactified moduli space M JE (1 , δ ) \ εE ( a,b ) (Γ + ; ( γ − )) consists entirely of regularcurves and coincides with the uncompactified moduli space M JE (1 , δ ) \ εE ( a,b ) (Γ + ; ( γ − )) .Moreover, the count M JE (1 , δ ) \ εE ( a,b ) (Γ + ; ( γ − )) is finite and independent of the choiceof δ, ε and generic J .Remark . (1) In Lemma 5.2.3, the condition m < b/a means we can replace E ( a, b ) with theskinny ellipsoid E sk . These are precisely the types of counts appearing in thedefinition of g b ( E ( a, b )) .(2) In Lemma 5.2.4, we can equivalently count degree d J -holomorphic planes in CP \ E ( a, b ) with negative end asymptotic to γ − . These are precisely the typesof counts appearing in the works [HK, CGH, CGHM, McD2] on the restrictedstabilized ellipsoid embedding problem.The following lemma, which is essentially a special case of Proposition 1.1.8 from theintroduction, gives a useful setting in which Lemma 5.2.4 applies. We provide a proof forthe sake of completeness. Lemma 5.2.6.
Suppose that p + q = 3 d for some p, q, d ∈ Z ≥ . Assume also that thereis no partition k + · · · + k m = p of p for some m ∈ Z ≥ and k , . . . , k m ∈ Z ≥ such that (cid:80) mi =1 ( k i + (cid:100) k i q/p (cid:101) ) = 3 d . Then for x = p/q + δ with δ > sufficiently small, γ short; p ismaximal in B CH lin ( E (1 , x )) with respect to the partial order from Definition 5.1.1.Remark . Note that the hypothesis about no such partitions existing holds forexample if gcd( p, q ) = 1 . Indeed, such a partition can only exist if k i q/p is an integer for i = 1 , . . . , m , since otherwise we have m (cid:88) i =1 ( k i + (cid:100) k i q/p (cid:101) ) > m (cid:88) i =1 ( k i + k i q/p ) = p + q = 3 d. If p and q are relatively prime, this necessitates k . . . , k m ≥ p , and hence m = 1 . Proof of Lemma 5.2.6.
The actions of Reeb orbits in ∂E (1 , x ) are given by A ,x ( γ short; k ) = k, A ,x ( γ long; k ) = kx, k ∈ Z ≥ . OMPUTING HIGHER SYMPLECTIC CAPACITIES I 35
With our conventions, the degree of the element in V can a,b corresponding to the Reeb orbit γ is given by n − CZ( γ ) − with n = 2 , and we have: CZ( γ short; k ) = 1 + 2( k + (cid:98) k/x (cid:99) )CZ( γ long; k ) = 1 + 2( k + (cid:98) kx (cid:99) ) for k ∈ Z ≥ .Now suppose that, for some a, b ∈ Z ≥ and k , . . . , k a , l , . . . , l b ∈ Z ≥ , the followingindex and action conditions hold:(1) (cid:80) ai =1 ( k i + (cid:98) k i /x (cid:99) ) + (cid:80) bj =1 ( l i + (cid:98) l i x (cid:99) ) + a + b − d − (2) (cid:80) ai =1 k i + (cid:80) bj =1 xl j ≥ p It suffices to show that we must have b = 0 and a = 1 with k = p .We have d = a (cid:88) i =1 ( k i + (cid:100) k i /x (cid:101) ) + b (cid:88) j =1 ( l j + (cid:100) l j x (cid:101) ) > (1 + 1 /x ) a (cid:88) i =1 k i + b (cid:88) j =1 xl j ≥ (1 + 1 /x ) p = p + (1 + δq/p ) − q. Observe that l j ( p/q + δ ) is not an integer for δ sufficiently small, and so we have (cid:100) l j x (cid:101) − l j x = 1 + (cid:98) l j x (cid:99) − l j x, which approaches (cid:98) l j p/q (cid:99) − l j p/q > as δ → . This shows that the strict inequalityabove is false for δ > sufficiently small, unless we have b = 0 . (cid:3) To end this subsection, we give some examples which help clarify the necessity of theassumptions in Lemmas 5.2.3 and 5.2.4.
Example 5.2.8.
Consider the moduli space M JE (1 , δ ) \ εE (1 , δ (cid:48) ) (Γ + ; Γ − ) with Γ + =( γ short , γ short ) and Γ − = ( γ short;2 ) for δ, δ (cid:48) > sufficiently small, which has expecteddimension zero. Since there is a cylinder in E (1 , δ ) \ εE (1 , δ (cid:48) ) positively asymp-totic to γ short and negatively asymptotic to γ short , any double branched cover with onebranched point at the negative puncture and one branched point in the interior givesrise to an element of M JE (1 , ε ) \ εE (1 , ε (cid:48) ) (Γ + ; Γ − ) . Since these covers appear in a two-dimensional family due to the moveable branch point, this shows that the moduli space M JE (1 , δ ) \ εE (1 , δ (cid:48) ) (Γ + ; Γ − ) appears with higher-than-expected dimension. Example 5.2.9.
Consider the moduli space M JE (1 , δ ) \ εE (1 , δ (cid:48) ) (Γ + , Γ − ) with Γ + =( γ short , γ short ) and Γ − = ( γ short;3 ) , for δ, δ (cid:48) > sufficiently small, which has expecteddimension zero. A sequence of curves in this moduli space could in principle degenerateinto a two level pseudoholomorphic building with • top level in the symplectization R × ∂E (1 , δ ) consisting of a rational curvewith two positive ends both asymptotic to γ short and one negative end asymptoticto γ short;2 Figure 5.1.
Perturbing the ellipsoid E ( a, b ) to the fully rounded convextoric domain (cid:101) E ( a, b ) . • bottom level in E (1 , δ ) \ εE (1 , δ (cid:48) ) consisting of a cylinder with positiveend asymptotic to γ short;2 and negative end asymptotic to γ short;3 .Note that the curves in both levels have index zero. The starting point for the proofof Theorem 1.2.2 is to replace the convex toric domain X Ω with a “fully rounded” convextoric domain X (cid:101) Ω of the same dimension. The following definition appears implicitly in[GH, Lem. 2.7]: Definition 5.3.1.
Let X Ω ⊂ C n be a convex toric domain, and put Σ := ∂ Ω ∩ R n> . Wesay that X Ω is ε fully rounded if(1) Σ is a smooth hypersurface in R n (2) the Gauss map Ga : Σ → S n − is a smooth embedding(3) ∂X Ω is a smooth hypersurface(4) for i ∈ { , . . . n } and any point p ∈ Σ ∩ { ( z , . . . , z n ) ∈ C n : z i = 0 } , the i thcomponent of G ( p ) is less than ε .We typically take ε > to be some sufficiently small constant and suppress it from thenotation, in which case we simply say that X Ω is fully rounded . Given any convex toricdomain X Ω , we can replace its moment map image Ω with a C -small perturbation (cid:101) Ω such that X (cid:101) Ω is fully rounded. Figure 5.3 illustrates this in the case of a four-dimensionalellipsoid X Ω = E ( a, b ) , with its fully rounded version denoted by (cid:101) E ( a, b ) . From thepoint of view of capacities or symplectic embedding obstructions this perturbation hasessentially no effect. On the other hand, the fully rounding process can have a drasticeffect on the Reeb dynamics of ∂X Ω . Indeed, whereas ∂E ( a, b ) has just two simple Reeborbits (assuming a and b are rationally independent), ∂ (cid:101) E ( a, b ) has infinitely many simpleReeb orbits. In general, for each p ∈ Σ such that Ga( p ) is a rational direction, themoment map fiber µ − ( p ) is fibered by simple Reeb orbits forming a T n − -family (see[Hut3, GH] for more details). Here we say that a vector in S n − is a rational direction if it is a positive rescaling of a vector in Z n , and we denote the set of rational directionsin S n − by S n − Q .We fix a preferred perfect Morse function f T n − : T n − → R . Using the perturbationscheme described in [Hut3, Lem. 5.4], we can find a C -close Liouville domain whosesimple Reeb orbits (up to an arbitrarily high action cutoff) are of the form γ ( q,c ) , indexedby pairs ( q, c ) such that:(1) q = ( q , . . . , q n ) ∈ Z n ≥ is nonzero and primitive Alternatively, we could directly apply Morse–Bott techniques to these families of generators.
OMPUTING HIGHER SYMPLECTIC CAPACITIES I 37 (2) c is a critical point of f T n − .Moreover, up to a small discrepancy, the action of γ ( q,c ) is given by || q || ∗ Ω , and its Conley–Zehnder index is given by CZ( γ ( q,c ) ) = n − (cid:80) ni =1 q i . We can formally extend thisdescription of the simple Reeb orbits to all Reeb orbits by allowing pairs ( q, c ) with q ∈ Z n ≥ nonzero but not necessarily primitive, and we still have A ( γ ( q,c ) ) = || q || ∗ Ω and CZ( γ ( q,c ) ) = n − (cid:80) ni =1 q i . From now on we will assume that such a perturbationhas been performed and we suppress it from the notation.Although the fully rounding process can introduce additional Reeb orbits, it turns outthat the relevant curves become easier to describe. Indeed, at least in the four-dimensionalcase, the above description gives a natural bijective correspondence between Reeb orbitsof ∂X (cid:101) Ω (up to an arbitrarily large action cutoff) and basis elements of V Ω . Moreover, forany X Ω this correspondence preserves degree and (up to small discrepancies) action. Theproof of Theorem 1.2.2 in [Sie2] extends this identification on basis elements to the levelof curves by deforming X (cid:101) Ω to a situation where curves can be explicitly enumerated.Up to minor rescalings, we have inclusions X Ω ⊂ X (cid:101) Ω and X (cid:101) Ω ⊂ X Ω . These inducefiltered L ∞ homomorphisms CH lin ( X (cid:101) Ω ) → CH lin ( X Ω ) and CH lin ( X Ω ) → CH lin ( X (cid:101) Ω ) which are identified with the maps F Ω and G Ω respectively from §5.1.Now suppose that we have an ellipsoid embedding E ( a, b ) s (cid:44) → E ( a (cid:48) , b (cid:48) ) . For enumerativepurposes, we can assume this is an inclusion, after possibly replacing E ( a, b ) with εE ( a, b ) for ε > sufficiently small. Up to the ambiguities considered in Lemma 5.1.4, the L ∞ homomorphism Φ a,b ◦ Ψ a (cid:48) ,b (cid:48) : V can a (cid:48) ,b (cid:48) → V can a,b coincides with the induced cobordismmap Ξ : CH lin ( E ( a (cid:48) , b (cid:48) )) → CH lin ( E ( a, b )) , provided that we choose the constants C q ; a,b and C q ; a (cid:48) ,b (cid:48) in Construction 3.2.1 so that we have agreement at the linear level, i.e. (Φ a,b ◦ Ψ a (cid:48) ,b (cid:48) ) = Ξ . Since Φ s,t and F s,t are L ∞ homotopy inverses of Ψ s,t and G s,t respectively, and A s,t ( β i,k − i ) > A s,t ( A k ) unless β i,k − i = β i ( k ) , j ( k ) , it suffices to choose theconstants C q ; s,t such that (cid:104) G s,t ( A k ) , β i ( k ) , j ( k ) (cid:105) = (cid:104) Ψ s,t ( A k ) , β i ( k ) , j ( k ) (cid:105) for all s, t ∈ R > and q ∈ Z ≥ .In §5.4, we show that there is a unique cylinder u (ignoring asymptotic markers) inthe symplectic cobordism E ( s, t ) \ ε (cid:101) E ( s, t ) which is positively asymptotic to A k andnegatively asymptotic to β i ( k ) , j ( k ) . We denote by κ A k the covering multiplicity of theReeb orbit corresponding to A k in ∂E ( s, t ) . We have: • the covering multiplicity of the Reeb β i ( k ) , j ( k ) at the negative end of u is gcd( i ( k ) , j ( k )) • the covering multiplicity of u is gcd( i ( k ) , j ( k ) , κ A k ) .Using the conventions described in Remark 5.2.1, this translates into (cid:104) G s,t ( A k ) , β i ( k ) , j ( k ) (cid:105) = gcd( i ( k ) , j ( k ))gcd( i ( k ) , j ( k ) , κ A k ) . We have thus proved the following theorem. Recall that we are identifying each Reeborbit γ in ∂E ( s, t ) with the corresponding generator A i ∈ V can s,t with | A i | = n − − CZ( γ ) ,and that the structure coefficients of Ξ are given by (5.2.2). Theorem 5.3.2.
In Construction 3.2.1, put C q ; s,t := gcd( i s,t ( q ) , j s,t ( q ))gcd( i s,t ( k ) , j s,t ( q ) ,κ As,tq ) for all s, t ∈ R > and q ∈ Z ≥ . Then in the context of Lemma 5.2.3 we have κ Γ+ M JE ( a (cid:48) ,b (cid:48) ) \ εE ( a,b ) (( γ +1 , . . . , γ + k ); ( γ short; m )) = (cid:104) (Φ a,b ◦ Ψ a (cid:48) ,b (cid:48) ) k ( γ +1 , . . . , γ + k ) , A m (cid:105) . Similarly, in the context of Lemma 5.2.4, we have M JE (1 , δ ) \ εE ( a,b ) (( γ long , . . . , γ long (cid:124) (cid:123)(cid:122) (cid:125) d ); ( γ − )) = (cid:104) (Φ a,b ◦ Ψ , δ ) d ( A , . . . , A ) , γ − (cid:105) . Our goal is to characterize the cylinders in thesymplectic cobordism E ( s, t ) \ ε (cid:101) E ( s, t ) . For each k ∈ Z ≥ , a Fredholm index zero cylinderwith positive asymptotic A k must have negative asymptotic β i ( k ) , j ( k ) by action and indexconsiderations. We will show that there is a unique such cylinder u for each k ∈ Z ≥ ,and this is a multiple cover if gcd( i ( k ) , j ( k ) , κ A k ) > .The basic idea is to apply the relative adjunction formula for a punctured curve u : c τ ( u ) = χ ( u ) + Q τ ( u ) + w τ ( u ) − δ ( u ) . Here the subscripted invariants depend on a choice τ of trivialization of the contactvector bundle over each asymptotic Reeb orbit of u : c τ ( u ) is the relative first Chernclass of u , χ ( u ) is the Euler characteristic of u , Q τ ( u ) is the relative intersection pairingof u , w τ ( u ) is the difference of writhes at the top and bottom of u , and δ is a count ofsingularities of u . We refer the reader to [Hut2, §3.3] for more details.Using writhe bounds (see [MSie, §3.2] and the references therein), we first show inLemma 5.4.1 that gcd( i ( k ) , j ( k ) , κ A k ) > contradicts the relative adjunction inequality,meaning that u cannot be somewhere injective. We then consider the somewhere injectivecase, and in Lemma 5.4.2 we argue similarly for a union of two cylinders with the sameasymptotics to conclude that u must be unique.Suppose that u is a cylinder as above. We put m := κ A k , so that the Reeb orbit γ in ∂E ( s, t ) corresponding to A k is either γ short; m or γ long; m . Let θ denote the rotationangle of γ (see [Hut2, §3.2]). Since ind( u ) = 0 , the negative asymptotic is then either β i,j = β m, (cid:98) mθ (cid:99) or β i,j = β (cid:98) mθ (cid:99) ,m respectively. According to [Hut3, §5.3], we can take therotation angle of β i,j to be positive and arbitrarily close to zero. Note that β i,j is a g -foldcover of its underlying simple orbit, where we put g := gcd( i, j ) = gcd( m, (cid:98) mθ (cid:99) ) . Lemma 5.4.1.
If the cylinder u is somewhere injective, then the Reeb orbit β i,j must besimple.Proof. Using the “split” trivialization τ sp for ∂E ( s, t ) from [MSie, §3.2] and the trivializa-tion τ for (cid:101) E ( s, t ) from [Hut3], we have • c ( γ ) = m • c ( β i,j ) = i + j = m + (cid:98) mθ (cid:99)• χ ( u ) = 0 • Q ( γ ) = 0 • Q ( β i,j ) = ij = m (cid:98) mθ (cid:99) We emphasize that κ A k and the pair ( i ( k ) , j ( k )) depend sensitively on s, t . More precisely, theydepend on the ratio t/s . Here we add the s, t superscripts into the notation as a reminder. OMPUTING HIGHER SYMPLECTIC CAPACITIES I 39 • w + ( u ) ≤ (cid:98) mθ (cid:99) ( m − • w − ( u ) ≥ g − .The relative adjunction inequality then gives w + ( u ) − w − ( u ) ≥ c ( γ ) − c ( β i,j ) − χ ( u ) − Q ( u ) , so we must have (cid:98) mθ (cid:99) ( m − − ( g − ≥ −(cid:98) mθ (cid:99) + m (cid:98) mθ (cid:99) , which is a contradiction unless g = 1 . (cid:3) Similarly, we prove that when u is somewhere injective, it is the unique representativeof its moduli space. Lemma 5.4.2.
Suppose we have g = 1 , and that u and u (cid:48) are two cylinders with thesame asymptotics as above. Then u = u (cid:48) .Proof. Let C denote the union of u and u (cid:48) . For this disconnected curve we have • c ( C ) = 2 m − m + (cid:98) mθ (cid:99) ) • Q ( C ) = 0 − m (cid:98) mθ (cid:99) (see [Hut3, §5.3]) • χ ( C ) = 0 • w + ( C ) ≤ m (cid:98) mθ (cid:99) − (cid:98) mθ (cid:99)• w − ( C ) ≥ g − From the relative adjunction inequality we get m (cid:98) mθ (cid:99) − (cid:98) mθ (cid:99) − (4 g − ≥ m − m + (cid:98) mθ (cid:99) ) + 4 m (cid:98) mθ (cid:99) , i.e. g − ≤ , which is a contradiction. (cid:3) In this somewhat speculative subsection,we give an alternative description of the filtered L ∞ algebra V Ω and extend its definitionto arbitrary dimensions. We also give a sketch proof of Theorem 1.2.2 based on someexpected structural properties of symplectic cohomology. This approach could be viewedas a version of quantitative closed string mirror symmetry for convex toric domains in C n . At the end we arrive at an explicit algebraic description of V Ω . However, the proofof Theorem 1.2.2 in [Sie2] instead computes CH lin ( X Ω ) by a more direct curve countingargument.Let X Ω ⊂ C n be a convex toric domain. We work with SC S , + ( X Ω ) in place of CH lin ( X Ω ) . Our sketch computation of SC S , + ( X Ω ) is based on the following steps: Step 1
Compute
SC( D Ω ) as a filtered homotopy Batalin–Vilkovisky (BV) algebra, where D Ω is a smooth Lagrangian torus fibration over Ω which partially compactifiesto X Ω . Step 2
Quotient out the action zero generators and perform an algebraic S -quotient toarrive at the filtered L ∞ algebra SC S , + ( D Ω ) . Step 3
Deform SC S , + ( D Ω ) by the Cieliebak–Latschev Maurer–Cartan element m ∈ SC S , + ( D Ω ) to obtain SC S , + ,m ( X Ω ) .We now elaborate on each of these steps. D Ω and its BV algebra struc-ture. Firstly, in order to define D Ω , consider the translation Ω + t of Ω by a vector t ∈ R n> which is small in each coordinate, and put D Ω := µ − (Ω + t ) . Note that D Ω is fibered by the smooth Lagrangian tori µ − ( p ) for p ∈ Ω + t , and (aftera slight shrinkening) we have a natural inclusion D Ω ⊂ X Ω (see e.g. [LMT]).The Liouville domain D Ω can be viewed as a unit disk cotangent bundle of T n withrespect to a Finsler norm determined by Ω . In particular, it is Liouville deformationequivalent to the standard unit disk cotangent bundle D ∗ T n , and by Viterbo’s theorem[Abo] we have an isomorphism of BV algebras SH ∗ ( D Ω ) ∼ = H n −∗ ( L T n ) . We note that T ∗ T n is symplectomorphic to ( C ∗ ) n , whose SYZ mirror is itself, andaccordingly we have an isomorphism SH ∗ ( D Ω ) ∼ = T poly (( C ∗ ) n ) , where T poly (( C ∗ ) n ) denotes the BV algebra of algebraic polyvector fields on ( C ∗ ) n withcoefficients in K . Here the BV operator on T poly (( C ∗ ) n ) is of the form ∆ = vol( − ) − ◦ d ◦ vol( − ) , where vol denotes the volume form z ...z n dz ∧ · · · ∧ dz n on ( C ∗ ) n , vol( − ) denotes theinduced isomorphism sending a k -vector field on ( C ∗ ) n to an ( n − k ) -form, and vol( − ) − denotes the inverse isomorphism. Recall that the product and BV operator togetherdetermine a Lie bracket (namely the Schouten bracket on polyvector fields) via the BVrelation { a, b } = ∆( a · b ) − ∆( a ) · b − ( − | a | a · ∆( b ) . In the case n = 1 , T poly ( C ∗ ) has a basis of the form { z k , z k ∂ z : k ∈ Z } , where z isthe coordinate on C ∗ . With our L ∞ conventions from §2.1, the functions z k have degree − and the vector fields z k ∂ z have degree − . The product is characterized by ( z k ) · ( z l ) = z k + l , z k · z l ∂ z = z l ∂ z · z k = z k + l ∂ z , z k ∂ z · z l ∂ z = 0 . The BV operator ∆ : T poly ( C ∗ ) → T poly ( C ∗ ) is characterized by ∆( z k ∂ z ) = ( k − z k − , ∆( z k ) = 0 . For general n , the basis elements of T poly (( C ∗ ) n ) are of the form z k . . . z k n n ∂ z i ∧· · ·∧ ∂ z im of degree m − , for k , . . . , k n ∈ Z and { i , . . . , i m } a subset of { , . . . , n } of size m ∈ { , . . . , n } . In accordance with the Künneth theorem, we can also view the BValgebra T poly (( C ∗ ) n ) as the tensor product of n copies of the BV algebra T poly ( C ∗ ) : T poly (( C ∗ ) n ) ∼ = T poly ( C ∗ ) ⊗ · · · ⊗ T poly ( C ∗ ) (cid:124) (cid:123)(cid:122) (cid:125) n . Remark . Alternatively, we have the description (see [Ton, §6.2]) H n −∗ ( L T n ; K ) ∼ = K [ z ± , . . . , z ± n ] ⊗ H ∗ ( T n ; K ) , where L T n denotes the free loop space of T n , the first factor roughly corresponds to thebased loop space of T n and the second factor corresponds to the constant loops. After OMPUTING HIGHER SYMPLECTIC CAPACITIES I 41 identifying H ∗ ( T n ; K ) with the exterior algebra on the dual space of K n , we get succinctformulas for the product and BV operator: ( z (cid:126)k ⊗ λ ) · ( z (cid:126)l ⊗ µ ) = z (cid:126)k + (cid:126)l ⊗ ( λ ∧ µ ) and ∆( z (cid:126)k ⊗ λ ) = z (cid:126)k ⊗ ι (cid:126)k λ. If e , . . . , e n denotes the standard basis of K n and e ∨ , . . . , e ∨ n denotes its dual basis, thenthe identification T poly (( C ∗ ) n ) ≈ H n −∗ ( L T n ; K ) sends z (cid:126)k ∂ z i ∧· · ·∧ ∂ z im to z (cid:126)k z − i . . . z − i m ⊗ e ∨ i ∧ · · · ∧ e ∨ i m . For any Liouville domain X , we expect theBV algebra SH( X ) to admit a natural filtered chain-level refinement, giving SC( X ) thestructure of a filtered homotopy BV algebra (see [GCTV]). Moreover, we expect theViterbo isomorphism SH( T ∗ T n ) ∼ = T poly (( C ∗ ) n ) to extend to a homotopy equivalenceof filtered homotopy BV algebras. At least in the absence of filtrations, a significantstep in this direction appears in the work [CG1]. By a version of Kontsevich’s formalitytheorem (see e.g.[TT, Cam]), it makes sense to view the BV algebra T poly (( C ∗ ) n ) asa homotopy BV algebra which happens to be a differential graded Batalin–Vilkoviskyalgebra (DGBV) with trivial differential. If X Ω ⊂ C n is a convex toric domain, we endowthe DGBV algebra T poly (( C ∗ ) n ) with its filtration by putting A Ω ( λ ⊗ z (cid:126)k ) := || (cid:126)k || ∗ Ω . Recall that we are most interested in theinvariant SC S , + . To pass from SC( D Ω ) to SC + ( D Ω ) , we simply by quotient out thebasis elements of zero action, namely those of the form λ ⊗ z (cid:126) . As for the S -quotient,this appears to be rather complicated for a general homotopy BV algebra, since thequotient must behave like a homotopy quotient in order to have appropriate functorialityproperties. The general formulation should be given in terms of cyclic homology (see e.g.[Gan, §2] for the linear case and [Wes, CW] for higher structures).If X is a Liouville domain, the homotopy quotient SC S , + ( X ) should inherit thestructure of a filtered homotopy gravity algebra (see e.g. [Get1, Get2, Wes, CW]).However, we are only concerned with a small part of this structure, namely the filtered L ∞ structure on SC S , + ( X ) . Fortunately, since our model for SC S , + ( D Ω ) happens tobe DGBV such that the BV operator ∆ is acyclic, we can instead realize the S -quotientas a naive quotient by restricting to the image of ∆ . In other words, we get a modelfor the filtered DGBV algebra SC S , + ( D Ω ) by simply restricting the differential, BVoperator, and filtration of SC + ( D Ω ) to im (∆) ⊂ SC + ( D Ω ) . The inclusion D Ω ⊂ X Ω gives risea Cieliebak–Latschev Maurer–Cartan element m ∈ SC S , + ( D Ω ) ≈ CH lin ( D Ω ) as in[Sie1, §4]. Using this Maurer–Cartan element, we can define the deformed L ∞ algebra SC S , + ,m ( D Ω ) . This is the target of the induced transfer map Π : SC S , + ( X Ω ) → SC S , + ,m ( D Ω ) , and since the action of m is arbitrarily close to zero, this is in fact afiltered L ∞ homomorphism. We show in [Sie2] that up to filtered gauge equivalence wehave m = z − + · · · + z − n . Note that this resembles the superpotential of the Cliffordtorus in C n , and indeed the partial compactification D Ω (cid:32) X Ω is mirror to turning on asuperpotential (c.f. [Aur]). In general, the cobordism map Π need not be an L ∞ homotopy equivalence. However,in our case we also have (up to small shrinkenings) a symplectic embedding X Ω s (cid:44) → D Ω (see [LMT]). The induced cobordism map gives a left filtered L ∞ homotopy inverse for Π , which is also a right inverse by finite dimensionality considerations. V Ω . The above discussion motivates the following.
Definition 5.5.2.
Let X Ω ⊂ C n be a convex toric domain.(1) We define W Ω to be the DGBV algebra T poly (( C ∗ ) n ) , endowed with the filtrationgiven by A ( z (cid:126)k ⊗ λ ) = || (cid:126)k || ∗ Ω .(2) We define V Ω to be the filtered DGLA given by restricting the filtration, differen-tial, and bracket of W Ω to the subspace im (∆) ⊂ W Ω . Remark . In the case n = 2 , the above definition is equivalent to Definition 2.3.3.Indeed, the BV operator in T poly (( C ∗ ) ) is given by ∆( z k z l ) = 0∆( z k z l ∂ z ) = ( k − z k − z l ∆( z k z l ∂ z ) = ( l − z k z l − ∆( z k z l ∂ z ∧ ∂ z ) = ( k − z k − z l ∂ z − ( l − z k z l − ∂ z , and we put α i,j := iz i z j +12 ∂ z − jz i +11 z j ∂ z and β i,j := z i z j . We then have for example ∂ ( α i,j ) = [ α i,j , m ]= [ iz i z j +12 ∂ z − jz i +11 z j ∂ z , z − + z − ]= jz i − z j − iz i z j − = jβ i − ,j − iβ i,j − . One can similarly check that we have [ α i,j , β k,l ] = ( il − jk ) β i + k,j + l and so on. The computations presented in Table 5.1 were performed withthe aid of a computer program. Recall that we put S d ;1 ,x := d ! k (cid:104) Φ ,x ◦ Ψ , ( (cid:12) d A ) , A d − (cid:105) . Here we consider the case x = p/q > τ with p + q = 3 d for small d , which is relevant forthe restricted stabilized ellipsoid embedding problem in light of Lemma 1.1.11. References [Abo] Mohammed Abouzaid. Symplectic cohomology and Viterbo’s theorem. arXiv:1312.3354 (2013).5.5.1[Aur] Denis Auroux. Mirror symmetry and T-duality in the complement of an anticanonical divisor.
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