The ghost stairs stabilize to sharp symplectic embedding obstructions
TTHE GHOST STAIRS STABILIZE TO SHARP SYMPLECTICEMBEDDING OBSTRUCTIONS
DAN CRISTOFARO-GARDINER, RICHARD HIND, AND DUSA MCDUFF
Abstract.
In determining when a four-dimensional ellipsoid can be symplecticallyembedded into a ball, McDuff and Schlenk found an infinite sequence of “ghost” ob-structions that generate an infinite “ghost staircase” determined by the even indexFibonacci numbers. The ghost obstructions are not visible for the four-dimensionalembedding problem because strictly stronger obstructions also exist. We show that incontrast, the embedding constraints associated to the ghost obstructions are sharp forthe stabilized problem; moreover, the corresponding optimal embeddings are givenby symplectic folding. The proof introduces several ideas of independent interest,namely: (i) an improved version of the index inequality familiar from the theoryof embedded contact homology (ECH), (ii) new applications of relative intersectiontheory in the context of neck stretching analysis, (iii) a new approach to estimatingthe ECH grading of multiply covered elliptic orbits in terms of areas and contin-ued fractions, and (iv) a new technique for understanding the ECH of ellipsoids byconstructing explicit bijections between certain sets of lattice points.
Contents
1. Introduction 21.1. Background 21.2. The ghost stairs 31.3. Methods and relationship with embedded contact homology 42. Preliminaries 72.1. Weight sequences and best approximations 72.2. Basics of embedded contact homology 143. Stabilizable curves 223.1. The setup 223.2. The low action curves 273.3. Analysing C U when n ą C L has no connectors 353.5. The case n “ C L has connectors. 51 Date : February 10, 2017.DCG partially supported by NSF grant DMS-1402200.RH partially supported by the Simons Foundation under grant a r X i v : . [ m a t h . S G ] F e b DAN CRISTOFARO-GARDINER, RICHARD HIND, AND DUSA MCDUFF
Introduction
Background.
Let p M , ω q , p M , ω q be symplectic manifolds. A symplectic em-bedding p M , ω q s ã Ñ p M , ω q is a smooth embedding Ψ : M Ñ M such that Ψ ˚ ω “ ω . It can be a difficultproblem to determine whether or not one symplectic manifold can be embedded intoanother; this is particularly true when the manifolds have the same dimension .In fact, even deciding when one symplectic ellipsoid E p a , . . . , a n q “ " π | z | a ` . . . ` π | z n | a n ă * Ă C n , can be embedded into another is largely open. Hofer conjectured a purely combina-torial criteria for settling the n “ c k p x q “ inf t µ | E p , x q ˆ C k s ã Ñ B p µ q ˆ C k u , where B p µ q “ E p µ, µ q is the 4-ball of capacity µ . This is the stabilized version of thefunction c p a q , which was computed by McDuff and Schlenk in [MS]. It is a version ofthe ellipsoid embedding problem in which most of the arguments are infinite.The main theorem of [CGHi] states that c k p x q “ c p x q , ď x ď τ , where τ : “ `? denotes the Golden Mean. The function c k p x q is currently unknownfor x ą τ ; in fact, it is not even known whether or not this function depends on k ě is known, however, that(1.1.1) c k p x q ď xx ` , x ą τ , If dim p M q ě dim p M q ` M is open, versions of Gromov’s h -principle apply. In fact, an optimal embedding can be realized in all cases where the value of c k p x q is known.Optimal 4-dimensional embeddings exist by Corollary 1.6 in [M1] and this covers the cases when c k p x q “ c p x q . The folding maps which give our other cases are quite explicit, and we can get optimalembeddings from Theorem 4.3 in [PV]. HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 3 because of an explicit “symplectic folding construction” given by Hind in [Hi] . Inparticular, because the volume bound gives c p x q ě ? x , we have c k p x q ă c p x q for x ą τ . It is then natural to ask the following: Question 1.1.1.
Is it the case that (1.1.2) c k p x q “ xx ` for x ą τ and k ě ? An affirmative answer to Question 1.1.1 would imply that the stabilized embeddingproblem is quite rigid: all of the optimal embeddings would be given either by stabilizingfour-dimensional embeddings as in [CGHi], or by Hind’s generalization of symplecticfolding. On the other hand, an outcome in the negative would require the existenceof as yet unknown embeddings. In their proof [HiK2] that c k p x q is asymptotic to 3 as x Ñ 8 , Hind and Kerman showed that (1.1.2) holds for all integers of the form 3 g n ´ g n is an odd index Fibonacci number.1.2. The ghost stairs.
One of the more mysterious aspects of McDuff and Schlenk’scomputation of c p x q is the “ghost stairs”, which we now review.Recall first of all that the function c p x q is particularly intricate for 1 ď x ď τ .Here, it is given by an infinite staircase determined by the odd index Fibonacci numbers g ‚ “ p , , , , , . . . q , called the “Fibonacci staircase”. For τ ď x ď
7, the function c p x q is seemingly simpler — it turns out that(1.2.1) c p x q “ x ` x in this range. Nevertheless, McDuff and Schlenk show that there is a kind ofanalogue of the Fibonacci staircase underlying (1.2.1), which they call the ghost stairs .The idea is that the deviation of the Fibonacci staircase from the classical volumeconstraint when x ă τ comes from a sequence of sharp obstructions, one for each ofthe embedding problems E ˆ , g n ` g n ˙ Ñ B p µ q . These obstructions imply that c ˆ g n ` g n ˙ “ g n ` g n ` “ g n ` g n g n ` g n ` , where the second equality holds by the Fibonacci identity 3 g n ` “ g n ` g n ` . One canwrite down analogous obstructions for the problem(1.2.2) E p , b n q Ñ B p µ q , where the b n , n ě , are determined by the even index Fibonacci numbers(1.2.3) h ‚ : “ p , , , , , . . . q Actually the construction in [Hi] only applied to compact subsets of the stabilized ellipsoid. Toembed the whole product we are appealing to [PV].
DAN CRISTOFARO-GARDINER, RICHARD HIND, AND DUSA MCDUFF via the formula b n “ h n ` h n ` . Thus b “ , b “ , b “ , and so on; the b n aredecreasing, with limit τ . Since h n ` “ h n ` ´ h n ` , we again obtain the estimate c p x q ě xx ` for x “ b n , n ě
0; moreover, as explained in [MS, § b n fit together to form an infinite staircase converging to τ from the right.However, in dimension 4 these obstructions are not sharp at b n , since as mentionedabove they are weaker than the volume obstruction, and so do not influence c p b n q directly. It is for this reason that McDuff and Schlenk call them ghost stairs .Our main result is that the embedding obstructions at the b n persist under stabi-lization. Because of the symplectic folding bound (1.1.1) they are sharp, so that weobtain the following. Theorem 1.2.1. c k p b n q “ b n b n ` for all k ě and n ě . Thus c k p q “ , c k p q “ and so on.1.3. Methods and relationship with embedded contact homology.
In view ofthe upper bound in (1.1.1), Theorem 1.2.1 will follow if we can establish a suitablelower bound for c k p x q at the given values of x . In other words, we must find embeddingobstructions for these x . As in [HiK, CGHi] this is accomplished by a two-step process: ‚ first, we find suitable J -holomorphic curves that obstruct the existence of afour-dimensional embedding E p , x q s ã Ñ B p µ q where µ ă xx ` , and ‚ second, we show that these obstructions persist for stabilized embeddings E p , x q ˆ C k s ã Ñ B p µ q ˆ C k . Although we are interested here in calculating c k p x q for the rational numbers b n , it isconvenient to increase b n slightly to x “ b n ` ε “ pq ` ε , where ε ą B p µ q to C P p µ q by adding theline at infinity. Thus, for the first step we consider the negative completion X : “ X µ,x of C P p µ q (cid:114) imΦ, where(1.3.1) Φ : E p , x q s ã Ñ C P p µ q is a symplectic embedding for some µ , and look for J -holomorphic curves C in X ofdegree d that are negatively asymptotic to the short orbit β on Φ pB E p , x qq with some The numerical properties of the ratios b s “ h s ` h s are not the same — for example, the evenindex terms in the sequence h ‚ are all divisible by 3 — and so their role in [MS] is somewhat different.However, the b s do come up in our arguments in §
4, since the class z M is determined by the ratio (cid:96) n (cid:96) n ´ “ b n . In fact, the graphs of y “ ? x and y “ xx ` cross at x “ τ . In fact if the domain is an ellipsoid the two embedding problems are equivalent: see [M1]. Here C P p µ q denotes C P with symplectic form ω scaled so that ω takes the value µ on the line L . Here we assume that J is admissible, i.e. adapted to the negative end of X : see § HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 5 multiplicity m . If such a curve exists for generic J and all sufficiently small ε ą
0, thenthe fact that it must have positive symplectic area gives the inequality(1.3.2) µ ą md . When proving the existence of C we will restrict to the case when C has Fredholmindex zero, since these are the curves that can potentially be counted, and will alsowork with a fixed value µ ‹ of µ . As we explain in more detail below, it turns out thatthe second stabilization step works for curves C that have genus zero, Fredholm indexzero and just one negative end of multiplicity m “ p .The second step is accomplished by the method of [HiK, CGHi], who prove a resultthat can be stated as follows. It will be convenient to denote by M ` X µ,x , dL, s, tp β , m q , p β , m qu ˘ the moduli space of genus zero J -holomorphic curves with degree d and s negativeends, that cover the short orbit β a total of m times and the longer orbit β a totalof m times. Here we assume that x “ pq ` ε as above.It turns out (see (2.2.29)) that if C has just one negative end of multiplicity m “ m then its Fredholm index is(1.3.3) ind p C q “ ` d ´ m ´ r mx s ˘ . Hence, if ind p C q “
0, we have 3 d ą m ` mx . If we now let ε Ñ
0, we obtain 3 d ě m ` mqp with equality exactly if p | m , since gcd p p, q q “ md ď pp ` q “ bb ` , with equality exactly if p | m . In other words, the obstruction that index 0 curves asabove give through (1.3.2) is never stronger than the folding bound, and so such curvescould potentially persist for the stabilized embedding. Our main stabilization resultproves this when m “ p . (See Remark 3.6.5 for some generalizations.) Proposition 1.3.1.
Let x : “ b ` ε , where b “ pq with gcd p p, q q “ and ε ą irrationaland very small, and fix µ ˚ ą . Suppose that for all sufficiently small ε ą and genericadmissible J there is a genus zero curve C in X µ ‹ ,x with degree d , Fredholm index zero,and one negative end on tp β , p qu , where gcd p d, p q “ . Then, d “ p ` q , and for all k ě , we have c k p b q ě pd “ bb ` . The proof is given in § C has genus zero, Fredholm index zero, and one negative end, then its Fredholm indexremains zero under stabilization. Moreover the arguments in [CGHi] guarantee thatits contribution to the count of curves in the stabilization cannot be cancelled by someother curve even when one varies µ and the almost complex structure.We end with some comments on the proof of the first step. In [CGHi], the authorsshow that when x “ g n ` g n ă τ the embedding obstruction coming from “embeddedcontact homology” (ECH) is carried by a curve with genus zero and one negative end DAN CRISTOFARO-GARDINER, RICHARD HIND, AND DUSA MCDUFF as above. However, for x “ b n ą τ , it cannot be the case that the obstruction comingfrom ECH stabilizes. Indeed, [CGHR] shows that embedded contact homology alwaysat least recovers the volume obstruction c p x q ě ? x , which as already mentioned isstrictly above xx ` for x ą τ . Further, curves C in four-dimensional cobordisms thatare detected by the ECH cobordism map generally have ECH index and Fredholm indexequal to zero. However, in our case we will see that the relevant curves have ECH index two and Fredholm index zero, and hence cannot be expected to be embedded. Thus, new methods are needed. The basic idea in the present work is to stretch acollection of nodal curves that are modified forms of the McDuff–Schlenk obstructionsthat give the ghost stairs, and look at the top part of the resulting buildings. Our aimis to show that at least one of the resulting buildings has a top level with Fredholmindex zero and one negative end, and so by Proposition 1.3.1 stabilizes to an index zerocurve that gives an obstruction in higher dimensions. We therefore need to analyzethe possible buildings that can arise when we stretch. This analysis is complicatedby the possible presence of negative index multiple covers — configurations that mostprobably do occur, see Remark 4.4.3 and Remark 3.5.4. To get around this difficulty,we use the fact that the modified McDuff-Schlenk obstructions lie in classes that haveprecisely 12 genus zero representatives, and we show that at most 9 of these break ina problematic way.To this end, we develop the tools used to analyze relative intersections and ECHindices. Specifically, we use a refined index inequality (Proposition 2.2.2), which is areformulation of results in Hutchings [H], together with a new approach to estimatingthe grading of elliptic orbits (which contributes to the ECH index in subtle ways)in terms of areas rather than lattice point counts, see Lemma 4.1.2. We also use atechnique pioneered by Hutchings and Nelson [HN1] that calculates writhes of curvesthat are close to breaking as the neck is stretched. Situations requiring the analysisof potentially complicated holomorphic buildings are quite common in applications ofholomorphic curve theory, see for example [H2, HN1, HT1, N], and so we expect ourstrategy to be potentially useful in other contexts.This analysis of the limiting buildings forms the bulk of the paper. It is describedin §
3, with the hardest computations deferred to § §
4. As we point out inRemark 3.1.8 (ii), the same method works rather easily for the Fibonacci stairs b “ g n ` g n ,while in the case of the ghost stairs it is complicated by the presence of the obstructioncurve that determines c p x q for τ ă x ă c k p x q for other values of x , as we explain in Remark 3.5.4 this is probablyneither efficient nor the best approach for general x . Indeed, many of the calculationshere are simplified because of special properties of the Fibonacci numbers, and even In fact, the ECH cobordism map detects buildings with ECH index zero that may (and often do)consist of curves with both positive and negative ECH index. Further, it may not always be the casethat curves with ECH index two and Fredholm index zero must have double points, but as mentionedin Remark 2.2.3 (ii) this is known in some situations.
HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 7 with this the computations are quite involved. We intend to explore other ways toconstruct suitable curves C in a later paper.2. Preliminaries
This section reviews basic background material on continued fractions, Fibonacciidentities, and the ECH index formulas.2.1.
Weight sequences and best approximations.
Beside the even index Fibonaccinumbers h ‚ in (1.2.3), the following auxiliary sequences will be useful, where Q n “ P n ´ “ h n ` : Q “ , Q “ , Q “ , Q “ , Q “ , Q “ , . . . (2.1.1) (cid:96) ´ : “ , (cid:96) “ , (cid:96) “ , (cid:96) “ , (cid:96) “ , (cid:96) “ , . . . (cid:96) n “ h n ` ,t “ , t “ , t “ , t “ , t “ , t k : “ (cid:96) k ´ (cid:96) k ´ . We also write b n : “ h n ` h n ` “ : P n Q n “ Q n ` Q n . We will use the following Fibonacci identities:3 ‹ n ` “ ‹ n ` ‹ n ` , ‹ “ g, h ;(2.1.2) t n ´ t n ´ “ (cid:96) n ´ , (2.1.3) Q n “ (cid:96) n ` (cid:96) n ´ , (2.1.4) h n ` ´ h n ` h n ` “ h n ` ´ p h n ` ´ h n ` q “ h n ` p h n ` ´ q ` . (2.1.6)Further, the Q n , (cid:96) n and t n are all linear combinations of certain Fibonacci numbers,and satisfy the recursion(2.1.7) ‹ n “ ‹ n ´ ´‹ n ´ , Hence their ratios ‹ n ‹ n ´ converge to τ . Moreover, the above identities may be provedby checking them on two or three low values of n : because the Fibonacci numberssatisfy a two step linear recursion, one only needs to check linear identities for twovalues of n , and quadratic identities such as (2.1.5) for three values of n : see [MS,Prop. 3.2.3]. Lemma 2.1.1. (i)
Let x n and y n , n ě , be two sequences that satisfy (2.1.7) .Then the quantity x n y n ´ x n ´ y n ` is independent of n . Moreover the followingidentities hold: Q n ` (cid:96) n “ Q n (cid:96) n ` ` , (2.1.8) (cid:96) n (cid:96) n “ (cid:96) n ´ (cid:96) n ` ` ,(cid:96) n P n “ (cid:96) n ´ P n ` ` . DAN CRISTOFARO-GARDINER, RICHARD HIND, AND DUSA MCDUFF (ii)
The following sequences (and their products) are increasing with n : Q n P n , (cid:96) n (cid:96) n ` , t n ` t n , (cid:96) n P n , (cid:96) n P n ` , Q n (cid:96) n , Q n ` k t n , ď k ď . Proof. (i) holds because x n y n ´ x n ´ y n ` “ p x n ´ ´ x n ´ q y n ´ x n ´ p y n ´ y n ´ q “ x n ´ y n ´ ´ x n ´ y n . Hence x n y n ´ x n ´ y n ` “ x y ´ x y “ : κ is constant. Thus the quotient x n y n ` increasesor decreases according to whether the constant is positive or negative. Alternatively,(i) implies that to check whether one of these sequences is increasing or decreasing,one just has to look at the first two terms. The rest of the lemma now holds by directcalculation. Note that it suffices to check that Q n ` t n increases because, if 0 ď k ă Q n ` k t n “ Q n ` t n ¨ Q n ` k Q n ` is the product of two increasing sequences. (cid:3) Because the sequence P n Q n converges to τ “ p ` ? q which is a solution of theequation τ ` τ ´ “
7, one can check thatlim n Ñ8 (cid:96) n P n “ lim n Ñ8 ˆ P n P n ` Q n P n ˙ “ σ “ : 16 p ´ ? q ă . , (2.1.9) lim n Ñ8 (cid:96) n Q n “ lim n Ñ8 (cid:96) n P n P n Q n “ ´ σ, lim n Ñ8 t n Q n “ lim n Ñ8 (cid:96) n Q n ´ lim n Ñ8 (cid:96) n ´ P n ´ “ ´ σ ą . . Weight sequences:
As explained in [MS, Lem.1.2.6] for example, the weight sequencefor b “ pq is a nonincreasing finite sequence of positive numbers in q Z such that w p pq q “ p w , . . . w m q , W p pq q : “ q w p pq q “ p W , . . . , W m q , where(2.1.10) W m “ , ÿ i W i “ pq, ÿ i W i “ p ` q ´ . If b has continued fraction expansion r a , a , . . . , a k s , then the weights W p b q occur inblocks of lengths a , a , . . . , a k . Hence m “ ř a i and we may write W p b q “ p X ˆ a , X ˆ a , . . . , X ˆ a k k q , X ˆ a : “ X, . . . , X loooomoooon a . (2.1.11)In this notation, given b : “ pq with gcd p p, q q “
1, the corresponding X i and a i aredetermined for increasing i by the recursion X ´ “ p, X “ q, X i ` “ X i ´ ´ a i X i i ă k, where the a i ą ď X i ` ă X i . On the other hand, given acontinued fraction r a , a , . . . , a k s , we can calculate which number b “ pq it represents HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 9 by using the same recursion but starting at the end with X k “
1. This recursion impliesthat p “ a X ` X and hence that W p b q “ ˆ X ˆ a , . . . , X ˆ a r r , W ˆ X r X r ` ˙˙ , @ r ě . (2.1.12)The relevance of weight expansions to our embedding problem is this result from [M1]. Proposition 2.1.2.
Let w p b q be the weight expansion of b “ pq . Then for any ε ą one can embed m disjoint balls of capacities p ´ ε q w p b q into int E p , b q , and henceremove almost all of the interior of an ellipsoid E p , b q by blowing it up m times withweights p ´ ε q w i . The following lemma is helpful when finding continued fraction expansions.
Lemma 2.1.3.
Let S , S , . . . be a (strictly) increasing sequence of positive integerswith ą S S ą τ , that satisfy the recursion (2.1.7) . Then there are positive integers a , . . . , a k for some k ě such that S n ` S n “ r p , q ˆ n , a , . . . , a k s , @ n ě . Proof.
We give an inductive argument. The case n “ ă S S ă S n ` S n is decreasing with limit τ . This holds byapplying part (i) of Lemma 2.1.1 with x n “ y n “ S n , and using the fact that S S : “ x ą τ so that x ´ x ` ą
0. In particular, for all n , 6 ă S n ` S n ă
7, so the first entryof its continued fraction expansion is 6, and S n ` S n ě , hence 2 p S n ` ´ S n q ą S n sothat the continued fraction expansion of S n ` S n has the form r , , . . . s . Thus by (2.1.12)we have W ´ S n ` S n ¯ “ ´ S ˆ n , S n ` ´ S n “ S n ´ S n ´ , W p S n ´ S n ´ S n ´ q ¯ . (2.1.13)But, by induction, we may assume that S n S n ´ “ r , p , q ˆp n ´ q , a , . . . , a k s . Therefore, S n ´ S n ´ S n ´ “ r , p , q ˆp n ´ q , a , . . . , a k s . Hence, because the continued fraction for S n ` S n is given by the length of the blocks in its weight expansion, we find that S n ` S n “ r , p , q ˆ n , a , . . . , a k s , as claimed. (cid:3) Corollary 2.1.4.
For n ě , we have the following weight expansions. p a q b n “ Q n ` Q n “ r p , q ˆp n ´ q , , s , p b q (cid:96) n ` (cid:96) n “ r p , q ˆ n , s “ r p , q ˆp n ´ q , , s , p c q t n ` t n “ r p , q ˆ n s . Proof.
Since the sequences Q n , (cid:96) n , t n satisfy (2.1.7), this an immediate consequence ofLemma 2.1.3. (cid:3) In § ř W i “ pq in(2.1.10). The first involves a vector z M p n q that is part of the data of a “model curve”that we will study. Lemma 2.1.5.
For n ě , define z M p n q to be the vector W p (cid:96) n (cid:96) n ´ q with ones appendedat the end. Thus z M p n q has the same length as W p b n q , and has the following expansion z M p n q “ ` p (cid:96) n ´ q ˆ , t n ´ , p (cid:96) n ´ q ˆ , . . . , t , ˆ ˘ . Then z M p n q ¨ W p b n q “ (cid:96) n ´ Q n ` t n Q n ` ´ . (2.1.14) Proof.
When n “ z M p q “ p ˆ , , ˆ q and the claim is that z M p q ¨ W p q “ ` ˆ ´ “ . But z M p q ¨ W p q “ p ˆ , , ˆ q ¨ p ˆ , , ˆ q “ ˆ ` ` “ . Thus we may assume inductively that the result is known for n ´ ě n . As in (2.1.13), we may write z M p n q “ ` p (cid:96) n ´ q ˆ , t n ´ , c M p n ´ q ˘ , where c M p n ´ q is the truncated version of z M p n ´ q in which the first entry 6 (cid:96) n ´ isremoved. Since W p b n q has an analogous expression, we find that z M p n q ¨ W p b n q “ (cid:96) n ´ Q n ` t n ´ p Q n ´ Q n ´ q ` c M p n ´ q ¨ W ` P n ´ ´ Q n ´ Q n ´ ˘ “ (cid:96) n ´ Q n ` t n ´ p Q n ´ Q n ´ q ` p (cid:96) n ´ Q n ´ ` t n ´ Q n ´ q´ (cid:96) n ´ Q n ´ “ Q n p t n ´ ` (cid:96) n ´ q ´ Q n ´ p (cid:96) n ´ ´ t n ´ q , where the second equality is obtained using the inductive hypothesis. Hence we mustshow that the right hand side of the last equation equals (cid:96) n ´ Q n ` t n Q n ` ´
1. If wewrite Q n ` “ Q n ´ Q n ´ and gather the terms in Q n , Q n ´ on different sides of theequation, we find that it suffices to show Q n ` t n ´ (cid:96) n ´ ´ t n ´ ˘ “ Q n ´ ` t n ´ t n ´ ´ (cid:96) n ´ ˘ . But the coefficient on the left vanishes because t n ´ t n ´ “ (cid:96) n ´ by (2.1.3), while thesame identity shows that the coefficient on the right also vanishes because t n ´ t n ´ “ t n ´ ´ t n ´ ´ (cid:96) n ´ “ (cid:3) HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 11
Remark 2.1.6.
We can instead write(2.1.15) z M p n q ¨ W p b n q “ (cid:96) n ` (cid:96) n (cid:96) n ´ ´ (cid:96) n ´ ` . For our purposes, the identity (2.1.14) is more geometrically natural — later, we willsee that it directly implies that the model curve has the area we expect. However, wewill need (2.1.15) as well. To prove (2.1.15), it is equivalent by Lemma 2.1.5 to showthat the right hand sides of (2.1.15) and (2.1.14) are equal. We can rewrite the righthand side of (2.1.14): (cid:96) n ´ Q n ` t n Q n ` ´ “ p (cid:96) n ´ ` t n ` q Q n ` “ p (cid:96) n ` ´ (cid:96) n ` (cid:96) n ´ q Q n ` “ (cid:96) n Q n ` “ (cid:96) n ` (cid:96) n (cid:96) n ´ ` , where in the first line we have used Lemma 2.1.1, and in the last we have used (2.1.4).So, it is equivalent to show p (cid:96) n ` (cid:96) n (cid:96) n ´ q ´ p (cid:96) n ` (cid:96) n (cid:96) n ´ ´ (cid:96) n ´ q “ , or equivalently that(2.1.16) 5 p (cid:96) n ´ (cid:96) n (cid:96) n ´ ` (cid:96) n ´ q “ . Since (cid:96) n ´ (cid:96) n (cid:96) n ´ ` (cid:96) n ´ “ (cid:96) n ´ ´ (cid:96) n (cid:96) n ´ “ , where the last equality follows by (2.1.8), equation (2.1.16) holds. (cid:51) The other identity that we will use gives a convenient way for studying sequencessatisfying a certain recursion closely related to (2.1.7).
Lemma 2.1.7.
Given positive integers
A, B ą , define R p A, B q : “ ´ R ˆ , R , R ˆ , . . . , R n ´ , R ˆ n ´ , R n ´ ¯ by the recursion R “ A, R “ B,R k “ R k ´ ´ R k ´ , R k ` “ R k ´ ´ R k , k ă n. (2.1.17) Then we have: (i) R p A, B q “ A ¨ R p , q ` B ¨ R p , q . (ii) R p , q “ ´ ˆ , , p´ q ˆ , , p´ q ˆ , . . . , p´ (cid:96) k ´ q ˆ , t k , . . . , p´ (cid:96) n ´ q ˆ , t n ´ ¯ . (iii) Let Ă W be the vector obtained from W p b n q by deleting the last block of length .Then Ă W “ ´ Q ˆ n , P n ´ Q n , p Q n ´ P n q ˆ , . . . ¯ ” ´ Q n ´ R p , q p mod Q n q . (iv) If ∆ “ R p x , x q “ p x ˆ , x , x ˆ , . . . , x n ´ q for some n ě , then ∆ ¨ R p , q “ (cid:96) n ´ x n ´ . Proof.
The sequences R p A, B q and A ¨ R p , q ` B ¨ R p , q both have the same initialconditions, and any linear combination of sequences satisfying (2.1.17) also satisfies thisrecursion. This proves (i). To prove (ii) one just has has to check that the recursion issatisfied, and this follows because (cid:96) n and t n : “ (cid:96) n ´ (cid:96) n ´ both satisfy (2.1.7). To prove(iii), notice first that Ă W does satisfy the recursion (2.1.17) by part (a) of Corollary 2.1.4.Further P n ´ Q n “ Q n ` ´ Q n “ Q n ´ Q n ´ by (2.1.7). Hence, (iii) follows from (i).We prove (iv) by induction on n . It is clear when n “
1, and the inductive step holdsbecause by (ii) and (2.1.17), we have (cid:96) n ´ x n ´ ´ (cid:96) n ´ x n ` t n x n ` “ (cid:96) n ´ p x n ´ ´ x n q ` t n x n ` “ (cid:96) n ´ x n ` ` t n x n ` “ (cid:96) n x n ` , as required. (cid:3) Best approximations:
Let θ be any irrational number. We will need to use some facts about rationalnumbers p { q that best approximate θ from below.Recall that a rational number p { q in lowest terms is a best rational approximation to θ if | θ ´ p { q | ă | θ ´ m { n | for all n ă q , while it is a best rational approximationfrom below if 0 ă θ ´ p { q ă θ ´ m { n, @ n ă q, m ă θn. To elaborate, let θ “ a ` a ` ... have continued fraction expansion θ “ r a , a , . . . s . The convergents of θ are the rational numbers c k : “ p k { q k : “ r a , a , . . . , a k s . (2.1.18)For any k , they satisfy c ă c ă . . . ă c k ă θ ă c k ` ă . . . ă c ă c . Any convergent is a best approximation to θ . To get all possible best approximations,we must also consider the semiconvergents of θ . A semiconvergent is a fraction ofthe form c k ´ ‘ r ¨ c k ´ , where ‚ ă r ă a k , ‚ the operation ‘ is defined by the rule pq ‘ p q “ p ` p q ` q , and ‚ the multiplication by r denotes repeated addition with the ‘ operation.For motivation, note that the convergents satisfy(2.1.19) c k “ c k ´ ‘ a k ¨ c k ´ . HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 13 If k is even, then the fractions c k ´ ‘ r ¨ c k ´ increase with r ě r ď a k are allsmaller than θ , while if k is odd these fractions are bigger than θ . Another useful factis that the convergents satisfy(2.1.20) q n p n ´ ´ q n ´ p n “ p´ q n . The following well known fact will be very useful; for a proof see [HW] or the proofof [HT2, Lem. 3.3].
Lemma 2.1.8.
Suppose that θ ą is an irrational number. Then the rational numbersthat best approximate θ from below are the even convergents c k , and the semiconver-gents c k ´ ‘ r ¨ c k ´ , with k ě and ď r ă a k . The following two examples are key.
Example 2.1.9.
Let θ “ θ n : “ b n ` ε for some very small irrational ε ą (cid:96) n . By Corollary 2.1.4 we have b n “ P n { Q n “ r p , q ˆp n ´ q , , s We know that Q n ą (cid:96) n . If ε ą b n is an even convergentof θ , and the even convergents with denominator less than Q n have continued fractionexpansion c k : “ r p , q ˆ k s “ t k ` t k , ď k ă n, while the odd convergents with denominator less than Q n have continued fractionexpansion(2.1.21) c k ` : “ r p , q ˆ k , s “ r p , q ˆp k ´ q , , s “ (cid:96) k ` (cid:96) k , ď k ă n. Further, there are 6 semiconvergents of the form t n ` r ¨ (cid:96) n t n ´ ` r ¨ (cid:96) n ´ , ď r ď θ , and for each k ă n , there are 4 semiconvergents of the form t k ` r ¨ (cid:96) k t k ´ ` r ¨ (cid:96) k ´ , ď r ď θ .The even convergents, and the semiconvergents mentioned above are all of the bestpossible approximations to θ from below with denominator no more than Q n . (cid:51) Example 2.1.10.
Now let θ “ ˜ θ n : “ θ n . We want to know those best approximationsfrom below with denominator no more than (cid:96) n . We know1 { b n “ r
0; 6 , p , q n ´ , , s . If ε ą θ are of the form c k : “ r
0; 6 , p , q ˆp k ´ q , s “ r
0; 6 , p , q ˆp k ´ q , , s “ (cid:96) k ´ (cid:96) k for 1 ď k ď n , except for the convergent c : “ r s “ (cid:96) ´ (cid:96) . Thus, in this case the bestrational approximations from below all have denominators (cid:96) k for 0 ď k ď n . (cid:51) Basics of embedded contact homology.
Let J be an almost complex structureon a completed symplectic cobordism X . We will assume throughout the paper that J is admissible . This means that on any symplectization end ` Y ˆ I, d p e s λ q ˘ of X (where I “ p´8 , ´ N q or p N, and s denotes the coordinate on R ), J is translationinvariant, rotates the contact structure ker p λ q positively with respect to dλ , and sends B s to the Reeb vector field R . We will want to consider J -holomorphic curves withdisconnected domain. So in the following we will call a curve with connected domain irreducible , and call it reducible otherwise. Further a curve is called somewhereinjective if each of its irreducible components is somewhere injective and no two havethe same image. All of the curves throughout the paper will have punctured domain,and are asymptotic to closed Reeb orbits near the punctures, see for example [H2, § Relative intersection theory:
Consider two distinct somewhere injective, J -holomor-phic curves C, C in a four-dimensional completed symplectic cobordism X . In ourproof, we will frequently want to compute C ¨ C. This is an algebraic count of intersection points of C with C . By positivity of inter-sections, each point counts positively.Because X is noncompact, the quantity C ¨ C is not purely homological. Rather, wehave(2.2.1) C ¨ C “ Q τ pr C s , r C sq ` L τ p C, C q . Here, τ denotes a trivialization of ξ “ Ker p λ q over all embedded Reeb orbits, and r C s denotes the relative homology class of C . This is defined regardless of whether ornot C is somewhere injective, and takes values in H p X, α, β q , where α and β are orbitsets , namely finite sets tp γ i , m i qu , where the γ i are embedded Reeb orbits, and the m i are positive integers. The orbit set α is given by the positive asymptotics of C . Wesay that C is asymptotic to an orbit set Θ “ tp γ i , m i qu at `8 if, for each i , the sumof the multiplicities of the positive ends of C at γ i is exactly m i , and C has no positiveends at any other orbit other than the γ i . The orbit set β is given by the negativeasymptotics. The fact that Q τ pr C s , r C sq is homological is proved in [H].Equations (2.2.28) and (2.2.31) show how to compute it in the situations relevant tous.If C partitions m i as m i , . . . , m in i , then we also denote the orbit set as(2.2.2) tp γ i , m i qu “ tp γ m i i , . . . , γ m ini i qu , m i “ n i ÿ j “ m ij , This means in particular that C and C have no irreducible components in common. HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 15 where γ ri denotes a single end on γ i of multiplicity r .For future use, we define M p X, J, α, β q “ M p α, β q to be the moduli space of J -holomorphic curves in X that are asymptotic to the orbitset α at `8 and asymptotic to the orbit set β at ´8 .The term L τ p C, C q is the asymptotic linking number of C and C . To defineit, first fix an embedded orbit γ i at which both C and C have positive ends. Byintersecting C with an s “ R slice in the positive end of X for sufficiently large R , thepositive ends of C at γ i form a link ζ ` i,C , which we can regard as a link in R via thetrivialization τ as in [H2, § ζ ` i,C similarly, and we can definethe linking number L τ p ζ ` i,C , ζ ` i,C q of these two links to be their linking number in R ,using the identification τ . If R is sufficiently large, then this number does not dependon the choice of R . We can define links ζ ´ i,C , ζ ´ i,C and linking numbers for orbits atwhich C and C both have negative ends analogously.We now define(2.2.3) L τ p C, C q “ n ÿ i “ L τ p ζ ` i,C , ζ ` i,C q ´ m ÿ j “ L p ζ ´ j,C , ζ ´ j,C q where the first sum is over the embedded orbits at which both C and C have positiveends, and the second sum is over the embedded orbits at which both C and C havenegative ends. The ECH index and the partition conditions:
Let C P M p α, β q be a somewhereinjective curve in X . Part of our proof will involve estimating the ECH index of sucha curve. We now review what we need to know about the ECH index.Recall first the Fredholm index for curves in 4-dimensions takes the form(2.2.4) ind p C q “ ´ χ p C q ` c τ p C q ` CZ indτ p C q . Here, c τ p C q denotes the relative first Chern class of C (see [H2, § CZ indτ p C q denotes the Conley–Zehnder index(2.2.5) CZ indτ p C q “ ÿ i CZ τ p γ i q ´ ÿ j CZ τ p γ j q , where the first sum is over the (possibly multiply covered) orbits given by the positiveends of C , the second sum is over the (possibly multiply covered) orbits given by thenegative ends of C , and CZ τ of a Reeb orbit γ denotes its Conley–Zehnder index: see(2.2.24) below for the elliptic case.If C is somewhere injective, we can bound ind p C q from above by the ECH index of C . The ECH index depends only on the relative homology class of C , and is definedfor mutiply covered curves as well by the formula(2.2.6) I pr C sq “ c τ pr C sq ` Q τ pr C sq ` CZ Iτ pr C sq , See Lemma 3.6.2 for higher dimensions. where Q τ denotes the relative intersection pairing from (2.2.1), and CZ Iτ is the totalConley-Zehnder index CZ Iτ pr C sq “ ÿ i m i ÿ k “ CZ τ p α ki q ´ ÿ j n j ÿ k “ CZ τ p β kj q , where α “ tp α i , m i qu , β “ tp β j , n j qu , and γ x denotes the x -fold cover of γ . The precisestatement of this bound is the index inequality (2.2.7) ind p C q ď I pr C sq ´ δ p C q for somewhere injective curves, proved in [H]; see also Proposition 2.2.2 below. Here, δ p C q ě C .When equality holds in (2.2.7), for example if ind p C q “ I p C q , then we can saymuch more about the asymptotics of C . Indeed, if such a curve C has ends at anembedded orbit α i with total multiplicity m i , then the multiplicities of the ends of C at α i give a partition of m i that is called the ECH partition . This partition dependsonly on whether α i is at the positive or negative end of C , and can be computed purelycombinatorially, as is shown in [H] and reviewed in the proof of Proposition 2.2.2 below.For positive and negative ends, it is denoted respectively as p ` α i p m i q , p ´ α i p m i q . The next remark explains what we will need.
Remark 2.2.1. (Computation of p ˘ α p m q for elliptic ends) (i) Consider a positive end along an elliptic orbit α with mod 1 monodromy angle of θ P p , q . Let Λ be the maximal concave piecewise linear path in the first quadrantthat starts at p , q , ends at p m, t mθ u q , has vertices at lattice points, and stays belowthe line y “ θx . It is shown in [H] that p ` α p m q is given by the horizontal displacementsof this path. Here the word “maximal” includes the assumption that the edges of Λhave no interior lattice points, in other words that for each segment of the path thehorizontal and vertical displacements are mutually prime. Thus instead of a singlesegment labelled by p , q , for example, we have two segments each with labels p , q .(ii) For a negative end the procedure is analogous, except that Λ is now the minimalconvex lattice path that lies above the line y “ θx . For example, if α is the long orbitof E p , x q where x “ PQ ` ε , then the monodromy angle of α is x , so p ´ α p Q q “ p Q q onlyif P ` Q is the best approximation to x from above. In the case PQ “ b n , it follows from(2.1.21) that this best approximation is (cid:96) n (cid:96) n ´ . Since 7 (cid:96) n ´ ă Q “ Q n , the path Λ startswith 7 segments along the line of slope (cid:96) n (cid:96) n ´ ; see Lemma 4.2.2 below. Thus in this casethe partition p ´ α p Q q must have at least eight terms. (cid:51) Since p ` α p m q only depends on the mod 1 monodromy angle θ of α , we will some-times write p ` θ p m q instead. A useful fact about the positive partition is if p ` θ p m q “p a , . . . , a s q , then(2.2.8) t p a i ` a j q θ u “ t a i θ u ` t a j θ u HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 17 for any 1 ď i ‰ j ď s , see [H2, Ex. 3.13.]. Similarly, if p ´ θ p m q “ p b , . . . , b s q then r p ř i b i q θ s “ ř i r b i θ s . For example, if θ ” PQ ` ε p mod 1 q and p ´ θ p Q q “ p b , . . . , b s q then(2.2.9) ÿ i t b i θ u “ ÿ i p r b i θ s ´ q “ P ` ´ s. The relative adjunction formula:
The index inequality (2.2.7) is related to anadjunction formula that we will also need. Namely, recall the relative adjunctionformula from [H2]. This says that if C is somewhere injective then(2.2.10) c τ pr C sq “ χ p C q ` Q τ pr C sq ` w τ pr C sq ´ δ p C q . The term here that has not already been introduced, w τ p C q , is called the asymptoticwrithe of C . Its definition is similar to the definition of the asymptotic linking numberin (2.2.3). Namely, fix an embedded orbit γ i at which C has positive ends, and regardthe links ζ ` i,C and ζ ´ i,C as links in R via the trivialization τ . Let w τ p ζ ` i,C q and w τ p ζ ´ i,C q denote the writhes of these links. If R is sufficiently large, then this does not depend onthe precise choice of R . We can define writhes associated to negative ends analogously.We can now define w τ p C q : “ n ÿ i “ w τ p ζ ` i,C q ´ m ÿ j “ w τ p ζ ´ j,C q , where the first sum is over the orbits at which C has positive ends, and the second sumis over the orbits at which C has negative ends. An improved index inequality:
There is a refined version of (2.2.7) that will berelevant to what follows. To state it, suppose first that γ is an elliptic orbit at which C has positive ends of total multiplicity m ą
0, and let θ be the mod 1 monodromyangle of γ , normalized to be in p , q . The ends of C give a partition p a , . . . , a n q of m .Order the numbers a , . . . , a n so that(2.2.11) θ ą t a θ u a ě . . . ě t a n θ u a n , and let Λ C be the concave lattice path in the first quadrant that starts at p , q , endsat p m, ř ni “ t a i θ u q , and has edge vectors p a i , t a i θ u q , appearing in the same order as the a i . Define(2.2.12) A C p γ, m q “ L p Λ C q ` b p Λ C q , where: ‚ L p Λ C q is the number of lattice points in the region bounded by the line y “ θx and the vertical line from p m, ř ni “ t a i θ u q to p m, mθ q that lie strictly above thepath Λ C ; ‚ b p Λ C q is the sum over all edges of Λ C of the number of interior lattice pointsin each edge. Now define(2.2.13) A p C q “ ÿ p γ,m q A C p γ, m q , where the sum is over pairs p γ, m q for which γ is elliptic and C has at least one positiveend. There is a similar definition for negative ends, that we do not give since we donot need it. Note that if C has ECH partitions, then A p C q “
0. This holds by themaximality condition in Remark 2.2.1 (i) together with (2.2.8).Here is the refined index inequality.
Proposition 2.2.2.
Let C be a somewhere injective curve. Then I p C q ´ ind p C q ě δ p C q ` A p C q . (2.2.14) Proof.
This is implicit in the work of Hutchings, but for completeness we give the proof.In this proof we will assume that C has no negative ends, since that is the case weneed.By combining the definition of the ECH index (2.2.6), the definition of the Fredholmindex (2.2.4), and the relative adjunction formula (2.2.10), we get(2.2.15) I p C q ´ ind p C q “ CZ Iτ p C q ´ CZ indτ p C q ´ w τ p C q ` δ p C q . The terms w τ p C q , CZ Iτ p C q and CZ indτ p C q are all sums over terms corresponding to eachorbit at which C has ends.So, let p γ, m q be a pair corresponding to an orbit at which C has positive ends,assume that γ is elliptic, and let ζ be the braid coming from the ends of C at γ . By[H2, Eq. 5.4] and [H2, Lem. 5.5], we have (2.2.16) w τ p ζ q ď n ÿ i,j “ max p p i a j , p j a i q ´ n ÿ i “ p i , where p i “ t a i θ u in the notation of (2.2.11). We also know that(2.2.17) CZ Iτ pp γ, m qq ´ CZ indτ pp γ, m qq “ n ÿ i “ p t iθ u ` q ´ n ÿ i “ p p i ` q , where CZ Iτ pp γ, m qq denotes the contribution of the pair p γ, m q to CZ Iτ , and similarlyfor CZ indτ pp γ, m qq . Consider 2 A : “ n ÿ i,j “ max p p i a j , p j a i q . This is twice the area of the region P bounded by the path Λ C defined above, thevertical line from p m, q to p m, ř ni “ p i q , and the x -axis. Pick’s theorem gives(2.2.18) 2 A “ T ´ B ´ , There is a similar lower estimate for the writhe of a negative end; see Remark 2.2.3 (ii).
HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 19 where T is the total number of lattice points in P , and B is the number of boundarylattice points. We have(2.2.19) T “ m ` ` n ÿ i “ t iθ u ´ L p Λ C q , and(2.2.20) B “ m ` n ` n ÿ i “ p i ` b p Λ C q . Combining (2.2.16) with (2.2.18), (2.2.19), and (2.2.20) gives w τ pp γ, m qq ď m ´ k ` n ÿ i “ t iθ u ´ L p Λ C q ´ b p Λ C q ´ n ÿ i “ p i . Combining this inequality with (2.2.17) gives w τ pp γ, m qq ď CZ Iτ pp γ, m qq ´ CZ indτ pp γ, m qq ´ L p Λ C q ´ b p Λ C q . Now sum this final equation over all elliptic orbit sets, use the bound [H2, Lem. 5.1]for the hyperbolic orbit sets, and combine the resulting equation with (2.2.15). (cid:3)
Remark 2.2.3. (i) If γ is a negative end of C of total multiplicity m , then the analogof (2.2.16) is the lower bound(2.2.21) w ´ τ p ζ q ě n ÿ i,j “ min p p i a j , p j a i q ´ n ÿ i “ p i , where C has ends of multiplicities p a , . . . , a n q on γ and p i “ r a i θ s , see [H2, § C near the limiting orbit γ . As pointed out to us by Hutchings, if C has only one positive end on γ with multiplicity m and if p ` θ p m q “ p m q so that C hasthe ECH partition at this end, then this estimate is in fact an equality. To see this, notethat by [H01, Lemma 6.4] this is equivalent to claiming that the asymptotic expansionof the trajectory has a term corresponding to the smallest possible elgenvalue. But thisholds by the argument outlined in [HT1, Remark 3.3]. A similar statement holds fornegative ends.Note also that exactness of the writhe bounds at both ends of a curve implies equalityin (2.2.14). Therefore, if a curve has ECH partitions (so that A p C q “ I p C q ´ ind p C q “
2, then it must have a double point.(iii) The proof of Proposition 2.2.2 above shows that if C is a simple curve such that I p C q “ ind p C q ´ δ p C q , then the path Λ C must be maximal, and so C must have theECH partitions. This proves (2.2.7). (cid:51) ECH index computations:
For our purposes, X will always be the completion ofa symplectic cobordism X with two boundary components B ˘ X that are either emptyor are ellipses Y “ B E p a, b q ; later we will refer to the ends that we add to complete X as symplectization-like ends . We now review the relevant formulas for c τ , Q τ , and CZ for these X . Recall that if b { a is irrational, then the Reeb vector field for B E p a, b q has exactly two embedded orbits, γ “ t z “ u and γ “ t z “ u . They are bothelliptic. It is convenient to keep track of their action defined by A p γ q “ ż γ λ. (2.2.22)We have A p γ q “ a and A p γ q “ b .First assume that X “ Y ˆ R . Then, as explained in eg [H2, § C ,both c τ pr C sq and Q τ pr C sq depend only on the asymptotics of C . Assume, then, that C P M p Y ˆ R , α, β q , where α “ tp γ , m q , p γ , m qu and β “ tp γ , n q , p γ , n qu .Then we define the action of C to be A p C q “ A p α q ´ A p β q “ ż α λ ´ ż β λ. (2.2.23)It is shown in [H2, § τ so that: ‚ The monodromy angle of γ is a { b and the monodromy angle of γ is b { a . ‚ c τ p C q “ p m ` m q ´ p n ` n q . ‚ Q τ p C q “ p m m ´ n n q .To compute the relevant CZ terms, recall that if γ is an elliptic orbit, with monodromyangle θ with respect to τ , then(2.2.24) CZ τ p γ q “ t θ u ` . Since in the current situation the terms in the ECH index for C only depend on theasymptotics of C , it is convenient to define a grading (2.2.25) gr ptp γ , m q , p γ , m quq “ m ` m ` m m ` m ÿ i “ CZ τ p γ i q ` m ÿ i “ CZ τ p γ i q associated to any orbit set on the ellipsoid B E p a, b q , so that if C P M p Y ˆ R , α, β q then(2.2.26) I p C q “ gr p α q ´ gr p β q . One can check that when b { a is irrational,(2.2.27) gr ptp γ , m q , p γ , m quq “ p N p m , m q ´ q , where N p m , m q is the number of integral points in the first quadrant triangle withslant edge x ` ba y “ m ` ba m .There are two other 4-dimensional cobordisms X for which we will want to under-stand these calculations.The first comes from removing the interior of an irrational ellipsoid E p , x q from C P p µ q and completing at the negative end, where the symplectic form on C P p µ q is theFubini–Study form scaled so that the line has size µ . In this case, any relative homologyclass is determined by its coefficient along the line class L and its negative asymptotics HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 21 in H p X, H , β q . Continuing with the trivialization from above, if C P M p X, H , β q , r C s “ dL , and β “ tp γ , n q , p γ , n qu then(2.2.28) c τ pr C sq “ d ´ p n ` n q , Q τ pr C sq “ d ´ n n . Further, if n is partitioned as p a , . . . , a r q while n is partitioned as p b , . . . , b s q , thenequations (2.2.4), (2.2.5), and (2.2.24) imply that if C has k connected componentsthen(2.2.29) ind p C q “ ´ k ` d ´ ÿ i ` a i ` t a i x u ˘ ´ ÿ j ` b j ` t b j x u ˘ . Finally we define the action (or ω -energy ) of C to be A p C q “ d µ ´ A p β q “ d µ ´ n ´ n x. (2.2.30)The second cobordism comes from performing a sequence of blowups in the interior ofan irrational ellipsoid and then completing at the positive end; call it p E . Let E , . . . , E k denote the exceptional classes associated to these blowups. In this case, any relativehomology class is determined by its coefficients along these classes. Using the sametrivialization as above, if C P M p p E , α, Hq , r C s “ ´p m E ` . . . m k E k q , and α “tp γ , n q , p γ , n qu , then(2.2.31) c τ pr C sq “ p n ` n q ´ p m ` . . . ` m k q , Q τ pr C sq “ n n ´ p dm ` . . . m k q , so that by (2.2.6) we have(2.2.32) I p C q “ gr p α q ´ ÿ i p m i ` m i q . Further, if C is connected and its positive end is partitioned as above then (2.2.4)implies that(2.2.33) ind p C q “ ´ ` r ` s ` ÿ i ` a i ` t a i x u ˘ ` ÿ j ` b j ` t b j x u ˘ ´ ÿ m i . Notice that in this formula the ` n , n in c τ pr C sq have been rewritten as sums ř i a i , ř j b j . Finally, if the symplectic area of E i is w i , we define the action of C to be A p C q “ A p α q ´ ÿ m i w i “ n ` n x ´ ÿ m i w i . (2.2.34)Note that in all these situations the action is nonnegative. This is clear in the case ofa symplectization since the condition that J is admissible implies that dλ is pointwisenonnegative on C , with equality at p P C if and only if the tangent space to C at p is the span of the Reeb vector field and B s . A similar argument works for any exactcobordism. It remains to note that the cobordisms in the second two examples can bemade exact by removing the line from C P p µ q and the exceptional divisors from p E , inwhich case the contributions d µ and ř d i w i to A p C q can be interpreted as actions ofthe corresponding Reeb orbits. In § N the analog of (2.2.4) is(2.2.35) ind p C q “ p N ´ q χ p C q ` c τ p C q ` CZ indτ p C q , where CZ indτ p C q is now calculated as follows. As before CZ indτ p C q is a sum of contri-butions from each end of C , where a single end of multiplicity a k on the k th orbit γ k ofa generic ellipsoid E p b , . . . , b N q contributes a sum of N ´ t θ u ` θ “ b i b k for 1 ď i ď N, i ‰ k . If some ratios b i { b j , i ‰ j, are rational, then we are in a Morse-Bott situation and the index dependson whether the end is positive or negative. In particular, if E p , x, S, . . . , S q Ă C ` k where 1 ă x ă S so that N “ ` k , then the contribution to the index of a negativeend of multiplicity a on an orbit of action S is the same as that of the third orbit γ onthe ellipsoid E p , x, S , . . . , S k q where S ă S ă ¨ ¨ ¨ ă S k are slight perturbations of S . Thus the monodromy angles are S , xS , S S , . . . , S k S , where all but the first two termsare slightly ą
1. 3.
Stabilizable curves
We now explain how to establish the existence of a curve C satisfying the conditionsin Proposition 1.3.1 for suitable values of d, p . The argument is slightly different forthe two cases n “ b “
8) and n ą The setup.
Recall from [MS] that for n ą E p , b n ` ε q s ã Ñ int p B p µ n ` ε qq , µ n : “ ` b n “ h n ` h n ` , n ą , where the numbers ε, ε ą Let E : “ E n be the image of this embedding, and write β for the short orbit on itsboundary B E n and β for the long one. We complete int p B p µ n ` ε qq to C P p µ n ` ε q and then define(3.1.2) X : “ negative completion of ` C P p µ n ` ε q (cid:114) E n ˘ . When n “
0, we make similar definitions with µ “ “ c p q .This section and the next explain the proof of the following result. Proposition 3.1.1.
For n ě and generic admissible J on X , the moduli space M p X, h n ` L, , β h n ` q , of genus zero curves C in class h n ` L with a single negativeend on β h n ` , is nonempty. This immediately implies our main result.
Corollary 3.1.2. c k p b n q “ b n b n ` for all n ě . i.e. the ratios b i { b j , i ‰ j, are irrational See Remark 3.2.1 for a more precise description. For now, we work with these small perturbationsby simplifying via approximate identities such as µ n ` ε « µ n . HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 23
Proof.
Since 3 h n ` “ h n ` ` h n ` by (2.1.2), it follows from (1.3.3) that the curves C in Proposition 3.1.1 have index 0. Hence we may apply Proposition 1.3.1, whichgives c k p b n q ě h n ` h n ` “ b n b n ` . The result follows from this together with the foldingbound (1.1.1). (cid:3) To prove Proposition 3.1.1, we first blow up C P in the interior of the ellipsoid E n ,denoting the blown-up manifold by y C P . For each n we consider a class B (describedbelow) that is represented in y C P by a finite number of genus zero curves with onedouble point, and then consider what happens to these representatives when we stretchthe neck along the boundary of the ellipsoid B E n . When we do this, we get a sequenceof curves that converge in a suitable sense to a limiting building, with top level in X , bottom level in p E : “ p E n , the positive completion of the blown up ellipsoid, andperhaps also some intermediate levels in the symplectization B E n ˆ R (usually calledthe “neck”). Our aim is to show that at least one of the resulting top level curveslies in M p X, h n ` L, , β h n ` q . We will assume that the reader is familiar with thisstretching process; for details see for example [HiK, § breaking of the B -curve.Here are more details. For each n , consider the weight sequence w p b n q : “ p w , . . . , w m q defined in (2.1.10); it satisfies ř w i “ b n . Also recall the normalized weight sequence W p b n q “ p W , . . . , W m q : “ h n ` w p b n q . By Proposition 2.1.2, it is possible to removealmost all of the interior of the ellipsoid E n “ Φ p E p , b n ` ε qq by a sequence of blowupsof weights almost equal to w , . . . , w m , to obtain a manifold y C P that contains theboundary B E n and has symplectic form r ω such that r ω p E i q « w i . The elements of thenormalized weight sequence are integers, so we can consider the homology class B “ h n ` L ´ W E ´ . . . ´ W m E m “ : 3 (cid:96) n ` L ´ E p b n q P H p y C P q . (3.1.3)We will want to record some information about the class B .Using (2.1.10) and (2.1.5), we find that B ¨ B “ h n ` ´ h n ` h n ` “ , (3.1.4) c p B q “ h n ` ´ h n ` ´ h n ` ` “ , where the last equality holds by (2.1.2). Thus, spheres in class B have Fredholm indexzero.Finally, note that for each n ą ω p B q “ h n ` ω p L q ´ ÿ i W i ω p E i q « h n ` ´ h n ` h n ` h n ` “ h n ` . We will frequently use the fact that when we stretch, the symplectic area ω p B q isthe sum of the action of each curve in any level of the resulting building. Here wedefine the action for each part of the building using the formulas (2.2.23), (2.2.30), and The actual weights of the blowup are p ´ ε q w i where ε ą (2.2.34); the claim about the action of the limit follows immediately from the fact thatcontributions to the action of the building from matching pairs of ends cancel.We now claim that there is a sequence of Cremona transforms taking the class B to the class 3 L ´ E ´ . . . ´ E . This essentially follows from [MS, Prop. 4.2.7]. Toelaborate, there the authors consider a vector v which is given by modifying r v : “p h n ` ; W , . . . , W (cid:96) q by replacing two of its entries of 1 by a single entry with value 2(note that by [MS, Eq. 4.12], there are 7 ones at the end of r v ; in this regard, it ishelpful to note that the b n ` in our notation correspond to the “ v n p q ” in the notationused there.) They then show that there is a sequence of Cremona transforms taking v to the vector p
1; 1 , q . The sequence of moves they describe first transforms this vectorto the vector p
3; 2 , , , , , , q “ : p
3; 2 , ˆ q , and then one reduces further to p
1; 1 , q ;see [MS, Lem. 4.2.9]. Since the first set of moves does not affect any of the last 7entries in v , when we apply these moves to r v we obtain p
3; 1 ˆ q , as required.Since Cremona transforms preserve the deformation class of the symplectic formon a blow-up y C P , the classes B and 3 L ´ E ´ . . . ´ E have the same (genus 0)Gromov-Witten invariant. Thus, the Gromov-Witten invariant of the class B is 12.For a generic choice of compatible J , the relative adjunction formula then implies thatthe class B is represented by 12 immersed spheres, each with one nodal point. Theidea is now to stretch these curves, and show that some of them must break in such away that the moduli space M in Proposition 3.1.1 is nonempty. Definition 3.1.3.
We denote by C U the top level of the building that arises when westretch, and by C L its lower part , i..e the union of all the other levels of the limitingbuilding. Further we denote by C LL its lowest level. Thus C LL Ă p E . Since the blowing up operations all take place inside the ellipsoid E n , the curve C U lies in the negative completion X of C P p µ n ` ε q (cid:114) E n , while the lowest level C LL liesin the positive completion p E n of the blown up ellipsoid. The building C L consists of C LL together (possibly) with some curves lying in the neck , i.e. in the symplectizationof B E n . Those of our arguments that involve C L will only consider its topologicalproperties. Hence later we will consider it to be a union of matched components : seeDefinition 3.3.2. Lemma 3.1.4.
When n ą and ε, ε ą are sufficiently small, there are only threepossibilities for the lower end of C U , namely the orbit sets tp β , h n ` qu , tp β , h n ` qu , and tp β , (cid:96) n q , p β , (cid:96) n qu .Proof. Note that by (2.1.1), (3.1.1) and (2.1.5), the maximal action (i.e. symplecticarea) of the lower end of C U is (cid:96) n p µ n ` ε q « h n ` h n ` h n ` “ h n ` ` h n ` . Now the action of β is 1, while that of β is b n ` ε « h n ` h n ` . Because b n is rational,we may choose ε, ε so small that the orbit set tp β , s q , p β , t qu at the bottom of C U HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 25 satisfies s ` tb n “ s ` th n ` h n ` ď h n ` ` h n ` . On the other hand, the estimate for ω p B q in (3.1.5) implies that (modulo ε, ε ) theaction of the bottom of C U must be at least h n ` . Thus the proof of the lemma iscompleted by Lemma 3.1.5 below. (cid:3) Lemma 3.1.5. (i)
There are precisely two orbit sets of action « h n ` , namely tp β , h n ` qu and tp β , h n ` qu . (ii) There is a unique orbit set of action « h n ` ` h n ` , namely tp β , (cid:96) n q , p β , (cid:96) n qu . (iii) For any ď x ă h n ` , there is at most one orbit set of action x .Proof. Assume that we have two distinct orbit sets of the same action, and write a ` b h n ` h n ` “ a ` b h n ` h n ` , for a, a , b, b nonnegative integers with b ě b . We can assume without loss of generalitythat b ą b , else b “ b , then a “ a . We know from above that p b ´ b q h n ` h n ` is an integer. We also know that h n ` and h n ` are relatively prime. Hence, p b ´ b q must be divisible by h n ` , and so a ` b h n ` h n ` must be at least h n ` . This proves (iii).Moreover, the equation k ` (cid:96) h n ` h n ` “ h n ` does have precisely two solutions, namely p h n ` , q and p , q , which proves (i).This argument also shows that the orbit set of action h n ` ` h n ` must be unique.Otherwise, there would be a solution to a ` b h n ` h n ` “ h n ` ` h n ` with b ě h n ` ,which is impossible. This proves (ii). (cid:3) To compute the gradings of these three orbit sets as in (2.2.27), note that by Pick’sTheorem, the number of lattice points in the triangle with vertices p , q , p h n ` , q and p , h n ` q is p h n ` ` qp h n ` ` q `
1. This implies that we havegr ptp β , h n ` quq “ p h n ` ` qp h n ` ` q ´ , (3.1.6) gr ptp β , h n ` quq “ p h n ` ` qp h n ` ` q , gr ptp β , (cid:96) n q , p β , (cid:96) n quq “ p h n ` ` qp h n ` ` q ` . To prove Proposition 3.1.1 in the case n ě
1, we want to show that when we stretchwe get at least one building such that C U has negative asymptotics tp β , h n ` qu . Tothis end, we prove the following: Proposition 3.1.6. If n ě , then when we stretch the curves in class B , there are atmost such that C U has negative asymptotics tp β , (cid:96) n q , p β , (cid:96) n qu or tp β , h n ` qu . This says that the area of a lattice triangle is i ` b ´ i is the number of interior latticepoints and b is the number of lattice points on the boundary. To be precise, for a sequence of almost-complex structures stretched to length R i wecan label the holomorphic curves in class B by C ik for k “ , . . . ,
12. By choosing asubsequence of i Ñ 8 we may assume that each of the C ki converge to a holomorphicbuilding. The proposition claims that at most 9 of these 12 buildings have C U withasymptotics tp β , (cid:96) n q , p β , (cid:96) n qu or tp β , h n ` qu . Corollary 3.1.7.
Proposition 3.1.1 holds when n ą .Proof of Corollary. Since there are 12 curves in class B , Proposition 3.1.6 implies thatthere are at least three curves with negative asymptotics tp β , h n ` qu . We show inProposition 3.3.4 that these curves must have exactly one end. Thus Proposition 3.1.1holds. (cid:3)
The proof of Proposition 3.1.6 is complicated and occupies most of the rest of thispaper. This section considers the easier parts of the proof, that investigate whathappens when C U has ends either just on β or just on β . Also, we show in § τ ă x ă E p , x q s ã Ñ B p µ q is a curve C in X of degree 3 with two ends of multiplicity 1, one on β and the other on β ; see Remark 3.2.5. This has essentially zero action (i.e. it is alow action curve in the sense of § C U mightwell be one of its multiple covers. Understanding the structure of limiting buildingswhose top is a multiple cover of C requires much of the ECH machinery explained in § § § Remark 3.1.8. (i) (The case n “ ) The formulas (3.1.6) hold for all n ě
0. Furtherif n “ B , i.e. we have B : “ L ´ E ´ ¨ ¨ ¨ ´ E .However, the calculation for the action (or symplectic area) no longer works because wenow can only embed E p , ` ε q into B p ` ε q . Thus the class B : “ L ´ E ´ ¨ ¨ ¨ ´ E has area « . More significantly, the curve C no longer exists generically (since it hasnegative Fredholm index; see Remark 3.2.5), and the proof of Proposition 3.3.4 fails:indeed there can now be curves C U with more than one end on β . For further detailsof this case see § (The Fibonacci stairs) The proof outlined above is markedly easier at the points a n : “ g n ` g n considered in [CGHi], where g ‚ “ p , , , , , . . . q is the sequence ofodd-placed Fibonacci numbers. In this case, the curve to be stretched lies in class B “ g n ` L ´ W p g n ` g n q . Since this is the class of an exceptional curve, it is representedby an embedded sphere, with ECH index equal to zero. Hence ECH theory applies toshow that the top curve C U has just one end on β g n ` , since the partition conditionsimply that p ´ β p g n ` q “ p g n ` q . Further details are left to the interested reader. All thecomplications in our argument are caused by the fact that we start with a curve withone double point which may well disappear into the neck or lower part of the curvewhen we stretch, yielding limits whose top is a multiple of C . Notice also that when x ă τ , the calculation of the embedding function c in [MS] shows that we can choose HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 27 µ so small that a curve in class C would have negative action, and so could not exist. (cid:51) The low action curves.
We begin by classifying the top level curves with actionon the order of ε , which we will call low action curves . (In § light curves.) The discussion after (3.1.5) implies that in any building in class B all but one curve has low action in this sense. This holds because ω p B q « Q n (where Q n : “ h n ` ) and the symplectic areas of the classes L, E i as well as the actions of theorbits on E p , x q also are approximately equal to multiples of Q n . Remark 3.2.1.
For each n , the proof of Theorem 1.2.1 given below involves a finitenumber of strict inequalities. Throughout we assume that ε, ε (and any other similarconstant) are so small that each of these finite number of inequalities that holds when ε, ε “ θ is an approximation to some (usually rational) quantity b by writing θ “ ε b ratherthan θ « b as in § (cid:51) Throughout we assume that J is admissible and generic as explained at the beginningof § Proposition 3.2.2.
When n “ there are no low action curves in class dL with d ă h n ` . If n ą the only such curves occur when d “ m for some positive integer m and have negative end on the orbit set tp β , m q , p β , m qu . Proof. If n “
0, we have h n ` “ E p , ` ε q s ã Ñ B p { ` ε q as in Remark 3.1.8. Hence d ă dL is not an integer,while the action of the negative end is (approximately) an integer. Hence there can beno low action curves in this case.From now on we suppose n ą
0. It suffices to prove the proposition when C is some-where injective and hence has nonnegative ECH index. If C is in class dL , asymptoticto β “ tp β , (cid:96) q , p β , m qu then, by (2.2.26), the ECH index I p C q and action of C aregiven by(3.2.1) I p C q “ d ` d ´ gr p β q , action p C q “ ε d h n ` h n ` ´ (cid:96) ´ m h n ` h n ` . To better understand the relationship between the grading and the action it is conve-nient to introduce the N p a, b q sequence, that has the following important property: p‹q if N s p a, b q “ (cid:96)a ` mb where ba is irrational, then the orbit set tp β , (cid:96) q , p β , m qu on B E p a, b q has grading 2 s .The following combinatorial lemma is now key: Lemma 3.2.3.
Let n ą , d ă h n ` , and k “ p d ` d q . Then N k ` , h n ` h n ` ˘ “ h n ` h n ` N k p , q This is the sequence obtained by listing the numbers (cid:96)a ` mb, (cid:96), m ě ba is irrational then the numbers (cid:96)a ` mb are all distinct. only if n ą and N k p , h n ` h n ` q “ m ` m h n ` h n ` for some integer m ą . Remark 3.2.4.
The following observation is helpful for understanding this lemma. If n ą k, k ď p d ` d q where d “ h n ` ´
1, then N k ` , h n ` h n ` ˘ “ N k ` , h n ` h n ` ˘ ùñ k “ k . In other words there are no repeated entries in this sequence until we get to the twopoints on the line x ` h n ` h n ` y “ h n ` that each give entries N ‚ p , h n ` h n ` q “ h n ` andthese occur at places k ą p d ` d q . We saw in Lemma 3.1.5 that there are no repeatedentries in N p , h n ` h n ` q until we get to the two entries of h n ` . Hence all we need tocheck is that these points occur for sufficiently large k .To see this, note that N p , q “ p , , , , , , . . . , (cid:96), . . . , (cid:96), . . . q , with (cid:96) ` (cid:96) for each (cid:96) . Therefore for each s there are ř s(cid:96) “ p (cid:96) ` q “ p s ` s q ` ď s . Hence, for any k ď p s ` s q , we have N k p , q ď s (recall that ourconvention is that the sequence N k p a, b q is indexed starting at k “ s “ h n ` ´ we find that for k in this range(3.2.2) N k ` , h n ` h n ` ˘ ď h n ` h n ` N k p , q ď h n ` h n ` p h n ` ´ q ă h n ` as required. (cid:51) Proof of Proposition 3.2.2 for the case n ą , assuming Lemma 3.2.3. If C has low action and bottom asymptotic to the orbit set β : “ tp β , (cid:96) q , p β , m qu ,then by (3.2.1) we must have d h n ` h n ` “ (cid:96) ` m h n ` h n ` . The term on the right is an entryin N p , h n ` h n ` q , while the term on the left is an entry in the sequence h n ` h n ` N p , q which as explained in Remark 3.2.4 we may take to occur at the place k “ p d ` d q .Therefore, with k “ p d ` d q , there is k such that h n ` h n ` N k p , q “ N k ` , h n ` h n ` ˘ . But, if n ą N ‚ ` , h n ` h n ` ˘ ď h n ` h n ` N ‚ p , q , so that N k p , q ď N k p , q . Because the sequence N p , q has d ` d withthe last one at place k “ p d ` d q , we could have k ă k , but if we do we find that h n ` h n ` N k ` p , q “ h n ` h n ` N k p , q “ N k ` , h n ` h n ` ˘ ă N k ` ` , h n ` h n ` ˘ , This says that if int p E p a, b qq s ã Ñ E p c, d q then N p a, b q is termwise no larger than N p c, d q ; we applythis to the embedding int p E p , b n qq s ã Ñ B p h n ` h n ` q from (3.1.1). HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 29
Figure 3.1.
The injection of lattice points used to prove Lemma 3.2.3.We illustrate the case where (cid:96) is divisible by 3; the other cases aresimilar.where at the last step we use the result in Remark 3.2.4. But this contradicts theMonotonicity Axiom. Hence k ě k . On the other hand, property p‹q for N impliesthat gr p β q “ k and we know I p C q “ d ` d ´ k “ k ´ k ě
0. Hence k “ k . Wenow apply Lemma 3.2.3 to deduce that N k ` , h n ` h n ` ˘ “ m ` m h n ` h n ` . Hence the negative asymptotics of C are tp β , m q , p β , m qu . It follows that its action is “ ε d p h n ` h n ` q ´ m p ` h n ` h n ` q . But by (2.1.2) this is zero only if m “ d P Z . (cid:3) It remains to prove Lemma 3.2.3.
Proof of Lemma 3.2.3.
The assumption on k implies that N p , q k ` ą N p , q k . De-fine (cid:96) “ N p , q k . Then k is equal to the number of lattice points in the triangle withvertices p , q , p (cid:96), q and p , (cid:96) q , minus 1. Since n ą τ ă b n ă
7, so that wemay apply [CGLS, Prop 2.4] which states that k is exactly the number of lattice pointsin the triangle T with vertices p , q , p , (cid:96) { τ q , p (cid:96)τ , q , where τ “ p ` ? q . Denoteby T the triangle with vertices p , q , p , (cid:96) p a ` q a q , p (cid:96) a ` , q , thus with slant edge of slope ´ a , where a : “ b n . Claim:
The number of lattice points in the triangle T is less than or equal to thenumber of lattice points that are both within T and strictly underneath the part of theupper boundary of T that is to the right of the line y “ x (this is the line L in Fig. 1). To see that this fits into the discussion in [CGLS] note that τ ` τ “ T is a rescalingof the triangle with α, β “ p , q considered in [CGLS, Thm 1.1]. Proof of Claim.
The line y “ x intersects the line a ` x ` aa ` y “ (cid:96) at P “ p (cid:96) { , (cid:96) { q . Note that the point P is independent of a ą τ ; hence, the lines L and L given by a ` x ` aa ` y “ (cid:96) and p { τ q x ` τ y “ (cid:96) respectively intersect at the point P . Wetherefore have(3.2.3) t Z X T a p (cid:96) qu “ t Z X T τ p (cid:96) qu ` D ´ U, where D is the number of points in the region R bounded by the lines L , L and the x -axis (the blue region), not including lattice points on the left boundary, and U is thenumber of lattice points in the region R bounded by the lines L , L , and the y -axis(the green region), not including lattice points on the lower boundary, see Figure 3.2.We must prove that U ď D . We now show that there is a simple explicit injectionof the lattice points counted by U into the lattice points counted by D , as illustratedin the figure. This is easiest to see in: Case 1. (cid:96) ” , mod . In this case, the point P is a lattice point. So, if Q is a lattice point counted by U ,let V “ P ´ Q and consider Q “ P ` V . This is also a lattice point, see Fig. 3.2.It lies on the line of slope V passing through P , so to see that Q is counted by D ,we need to show that the y -coordinate of Q is nonnegative, and Q is not on the leftmost boundary of R . The second condition is immediately verified, since the line L has irrational slope, hence P is the unique lattice point on it. For the first condition,observe that if Q is in R , then the y -coordinate of Q is no larger than (cid:96) { τ , hence the y -coordinate of Q is bounded from below by (cid:96) p { ´ { τ q ą Q Ñ Q is an injection, the claim holds in this case. Case 2. (cid:96)
P t , u , mod 3.In this case, the point P “ p x, x q for x satisfying frac p x q P t { , { u (here, frac p x q denotes the fractional part). We now argue similarly as in the previous case. Let Q bea lattice point counted by U , let V “ P ´ Q , and consider Q “ P ` V . This is a latticepoint, on the line of slope V passing through P ; it is not on the leftmost boundaryof R , because the line L has no lattice points on it at all, being a line of irrationalslope passing through a nonintegral rational point. To see that the y -coordinate of Q is nonnegative, we observe similarly to above that if Q is in R , then the y -coordinateof Q is no larger than (cid:96) { τ , hence the y -coordinate of Q is bounded from below by (cid:96) p ´ { τ q ą
0. Since the assignation Q Ñ Q is an injection as above, the claim holdsin this second case as well.This completes the proof of the Claim. (cid:3) Now if N k p , h n ` h n ` q “ h n ` h n ` N k p , q “ h n ` h n ` (cid:96) , then by Remark 3.2.4, we can find aunique lattice point p m, n q satisfying m h n ` h n ` ` n h n ` h n ` “ (cid:96). Thus p m, n q is the unique lattice point on the upper boundary of the triangle T . Toprove the lemma we must show that p m, n q is on the line y “ x . HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 31
To see this, assume that p m, n q is strictly to the right of the line y “ x . Then bywhat was said previously, there are strictly more lattice points in the triangle T thanin the triangle T . This implies that N k ` , h n ` h n ` ˘ “ h n ` h n ` N k p , q “ N k ` , h n ` h n ` ˘ for some k ą k, a contradiction.Now assume that p m, n q is strictly to the left of the line y “ x . In fact, there are nosuch lattice points. To see this, observe that the intersection of the line y “ x with theslant edge of the triangle T occurs at the point p (cid:96) , (cid:96) q . Since the slant edge has slope ´ h n ` h n ` , if there is such a point p m, n q , it has integral nonnegative coordinates of theform ` (cid:96) ´ δ, (cid:96) ` δ h n ` h n ` ˘ for some δ ą . But then 3 δ h n ` h n ` P Z , which, because gcd p h n ` , h n ` q “
1, implies δ ě h n ` ą (cid:96) .Thus (cid:96) ´ δ must be negative, a contradiction.It follows that p m, n q must be on the line y “ x , which completes the proof ofLemma 3.2.3 and hence also of Proposition 3.2.2. (cid:3) Remark 3.2.5.
The numerical arguments in the above proof of Proposition 3.2.2 leaveopen the possibility that C U could be a multiple cover of a curve C in class 3 L with twoends, one on β and the other on β , both of multiplicity one. When τ ă x ă
7, such acurve has index 2 ` ´ ` ´ ´p ` t x u q ˘ “ ă x ă
7, and one can check that its ECHindex is also 0 for these x . Further, one can show that C exists either by consideringwhat happens to an exceptional sphere in the class A “ L ´ E ´ E ´ . . . E whenthe neck is stretched, or by considering the ECH cobordism map from the boundary ofthe (distorted) ball B B p µ q to the ellipsoid B E p , b n q as in [CGHi]. The correspondingembedding obstruction is 3 µ ą ` x , which gives c p x q ě ` x . Hence by [MS], thisobstruction is sharp in dimension 4.But because C has two negative ends, the stabilization arguments in Proposi-tion 3.6.1 do not apply; and indeed this obstruction cannot persist in higher dimensionsbecause we know from [Hi] that c k p x q ď x ` x for k ě (cid:51) Remark 3.2.6.
In the proof of the Claim above, the only relevant property neededof the b n is the inequality b n ą τ . Hence, the argument also shows that N p , a q ď a ` N p , q , which implies that there is a symplectic embedding E p , a q s ã Ñ B p a ` q byMcDuff’s proof of the Hofer conjecture in [M2, Thm. 1.1]. This gives a new andprobably simpler proof of the computation of c p x q for τ ď x ď § N p , a q ď λ ¨ N p , q implies λ ě a ` , follows directly by looking at the tenth term in each sequence.)3.3. Analysing C U when n ą . This section contains the proof of Proposition 3.1.6when n ą
0, modulo some results that are deferred to § §
4. By Lemma 3.1.4there are three possibilities for the negative asymptotics of C U , namely tp β , h n ` qu , tp β , (cid:96) n q , p β , (cid:96) n qu and tp β , h n ` qu . We discuss these cases in turn. Lemma 3.3.1.
The negative end of C U cannot be tp β , h n ` qu .Proof. If C U were reducible, then one of its irreducible components would have tohave low action so that by Proposition 3.2.2 its lower end would have to involve theorbit β as well as β . So this cannot happen. Hence, since C U cannot be irreducibleand multiply covered because gcd p h n ` , h n ` q “
1, we conclude that C U must beirreducible and somewhere injective. By the grading calculations in (3.1.6), we alsomust have I p C U q “
0, and thus ind p C U q “
0; so the negative ends of C U must satisfythe partition conditions. These are described in Remark 2.2.1 (ii). By (2.2.9), whenwe compute ind p C U q using the index formula (2.2.29), we find ind p C U q “ ´ ` (cid:96) n ` ´ p h n ` ` h n ` ` ´ s q “ s ´ , where s denotes the number of negative ends and we have applied (2.1.2). But Re-mark 2.2.1 (ii) shows that s ě
8, contradicting the fact that ind p C U q “ (cid:3) The case with negative end tp β , (cid:96) n q , p β , (cid:96) n qu : In this case C U has ECH index ´ J onlyif it is a multiple cover. Since it could be a multiple cover of the curve C mentionedin Remark 3.2.5, we cannot ignore this possibility. This case is analyzed by looking atthe structure of the building C L formed by all but the top level of the limiting building(see Definition 3.1.3). Thus C L consists of some curves in the neck together with somecurves C LL in the completion p E n of the blown up ellipsoid. Note that, for generic J , C L intersects the exceptional divisors transversally in ř i W i “ h n ` ` h n ` ´ constraint .Many of our arguments are essentially topological rather than analytical in nature,and it is convenient to introduce the following terminology. By a curve or trajectory we mean the image of some J -holomorphic map u ; thus curves lie in a single levelof the building and need not be connected. A curve with a connected domain iscalled irreducible . We may join these curves along pairs of positive and negativeends that match in the sense that each limits on the same multiple β ri of the sameorbit, thereby decomposing the building into a union of connected pieces that we call matched components , or components for short.Here is the key definition. Definition 3.3.2.
A matched component in C L is called a connector if its top hasends both on β and on β . Thus a connector might have two levels, the top (lying in the neck) consisting of atrivial cylinder over β r together with a cylinder from β s to β m , joined by an irreduciblecurve in p E n with two top ends, one on β r and the other on β m . Proof of Proposition 3.1.6:
By Lemma 3.3.1, Proposition 3.1.6 will hold if we showthat there are at most 8 B -curves that limit on a building such that C U has negativeend tp β , (cid:96) n q , p β , (cid:96) n qu and C L has no connectors, and at most one more in which C L has a connector. The first claim is proved in § § l HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 33
Example 3.3.3. (The case n “ , i.e. with b “ ) We illustrate what can happenin the first nontrivial case. Here (cid:96) “ E p q “ p E ` ¨ ¨ ¨ ` E q ` E ` E ` ¨ ¨ ¨ ` E “ : 8 E ... ` E ` E ... p q . If C L has no connectors, we may divide its matched components into two groups D , D ,where, for i “ , D i is a union of planes with top on some multiple of β i . We will seein § C L , one for each k “ , . . . , k “ D goes through E , . . . , E (and has action “ ε ), while D goes through all the othersand has low action. On the other hand, if k ą D goes through the 8 constraints E , . . . , E , E k and has low action, while D goes through all the others. Notice thatin the first case in order for D to have nonnegative index it must be connected with asingle top end on β , while in the second case D must be connected with one top endon β . Thus in both cases C U is a connected 7-fold cover of C , with at least one endof multiplicity 7.If C L has a connector, then we show in § C U is the union of C with a 6-foldcover of C , and that there is a unique connector D , with two top ends on β , β ,and action “ ε . In this case D consists of the 6 planes with top β , each through asingle constraint E i , ď i ď
6, while D has top β and goes through E , . . . , E . Allthe other constraints 6 E ... ` E ... p q lie on D .As we will see, there are analogous decompositions for all n ą
1. In particular, ifthere is a connector it is unique and has essentially all the action. It has two top ends,one on β (cid:96) n ´ and the other on β (cid:96) n ´ (cid:96) n ´ , and there is a formula for its constraints interms of weight expansions; see Proposition 4.4.1. As in the proof of Proposition 3.5.1below, we use the fact that B ¨ B “ B -curve that breaks this way. (cid:51) The case with negative end tp β , h n ` qu :Proposition 3.3.4. Let n ą . Then, if C U has negative ends on tp β , h n ` qu , itmust have a single negative end.Proof. First note that because the bottom constraint involves only β it follows asin the proof of Lemma 3.3.1 that C U is irreducible and hence somewhere injectivesince 3 (cid:96) n ` “ h n ` and h n ` are mutually prime. By the relative adjunction for-mula (2.2.10) and the formulas in (2.2.28), we must have w τ p C U q “ ´ w ´ τ p C U q “ ´ s ` h n ` ´ p h n ` ´ h n ` q ´ δ p C q , (3.3.1) “ ´ s ` h n ` p h n ` ´ q ` ´ δ p C q , where s denotes the number of negative ends, and we have used the Fibonacci iden-tity (2.1.6). On the other hand, if p a , . . . , a s q is the partition given by the negativeends of C U , then by Remark 2.2.3 (i) we have(3.3.2) w ´ τ p C U q ě ÿ i ‰ j min p a i p j , a j p i q ` s ÿ i “ p a i ´ q p i where p i “ r a i h n ` h n ` s . We can therefore bound the right hand side of (3.3.2) from belowby h n ` h n ` `ÿ i ‰ j a i a j ` s ÿ i “ p a i ´ a i q ˘ “ h n ` h n ` ˜ p ÿ i a i q ´ p ÿ i a i q ¸ (3.3.3) “ h n ` p h n ` ´ q , since ř a i “ h n ` .So regardless of the a i , the right hand side of (3.3.2) is some integer bounded frombelow by h n ` p h n ` ´ q . In fact, (3.3.2) must be strictly greater than h n ` p h n ` ´ q as long as s ě
2, since the bound (3.3.3) comes from throwing away some fractionalparts, which must be positive.So, assume for the sake of contradiction that s ě
2. Then the right hand side of(3.3.2) must be at least h n ` p h n ` ´ q `
1, so (3.3.1) implies that we must have s “ δ p C q “ s “ ÿ i “ p a i ´ q p i ě h n ` h n ` ÿ i “ p a i ´ a i q ` . By (3.3.2) this implies that w ´ τ p C U q ě ÿ i ‰ j min p a i p j , a j p i q ` h n ` h n ` ÿ i “ p a i ´ a i q ` ą h n ` p h n ` ´ q ` . Hence, because w ´ τ p C U q is an integer, we must have w ´ τ p C U q ě h n ` p h n ` ´ q ` a and a are additiveinverses modulo h n ` , without loss of generality we have frac p a h n ` h n ` q ă . Thisimplies that p ´ h n ` h n ` a ą so that p a ´ q p ´ h n ` h n ` p a ´ a q ą p a ´ q ě a ď
2. If a ď
2, then h n ` ą a ě h n ` ´ , and 1 ´ frac ` a h n ` h n ` ˘ ě h n ` h n ` , To get strict inequality, we also use the fact that r θ is slightly larger than h n ` h n ` . HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 35 which implies that p a ´ q p ´ h n ` h n ` ` a ´ a ˘ “ p a ´ q ` p ´ a h n ` h n ` ˘ “ p a ´ q ` ´ frac p a h n ` h n ` q ˘ ě p a ´ q h n ` h n ` ą . In either case, then, the claim is true, hence the proposition. (cid:3)
The case when C L has no connectors. We now consider what happens when n ą
0, the negative end of C U is tp β , (cid:96) n q , p β , (cid:96) n qu , and there are no connectors inthe sense of Definition 3.3.2. We write C L “ D Y D where D denotes the union ofmatched components with ends only at β , and D denotes the union of componentswith ends only at β . Thus a component of D might consist of a cylinder in the neckwith top end on some multiple of β and bottom on a multiple of β , completed by aunion of planes in the blown up ellipsoid. Notice that if a B -curve is close to breakinginto a building whose top is an (cid:96) n -fold cover of C and whose bottom has no connectors,then we may cut the B -curve just above the neck into three pieces that approximate (cid:96) n C , D , and D .We record the homology classes of D and D via integer vectors. Here, as in (3.1.3),we order the exceptional classes in decreasing order of size, and we write p z , . . . , z k q for the homology class ´p z E ` . . . ` z k E k q . So, with this notation the homology class of C L is given by the normalized weightexpansion W p b n q “ ` W ˆ , W , W ˆ , . . . , , ˆ ˘ , which by (2.1.10) is a vector of total length 6 n ` B “ p B , B , . . . q whose lengths are given by the entries in the continued fractionexpansion r , p , q ˆp n ´ q , , s of b n “ P n Q n . The symplectic areas (or actions) of theexceptional classes are given (modulo ε ) by the vector w p b n q .Consider some representative of B , and let z denote the homology class of D and y the homology class of D . Thus, since we are assuming that there are no connectors, z ` y “ W p b n q . Proposition 3.4.1.
In any neck stretching, at most representatives of B limit on abuilding with no connectors and where C U has negative end tp β , (cid:96) n q , p β , (cid:96) n qu . Proposition 3.4.1 will follow easily from the next two lemmas. Before reading theproof of Lemma 3.4.2, it might be useful to refer back to Example 3.3.3.
Lemma 3.4.2.
There are at most elements in the collection C of vectors z that occuras possible homology classes for D . Proof.
Step 1.
Two elements z, z P C differ in at most two places. Moreover any twoentries differ by at most one. Let z and z be two homology classes with representatives D , D , and consider p z ´ z q ¨ p y ´ y q ; let D and D denote the representatives for y and y . We have(3.4.1) p z ´ z q ¨ p y ´ y q “ z ¨ y ` z ¨ y ´ z ¨ y ´ z ¨ y. We can understand the right hand side of (3.4.1) by observing that, because theorbits β , β on B E can be filled by discs that intersect once, the number of intersectionsbetween D and D is given by (cid:96) n ´ z ¨ y , and similarly for the other relevant pairings ofthe D i . On the other hand, as we pointed out above there are J -holomorphic B -curveswhose lower parts approximate D Y D and D Y D arbitrarily closely. Hence all theseintersection numbers are nonnegative, and bounded from above by 1 because B ¨ B “ |p z ´ z q ¨ p y ´ y q| ď z ` y “ z ` y “ W the vectors z ´ z , y ´ y are equal. Thus,if z ´ z : “ p ε i q we have(3.4.2) 0 ď ÿ ε i “ p z ´ z q ¨ p y ´ y q ď . Step 1 follows readily.
Step 2.
Completion of the proof.
Now recall that each of the classes E i have a definite area, and these areas come inblocks; no block has length greater than 7. We also know that the total action of each D i must be close to either 0 or Q n .Also recall that only the last block of the weight vector w has entries of size Q n ;further, the only two blocks whose entries differ by Q n are the second and third to last.We now prove that C has at most 8 elements by a case by case analysis. Claim.
If any two z and z in C differ on any block B other than the last three, thenall the representatives in C differ on this block.Proof of Claim. Because the entries in B have size ą Q n and the total area of D is atmost Q n it follows from Step 1 that the difference vector z ´ z must have exactly twoentries, one ` ´
1. Further these must occur on the same block because B isnot one of the last three blocks. So, if z is any other representative in C , z must havethe same entries as z and z on all other blocks: otherwise, either z would differ fromone of z and z in more than 2 places, violating Step 1, or it would have the wrongarea. (cid:3) Case 1:
With this claim in mind, consider the case where all of the elements in C differon B , which by assumption has length (cid:96) , where (cid:96) “ B (i.e. the entries in any vector in C corresponding to the block B ) p , . . . , p (cid:96) , and fix anelement z , with entries a , . . . , a (cid:96) on this block. Now consider another element z P C .Then z and z must differ in two places, and without loss of generality we may assumethat z “ p a ` ε , a ´ ε , . . . , a (cid:96) q , where | ε | “
1. A third element z can differ fromboth z, z in the first place by at most 1. Hence the first place of z must be either HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 37 a or a ` ε ; and similarly, its second place is either a or a ´ ε . Further, if z isdifferent from both z, z we may assume it differs from z (and hence also z ) in place3, and hence is either p a ` ε , a , a ´ ε , a , . . . , a (cid:96) q or p a , a ´ ε , a ` ε , a , . . . , a (cid:96) q . Since the two third entries above differ by 2, only one of these possibilities can occur.Thus there are at most three elements in C whose entries differ only in the places1 , , z, z we may assume that z “ z inits first place, so that z, z differs only in the first two places, while z , z differ only inthe second and third places. If there is another element z P C we may assume thatit differs from z, z , z in the fourth place. Again there are two possibilities for thiselement, only one of which occurs. Finally there might be one or two more elements in C that differs from the previously found elements in places 5 and or 6. Thus | C | ď Case 2:
The other case to consider, then, is the case where all the representatives in C differ only on the last three blocks.Note first of all that in this case, by area considerations, all of the elements in C either differ on the second and third to last blocks, or on just the last block. In thecase where the elements differ on the second and third to last blocks, we can repeat theargument from Case 1, to conclude that there are no more representatives in C thanthe sum of the lengths of these two blocks. Since the sum of these lengths is no morethan 7, this proves Lemma 3.4.2 in this case.We can assume, then, that all of the entries in C differ on the last block. With this inmind, label the places in this block p , . . . , p , as above. As above, first fix an element z P C , and consider another element z P C . Note that because z ` y “ W has entries1 on the last block, all entries of z are 0 or 1. Since this is true for all elements in C ,any other element z P C is determined by the places at which it differs from z . Wechoose z so that it has the smaller of the two areas of the elements in C , and then writeany z P C as z ` ε , where ε is a vector of length 7 with entries 0 or 1 and we addmodulo 2. Thus ε belongs to a collection E Ă Z of vectors that each have at most twononzero entries. Let E Ă E be the subset of vectors of length 1. If E “ H , then theargument in Step 1 shows that | E | “ | C | ď
7. If E ‰ H , then our choice of z impliesthat area p z ` ε q ą area p z q for ε P E . Therefore the places where ε ‰ z “
0. Therefore if z has k zero places, assumed w.l.o.g. to bethe first k , there is a subset I Ă t , . . . , k u such that for all i P I there is an element z i in C that agrees with z except at the i th place where z has zero while z i has one.Any another element z in C must have a zero in some place k where z is one. Hencebecause z has minimal area, z also has to have a one in some place j that z is zero.But if i P I (cid:114) t j u , then z i differs from z in all three places i, j, k . Therefore, if | I | ą z , which implies that | C | “ | I | ` ď
8. On the other handif | I | “
1, there are at most 7 ´ | I | elements z , one for each place where z has entry1. Thus in all cases there are at most 8 elements in C . This completes the proof ofLemma 3.4.2. (cid:3) Lemma 3.4.3. D and D must intersect.Proof. We know that z ` y “ W , the normalized weight vector; below, we also makeuse of the unnormalized weight vector w . Observe that(3.4.3) P n Q n “ W ¨ W “ p z ` y q ¨ p z ` y q “ z ¨ z ` y ¨ y ` z ¨ y. Step 1.
The following key estimate holds: (3.4.4) z ¨ z ` y ¨ y ě (cid:96) n ` Q n P n ` P n Q n ´ ˘ . We prove this using an optimization argument. Recall that one of the D i has areaclose to 0, and the other has area close to 1 { h n ` . If D has area close to 0, then z issubject to the area constraint(3.4.5) z ¨ w “ (cid:96) n . Hence z ¨ z is minimized when z “ λw , where λ is a scalar satisfying(3.4.6) λ “ (cid:96) n w ¨ w “ (cid:96) n Q n P n so that(3.4.7) z ¨ z “ (cid:96) n Q n P n Similarly, the same argument gives that if D has area close to 0, then y ¨ y is minimizedwhen(3.4.8) y ¨ y “ (cid:96) n P n Q n . Thus, if D and D both had area close to 0, then combining (3.4.7) and (3.4.8) wouldgive (3.4.4). The only difference when D has area close to Q n is that the right handside of (3.4.5) is smaller by Q n and analogously for D . This will weaken our estimatefor z ¨ z ` y ¨ y . The difference is greatest when D has area close to Q n In this case,the right hand side of the analog of (3.4.5) is smaller by Q n so that the right hand sideof the analog of (3.4.6) is smaller by P n . Hence, the right hand side of the analog of(3.4.8) is smaller by an amount that is bounded from below by2 (cid:96) n P n Q n ¨ P n “ ¨ (cid:96) n Q n ă . Since z ¨ z ` y ¨ y is an integer, we obtain (3.4.4). Step 2.
Completion of the proof.
Given (3.4.4), we can now prove the claim by using (3.4.3). We know that D and D must intersect (cid:96) n ´ z ¨ y HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 39 times. By using (3.4.4) and (3.4.3), we get that z ¨ y is bounded from above by κ : “ ˆ P n Q n ´ (cid:96) n ` Q n P n ` P n Q n ˘˙ ` . Thus, the number of intersection points is bounded from below by (cid:96) n ´ κ . But P n ` Q n ´ P n Q n “
9, which implies Q n P n ` P n Q n ą
7. Thus2 p (cid:96) n ´ κ q ą (cid:96) n ´ P n Q n “ pp P n ` Q n q ´ P n Q n q “ . Thus the intersection number is positive, as claimed. (cid:3)
Proof of Proposition 3.4.1.
For generic J on the blow up z C P , the class B is representedby precisely 12 disjoint embedded curves C α , ď α ď
12. When we stretch the neckvia a generic family J R , R Ñ 8 , of almost complex structures, we may choose anincreasing unbounded sequence R i and choose labels for these B -curves so that, foreach α , the sequence p C iα q i ě converges to some limiting building as i Ñ 8 . In viewof Lemma 3.4.2, it suffices to show that for each splitting z, y of the constraints thereis at most one such sequence p C iα q i ě whose limiting building has these constraints.Suppose to the contrary that there were two such sequences. Then as we remarkedearlier for very large i we may cut the spheres C i and C i just above the neck in such away that the lower parts of these curves are unions D i Y D i and D i Y D i , where D i and D i are compact curves with constraints z and boundaries that are both are veryclose to β , while D i and D i have constraints y and boundaries very close to β . ButLemma 3.4.3 implies that for large i both intersections D i X D i and D i X D i arenonempty. Since these are intersections of J -holomorphic curves, both intersectionsare positively oriented. It follows that C i ¨ C i ě
2. Since these curves both representthe class B which has B ¨ B “
1, this is impossible. (cid:3)
The case n “ . As in Remark 3.1.8, we take(3.5.1) µ ˚ “ ` ε , and start from the embeddingΦ : E p , ` ε q s ã Ñ C P p ` ε q , defining X to be the completed complement of its image. The class B is now 3 L ´ E ´¨ ¨ ¨ ´ E , and we are interested in the structure of the top part C U of the limits of genus0 representatives of B as the neck Φ pB E p , ` ε qq is stretched. One main differencefrom the case n ą C U must limit on the orbit set tp β , qu . Indeed, since C U must have positive action it cannot be asymptotic to tp β , q , p β , qu ; because itsaction is ă tp β , r qu , r ă
8; while if it were asymptotic to p β , q itwould by (2.2.29) have index 2 p´ ` ´ ´ q ă C U to have two negative ends on the orbitset tp β , qu . To see this, note that if C U has a single negative end on tp β , qu then, by Remark 2.2.3 (ii) the writhe bound in (2.2.16) is an equality, and one can use it togetherwith Proposition 2.2.2 to calculate that 2 δ p C U q “
2, i.e. C U must have a double point.On the other hand, the top part C AU of the building obtained from the exceptionalsphere in class A “ L ´ E ´ E ´ . . . E by stretching the neck B E p , ` ε q must beembedded. Therefore, C AU must have more than one end on tp β , qu ; cf. Remark 3.5.4below.Here is our main result. Proposition 3.5.1.
When n “ all representatives of B break into a building whosetop C U is connected and has negative end on the orbit set tp β , qu . There are at most representatives with more than one end, and hence at least with just one end. The proof below shows that if C U has more than one end, it must have two endsof multiplicities 1 ,
7, and that there are at most 8 such possibilities; cf. Remark 3.5.4.Our main tool is a writhe calculation for neck components.
Lemma 3.5.2.
When n “ , C U is connected and simple with negative end on tp β , qu .Moreover, the bottom level C LL consists of eight disjoint and embedded components,each with top p β , q and going through one constraint.Proof. We saw above that C U must have negative end on tp β , qu . A curve of nonnega-tive index in X of degree 1 has bottom end on tp β , m qu for m ď tp β , m qu for m ď
5. Hence C U must be connected,and hence somewhere injective because gcd p , q “ C LL of the limiting building lies in the completed blown up ellip-soid, has top on tp β , qu with grading 16 and goes through the constraints E , . . . , E .Therefore, (2.2.32) implies that I p C LL q “ “ ind p C LL q . Therefore by Proposi-tion 2.2.2, C LL is embedded with ECH partitions. Since p ` ` p β , q ˘ “ p ˆ q , C LL must have eight positive ends. But no component of C LL can have more than onepositive end because C U is connected and the original curve in class B has genus zero.Hence C LL must have 8 components, which are disjoint, since their union has no doublepoints. (cid:3) Lemma 3.5.3.
When n “ , C U has either one negative end or two negative ends ofmultiplicities , . Moreover, in the latter case there is one double point in the neckjust before breaking.Proof. Consider the part in the neck region of a stretched B -curve that is very close tobreaking into a curve with top C U . This neck part is a union of connected curve piecesthat for short we call components, one for each negative end of C U . By Lemma 3.5.2,its bottom ends all have multiplicity one, i.e. they approximate the simple orbit β .We denote by C necks a component with one positive end very close to β s , and therefore s negative ends each of multiplicity 1. Consider the union C neck : “ C necks Y C necks of two such neck components where s ď s . Because C U and C LL are simple, wemay estimate the writhe w ` p C neck q at the top of C neck by using the writhe estimatefor the appropriate negative ends of C U (which is a lower bound), and the writhe HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 41 w ´ p C neck q at its bottom by the formula for the positive end of C LL (which is an upperbound). Thus at the positive end we have a “ s , a “ s and p “ p “ w ` p C neck q ě p s ´ q ` p s ´ q ` s , while w ´ p C neck q ď a i “ , p i “
0. Thus the adjunction formula gives(3.5.2) 2 δ p C necks Y C necks q ě ´ ´ p s ` s q ` p s ´ q ` p s ´ q ` s “ s , i.e. there are at least s double points in the neck. More generally, if the bottom of C U has r ends with multiplicities a ď a ď ¨ ¨ ¨ ď a r , then we have(3.5.3) n ě p r ´ q a ` p r ´ q a ` ¨ ¨ ¨ ` a r ´ double points in the neck. Since we must have n ď r “ a “ (cid:3) Proof of Proposition 3.5.1.
Let us suppose that C U has two negative ends. By Lemma 3.5.2these have multiplicities 1 ,
7, and the curve C neck Y C neck has one double point. Weclaim that each neck component is embedded. (Recall these are components of ourcurves mapping to the neck region just before breaking.) To see this, notice that theright hand expression in formula (3.5.2) decomposes as a sum of three terms, the terms2 ´ ´ s i ` p s i ´ q that give lower bounds for 2 δ p C necks i q and the term 2 s that boundstwice the intersection number of the components. Since there is at most one doublepoint, we must have 2 δ p C necks i q “ i .To complete the argument, observe now that because there is a distinguished con-straint that is attached to C neck , there are eight ways to assign the constraints. More-over, as in the proof of Proposition 3.4.1 given at the end of § B -curve that is close to splitting in this way. To checkthe latter statement, note that any two such distinct curves C, C which are close tosplitting would have to intersect in the neck region in at least two points, namely p C q neck X C neck and C neck X p C q neck (which both consist of a single point by the cal-culation above). But this is impossible because all intersections count positively (sincethe curves are J -holomorphic) and B ¨ B “ (cid:3) Remark 3.5.4. (i) One can similarly study the possible splittings of the exceptionalsphere C A in class A “ L ´ E ´ E ´ ¨ ¨ ¨ ´ E . Again it must split along the orbitset tp β , qu , but now the grading of this orbit set is smaller than the contribution tothe ECH index of the bottom constraint ř m i E i “ E ` E ` ¨ ¨ ¨ ` E ; indeed, by(2.2.32), this is 2 ř i m i ` m i .Hence C LL cannot be simple: indeed it consists of a two-fold cover of a cover of acomponent C LL through E together with six other components through E i , ď i ď C A splits into a building whose top level has two negative ends ofmultiplicities 1 ,
7, and that the component of C LL attached to C neck is C LL and sogoes though α , while the 7 components attached to the 7 negative ends of C neck gothrough α , . . . , α . For in this case one can imagine that the bottoms of C neck and C neck are distorted by the attached copies of C LL in such a way that the correspondingneck components of an approximating A -curve no longer intersect. Although we do notattempt to prove here that this is what happens, the fact that C LL is not simple does mean that the writhe calculation in Lemma 3.5.3 is no longer valid. This situation wasmisunderstood in [HiK]; see [HiK2].(ii) One can try to generalize this argument to ellipsoids of the form E p , k ´ q , k ą . One would now start from a curve in class B “ k ´ E ´ ¨ ¨ ¨ ´ E k ´ , with genus zeroand δ p B q “ p k ´ qp k ´ q double points, stretch the neck, and then hope to findamong the resulting buildings at least one whose top level is a genus zero trajectory C U with just one negative end on β k ´ . One can argue as above that the top level ofeach such building must have negative ends on the orbit set tp β , k ´ qu . However,because B -curves have more double points, there are now more possibilities for thepartitions of 3 k ´ C U for each possible partition of 3 k ´ A in (i)above. For example, if k “ c “ B ¨ A i “
5) arerelevant: A : “ L ´ E ´ E ... A : “ L ´ E ´ E ... with A “ , A “ A : “ L ´ E ´ E ... A : “ L ´ E ´ E ... with A “ A “ ´ . Here A , A are classes of exceptional spheres, while A has one double point and A has three. This approach is barely possible when k “ C U with just one negative end. The argument uses the fact thatthere are 620 B -curves, 12 A -curves and 96 A -curves through a generic set of points.Since the genus zero Gromov–Witten invariant of the class B and the relevant classes A grows very rapidly with k , this method does not seem feasible for large k . (cid:51) The stabilization process.
We conclude this section with a proof of the fol-lowing sharpened version of Proposition 1.3.1, which gives conditions under whichembedding obstructions in dimension 4 persist under stabilization.
Proposition 3.6.1.
Let X µ ˚ ,x be the completion of C P p µ ˚ q (cid:114) Φ p E p , x qq , where x “ b ` ε , where b “ pq for some relatively prime integers p, q and very small and irrational ε ą . Suppose that for all sufficiently small ε ą and generic admissible J there isa genus zero curve C in X µ ˚ ,x with degree d , Fredholm index zero, and one negativeend on p β , p q , where gcd p d, p q “ . Then there is a constant S p d, x q such that for any S ě S p d, x q , k ě , and µ ą , the existence of a symplectic embedding (3.6.1) E p , x, S, . . . , S looomooon k q s ã Ñ C P p µ q ˆ R k implies that µ ě dp “ bb ` Proof.
This is proved by slightly extending the arguments given in [CGHi, § b “ g n ` g n . We start fromthe stabilized embedding r Φ : “ Φ ˆ ι : E p , x, S, . . . , S q s ã Ñ C P p µ ˚ q ˆ R k HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 43 where Φ is as in (1.3.1), µ ˚ is given by (3.1.1) or (3.5.1), and ι is the obvious inclusionon the last 2 k -dimensions. Consider the space M obtained by removing the ellipsoidim p r Φ q and completing at the negative end. By the argument in [HiK, Lem. 3.1], anyembedding r Φ of r ¨ E p , x, S, . . . , S q into C P p µ ˚ q ˆ R k may be connected to r Φ by a1-parameter family r Φ t : λ p t q E p , x, S, . . . , S q Ñ C P p µ ˚ q ˆ R k , t P r , s , of embeddings, for a suitable function λ p t q P p , max p r, qs with λ p t q “ t near0 and λ p t q “ r for t near 1. In particular, we may apply this to the embedding(3.6.1), rescaled by µ ˚ µ . The idea is now to translate this family of embeddings into afamily of compatible almost complex structures on a fixed manifold, in order to find a J -holomorphic curve that gives a constraint on µ .To elaborate, choose an identification of the completed spaces obtained by removingim p r Φ t q with p M , ω t q , where ω t is a suitable family of symplectic forms; we assume thatthe identification is the identity outside C P p µ ˚ q ˆ B k p T { q for a suitably large T . Weconsider the space J p T q of almost complex structures J on M that are admissible forsome ω t , and have product form outside C P p µ ˚ q ˆ B k p T { q with projection to R k equal to the standard complex structure. Our aim is to show that for generic J P J p T q that is admissible with respect to ω there is at least one degree d curve in M with asingle end on β p , since then the required bound comes from the positivity of its action.We may enlarge T so that the monotonicity theory implies that every degree dJ -holomorphic curve lies inside C P p µ ˚ q ˆ B k p T q , and then compactify M at itspositive end, obtaining a family of symplectic manifolds p M , ω t q that are identifiedwith completions of subsets of C P p µ ˚ q ˆ C P k p T q : see [HiK, Lemma 3.3]. It therefore suffices to analyze curves in p M , ω t q for J P J p T q , where J p T q is theobvious analog of J p T q . Because im p r Φ q and the symplectic form ω on M are in-variant with respect to a suitable T k -action that rotates the second factor R k , thereis an ω -compatible element J M P J p T q which is also T k invariant and restricts to analmost-complex structure J X on X . Thus, the initial curve C in X whose existencewe assume may be considered as an element of the moduli space M J M p M , dL, β p q , ofsomewhere injective genus zero curves of degree d and with one negative end on theshort orbit β of multiplicity p . The index formulas in [CGHi, Prop. 13] show that C has index zero for any k ; cf. [CGHi, Lemma 14] and (3.6.2) below. The proof thenconsists of the following steps. ‚ We show that the moduli space Ť t Pr , s M J t p M , dL, β p q is compact, where J t P J , t P r , s , is a generic path of almost complex structures. ‚ We show that the count of curves in M J M p M , dL, β p q is positive.The proposition then follows immediately: for more details see [CGHi, § The compactness argument . In [CGHi], the divisor p line q ˆ C P k p T q is removed from the target, one slightly distorts C P p µ ˚ q (cid:114) p line q to E p µ ˚ , µ ˚ ` ε q , and then completes M at its positive end as well. However, how onetreats the positive end makes no difference to the index calculations or stabilization arguments. Consider a generic 1-parameter family J t P J p T q , t P r , s , of almost complex struc-tures on the completed p k ` q -dimensional manifold M . There are two possibilitiesfor loss of compactness.The first is convergence to a multiply covered curve in M . This can be excludedusing the fact that p and d are coprime by hypothesis. The other possibility is thatthere is a limiting building C . Any such limiting building has a top component in M and lower levels in the symplectization B E p , x, S, . . . , S q ˆ R . To deal with this casewe argue much as in [CGHi], the crucial point being that the limiting building musthave index zero. It is convenient to think of C as a union of matched components (seeDefinition 3.3.2) as follows. We first match all possible curves in symplectization levels.This defines a single component C ˚ with a negative end on β p , and other matchedcomponents in the symplectization with no negative ends. We then group the curvesin C (cid:114) C ˚ into matched components; in other words, we match each symplectizationcomponent with any compatible curves im p u i q in M forming larger components. Wedefine the index of a matched component to be the sum of the deformation indicesof the constituent curves minus the dimension of the orbit spaces where matchingoccurs (for ends asymptotic to covers β or β this dimension is just 0, otherwiseit is 2 p k ´ q , the dimension of the Morse-Bott family). By the index formulas forcurves in symplectizations, the Fredholm index of any component formed by joiningcurves in symplectizations can be calculated as if the component was a curve in asingle level, since the contributions to the Fredholm index of pairs of matching ends insymplectizations cancel. Similarly, if the component contains curves in the top level,then we can calculate its index as though all its components were in the top level.We begin with the following. Lemma 3.6.2. If k ě , S ě S p d, x q is sufficiently large and the path J t is generic,then any curve in the limiting building C that lies in M has the following properties. ‚ Any ends on B E p , x, S, . . . , S q must be asymptotic to covers of β or β ; ‚ Every end must have multiplicity less than S { x .Proof. In addition to β and β , we have a Morse-Bott orbit γ . To prove the first bulletpoint, it suffices to consider irreducible somewhere injective curves. Since J t is generic,any irreducible somewhere injective curve C in C asymptotic to γ has index ě ´ ě
0. If C has degree d , ends on β r i for 1 ď i ď n , on β s j for 1 ď j ď n , and on γ t (cid:96) for 1 ď (cid:96) ď n , we find using thediscussion following (2.2.35) that This is called S in [CGHi]. HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 45 ind p C q “ p k ´ qp ´ n ´ n ´ n q ` d ´ n ÿ i “ ´ r i ` p t r i { x u ` q ` k p t r i { S u ` q ¯ (3.6.2) ´ n ÿ j “ ´ s j ` p t s j x u ` q ` k p t p s j x q{ S u ` q ¯ ´ n ÿ (cid:96) “ ´ t (cid:96) ` p t t (cid:96) S u ` q ` p t t (cid:96) S { x u ` q ` p k ´ qp t (cid:96) ´ q ¯ . We must have d ď d , and, if there are any ends on γ , the term t t (cid:96) S u ě t S u is largewhile the very first term combined with the final term in the sum over (cid:96) together givea nonpositive contribution.Therefore if S ě d the right hand side of (3.6.2) must be negative if C has any endson γ . This gives a contradiction.Similarly, from (3.6.2) no irreducible somewhere injective curve of degree d in C canhave ends of multiplicity more than 3 d . Since an arbitrary degree d curve is a sum of m i -fold covers of irreducible somewhere injective degree d i curves, where ř m i d i “ d ,the maximum multiplicity of an end of such a curve is ř i m i d i “ d ď d which is ă Sx if S ě S p d, x q : “ max p xd , d q “ xd . Therefore we may take S p d, x q “ xd . (cid:3) The first point in the next lemma is a slight generalization of [CGHi, Lemma 18].Note that the proof uses the fact that we consider curves with bottom multiplicity p rather than a more general m . Lemma 3.6.3.
As above, let C ˚ be the unique matched component of C in the sym-plectization with a negative end on β p , and suppose that S ě S p x, d q Then index p C ˚ q ě with equality if and only if C ˚ is an unbranched cover of a trivial cylinder over β ,and hence has one positive end of multiplicity p .Proof. Every positive end of C ˚ is matched by the negative end of some curve in C that lies in M . Hence, because S ě S p d, x q Lemma 3.6.2 shows that C ˚ has no endson the Morse–Bott orbit γ .Suppose that the positive ends of C ˚ are asymptotic to β r i , ď i ď n , and β s j , ď j ď n . Then by [CGHi, Prop. 17] and [CGHi, Lemma 18] (see also the discussion after(2.2.35)), we have index p C ˚ q “ ´ ` n ` n ` n ÿ i “ p r i ` t r i x u q ` n ÿ j “ p s j ` t s j p x q u q ´ p ´ t px u (3.6.3) “ ÿ p r i ` r r i x s q ` ÿ p s j ` r s j x s q ´ p ´ r px s . As C ˚ has nonnegative area, we have ř r i ` ř xs j ě p . Hence ÿ r i ` ÿ r xs j s ě p, (3.6.4) and ÿ r r i x s ` ÿ s j ě r ÿ r i x s ` ÿ s j ě r px s . (3.6.5)These estimates establish the inequality.Since x is irrational, ř r xs j s ą xs j whenever s j ‰
0. Therefore, because ř r i ` ř xs j ě p , there is equality in (3.6.4) only if s j “ j and ř i r i “ p . As for thesecond estimate, first observe that because x “ pq ` ε , we have r px s “ q , so that r px s isjust larger that px . On the other hand if r ă p , then rb R Z so that r rx s is significantlylarger than rx . Hence there is equality in (3.6.5) only if n “ r “ p . (cid:3) We now complete the compactness argument by dividing into cases.
Case 1: C ˚ has positive index. In this case, Lemma 3.6.3 implies that at least onematched component of the building C (cid:114) C ˚ has negative index. But this is ruled outby the next lemma. Lemma 3.6.4.
Let C be a matched component of the limiting building C (cid:114) C ˚ . Then ind p C q ě , with equality only if C consists of a single curve (hence with a singlenegative end matched with C ˚ ).Proof. We first recall some index formulas. It follows from [CGHi, Prop. 17] that if S ě S p d, x q and v is a connected curve in the symplectization B E p , b ` ε, S, . . . , S q ˆ R with n ` n positive ends on β r i , ď i ď n , and on β s j , ď j ď n , and no negativeends, then(3.6.6) ind p v q “ k ´ ` n ` n ` n ÿ i “ ` r i ` t r i x u ˘ ` n ÿ j “ ` s j ` t s j x u ˘ , where k ě . In particular, this index is always positive.(This index formula can be obtained using the observations after (2.2.35); notice that t r j S u “ t s j xS u “ p u q is a curve in M with degree d and negative ends on β r i for 1 ď i ď n and negative ends on β s j for 1 ď j ď n , then similarly by [CGHi, Prop. 13],(3.6.7) ind p u q “ k ´ ` d ´ k p n ` n q ´ n ÿ i “ p r i ` t r i x u q ´ n ÿ j “ p s j ` t s j x u q . If u is an m -fold multiple cover of a curve r u with degree r d and negative ends on β r r i for 1 ď i ď r n and negative ends on β r s j for 1 ď j ď r n , then we have d “ m r d , Note that k “ N ´ Note that this index is the same as the index of a curve with (cid:96) positive ends, each of multiplicityone, on the larger orbit of B B p µ ˚ , µ ˚ ` ε q ˆ C P k p T q . HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 47 ř r i “ m ř r r i and ř s j “ m ř r s j . Therefore using equation (3.6.7) we obtain ind p u q ´ m ind p r u q “ p k ´ qp ´ m q ´ k p n ` n ´ m r n ´ m r n q (3.6.8) ´ n ÿ i “ t r i x u ´ n ÿ j “ t s j x u ` m r n ÿ i “ t r r i x u ` m r n ÿ j “ t r s j x u . We now divide C into the unique curve u with a negative end matching C ˚ and acollection of connected planar components. Such planar components necessarily havestrictly positive index. (Indeed, the matched index is given by (3.6.6) in the case wherethere is just one positive end, perhaps with an additional positive contribution if thematched component intersects M and hence has positive degree.)Suppose that u is a multiple cover m r u for some m ą
1. Assume first that u is attachedto C ˚ along β r . Since all the other ends of u are matched by planar components in thesymplectization, we may combine the indices of planar components calculated using(3.6.6) with the formula (3.6.8) for the multiply covered curve and use the fact thatind p r u q ě ind p C q ě p k ´ qp ´ m q ´ k p n ` n ´ m r n ´ m r n q (3.6.9) ´ n ÿ i “ t r i x u ´ n ÿ j “ t s j x u ` m r n ÿ i “ t r r i x u ` m r n ÿ j “ t r s j x u ` p n ´ q k ` n ÿ i “ p r i ` t r i x u q ` kn ` n ÿ j “ p s j ` t s j x u q“ p k ´ qp ´ m q ´ k ` km p r n ` r n q` n ÿ i “ r i ´ t r x u ` n ÿ j “ s j ` m r n ÿ i “ t r r i x u ` m r n ÿ j “ t r s j p x q u . Since r n ` r n ě
1, we see that index p C q ě p m ´ q ` n ÿ i “ r i ` m r n ÿ i “ t r r i x u ´ t r x u ě n ÿ i “ r i ` m r n ÿ i “ r r r i x s ´ r r x s . Now the end, say β r r , of r u that is covered by the end of u asymptotic to β r satisfies m r r ě r . Therefore m r n ÿ i “ r r r i x s ě r m r n ÿ i “ r r i x s ě r r x s which implies that index p C q ě
0. Moreover, ind p C q ą r n “ n “ , r n “ r “ m r r , i.e. unless u has just one negative end of multiplicity r , in which caseit is the unique curve in C .If u is attached to C ˚ along β , then the same argument shows that ind p C q ě C has one negative end. (cid:3) This completes the proof of compactness in Case 1.
Case 2: C ˚ has index . In this case, Lemma 3.6.3 shows that C ˚ is a multiplecover of β with just one positive end. Then, if the limit C is nontrivial, theremust be other nontrivial curves in the symplectization, which implies that the top ofend of C ˚ must attach to a curve im p u q with more than one negative end. But byLemma 3.6.4 a component C containing such im p u q must have positive index. Sinceagain no component has negative index, this scenario is impossible.This completes the proof of the compactness argument. The counting argument.
This is proved much as in [CGHi, § M with the completion of the complement of im p r Φ q so thatit supports a T k action, and consider the space J T k reg of all T k -invariant and admissiblealmost complex structures on M for which all somewhere finite action curves in both X and M are regular. One shows first that ‚ J T k reg is nonempty; and second that ‚ when J P J T k reg the count of J -holomorphic curves in M J p M , dL, β p q is nonzero.The second step (Proposition 10 in [CGHi]) is proved as in [CGHi, § J is T k -invariant and the elements of M J p M , dL, β p q have index zero, theelements of this moduli space must lie in the 4-dimensional manifold X , so that onecan appeal to Wendl’s automatic transversality results. In our case this argument isslightly easier than in [CGHi] since our curves have no positive ends.To establish the first step (Proposition 11 in [CGHi]), one first notes that it isimmediate provided that there is J P J T k reg such all elements in M J p M , dL, β p q that donot lie entirely in X are orbitally simple (i.e. intersect at least one T k orbit exactlyonce transversally), since then standard methods allow one to find an T k -invariant andregular perturbation of J . To show that there is a suitable J one considers a secondneck stretching as in [CGHi, § “ Φ ` Bp ` δ q E p , x q ˘ ˆ C P k p T q in M . (Thus, one extends the initial embedding Φ : E p , x q Ñ C P p µ ˚ q to a slightlylarger ellipsoid, extends it trivially to the product, and then stretches by an amount K along the corresponding product boundary Σ.) We consider T k -invariant almostcomplex structures J K on M that are products both near and outside the regionbounded by Σ, so that when one stretches the neck the top level is a product that wedenote X ˆ C P k p T q while the rest is a (possibly multi-level) cobordism from Σ tothe ellipsoid B E : “ r Φ pB E p , x, S, . . . , S qq .If for some K all J K -holomorphic curves C K in M J K p M , dL, β p q are orbitally simple,then we are done. Hence we only need to consider the case when there is a sequenceof non-orbitally simple curves C K for K Ñ 8 . In this case there is a limiting building C , whose top level C top lies in X ˆ C P k p T q . Consider the projection C of C top to the 4-dimensional space X . By construction, C is the limit of the projection to X of (pieces of) non-orbitally simple curves, and so has at least one multiply covered HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 49 component. We must show that this is impossible. The argument used to prove thisin [CGHi, Prop. 12] does not generalize since it exploits the fact that in their case C has essentially zero action. However, it is possible to prove this in our more generalsituation by using Lemma 3.6.4.The projected curve C cannot consist of a single component with end on the orbitset tp β , p qu which is an m -fold cover for m ą
1, because then it would have to havedegree d and we are assuming that gcd p p, d q “ C is nontrivial, i.e. it cannot consist just of anindex zero cylindrical cover of β (in the cobordism from Σ to the negative end of M )together with a single component in X ˆ C P k p T q . We now argue much as in thecompactness argument.Note first that the index arguments used above can be adapted essentially with-out change. Indeed, although Σ – B ` p ` δ q E p , x q ˘ ˆ C P k p T q is different from B E p , x, S, . . . , S q , the index formulas for curves positively asymptotic to the orbits β , β on Σ are the same as they are for B E p , x, S, . . . , S q , as one sees by comparingthe formulas in Propositions 13 and 17 in [CGHi]. Similarly, the contribution to theindex of negative ends on Σ is just as in (3.6.7), except that the index has an addi-tional positive contribution of 2 k p n ` n q (see Proposition 15 in [CGHi]) which takesinto account the fact that the negative ends lie on the product Σ so that each endlies in a 2 k -dimensional family. However, this additional contribution is cancelled outby the fact that when we match ends in this Morse-Bott situation we must subtract2 k . Therefore we may calculate the indices of matched components just as before. Inparticular, if we define C ˚ to be the component of C lower with bottom end on β p , thenLemma 3.6.3 and Lemma 3.6.4 both hold. Therefore all the components of C (cid:114) C ˚ have nonnegative index, and have positive index if they have more than one negativeend. In particular, if the building is nontrivial, either C ˚ or some component of C (cid:114) C ˚ has positive index. But this is impossible.Together, these two steps complete the proof of Proposition 3.6.1. (cid:3) Remark 3.6.5. (i) One might try to generalize Proposition 3.6.1 by considering curvesof genus zero C with one negative end on β m where m is chosen so that the index iszero. Thus, if C has degree d we assume gcd p d, m q “ d “ m ` r mx s ; see(1.3.3). However, in this case Lemma 3.6.3 might fail, so that compactness does nothold. For example, if one can decompose d “ d ` d and m “ m ` m in such a waythat m i ` r m i x s “ d i for i “ ,
2, then a curve in M of degree d and one negativeend on β m might split into a nontrivial building whose top has two components ofdegrees d , d and bottom (in the symplectization) has two positive ends of multiplicities m , m . In such a case, we would have r m x s ` r m x s “ r m ` m x s so that the curve inthe symplectization as well as the two components in M all have index zero. As anexample, take x “ , m “ , m “
18 and d “ d “ M p X µ ˚ ,x , dL, β p q where x “ b ` ε “ pq ` ε , provided we assumethat each of the pairs p p, q q , p d, p q and p , p ` q q is mutually prime. By (1.3.3), the index condition then implies that d “ p ` q so that we again get the sharp bound c k p b q ě dp “ bb ` .To prove the claim we must first establish compactness. First of all, any curve in thismoduli space cannot be multiply covered, since d and 3 p are coprime, and similarlyconvergence to a multiply covered curve in M can be excluded. For the rest of theargument, the key ingredients Lemmas 3.6.2 and 3.6.4 hold as before, but in the caseof equality in Lemma 3.6.3 we can only conclude that C ˚ is a branched cover of thetrivial cylinder over β , with a single negative end of multiplicity p and positive endswhich have multiplicities r p, . . . , r n p where ř n i “ r i “
3. But in the case when C ˚ isbranched, that is when n ą
1, the remainder of the limiting building consists of n ą d , . . . d n , each attached to one of the top endsof C ˚ . By Lemma 3.6.4, each such component has nonnegative index, and hence allmust have index 0. But if n ą r i “
1, and the corresponding degree d i must then satisfy 3 d i “ p ` q , contradicting our assumption that gcd p , p ` q q “ Proposition 3.6.6.
Suppose that for some triple p d, x, m q there is a genus zero curvein X µ ˚ ,x of degree d , index zero, and with one negative end on β m . Suppose furtherthat there are no decompositions d “ ř ni “ d i , m “ ř n “ m i , with n ą and d i ą forall i , and such that d i “ m i ` r m i x s @ i. Then c k p x q ě dm . To prove compactness here, we first observe that the hypothesis excludes the possi-bility that the moduli space M p X µ ˚ ,x , dL, β m q contains an n -cover of an index 0 curvefor some n , because such a cover would give rise to a decomposition with m i “ mn and d i “ dn for all i . Further, in this general situation, equality in Lemma 3.6.3 only impliesthat C ˚ is a branched cover of the trivial cylinder, with no restrictions on the positiveends except that their multiplicities m i must satisfy the condition 3 d “ ř i p m i ` r m i x s q .However Lemma 3.6.4 again implies that the remaining planar components have index0, and our hypotheses precisely exclude this if C ˚ has multiple positive ends.Notice that in such a situation the inequality c k p x q ě dm is in general no longer sharp.(iii) Proposition 3.6.1 implies that we can prove that c k p m q “ mm ` for an integer ofthe form 3 d ´ d and single end on β d ´ on the ellipsoid B E p , d ´ q . However, to find suitable obstructions at the other integers we need touse the generalizations in (ii) above. For example, to show that c k p q “ it wouldsuffice to find a degree 8 curve in the completion of C P p µ q (cid:114) Φ p E p , ` ε qq with oneend on β . Constructing such curves is the subject of ongoing work. HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 51 The case when C L has connectors. We now assume that the limiting building C has connectors (see Definition 3.3.2),and denote by D the union of all the matched components of C L that contain aconnector. This section is devoted to showing that there is at most one representativeof B that is close to breaking into a building with D nonempty. This result is statedin § § C has a connector then C U has negative end tp β , (cid:96) n q , p β , (cid:96) n qu ,and moreover, by Proposition 3.2.2, any curve in the upper level has negative end onthe orbit set tp β , m q , p β , m qu for some positive integer m . It is relatively easy to finda model candidate C M for the connector (see Lemma 4.4.2), and to show that thereis at most one B -curve that could limit on a building with connector C M (see § I , ind and writhe (which are all integers) thatis contained in the index inequality (2.2.14) into numerical terms that are easier tocalculate and manipulate. For example, the quantity A p C q in (2.2.14) that is a countof lattice points is converted via Pick’s formula into a sum of terms A p θ, s q that areareas. This formula uses the fact the constraints on C L as well as the symplectic formcome from related weight expansions and so have simple numerical descriptions. In § b n to gain information onthe relevant areas A p θ, s q .The next step is to use the fact that, because the total action of C L is (approximately) Q n , all but one of its constituent curves have low action and so are ‘light’; the connectedcurve with nontrivial action is called heavy . The heavy curve cannot be multiplycovered, since all connected curves in C L have action that is (approximately equalto) some multiple of Q n , so that its properties can be analyzed using the machinerydeveloped in § § § C M . The ratherelaborate proof compares the ECH and Fredholm indices, again using Proposition 4.1.4and the arithmetic properties of the numbers b n . The argument is completed in § The fundamental estimate.
The basic estimate needed for the proof of Propo-sition 4.5.1 is given by Proposition 4.1.4 below. To state it, we need to introduce somenotation.
Definition 4.1.1.
Given a pair p θ, t q , where θ is an irrational number and t is apositive integer, we define A p θ, t q to be the area of the region in the first quadrantformed by the line y “ θx , the vertical line from p t, t tθ u q to p t, tθ q , and the maximalconcave lattice path starting at the origin, ending at p t, t tθ u q , and staying below the line y “ θx . The significance of A p θ, t q to our problem is given by the following. Lemma 4.1.2.
Let γ be an elliptic orbit with monodromy angle θ , and let r be thelength of p ` θ p t q . Then (4.1.1) 2 A p θ, t q “ θt ´ gr p γ t q ` t ` t tθ u ` r. Proof.
Let M p θ, t q be the area underneath the maximal concave lattice path. Then(4.1.2) A p θ, t q “ θt ´ M p θ, t q . We can compute M p θ, t q by using Pick’s theorem. It is the area of a region R with gr p γ t q ` r ` t ` t tθ u lattice points on the boundary. Hence byPick’s theorem (cf. (2.2.18)), we have(4.1.3) 2 M p θ, t q “ gr p γ t q ´ t ´ r ´ t tθ u . The lemma follows from this together with (4.1.2). (cid:3)
Given a preglued holomorphic building C , we now define a vector diff C that will beimportant in our estimates. To motivate its definition, recall that for any vectors z, w the quantity z ¨ z is minimized subject to the constraint(4.1.4) z ¨ w “ κ if z “ λw , where λ “ κw ¨ w . Now let r C s be any homology class in H p p E n , α, Hq corresponding to a preglued holo-morphic building C . As explained in § we can identify the class r C s with a vector z . Then if w “ w p b n q , the weight vector of b n , the symplectic area of the constraintsis given by z ¨ w , modulo an arbitrarily small error caused by the fact that we cannotcompletely fill the ellipsoid by balls. Now consider the vector(4.1.5) diff C : “ λw ´ z, where λ : “ κw ¨ w , κ : “ z ¨ w. Then w ¨ diff C “
0, so that(4.1.6) z ¨ z “ p λw ´ diff C q ¨ p λw ´ diff C q “ λ w ¨ w ` diff C ¨ diff C . Since some of the quantities in our estimates are exact, while others are only ap-proximate it will be convenient to introduce the following notation.
Definition 4.1.3. If A p ε q , A p ε q are quantities that depend on a finite number ofarbitrarily small constants ε i ą , then we write A p ε q ď δ A p ε q if for all δ ą we have A p ε q ´ A p ε q ă δ for all sufficiently small ε, ε . Further, wewrite A p ε q ą δ A p ε q Our notation is such that this vector does not include the asymptotics of r C s ; it just records thecoefficients along the exceptional classes. HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 53 if there is δ ą so that A p ε q ´ A p ε q ą δ for all sufficiently small ε i . Further wewrite A p ε q “ ε A p ε q if A , A are continuous functions of the small parameters ε i thatare equal when all ε i “ . Note that the properties ď δ and ą δ are mutually exclusive; that is, it is impossiblethat A p ε q ď δ A p ε q and also A p ε q ą δ A p ε q .We can now state the crucial estimates. To simplify the notation for what will follow,define θ n : “ b n ` ε n , r θ n : “ { θ n , (4.1.7)where ε n is small and irrational. We will apply this result when s, t ď (cid:96) n , so that thequantity sP n ` tQ n on the RHS of (4.1.9) is at most (cid:96) n p P n ` Q n q , which by (2.1.9) is adecreasing sequence that converges to 1. Thus this RHS is approximately 3. Proposition 4.1.4.
Let C be a connected somewhere injective curve in p E n , asymptoticto tp β , s q , p β , t qu . Then: ‚ If C has low action, we have (4.1.8) 2 A p r θ n , s q ` A p θ n , t q ` diff C ¨ diff C ď δ . ‚ Otherwise, (4.1.9) 2 A p r θ n , s q ` A p θ n , t q ` diff C ¨ diff C ď δ ` p sP n ` tQ n q . Proof.
Let z “ r C s . We prove Proposition 4.1.4 in several steps. Step 1 : Applying the (improved) index inequality
By Proposition 2.2.2, we have(4.1.10) I p C q ´ ind p C q ě I p C q ´ ind p C q ě A p C q where A p C q is a certain count of lattice points. We also have(4.1.11) ind p C q “ ´ ` n ` n ` n ÿ i “ t s i r θ n u ` n ÿ j “ t t j θ n u ` s ` t ´ z ¨ , where p s , . . . , s n q is the partition of s given by the ends of C , and p t , . . . , t n q is thepartition of t . In addition, by (2.2.25) and (2.2.32) we have(4.1.12) I p C q “ st ` gr p β s q ` gr p β t q ´ z ¨ z ´ z ¨ . We can substitute for gr p β s q and gr p β t q in (4.1.12) using (4.1.1) to get(4.1.13) I p C q “ ´ A p r θ n , s q´ A p θ n , t q` s ` r ` t s r θ n u ` t ` r ` t tθ n u `p θ n t ` r θ n s ` st ´ z ¨ z q´ z ¨ , where r , r are the number of ends in the ECH partitions. Step 2:
Estimates
Let us first suppose that C has symplectic area (approximately) Q n . Then becausethe orbit β has action 1, while β has action θ n “ ε P n Q n , we have ω p C q “ ε s ` tθ n ´ z ¨ w “ ε Q n . so that by (4.1.5) we have κ “ z ¨ w “ ε s ` tθ n ´ Q n . Since w ¨ w “ P n Q n “ ε θ n we obtain from (4.1.6) that λ “ κw ¨ w “ ε s Q n P n ` t ´ P n ,z ¨ z ě δ θ n t ` r θ n s ` st ´ Q n p s r θ n ` t q ` diff C ¨ diff C . Moreover, we can improve this to z ¨ z ě δ θ n t ` r θ n s ` st ` diff C ¨ diff C in the case where C has low action.Substitute this into (4.1.13) to get(4.1.14) I p C q ě δ A p r θ n , s q` A p θ n , t q` s ` r ` t s r θ n u ` t ` r ` t tθ n u ´ z ¨ ` Q n p s r θ n ` t q´ diff C ¨ diff C . with the improvement to I p C q ě δ A p r θ n , s q ` A p θ n , t q ` s ` r ` t s r θ n u ` t ` r ` t tθ n u ´ z ¨ ´ diff C ¨ diff C . in the low action case. Now substitute (4.1.14) and (4.1.11) into (4.1.10). This gives1 ´ A p r θ n , s q ´ A p θ n , t q ´ diff C ¨ diff C ` Q n p s r θ n ` t q ` p r ´ n q ` p r ´ n q` p t s r θ n u ´ n ÿ i “ t s i r θ n u q ` p t tθ n u ´ n ÿ i “ t t j θ n u q ě δ A p C q , (4.1.15)with the improvement to1 ´ A p r θ n , s q ´ A p θ n , t q ´ diff C ¨ diff C `p r ´ n q ` p r ´ n q` p t s r θ n u ´ n ÿ i “ t s i r θ n u q ` p t tθ n u ´ n ÿ i “ t t j θ n u q ě δ A p C q , (4.1.16)in the low action case. HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 55
Step 3:
We prove A p C q ě max p r ´ n , q ` max p r ´ n , q` (4.1.17) p t s r θ n u ´ n ÿ i “ t s i r θ n u q ` p t tθ n u ´ n ÿ j “ t t j θ n u q . Proof.
Recall from (2.2.13) that A p C q “ A C p β , s q ` A C p β , t q “ L p Λ C q ` b p Λ C q is a certain count of lattice points. Consider A C p β , s q . Denote the concave pathdetermined by the ends of C at β by Λ C , and let Λ be the path determined by thepartition conditions. Counting in the vertical line x “ s gives p t s r θ n u ´ ř n i “ t s i r θ n u q lattice points that contribute to L p Λ C q .Further, if any part of the paths Λ and Λ C are geometrically the same (thoughperhaps with different subdivisions), then the maximality of the ECH path impliesthat it has at least as many vertices as Λ C , so that any extra vertices on this part of Λare interior lattice points that contribute to the term b p Λ C q in (2.2.12). On the otherhand any vertex in Λ that does not lie on Λ C contributes to the term L p Λ C q . Thereforethe vertices of Λ that do not lie on the line x “ s and are not vertices of Λ C contributeat least max p r ´ n , q to 2 A p C q . Combining this with the analogous analysis for β gives (4.1.17). (cid:3) Step 4: Completing the proof
Combine (4.1.17) with (4.1.15) to get1 ´ A p r θ n , s q ´ A p θ n , t q ` Q n p s r θ n ` t q ´ diff C ¨ diff C ě δ . In the low action case, combine (4.1.17) with (4.1.16). This gives1 ´ A p r θ n , s q ´ A p θ n , t q ´ diff C ¨ diff C ě δ . Since Q n p s r θ n ` t q “ ε sP n ` tQ n , this proves Proposition 4.1.4. (cid:3) Remark 4.1.5.
Although we will not use this in the current paper, we note here thatthe bounds in Proposition 4.1.4 can be improved if any segment of the concave pathsΛ C at the ends of C does not lie on a maximal concave path. Specifically, we candefine quantities A C p ˜ θ n , s q , A C p θ n , t q analogously to A p ˜ θ n , s q , A p θ n , s q , but using theconcave path Λ C formed from the ends of C instead. By maximality of the partitionpath, we always have A C p ˜ θ n , s q ě A p ˜ θ n , s q , A C p θ n , t q ě A p θ n , t q ; moreover, the proof ofProposition 4.1.4 shows that as in (4.1.8)2 A C p ˜ θ n , s q ` A C p θ n , t q ` diff C ¨ diff C ď δ β the paths Λ and Λ C have no common segments, and let i be the number of lattice points lying strictly between them. Then2 A C p β , t q “ r ´ ` ` t tθ u ´ ÿ j t t j θ u ˘ ` i ` b Λ , while Pick’s Theorem gives2 A C p θ n , t q ´ A p θ n , t q “ r ` n ` b ` ` t tθ u ´ ÿ j t t j θ u ˘ ` i ´ “ A C p β , t q ´ r ` n ´ ` t tθ u ´ ÿ j t t j θ u ˘ . This equality still holds if the paths Λ and Λ C do have common segments. Indeed,in this case the convexity condition implies that these occur at the beginning of thepaths. By additivity, it therefore suffices to consider the case when the two paths aregeometrically the same. But in this case the left hand side is clearly zero, while theright hand side also vanishes because 2 A C p β , t q “ b Λ “ r ´ n by the maximalitycondition on ECH partitions. Now substitute this, together with the analogous identityfor ˜ θ n , in the inequalities (4.1.16) and (4.1.15) to obtain the strengthened versions of(4.1.8) and (4.1.9).4.2. Area estimates.
In order to understand the asymptotics of the connector, wenow establish the following estimates for the area A p θ, t q defined in (4.1.2). The con-nector has top on the orbit set tp β , s qu , tp β , t qu where 0 ă s, t ă (cid:96) n . Proposition 4.1.4shows that the areas A p θ, ¨q at these ends must be rather small. As we explain in moredetail in Lemma 4.2.2 below, these areas are closely related to the partition conditionsand hence to best lower approximations to θ . We saw in Example 2.1.9 that the lowerconvergents to θ n have denominators t k , ď k ď n ´
1. Further the best approxi-mations for θ n from below whose denominator t satisfies (cid:96) n ´ ă t ă (cid:96) n are given bythe semiconvergents c n ´ ‘ r c n ´ “ r , p , q n ´ , , r s for 1 ď r ă
6. These havedenominators t n ´ ` r(cid:96) n ´ . In particular t n “ t n ´ ` (cid:96) n ´ , while (cid:96) n “ t n ´ ` (cid:96) n ´ .The following proposition summarizes the results we shall need when n ą
1. (Forthe case n “
1, see Example 4.2.5.) Recall the notation ď δ , ă δ from Definition 4.1.3. Proposition 4.2.1.
Let n ą . (i) We have A p θ n , t q ą δ τ ą . for all t ă (cid:96) n . (ii) If A p θ n , t q ă . and (cid:96) n ą t ą t n , then t “ t n ` t k for some k ă n . In thiscase, A p θ n , t q ě δ . ` τ ą . . (iii) 2 A p θ n , t n q ě . . (iv) 2 A p θ n , t n ` t n ´ q ě δ . If t n ´ (cid:96) n ´ ă t ă t n and t ‰ t n ´ (cid:96) n ´ , then A p θ n , t q ě δ . ` τ ą . . (vi) 2 A p θ n , t n ´ (cid:96) n ´ q ě δ . . (vii) 2 A p r θ n , s q ą δ for all ď s ă (cid:96) n . (viii) 2 A p r θ n , (cid:96) n ´ q “ (cid:96) n ´ P n ă δ στ . (xi) If A p r θ n , s q ă for some ď s ă (cid:96) n then s “ (cid:96) k for some ď k ă n. HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 57
We prove the proposition in several steps. As a first step, we investigate the rela-tionship of the area A p θ, t q with the partition conditions. Lemma 4.2.2.
Let θ P p , q be any irrational number, let m ě be an integer, and let Λ be the path corresponding to the partition conditions for p ` θ p m q ; see Remark 2.2.1. ‚ If m is the denominator of a best approximation m { m to θ from below, then p ` θ p m q “ m and Λ is a straight line from the origin to p m, m q . ‚ Otherwise, let k ă m be the largest possible denominator of a best approximation k { k from below. Then Λ is given by concatenating the straight line from theorigin to p k, k q with the maximal concave path for p m ´ k q , and p ` θ p m q “ p ` θ p m ´ k q \ p k q . Proof.
The first bullet point follows from the fact that this straight line is a latticepath, and it is maximal by the definition of a best approximation from below.To prove the second bullet point, we have to show that the claimed path Λ is concaveand maximal. This path is the concatenation of a line with a concave path, so to seethat it is concave, we just have to check that the second segment of Λ does not havestrictly greater slope than the first. Assume that it does, and translate this secondsegment to be at the origin. This translated segment is part of the concave pathgiving p ` θ p m ´ k q , so in particular it must be below the line y “ θx . It cannot have x -coordinate less than or equal to k , since k is assumed the denominator of a bestapproximation. And it cannot have x -coordinate more than k , since k was assumedthe largest denominator of a best approximation. This is a contradiction.To see that Λ is maximal, assume otherwise, and consider the actual maximal concavepath. As in the previous paragraph, the first segment of this path must agree with thefirst segment of Λ. Now let p k ` δ, t p k ` δ q θ u q be the lattice point that is the endpointof the first segment of this path that does not agree with Λ. Then p δ, t p k ` δ q θ u ´ t kθ u q is above the maximal concave path for p m ´ k q , and by (2.2.8), it is below the line y “ θx . This is a contradiction. (cid:3) Example 4.2.3.
Recall from Examples 2.1.9 and 2.1.10 that the even convergents of θ n have denominators t k , k ă n , while those of r θ n have denominators (cid:96) k , k ď n . Further,the best approximation to θ n with denominator ă Q n is (cid:96) n . Therefore p ` θ p m q “ p m q when m “ (cid:96) n , t k , k ă n , while p ` Ă θ n p m q “ p m q when m “ (cid:96) k , k ď n . (cid:51) This has the following consequences for estimating A p θ, s q . To simplify the notation,given θ and a positive integer s , define κ p θ, s q “ s ¨ p sθ ´ t sθ u q . (4.2.1)Then the discussion above implies: Lemma 4.2.4. ‚ If p ` θ p m q “ p m q , then A p θ, m q “ κ p θ, m q . ‚ If p ` θ p m q “ p a , . . . , a n q , then (4.2.2) 2 A p θ, m q ě n ÿ i “ κ p θ, a i q with strict inequality unless n “ . ‚ If p ` θ p m q “ p a, b q with a ě b then (4.2.3) 2 A p θ, m q “ κ p θ, a q ` κ p θ, b q ` ba κ p θ, a q . Proof.
The first two bullet points follow from Lemma 4.2.2 and the definition 4.2.1.The third follows by observing that 2 A p θ, m q ´ κ p θ, a q ´ κ p θ, b q is given by twice thearea of the parallelogram determined by the vectors p , aθ ´ t aθ u q and p b, t mθ u ´ t aθ u q , and computing this area with the two-dimensional cross product. (cid:3) Example 4.2.5.
When n “
1, we have b “ , (cid:96) “ t “ , t “
1. The integers m P t , . . . , u are all lower semiconvergents to θ , and p ` p θ qp m q “ p m q for 1 ď m ď p ` p r θ qp m q “ p ˆ m q for 1 ď m ď
6. Hence because b “ ´ we find that2 A p θ , m q “ κ p θ , m q “ m p ´ m q , ď m ď , (4.2.4) 2 A p r θ , m q “ κ p r θ , m q “ m , ď m ď . We use these calculations instead of Proposition 4.2.1 in the case n “ (cid:51) The next lemma estimates κ for n ą Lemma 4.2.6.
Let n ą . (a) If m ă (cid:96) n is the denominator of an even convergent of θ n : “ b n ` ε , then (4.2.5) 2 A p θ n , m q “ κ p θ n , m q ą { τ ą . . (b) If t n ´ ă m ă (cid:96) n is the denominator of a lower semiconvergent of θ n , then (4.2.6) 2 A p θ n , m q “ κ p θ n , m q ą . . (c) If m ă (cid:96) n is the denominator of any lower semiconvergent of θ n , then (4.2.7) 2 A p θ n , m q “ κ p θ n , m q ą . . (d) If m ă (cid:96) n is the denominator of an even convergent of r θ n then (4.2.8) 2 A p r θ n , m q “ κ p r θ n , m q ą ą . if n ą . (e) κ p r θ n , (cid:96) n ´ q “ (cid:96) n ´ P n ă τ .Proof. To prove (a), first let c k “ t k t k ´ “ p k q k be an even convergent with 0 ă k ă n ´ c k in (2.1.18).) We want to estimate q k p q k θ n ´ p k q . It suffices to estimate q k p q k b n ´ p k q “ q k p b n ´ c k q . By (2.1.19) we have c k ‘ c k ` “ c k ` ă b n . Thus, b n ´ c k ą c k ‘ c k ` ´ c k “ p k ` p k ` q k ` q k ` ´ p k q k “ q k p q k ` q k ` q , HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 59 where in the last equation we have used (2.1.20). However, by (2.1.19), we have q k ` q k ` “ q k ` . Hence, we have b n ´ c k ą q k p q k ` q so q k p b n ´ c k q ą q k q k ` “ t k t k ` . The fractions t k t k ` “ c k ` are decreasing with k by Lemma 2.1.1 (ii) and limit to τ .This proves the first bullet point in the case where 0 ă k ă n ´
1. The quantity thatwe want to estimate for c “ θ n ´ ą τ ´
6; this is also bigger than τ . The case k “ n ´ q n ´ p q n ´ b n ´ p n ´ q “ q n ´ p b n ´ c n ´ q . By (2.1.19), we have b n “ c n ´ ‘ c n ´ , which implies b n ´ c n ´ “ c n ´ ‘ c n ´ ´ c n ´ “ p n ´ ` p n ´ q n ´ ` q n ´ ´ p n ´ q n ´ “ q n ´ p q n ´ ` q n ´ q “ q n ´ q n , where the third equality uses (2.1.20). We know that t n ´ Q n is decreasing by part (ii) ofLemma 2.1.1, and (2.1.9) implies that t n ´ Q n “ (cid:96) n ´ P n ´ ´ (cid:96) n ´ P n ´ converges to σ ´ στ . Since 7 p σ ´ στ q ą τ , this proves (a). Proof of (b) and (c) . Let p { q “ c k ‘ r c k ` be a lower semiconvergent. As above, itsuffices to estimate q p b n ´ p { q q .Assume first that k “ n ´
1; this is the case for m ą t n ´ . To simplify the notation,let p : “ P n “ p n ´ ` p n ´ , q “ Q n “ q n ´ ` q n ´ . Since b n “ c n ´ ‘ c n ´ , we have b n ´ p c n ´ ‘ r c n ´ q “ p c n ´ ‘ c n ´ q ´ p c n ´ ‘ r c n ´ q “ ´ rqq . We are interested in the case q ă (cid:96) n . Since (cid:96) n “ t n ´ ` (cid:96) n ´ , it follows that r ă q k ´ “ q k ´ ` q k ´ , q k ´ “ q k ´ ` q k ´ ă q k ´ , ď k ď n. It follows that(4.2.9) q k ´ ě q k ´ , ď k ď n. Since we also have q n ´ ` q n ´ ă q n ´ , we have q q “ q n ´ ` rq n ´ q n ´ ` q n ´ ě ` r . Hence p ´ r q q q ě p ´ r qp ` r q . This is minimized over integers 1 ď r ď r “
1, in which case its value is largerthan 1 .
39. This proves (b).If 0 ă k ă n ´ b n ´ p c k ‘ r c k ` q “ p b n ´ c k ` q ` p c k ‘ c k ` ´ c k ‘ r c k ` q . As above, let p “ p k ` p k ` and q “ q k ` q k ` . As in the proof of (a) we canbound q p b n ´ c k ` q ą τ , so(4.2.10) q p b n ´ c k ` q ą τ q q . Similarily to above, we also have(4.2.11) p c k ‘ c k ` ´ c k ‘ r c k ` q “ ´ rqq . Since we also have q k ` q k ` ă q k ` , and since (4.2.9) still applies, we have q q “ q k ` rq k ` q k ` q k ` ą r ` . Putting this all together, we therefore have q p b n ´ c k ‘ r c k ` q ą τ p q q q ` p ´ r q q q ą τ p r ` q ` p ´ r qp r ` q . The quantity τ p r ` q ` p ´ r qp r ` q is minimized for r P t , , , u when r “ .
28. This completes the proof of (c) in all cases except k “ k “ q p b n ´ c k ‘ r c k ` q ą τ p q q q ` p ´ r q q q . Since q “ q “
1, we have q q “ ` r . Thus, we have5 τ p q q q ` p ´ r q q q “ τ ` r ` ˘ ` p ´ r q r ` . HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 61
This is minimized over r P t , , , u when r “
4, in which case it is greater than 1 . Proof of (d) : Now consider convergents to r θ n : “ b n ` ε , that we also denote by c i “ p i q i .We want to estimate q k p b n ´ c k q . Because m ă (cid:96) n , we know that k ă n . Assume firstthat 0 ă k . Then c k ‘ c k ` “ c k ` ă b n . Thus, 1 { b n ´ c k ą c k ‘ c k ` ´ c k “ p k ` p k ` q k ` q k ` ´ p k q k “ q k p q k ` q k ` q , where in the last equation we have used (2.1.20). However, by (2.1.19), we have q k ` q k ` “ q k ` , so q k p b n ´ c k q ą q k q k ` “ (cid:96) k (cid:96) k ` . The fractions (cid:96) k (cid:96) k ` increase with k by Lemma 2.1.1 and so are ě ě . k ě
1. When k “
0, because n ě κ p r θ n , k q “ p r θ n ´ t r θ n u q ě ε ą . Proof of (e) : We want to compute (cid:96) n ´ ` (cid:96) n ´ b n ´ t (cid:96) n ´ b n u ˘ , n ě . But (cid:96) n ´ P n ´ ´ (cid:96) n ´ P n “ t (cid:96) n ´ b n u “ t (cid:96) n ´ P n ´ P n u “ (cid:96) n ´ (4.2.12)and also gives the equality in (e). The estimate on (cid:96) n ´ P n follows by observing that thisis an increasing function of n by Lemma 2.1.1 (i) with limit στ ă τ . This completesthe proof of Lemma 4.2.6. (cid:3) Proof.
Proof of Proposition 4.2.1. Part (i) follows from additivity (see (4.2.2)), togetherwith the fact that by Lemma 4.2.6 (a), each of the convergents and semiconvergents of θ n contribute at least 5 { τ to the function κ .To prove (ii), note first that by Lemma 4.2.2, since t ą t n , the partition for t muststart with t n . Since t n is the denominator of a lower semiconvergent to θ n we have κ p θ n , t n q ą .
39 by Lemma 4.2.6 (b). Assume first that t ´ t n is not a convergent.If it is a semiconvergent, then p ` θ n p t q “ p t n , t ´ t n q , and, by part (c) of Lemma 4.2.6, κ p θ n , t ´ t n q is more than 1 .
28, so that by additivity, 2 A p θ n , t q ą . . If t ´ t n is thesum of at least two convergents or semiconvergents, then we still have 2 A p θ n , t q ą . by Lemma 4.2.6 (b,c) and additivity. Thus, if 2 A p θ n , t q ă .
67, then t ´ t n must bea convergent, hence t “ t n ` t k for some k ă n . In this case, we have 2 A p θ n , t q ą . ` { τ , since κ p θ n , t n q ą .
39, by Lemma 4.2.6. This proves (ii). The bound (iii)also follows from this, again by Lemma 4.2.6 (b).To prove (iii), note that p ` θ n p t q “ p t n , t n ´ q by Lemma 4.2.2. Now by (4.2.3) and thediscussions above, we have2 A p θ n , t n ` t n ´ q ą . ` { τ ` t n ´ t n . . But the fraction t n ´ t n is decreasing by Lemma 2.1.1, and limits to 1 { τ . Since p qp . q ` τ ` . ą . , this proves (iii).Parts (iv) and (v) follows from the fact that if t n ´ (cid:96) n ´ ă t ă t n , then t is asemiconvergent larger than t n ´ ; one now applies Lemma 4.2.6 and (4.2.2) as above.Parts (vi), (vii), and (viii) follow from Lemma 4.2.6 (d), (e), together with (4.2.2). (cid:3) Facts about the curves in the lowest level.
Assume throughout this sectionthat D is nonempty, i.e. there is at least one connector component. The main resultwe prove here is that the heavy curve must lie in C LL and have exactly one end on β ,of multiplicity t “ t n .We start by showing that all curves in any neck level are covers of trivial cylinders.For this, it is helpful to keep in mind that, as reviewed in § C ina symplectization level, the symplectic form dλ is pointwise nonnegative on C , withequality at a point y P C if and only if the tangent space to C at y is the span of theReeb vector field and B s . Therefore, any curve with zero action is trivial , i.e. a unionof covers of R -invariant cylinders. Further any low action curve in the neck must havetop and bottom with almost the same action, and hence, if its top is tp β , s q , p β , t qu with s, t ă (cid:96) n , must in fact have zero action by Lemma 3.1.5, and so be trivial. Lemma 4.3.1.
Any symplectization level of C L must be a union of covers of trivialcylinders; in particular C LL has top tp β , (cid:96) n q , p β , (cid:96) n qu .Proof. By equations (3.1.6), (2.2.32), and (2.1.10), a curve C in p E with top asymptoticto tp β , P n qu passing through the constraints W p P n Q n q has I p C q “
0. On the other hand,we know that I p C L q “
4, and because J is generic, we know from [H2, Prop. 3.7] thatthe ECH index I of every curve in the neck is nonnegative. Hence, becausegr ptp β , (cid:96) n q , p β , (cid:96) n quq “ gr tp β , Q n qu ` “ gr tp β , P n qu ` C LL must be asymptotic to one of the orbit sets tp β , (cid:96) n q , p β , (cid:96) n qu , tp β , Q n qu or tp β , P n qu . If the top level of C LL is tp β , (cid:96) n q , p β , (cid:96) n qu then by the actionconsiderations explained above we are done. (Recall that, since we have a connector, C U necessarily has negative ends tp β , (cid:96) n q , p β , (cid:96) n qu .) So, we can assume that the toplevel of C LL is either tp β , Q n qu or tp β , P n qu . In fact, our arguments for both of these HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 63 cases will only use the fact that the asymptotics for this top level are supported byonly one of the β i . Assume this, and without loss of generality assume that i “ C U must consist of at least 2 irreduciblecurves. Because we are stretching curves of genus 0, it then follows that any irreduciblecurve in the neck has upper asymptotics at an orbit set tp β , s q , p β , t qu with at leastone of s, t ă (cid:96) n . Hence, by Lemma 3.1.5, its action A ptp β , s q , p β , t quq is strictly lessthan P n “ A ` tp β , P n qu ˘ “ δ A ` tp β , Q n qu ˘ q . Therefore, if its top and bottom havedifferent asymptotics, part (iii) of Lemma 3.1.5 implies that it cannot have low action.In other words, any nontrivial irreducible curve in the neck is heavy, which implies thatthere can be only one nontrivial matched component in the neck.Call this component C neck . The curve C neck must have positive ends on β of themaximum multiplicity (cid:96) n , since otherwise there would have to be another componentin the neck with ends on both β and β , which is therefore nontrivial. Given this, C neck cannot have any positive ends on β at all since our building must have genus 0.These positive asymptotics for C neck are not possible, however. This is becausethe lowest level of any connector would have to meet C neck , and by definition theneck components for the connector would also have positive ends on β . But this alsocontradicts the fact that our building has genus 0. (cid:3) Corollary 4.3.2.
If an irreducible curve C in C LL has top on tp β , s q , p β , t qu then s, t ă (cid:96) n and s ` t ď (cid:96) n .Proof. If C has s, t ą C U that are attached via trivial cylinders to C along covers of β must be different fromthe components of C U that are attached to C via β . Hence s ` t ď (cid:96) n . Thus if C ispart of a connector it must have s, t ă (cid:96) n . But if C is not part of a connector, its tophas to be disjoint from the top of any part of a connector. Hence its top must havemultiplicity ă (cid:96) n . (cid:3) We next analyze curves in the lowest level, i.e. those in the completion p E of theblown up ellipsoid. Lemma 4.3.3.
If a curve C in p E has low action then it cannot go through any con-straints on the last block.Proof. By Corollary 4.3.2, we may suppose that C has top end on the orbit set tp β , s q , p β , t qu , where s, t ă (cid:96) n and s ` t ď (cid:96) n . As above, we denote its constraintvector by z . Case 1: C has ends just on β , i.e. t “ . If C has ends on tp β , s qu then (4.1.8) implies that2 A p r θ n , s q ` diff C ¨ diff C ď , (4.3.1)where diff C is as in (4.1.5). Since s ă (cid:96) n , when n ą C ¨ diff C ď ´ { . The same inequality holds when n “ z ¨ w ď s ă (cid:96) n , thefinal block of λw has value at most α : “ r θ n sQ n ď (cid:96) n P n ă . , where we use the fact that (cid:96) n P n is an increasing function of n by Lemma 2.1.1 thatconverges to σ ă .
128 by (2.1.9). If C passes through r of the constraints correspondingto the last block, the contribution to diff C ¨ diff C from this block is p ´ r q α ` r p ´ α q .Since α ă .
128 we have 2 p ´ α q ą
1, so that C can pass through at most one ofthese constraints. But the minimum of 6 α ` p ´ α q is taken when α “ and is .Since ` ą Case 2: C has ends just on β . The argument is essentially the same. As above we must have 2 A p θ n , t q ă
1. Supposefirst that n ą
1. Since t ă (cid:96) n , it follows from Proposition 4.2.1 (ii) that t ď t n . Hencethe entries α of λw on the final block are at most t n Q n “ t n P n ´ , which is a decreasingfunction of n by Lemma 2.1.1. Hence α ă t n Q n ď t Q “ . If α ě then diff C ¨ diff C is minimized if x “ A p θ n , t q ` diff C ¨ diff C ě τ ` ¨ p ´ . q ą , which is impossible. Hence we must have α ă . But now the argument in Case 1shows that the error is too large unless z “ n “ Case 3: C has ends on both β and β . In this case (4.1.8) implies thatdiff C ¨ diff C ď ´ A p r θ n , s q ´ A p θ n , t q ď ´ ´ τ « . . As in Case 2, since t ă (cid:96) n , we must in fact have t ď t n , since otherwise 2 A p θ n , t q ą λw on the last block is α “ Q n p s r θ n ` t q , which is a maximumwhen t “ t n and s “ (cid:96) n ´ t n . Hence α ď δ Q n ` p (cid:96) n ´ t n q r θ n ` t n ˘ “ (cid:96) n ´ P n ` t n Q n . By Lemma 2.1.1, (cid:96) n ´ P n increases with limit στ ă .
02 while t n Q n decreases and is ď t Q “ . Since the contribution of the new term (cid:96) n ´ P n is so small, we can complete theargument as in Case 2.This completes the proof. (cid:3) HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 65
Corollary 4.3.4.
The heavy curve lies in C LL and goes through all the constraints onthe last block. Moreover, it has some ends on β .Proof. The first claim holds because there is only one heavy curve. The second holdsby a slight generalization of Case 1 in the previous proof. Since the curve is heavy,we need to use (4.1.9) rather than (4.1.8), which means that we need to add the term2 (cid:96) n Q n r θ n “ (cid:96) n P n to the RHS of (4.3.1). As (cid:96) n P n is an increasing sequence with limit lessthan .
13, this term is also less than .
13. On the other hand, if the curve passed throughall 7 of the smallest constraints, then by the computations in Case 1, we would havethe contribution from diff C ¨ diff C at least 7 p ´ . q , which is far too large. (cid:3) We next establish the asymptotics of the heavy curve along β . Notice that we makeno claims about s , which could be zero. Proposition 4.3.5.
The heavy curve has just one end on β of multiplicity t n “ (cid:96) n ´ (cid:96) n ´ .Proof. We suppose as before that the heavy curve C is asymptotic to tp β , s q , p β , t qu .By Corollary 4.3.2 we know that s ` t ď (cid:96) n , with s, t ă t n and t ą
0. By Proposi-tion 4.1.4 (ii) and the bounds from Proposition 4.2.1, we have for n ą A p θ n , t q ď ` Q n p s r θ n ` t q ´ diff C ¨ diff C if s “ A p θ n , t q ď ` Q n p s r θ n ` t q ´ diff C ¨ diff C ´
748 if s ą . (4.3.3)We now consider various cases. We will first show that t “ t n and then will discusswhy it has just one end on β . Case 1: t ą t n : First observe that when n “ t “ (cid:96) “
7. Thereforewe cannot have t ă t ă (cid:96) . Hence we may assume n ě t ă (cid:96) n , the maximum possible value of Q n p s r θ n ` t q occurswhen t “ (cid:96) n ´ s “
1, so that α : “ Q n p s r θ n ` t q ă (cid:96) n Q n . The right hand side of this inequality is a decreasing function of n by Lemma 2.1.1, soit is no more than (cid:96) Q “ . Since C goes through all the constraints on the last blockwe find that diff C ¨ diff C is at least ą . . Thus, the RHS of (4.3.2) is at most 2 . t “ t n ` t k for some k ă n . Moreover,if s ą .
51 so that Proposition 4.2.1 (iii) implies that t “ t n ` t k for some 0 ă k ă n ´
1. In other words, if s ą t ď t n ` t n ´ . Case 1(A): t ą t n and s ą . We saw above that we must have t ď t n ` t n ´ . Themaximum value of α then occurs when t “ t n ` t n ´ and s “ (cid:96) n ´ t n ´ t n ´ ă (cid:96) n ´ t n “ (cid:96) n ´ . By Lemma 2.1.1 and (2.1.9), (cid:96) n ´ P n increases with limit στ ď .
02. On the other hand, both t n ´ Q n and t n Q n , decrease with n by Lemma 2.1.1. Hence because n ě α ď t n ` t n ´ Q n ` (cid:96) n ´ P n ă ` . ă . , so that the contribution to diff C ¨ diff C from the last block is at least 7 p ´ α q ą . A p θ n , t q ď ` α ´ . ´ . ă . , which is impossible for t ą t n by Proposition 4.2.1 (ii). Case 1(B): t n ` t n ´ ď t ă (cid:96) n and s “ . We saw above that we must have t “ t n ` t n ´ , so that α “ t n ` t n ´ Q n . Now t n ` t n ´ Q n decreases by Lemma 2.1.1. Moreover we saw in (2.1.9) that lim t n Q n “ ´ σ , so thatlim n Ñ8 t n ` t n ´ Q n “ p ´ σ q ´ στ . Therefore0 . ă ´ . ` ` ˘ ă t n ` t n ´ Q n “ α ď t ` t Q “ ă . . The values of λw on the third to last block are 8 α and so lie in the interval r . , . s ,while those on the penultimate block are 7 α « .
88. Therefore the contribution todiff C ¨ diff C from the last three blocks is at least7 p ´ α q ` p ´ α q ` p ´ α q “ ´ α ` α . Therefore the RHS of (4.3.2) is1 ` α ´ diff C ¨ diff C ď ´ ` α ´ α ď . α in the given interval, where the last inequality was obtained by evaluating thequadratic expression at since it increases over this interval. Hence this is impossibleby Proposition 4.2.1 (iv). Case 1(C): t n ă t ă t n ` t n ´ and s “ . We saw above that in this case t “ t n ` t k ď t n ` t n ´ . As above, Lemma 2.1.1implies that α “ t n ` t n ´ Q n decreases with n and so is ď « . t n Q n “ ´ σ ą . , by (2.1.9) the quantity α lies in the range 0 . ă α ă ,Hence 4 . ă α ă .
5, so that by looking at the last two blocks we find that1 ` α ´ diff C ¨ diff C ď ` α ´ p ´ α q ´ p ´ α q (4.3.4) “ ´ ` α ´ α ă .
02 if α P r , s . Hence again this scenario is impossible by Proposition 4.2.1 (ii).
Case 2: ă t ă t n .We begin with some general remarks.We may use the estimate in (4.3.4) for 1 ` α ´ diff C ¨ diff C if α P r , s . On theother hand, if α ă we get even better estimates. Indeed, if 3 . ď α ď . HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 67 use the estimate1 ` α ´ diff C ¨ diff C ď ` α ´ p ´ α q ´ p ´ α q (4.3.5) “ ´ ` α ´ α ď . , if α P r , s . where the maximum is taken precisely at α “ . Further if α ď . ` α ´ diff C ¨ diff C ď ` α ´ p ´ α q ď if α ď . (4.3.6)Since 2 A p θ n , t q ą in all cases with t ą
0, the case α ă never occurs.The following information will also be useful.For 1 ď r ď t n ´ r(cid:96) n ´ Q n “ t n Q n ´ r (cid:96) n ´ Q n decreases,(4.3.7) since, by Lemma 2.1.1, t n Q n decreases and (cid:96) n ´ Q n “ (cid:96) n ´ P n ´ increases.With these preliminaries in place, we can now analyze various cases. We begin withthe case n “ Case 2(A): n “ and t ă t “ . As we saw in Example 4.2.5, in this case we have exact formulas for the terms2 A p θ , m q and 2 A p r θ , m q . The maximum value for α “ p s r θ ` t q occurs when t “ s “
1, in which case it is approximately ` « . ą . We may estimatethe value of α at ` by calculating its value at 0 .
65, which is 1 .
24. Therefore,because 2 A p θ , q “ , this case does not occur. Similarly, the case t “ s “ α ă so that (4.3.5) implies we must have 2 A p θ , q ă . t ă α ă for all s , while 2 A p θ , t q ě A p θ , q except if t “ ,
2. But inthis case α ă for all s , which is also impossible as we explained above.As we now see, the argument for n ą Case 2(B): n ą , t n ´ (cid:96) n ´ ă t ă t n and any s . The maximum value of Q n p s r θ n ` t q occurs when t “ t n ´ s “ (cid:96) n ´ t n ` (cid:96) n ´ P n increases with n , and t n Q n decreases by Lemma 2.1.1, we have α ď Q n ´ p (cid:96) n ´ t n ` q r θ n ` t n ´ ¯ ă (cid:96) n ´ P n ` t n Q n ă στ ` t Q ă . ă . (4.3.8)Therefore, by equations (4.3.4), (4.3.5) and (4.3.6), we must have2 A p r θ n , s q ` A p θ n , t q ď . . Hence, by Proposition 4.2.1 (v) we must have t “ t n ´ (cid:96) n ´ . However, if t “ t n ´ (cid:96) n ´ then s ď (cid:96) n ´ t n ` (cid:96) n ´ “ (cid:96) n ´ , and (4.3.7), (4.3.8) imply t n ´ (cid:96) n ´ Q n ď α ď t n ´ (cid:96) n ´ Q n ` (cid:96) n ´ P n ď t ´ (cid:96) Q ` στ “ ` ă . . Since α ă ă , we may estimate 1 ` α ´ diff C ¨ diff C by evaluating the quadraticexpression in (4.3.4) (which increases with α for α ă ) at α “ . .
37, which is smaller than the allowed bound from Proposition 4.2.1 (vi).Hence this case does not occur.
Case 2(C): n ą , t ď t n ´ (cid:96) n ´ and any s . As in Case 2(B), it follows from (4.3.7) that α ď (cid:96) n ´ P n ` t n ´ (cid:96) n ´ Q n ă στ ` t ´ (cid:96) Q ă . ă . Therefore, by evaluating (4.3.5) at α “ .
55 we have1 ` α ´ diff C ¨ diff C ď . ă τ . Hence this case cannot occur by Proposition 4.2.1 (i).This completes the proof that t “ t n . It remains to show that this component C has just one end on β . To see this, first note that, because b n is a semiconvergentto θ n , Lemma 4.2.4 implies that p ` θ n p t n q “ p t n q , namely the length of the partitionconditions for this t is 1. If C has two or more ends, then the p r ´ n q term on theleft hand side of (4.1.15) is strictly negative, and hence is less than the correspondingterm max p r ´ n , q in (4.1.17). Therefore, we can improve (4.1.9) by subtracting 1from the right hand side, and so can improve all of the estimates in Case 2 above byat least 1 as well.However, there are no values of s for which p s, t n q satisfies these new estimates. Thisis because by (4.3.8), regardless of the value of s , we have that a strengthened versionof either (4.3.4), or a strengthened version of one of the stronger estimates (4.3.5),(4.3.6) holds, and this is impossible by Proposition 4.2.1 (iii). This completes the proofof Proposition 4.3.5. (cid:3) The asymptotics of the heavy curve.
By Proposition 4.3.5, the heavy curvemust pass through all the smallest constraints, and have a single end on β of multi-plicity t n . In this subsection we improve this result as follows. Proposition 4.4.1.
The heavy curve is a connector with exactly two ends asymptoticto tp β , (cid:96) n ´ q , p β , (cid:96) n ´ (cid:96) n ´ qu . It has homology class z M : “ z M p n q given by taking ¨ W ´ (cid:96) n (cid:96) n ´ ¯ , and appending the last block of s to the end. We begin the argument by showing that there is an ECH index zero candidate C M for C with the above properties, that we call the model curve . Thus the curve C M has s “ (cid:96) n ´ , t “ (cid:96) n ´ (cid:96) n ´ , and homology class (i.e. constraint vector) z M : “ ` W p (cid:96) n (cid:96) n ´ q , ˆ ˘ . (4.4.1)It follows from Lemma 2.1.5 that C M has action precisely Q n , so that it is a candidatefor the heavy curve. HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 69
The calculations in Lemma 4.4.2 show that it is consistent to require that C M havegenus zero and ECH partitions at its ends. In the second step, we show that theconnector C must have the same numerics as C M , i.e. the same homology class, genus,and multiplicities of ends. Lemma 4.4.2. I p C M q “ ind p C M q “ . Proof.
By (2.2.25), we have gr p β (cid:96) n ´ , β (cid:96) n ´ (cid:96) n ´ q “ gr p β (cid:96) n ´ q ` gr p β (cid:96) n ´ (cid:96) n ´ q ` (cid:96) n ´ p (cid:96) n ´ (cid:96) n ´ q . Since (cid:96) n ´ is a lower semiconvergent of r θ n , and (cid:96) n ´ (cid:96) n ´ is a lower semiconvergentof θ n , Lemma 4.2.2 shows that both ends have ECH partitions of length 1. Thus, by(4.1.3), in both cases M p θ, t q is the area of a triangle, and we have gr p β (cid:96) n ´ q “ p (cid:96) n ´ ` q t (cid:96) n ´ { b n u ` (cid:96) n ´ ` , and gr p β t n q “ p t n ` q t b n p t n q u ` t n ` . We saw in (4.2.12) that t (cid:96) n ´ { b n u “ (cid:96) n ´ , and we have t b n t n u “ t n ` , because of theidentity P n t n ´ P n ´ t n ` “ P t ´ P t “ , (4.4.2)see Lemma 2.1.1. Thus, we have gr p β (cid:96) n ´ , β (cid:96) n ´ (cid:96) n ´ q “ (cid:96) n ´ p (cid:96) n ´ ` q ` (cid:96) n ´ ` ` t n ` p t n ` q ` t n ` ` (cid:96) n ´ t n “ p (cid:96) n ´ ´ (cid:96) n qp (cid:96) n ´ ` q ` (cid:96) n ´ ` ` p (cid:96) n ´ (cid:96) n ´ qp (cid:96) n ´ (cid:96) n ´ ` q (4.4.3) ` (cid:96) n ´ (cid:96) n ´ ` ` (cid:96) n ´ p (cid:96) n ´ (cid:96) n ´ q“ (cid:96) n ´ (cid:96) n (cid:96) n ´ ` (cid:96) n ´ ` (cid:96) n ` (cid:96) n ´ ` , where in the second line we substituted for (cid:96) n ´ , t n ` , t n in terms of (cid:96) n , (cid:96) n ´ using(2.1.7). Finally, the identities (2.1.10) satisfied by weight expansions imply that(4.4.4) z M ¨ z M ` z M ¨ “ (cid:96) n (cid:96) n ´ ` p (cid:96) n ´ ` (cid:96) n ´ q ` . We now claim that the right hand sides of (4.4.4) and (4.4.3) are equal. To see this,subtract the right hand side of (4.4.4) from (4.4.3) to obtain6 p (cid:96) n ´ (cid:96) n (cid:96) n ´ ` (cid:96) n ´ ´ q “ p (cid:96) n ´ (cid:96) n ` (cid:96) n ´ ´ q “ I p C M q “ C M has ECH partitions by assumption, it has two ends. By (2.1.10), C M goesthrough c H ¨ “ p (cid:96) n ` (cid:96) n ´ q ` ind p C M q “ ´ ` ` (cid:96) n ´ ` p (cid:96) n ´ (cid:96) n ´ q ` t r θ n (cid:96) n ´ u ` t θ n t n u ´ p (cid:96) n ` (cid:96) n ´ q ´ “ (cid:96) n ` t (cid:96) n ´ Q n P n u ` t t n P n Q n u ´ p (cid:96) n ` (cid:96) n ´ q . (4.4.5) By (4.2.12) and (4.4.2) we have (cid:96) n ´ Q n Q n ` “ (cid:96) n ´ ` Q n ` , t n P n Q n “ t n ` ` Q n . Hence, the final line in (4.4.5) simplifies to (cid:96) n ` (cid:96) n ´ ` t n ` ´ p (cid:96) n ` (cid:96) n ´ q “ (cid:96) n ` ´ (cid:96) n ´ (cid:96) n ´ ` (cid:96) n ´ “ (cid:3) Remark 4.4.3. (i) Notice that if C M were represented by a J -holomorphic curve, thenit could not be multiply covered since it goes through some constraints with multiplicityone, and hence as proved by Hutchings the condition I p C M q “ p C M q “
0, see the index inequality Proposition 2.2.2.Since we have not shown that C M must exist, however, we must verify some of this bydirect computation.(ii) If one wanted to show that C M has a J -holomorphic representative, then one couldprobably prove this as follows. Suppose for simplicity that n “
1. Then the connectorhas top ends on β , β and has constraint vector x “ p ˆ , , ˆ q ; see Example 3.3.3.It can be built by starting with a curve C with top β through E and a curve C withtop β through 6 p E ` ¨ ¨ ¨ ` E q ` E ` E ` ¨ ¨ ¨ ` E . These curves must intersectonce (since a plane in E asymptotic to β intersects a plane asymptotic to β exactlyonce). One can check that these curves have I p C q “ ind p C q “
0; and can probablyconstruct them by stretching suitable classes as outlined in Remark 3.5.4. Resolvingthe point of intersection C ¨ C gives a 2-parameter family of curves with two positiveends, so that we can recover an index 0 curve by imposing the constraint that it gothrough one more constraint, namely E . (cid:51) We next investigate the homology class of C , that we write as z M ´ ∆, where z M isthe “model” set of constraints as in (4.4.1). We know from Proposition 4.3.5 that ∆must be zero on the last block. Let k “ w ¨ ∆ , (4.4.6)so that C is asymptotic to tp β , (cid:96) n ´ ´ k q , p β , (cid:96) n ´ (cid:96) n ´ qu . We interpret the case k “ (cid:96) n ´ as corresponding to C having no ends on β at all.Our main tool is the following dot product calculation, whose (rather technical)proof is deferred to the end of this subsection. Lemma 4.4.4. ∆ ¨ z M “ k(cid:96) n ´ . Granted this, we can now prove our main result.
Proof of Proposition 4.4.1.
We first show that C has the same numerics as C M , i.e.that(4.4.7) k “ ∆ “ . HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 71
To begin, we estimate the ECH index of C as follows. Since I p C M q “
0, we have: I p C q “ I p C M q ´ p I p C M q ´ I p C qq“ ´ ´ gr p β (cid:96) n ´ q ´ gr p β (cid:96) n ´ ´ k q ` t n k ` pp z M ´ ∆ q ¨ p z M ´ ∆ q ´ z M ¨ z M q ´ ∆ ¨ ¯ “ ∆ ¨ ´ ´ gr p β (cid:96) n ´ q ´ gr p β (cid:96) n ´ ´ k q ` t n k ´ z M ¨ ∆ ` ∆ ¨ ∆ ¯ , where the term 2 t n k comes from the term 2 m m in (2.2.25). We compute the difference p gr p β (cid:96) n ´ q ´ gr p β (cid:96) n ´ ´ k qq by applying (4.1.1), obtaininggr p β (cid:96) n ´ q ´ gr p β (cid:96) n ´ ´ k q “ r θ n ` (cid:96) n ´ ´ p (cid:96) n ´ ´ k q ˘ ` k ` t (cid:96) n ´ r θ n u ´ t p (cid:96) n ´ ´ k q r θ n u `p ´ r q ` A p r θ n , (cid:96) n ´ ´ k q ´ A p r θ n , (cid:96) n ´ q , where r is the length of the ECH partition p ` r θ n p (cid:96) n ´ ´ k q . Since ind p C M q “
0, we canalso write ´ ind p C q “ ` ind p C M q ´ ind p C q ˘ “ p ´ r C q ` k ` t (cid:96) n ´ r θ n u ´ r C ÿ i “ t s i r θ n , u ´ ∆ ¨ r C is the number of ends of C on β ; we interpret r C “ C has no ends on β , and any sum with indices from 1 to r C as equal to 0 as well. Furthermore, because k “ ∆ ¨ w ď ? ∆ ¨ ∆ ? w ¨ w “ ? ∆ ¨ ∆ a b n , we have(4.4.8) ∆ ¨ ∆ ě r θ n k . By (4.4.8), we know that ∆ ‰ k ě
1, thus if k ě w cannot be parallel because the last block of ∆ is identically 0. If k “ ‰
0. Thus, if (4.4.7) does nothold, and we set z M ¨ ∆ “ k(cid:96) n by Lemma 4.4.4, we obtain I p C q ´ ind p C q ă ´ ” k p t n ´ (cid:96) n ´ ` r θ n (cid:96) n ´ q ` A p r θ n , (cid:96) n ´ ´ k q ´ A p r θ n , (cid:96) n ´ q ı ´ ” r C ´ r ` r C ÿ i “ t p s i r θ n q u ´ t p (cid:96) n ´ ´ k q r θ n u ı . (4.4.9) Claim.
The first term in square brackets above is nonnegative.Proof of Claim.
This is immediate if k “
0. So assume that k ą
0. Notice first that t n ´ (cid:96) n ´ ` r θ n (cid:96) n ´ “ ε P n ą , (4.4.10)since t n “ (cid:96) n ´ (cid:96) n ´ , r θ n “ ε Q n P n and Lemma 2.1.1 implies that P n p (cid:96) n ´ (cid:96) n ´ q ` Q n (cid:96) n ´ “ Q n (cid:96) n ´ ´ P n (cid:96) n ´ “ Q (cid:96) ´ P (cid:96) ´ “ . If 0 ă k ă (cid:96) n ´ ´ (cid:96) n ´ , then (cid:96) n ´ ´ k ą (cid:96) n ´ so that 2 A p r θ n , (cid:96) n ´ ´ k q ě byProposition 4.2.1 (viii).Since ą στ because σ ă .
2, part (vii) of the same proposition shows that2 A p r θ n , (cid:96) n ´ ´ k q ´ A p r θ n , (cid:96) n ´ q ą , hence the claim. If (cid:96) n ´ ą k ě (cid:96) n ´ ´ (cid:96) n ´ , then(4.4.10) implies that k p t n ´ (cid:96) n ´ ` r θ n (cid:96) n ´ q ě δ t n ´ P n , so that 2 k p t n ´ (cid:96) n ´ ` r θ n (cid:96) n ´ q ` A p r θ n , (cid:96) n ´ ´ k q ´ A p r θ n , (cid:96) n ´ qą k p t n ´ (cid:96) n ´ ` r θ n (cid:96) n ´ q ` ´ A p r θ n , (cid:96) n ´ qě δ ˆ (cid:96) n ´ P n ´ A p r θ n , (cid:96) n ´ q ˙ ` p ´ (cid:96) n ´ P n q ą , where the last step uses Proposition 4.2.1 (vii), and the fact that (cid:96) n ´ P n is an increasingsequence with limit στ . If k “ (cid:96) n ´ , then the first term in square brackets is 0. Thusin all cases, the claim holds. (cid:3) Thus in all cases, if (4.4.7) does not hold, we have I p C q ´ ind p C q ă p r ´ r C q ` t p (cid:96) n ´ ´ k q r θ n u ´ r C ÿ i “ t s i r θ n u q . However, by (4.1.10) and (4.1.17) in Proposition 4.1.4, we have I p C q ´ ind p C q ě r ´ r C ` t p (cid:96) n ´ ´ k q r θ n u ´ r C ÿ i “ t s i r θ n u . This is a contradiction. Hence we must have (4.4.7).It remains to show that the connector has just one end on β . This holds because ofour initial assumption that there is a breaking with a connector, i.e. by assumption theconnector C does exist as a holomorphic curve. Since it has the same asymptotics as C M , it has I p C q “
0, and because is simple (because it goes through some constraintswith multiplicity one) it must therefore have ECH partitions; see Remark 2.2.3 (iii).The result now holds because p ` β p (cid:96) n ´ q “ p (cid:96) n ´ q . (cid:3) It remains to prove :
Proof of Lemma 4.4.4.
We must show ∆ ¨ z M “ k(cid:96) n ´ . We do this in several steps. Step 1: Applying the recursion for z M and w . Let r z M denote the homology class of z M , with the last block removed, and define r w analogously. Because ∆ is supported away from the last block, we have(4.4.11) ∆ ¨ z M “ ∆ ¨ r z M . HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 73
Here and below, to simplify the notation we truncate the vector ∆ without furthercomment by removing the last block of zeroes, so that expressions like ∆ ¨ ˜ z M aredefined. This is justified in view of (4.4.11).To simplify the discussion, we suppose for the moment that the entries of ∆ areconstant on the blocks of W p b n q . Thus,(4.4.12) ∆ “ ` x ˆ , x , x ˆ , . . . , x ˆ , . . . , x ˆ n ´ , x n ´ ˘ . This assumption does require a slight loss of generality, but below we will see that thisis justified. By the discussion after (4.4.1) the vector r z M “ W p (cid:96) n (cid:96) n ´ q has the sameblock decomposition, and in the notation of Lemma 2.1.7 we may write r z M “ p (cid:96) ˆ n ´ , t n ´ , . . . q “ (cid:96) n ´ ¨ R p , q ` t n ´ ¨ R p , q by Lemma 2.1.7(i). Hence∆ ¨ r z M “ (cid:96) n ´ ∆ ¨ R p , q ` t n ´ ∆ ¨ R p , q . Similarly, because the weight vector w “ w p b n q “ p ˆ , b n ´ , . . . q satisfies the samerecursion on all but the last block (on which ∆ “ k “ ∆ ¨ w “ ∆ ¨ R p , q ` p b n ´ q ∆ ¨ R p , q . Therefore, ∆ ¨ r z M “ (cid:96) n ´ k ` ` t n ´ ´ (cid:96) n ´ p b n ´ q ˘ ∆ ¨ R p , q . (4.4.13)It remains to show that ∆ ¨ R p , q “ Step 2: We prove | ∆ ¨ R p , q| ă Q n when ∆ satisfies (4.4.12) . Assume now that the entries x , . . . , x n ´ of ∆ satisfy the recursion in (2.1.17).Then Lemma 2.1.7 implies that∆ ¨ R p , q “ (cid:96) n ´ x n ´ . (4.4.14)It is possible that the x i do not satisfy (2.1.17). However, they differ from a sequence r x i that does by a small amount. Namely, recall that we may write r z M ´ ∆ “ λ r w ´ Ą diff C , λ : “ p r z M ´ ∆ q ¨ ww ¨ w . So, ∆ “ p r z M ´ λ r w q ` Ą diff C . Let the r x i be the entries of r z M ´ λ r w . Then the r x i for i ě z i of Ą diff C satisfy | z i | ă , @ i. (4.4.15)To see this, note that the sum of the two area terms 2 A p¨ , ¨q on the left hand side of(4.1.9) must be at least 1 .
39 by Proposition 4.2.1 (iii); the right hand side of (4.1.9) is no more than 2 .
54 by (4.3.8); and the contribution to diff C ¨ diff C from the last blockmust be at least 7 p ´ . q , again by (4.3.8). Thus we have ÿ i z i ď diff C ¨ diff C ´ p ´ . q ď . ´ . ´ p ´ . q ă , which proves (4.4.15). Hence if I denotes the vector all of whose entries are ˘ R p , q , we find by replacing ∆ in (4.4.14) with r z M ´ λ r w and using | z i | ă
1, that∆ ¨ R p , q ă (cid:96) n ´ r x n ´ ` I ¨ R p , q“ (cid:96) n ´ r x n ´ ` ` t ` t ` . . . ` t n ´ ` p (cid:96) ` (cid:96) ` . . . ` (cid:96) n ´ q ˘ “ (cid:96) n ´ r x n ´ ` (cid:96) n ´ ` t n ´ ´ , where the first equality uses Lemma 2.1.7 (ii) and the second the identities t k “ (cid:96) k ´ (cid:96) k ´ and 5 (cid:96) k “ t k ` ´ t k from (2.1.7).We next claim that | r x n ´ | ď . To see this, note that r x n ´ “ ´ λ Q n , where t n ď λ ď (cid:96) n ´ r θ n ` t n . Further, 7 t n Q n ą , since the fractions t n Q n are decreasing by Lemma 2.1.1, with limit σ p τ ´ q by (2.1.9).We also claim that 7 ˆ (cid:96) n ´ P n ` t n Q n ˙ ă , because (cid:96) n ´ P n are increasing with limit στ , while t n Q n decreases and7 ˆ στ ` ˙ ă . Thus, 0 ď r x n ´ ď
1, so that(4.4.16) | ∆ ¨ R p , q| ă (cid:96) n ´ r x n ´ ` (cid:96) n ´ ` t n ´ ´ ă (cid:96) n ´ ` t n ´ ´ ă Q n , as claimed. Step 3: Divisibility considerations:
We now claim that ∆ ¨ R p , q is divisible by Q n .To see this, note that ∆ ¨ r w “ ∆ ¨ w “ k is an integer, which implies that ∆ ¨ Ă W isdivisible by Q n . Therefore by Lemma 2.1.7 (iii)0 ” ∆ ¨ Ă W ” ´ Q n ´ ∆ ¨ R p , q p mod Q n q . Since Q n , Q n ´ are relatively prime, this implies that ∆ ¨ R p , q is a multiple of Q n .By (4.4.16), this implies ∆ ¨ R p , q “
0. Thus ∆ ¨ r z M “ (cid:96) n ´ k by (4.4.13). Step 4: Justifying the special form for ∆ : We have therefore proved Lemma 4.4.4,except for the assumption that ∆ can be written in the form (4.4.12), i.e. that its entriesare constant on each block. However, the above arguments only used information
HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 75 about the two dot products, ∆ ¨ r w , ∆ ¨ r z M , and the fact that if V “ R p A, B q “p V ˆ , V , V ˆ , . . . q for some integers A, B then∆ ¨ V “ ÿ i k i V i for some k i P Z . (4.4.17)If ∆ does not have the form in (4.4.12), rewrite it, taking the average value on eachblock of size m , where m “ ,
6. This does not change either of these dot products.Moreover, because the new values of x i have the form n i m (i.e. their denominator equalsthe length of the relevant block) the dot product ∆ ¨ V still satisfies (4.4.17). Hencethe argument goes through even if ∆ does not have this special form.This completes the proof of Lemma 4.4.4 and hence of Proposition 4.4.1. (cid:3) The rest of the proof.
The next result completes the proof of Proposition 3.1.6,and hence the proof of Theorem 1.2.1.
Proposition 4.5.1.
There is at most one breaking with a connector.
To prove this, we locate the unique double point in the limiting building and then usethe fact that B ¨ B “ § different connectedcomponents of the building: if it were internal to one component, then there would beno obvious way to control the number of nearly J -holomorphic representatives.Consider a limiting building C that has a connector. We saw in Lemma 4.3.1 thatthe curves in the neck are all multiple covers of trivial cylinders. Hence we can divide C in a slightly different way than before (cf. Definition 3.1.3), taking the top level toconsist of the curves in C U together with those in the neck (which are covers of trivialcylinders by Lemma 4.3.1), and then dividing the curves in C LL into three groupsaccording to the asymptotics at their top end. Thus, we now consider the top level tobe a union of matched components as follows. Definition 4.5.2.
In this section, D j , for j “ , , denotes the union of the curves in p E with top end on β j , while the connector D is also a curve (rather than a matchedcomponent). Further, we define U to be the matched component in the upper levelswhose negative end connects to the positive end of D at β ; similarly, U connects tothe positive end of D at β . Then U is at least an (cid:96) n ´ -fold cover of the low action curve C , and U is at leastan p (cid:96) n ´ (cid:96) n ´ q -fold cover of C . Hence, because U , U contain different curves sincethe building has genus zero, it follows that ‚ the upper level is U Y U , ‚ U is precisely an (cid:96) n ´ -fold cover of C (extended by trivial components in theneck), and ‚ U is precisely a p (cid:96) n ´ (cid:96) n ´ q -fold cover of C . In particular, U is in class 3 (cid:96) n ´ L and U is in class 3 t n L .We now argue much as in the proof of Proposition 3.4.1 at the end of § C with a connector as above, and chop the nearly broken curvebuilding close to the top of the lowest level, i.e. at the bottom of the neck region.This gives compact curves with boundary U ,i , U ,i , D ,i , D ,i and D ,i defined from U , U , D , D and D and holomorphic with respect to a sequence of almost-complexstructures J R i with R i Ñ 8 . Note that because every B -curve has exactly one doublepoint, there can be at most one intersection point between these compact curves. Wenow prove Proposition 3.1.6 in several steps.We begin with the following intersection alternative: Lemma 4.5.3.
Either U ,i intersects U ,i , or p D ,i Y D ,i q intersects D ,i .Proof. Because B -curves are simple, the curves U ,i Y U ,i , D ,i Y D ,i , and D ,i aresomewhere injective compact curves whose boundaries form links around the orbits β , β , so that their intersection number can be calculated using the intersection formula(2.2.1). Thus(4.5.1) U ,i ¨ U ,i “ Q τ pr U ,i s , r U ,i sq ` L τ p U ,i , U ,i q , and similarly(4.5.2) p D ,i Y D ,i q ¨ D ,i “ Q τ pr D ,i Y D ,i s , r D ,i sq ` L τ p D ,i Y D ,i , D ,i q , where L τ denotes the asymptotic linking number defined in (2.2.3), and Q τ is therelative intersection pairing whose formula is given in (2.2.28) and (2.2.31). Becausethe negative ends of U ,i Y U ,i are the same as the positive ends of D ,i Y D ,i Y D ,i we have(4.5.3) L τ p U ,i , U ,i q “ ´ L τ p D ,i Y D ,i , D ,i q . Now, by (2.2.28), we have(4.5.4) Q τ pr U ,i s , r U ,i sq “ (cid:96) n ´ t n ´ (cid:96) n ´ t n “ (cid:96) n (cid:96) n ´ ´ (cid:96) n ´ . To compute Q τ pr D ,i Y D ,i s , r D ,i sq , note that by Proposition 4.4.1, r D ,i s “ z M , andso r D ,i Y D ,i s “ W ´ z M . Thus by (2.2.31) and (2.1.15) we have: Q τ pr D ,i Y D ,i s , r D ,i sq “ (cid:96) n ´ ` t n ´ W ¨ z M ` z M ¨ z M (4.5.5) “ (cid:96) n ´ ` p (cid:96) n ´ (cid:96) n ´ q ´ p (cid:96) n ` (cid:96) n (cid:96) n ´ ´ (cid:96) n ´ ` q` (4.5.6) p (cid:96) n ´ (cid:96) n ` q“ ´p (cid:96) n (cid:96) n ´ ´ (cid:96) n ´ q ` . In §
2, we stated this formula for punctured curves. However, it is equally valid for curves withboundary obtained by truncating a completed cobordism by removing any Y ˆp R, or Y ˆp´8 , ´ R q region for large R . For a similar situation, see [HN1, Lem 3.5]. HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 77
By combining (4.5.4) and (4.5.5), it follows that(4.5.7) Q τ pr U ,i s , r U ,i sq “ ´ Q τ pr D ,i Y D ,i s , r D ,i sq ` . Now combine (4.5.7) with (4.5.1), (4.5.2). This gives U ,i ¨ U ,i “ ´ p D ,i Y D ,i q ¨ D ,i . Since 0 ď p D ,i Y D ,i q ¨ D ,i ď
1, the claim now follows. (cid:3)
Corollary 4.5.4. D ,i and D ,i do not intersect.Proof. Assume that they do intersect. Then neither can intersect D ,i , or else wewould have too many intersection points. Thus, by the previous step, U ,i and U ,i would have to intersect. This also gives too many intersection points. (cid:3) The next step is the following uniqueness claim.
Lemma 4.5.5.
The constraint z that is carried by the curve D ,i is independent of thebreaking. Note that this implies the same statement for D ,i , since the homology class of D ,i is fixed in view of Proposition 4.4.1. Proof.
Suppose given one breaking D ,i , D ,i with constraints z, y and another D ,i , D ,i with constraints z , y . Since z ` y “ z ` y , we have p z ´ z q¨p y ´ y q “ p z ´ z q¨p z ´ z q ě z, y as vectors as in the previous section). But because D ,i , D ,i areasymptotic to β while D ,i , D ,i are asymptotic to β , the contribution of the top endsto intersection numbers such as D ,i ¨ D ,i is fixed and equal to T : “ (cid:96) n ´ ¨ p (cid:96) n ´ (cid:96) n ´ q .Hence 0 ď p z ´ z q ¨ p y ´ y q “ z ¨ y ` z ¨ y ´ z ¨ y ´ z ¨ y “ ´ ´ p T ´ z ¨ y q ` p T ´ z ¨ y q ´ p T ´ z ¨ y q ´ p T ´ z ¨ t q ¯ “ ´ ´ p T ´ z ¨ y q ` p T ´ z ¨ y q ¯ where the final equality above follows from Corollary 4.5.4. But p T ´ z ¨ y q and p T ´ z ¨ y q both compute the number of intersections of J -holomorphic curves, and so arenonnegative. Thus p z ´ z q ¨ p y ´ y q “ , so that z “ z , y “ y as claimed. (cid:3) Now assume as above that C i and C i are two different B -curves that are close tobreaking into a building with a connector. The final step is the following variant ofLemma 4.5.3. Lemma 4.5.6. ‚ If U ,i does not intersect U ,i , then either D ,i intersects D ,i or D ,i intersects D ,i . ‚ If U ,i does not intersect U ,i , then either D ,i intersects D ,i , or D ,i inter-sects D ,i . Corollary 4.5.7.
Proposition 4.5.1 holds.
Proof.
Since C i and C i are B -curves, we have C i ¨ C i “
1. On the other hand, the inter-section alternative in Lemma 4.5.6 guarantees that there are at least two intersectionpoints between C i and C i . Hence this scenario cannot occur. (cid:3) The proof of Lemma 4.5.6 mimics that Lemma 4.5.3. One has to be precise to getthe relevant linking terms to cancel, however; hence the quite specific alternatives.
Proof of Lemma 4.5.6.
We begin by proving the first bullet point.By (2.2.1), we have:(4.5.8) U ,i ¨ U ,i “ Q τ pr U ,i s , r U ,i sq ` L τ p U ,i , U ,i q , where L τ denotes the asymptotic linking number defined in (2.2.3). Further, L τ p U ,i , U ,i q “ ´ L τ p D ,i , D ,i q ´ L τ p D ,i , D ,i q , (4.5.9)because, by Definition 4.5.2, the negative ends of U ,i and U ,i on β are the same asthe positive ends on β of D ,i and D ,i respectively, while the negative ends of U ,i and U ,i on β are the same as the positive ends on β of D ,i and D ,i . Note alsothat there is no linking between the end of D ,i at β and the ends of D ,i , nor anybetween the end of D ,i at β and the ends of D ,i . Hence(4.5.10) D ,i ¨ D ,i ` D ,i ¨ D ,i “ Q τ pr D ,i s , r D ,i sq` Q τ pr D ,i s , r D ,i sq´ L τ p U ,i , U ,i q . By Lemma 4.5.5, the constraints z, z on D ,i , D ,i are the same. Hence Q τ pr D ,i s , r D ,i sq “ t n ´ z ¨ z M , Q τ pr D ,i s , r D ,i sq “ (cid:96) n ´ ´ y ¨ z M , so that Q τ pr D ,i s , r D ,i sq ` Q τ pr D ,i s , r D ,i sq “ (cid:96) n ´ ` t n ´ p z ` y q ¨ z M “ (cid:96) n ´ ` t n ´ p W ´ z M q ¨ z M , (4.5.11) “ Q τ pr D ,i Y D ,i s , r D ,i sq , where the last equality holds by the first line of (4.5.5). Equation (4.5.7) now impliesthat Q τ pr U ,i s , r U ,i sq “ ´p Q τ pr D ,i s , r D ,i sq ` Q τ pr D ,i s , r D ,i sqq ` , since Q τ pr U ,i s , r U ,i sq “ Q τ pr U ,i s , r U ,i sq . Combine this with (4.5.8) and (4.5.10) as inthe proof of Lemma 4.5.3, we obtain U ,i ¨ U ,i “ ´ p D ,i ¨ D ,i ` D ,i ¨ D ,i q . Since all intersection numbers are nonnegative, this completes the proof of the firstbullet point.The proof of the second is identical, modulo switching the roles of C i , C i . (cid:3) HE GHOST STAIRS STABILIZE TO SHARP SYMPLECTIC EMBEDDING OBSTRUCTIONS 79
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Proc.London Math. Soc. , (2015), 787–804. Department of Mathematics, Harvard University and University of California, SantaCruz
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