On infinite staircases in toric symplectic four-manifolds
Dan Cristofaro-Gardiner, Tara S. Holm, Alessia Mandini, Ana Rita Pires
IINFINITE STAIRCASES AND REFLEXIVE POLYGONS
DAN CRISTOFARO-GARDINER, TARA S. HOLM, ALESSIA MANDINI, AND ANA RITA PIRES
Abstract.
We explore the question of when an infinite staircase describes part of theellipsoid embedding function of a convex toric domain. For rational convex toric domains infour dimensions, we conjecture a complete answer to this question, in terms of six familiesthat are distinguished by the fact that their moment polygon is reflexive. To understandbetter when infinite staircases occur, we prove that any infinite staircase must have a uniqueaccumulation point given as the solution to an explicit quadratic equation. We then providea uniform proof of the existence of infinite staircases for our six families, using two tools. Forthe first, we use recursive families of almost toric fibrations to find symplectic embeddingsinto closed symplectic manifolds. In order to establish the embeddings for convex toricdomains, we prove a result of potentially independent interest: a four-dimensional ellipsoidembeds into a closed symplectic toric four-manifold if and only if it can be embedded intoa corresponding convex toric domain. For the second tool, we find recursive families ofconvex lattice paths that provide obstructions to embeddings. Our work contrasts the workof Usher, who finds infinite families of infinite staircases for irrationally shaped rectangles.
Contents
1. Introduction 21.1. Accumulation points of infinite staircases 41.2. Reflexive polygons and infinite staircases 5Organization of the paper 82. Preliminaries and tools 92.1. Properties of the ellipsoid embedding function 92.2. ECH capacities 102.3. Obstructive classes 122.4. Toric manifolds and almost toric fibrations 153. Passing to closed symplectic manifolds 194. Pinpointing the location of the accumulation point 225. The existence of the Fano staircases 296. Conjecture: why these may be the “only” infinite staircases 38Appendix A. Lattice Paths 41Appendix B. ATFs 44Appendix C. Behind the scenes 49The Mathematica Code 50References 51
Date : April 29, 2020. a r X i v : . [ m a t h . S G ] A p r . Introduction
In the past two decades, there has been considerable interest in and progress on thequestion of whether there is an embedding ( M , ω M ) s ↪ ( N , ω N ) preserving symplectic structures, or whether the existence of such a map is in some wayobstructed. On the one hand, Local Normal Form theorems and clever constructions likesymplectic folding and symplectic inflation allow us to find embeddings. On the other hand,in dimension four there are well-developed tools involving pseudo-holomorphic curves thatprovide numerous obstructions to these maps.We examine this question when the target is a toric symplectic four-manifold associated toa lattice polygon in R . The answers we have found are governed by beautiful combinatoricsand number theory. We begin with toric domains. A 4-dimensional toric domain X is thepreimage of a domain Ω ⊂ R ≥ under the moment map µ ∶ C → R , ( z , z ) ↦ ( π ∣ z ∣ , π ∣ z ∣ ) . For example, if Ω is the hypotenuse-less triangle with vertices ( , ) , ( a, ) , and ( , b ) , then X Ω is the open ellipsoid E ( a, b ) : E ( a, b ) = {( z , z ) ∈ C ∶ π ( ∣ z ∣ a + ∣ z ∣ b ) < } . Note that B ( a ) = E ( a, a ) is an open ball of capacity a (and radius √ aπ ). Following thenotation set forth in [6, Definition 1.1], a convex toric domain is the preimage under µ of a closed region Ω ⊂ R ≥ that is convex, connected, and contains the origin in its interior.We denote this X Ω = µ − ( Ω ) and call the region Ω the moment polygon of X Ω , in analogywith the case of closed symplectic toric-manifolds.There is an extensive literature on symplectic embedding problems where the domain isan ellipsoid: [4, 5, 7, 8, 9, 10, 13, 17, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 40]. Even inthis seemingly simple situation, there is a subtle mix of rigidity and flexibility. Our workcontinues this theme. First, to fix notation, we write E s ↪ X to mean that there is a symplecticembedding of E into X , and define the ellipsoid embedding function of X by(1.1) c X ( a ) ∶ = min { λ ∣ E ( , a ) s ↪ λX } , for a ≥ , where λX represents the symplectic scaling ( X, λ ⋅ ω ) of ( X, ω ) . We could have defined thefunction for a >
0, but there is a symmetry across a =
1, making this redundant.The embedding capacity function makes sense for any symplectic manifold X , not justconvex toric domains. Indeed, one motivation for studying convex toric domains comes fromthe following result that we prove, which ties together the ellipsoid embedding functions forclosed toric manifolds and convex toric domains. This result features essentially in our proofof Theorem 1.14 as well. For a general symplectic manifold target, we should replace the min in (1.1) with an inf. For a closedtoric manifold, we will see in Theorem 1.2 that we can still use the min. heorem 1.2. Let Ω ⊂ R ≥ be a convex region that is also a Delzant polygon for a closedtoric symplectic four-manifold M . Then there exists a symplectic embedding (1.3) E ( d, e ) s ↪ M if and only if there exists a symplectic embedding (1.4) E ( d, e ) s ↪ X Ω . Thus, from the point of view of the function c X , convex toric domains significantly gen-eralize closed toric manifolds. In fact, we can relate embeddings into convex toric domainsto embeddings into closed manifolds in a slightly more general context, including some well-known examples, for example equilateral pentagon space: see Remark 3.3, Proposition 3.5,and the accompanying Remark 3.6.For a general convex toric domain X , the embedding function c X ( a ) has an interestingqualitative structure. For a fixed X , the volume curve is the curve y = √ a vol and theconstraint c X ( a ) ≥ √ a volholds because E ( , a ) s ↪ λX ⇒ volume ( E ( , a )) ≤ volume ( λX ) ⇔ a ≤ λ volume ( X ) ⇔ λ ≥ √ a vol . We will show in Proposition 2.1 that c X ( a ) is continuous and non-decreasing, but not gener-ally C . For sufficiently large a , we also show that the function c X ( a ) remains equal to thevolume curve: this is the phenomenon known as packing stability . Moreover, the function c X ( a ) is piecewise linear when not equal to the volume curve, except at points that are limitpoints of singular points of c X . We call these limit points accumulation points and theyare an important focus of this paper. We now codify this with the following definition, whichis the main topic of our investigation. Definition 1.5.
For a symplectic manifold X , we say that the ellipsoid embedding function c X ( a ) has an infinite staircase if its graph has infinitely many non-smooth points, i.e.infinitely many staircase steps. Remark 1.6.
In [4, Definition 1.1], Casals and Vianna work with a different concept, a sharp infinite staircase. This is an infinite staircase where infinitely many of the non-smooth points must be on the volume curve. That notion therefore excludes the J = c B ( a ) has an infinite staircase, the coordinates of its steps arerelated to the Fibonacci numbers, and there is a unique accumulation point at an appropriatepower of the Golden Mean; the portion of the graph corresponding to this phenomenon isoften called the “Fibonacci staircase.” In the paper [13], Cristofaro-Gardiner and Kleinmanstudied the ellipsoid embedding function of an ellipsoid c E ( ,b ) ( a ) and found infinite stair-cases when b = b = . Frenkel and M¨uller found an infinite staircase in the ellipsoid We call a non-smooth point of c X a singular point, and we use these terms interchangeably. mbedding function for a polydisc P ( , ) [17], where the function is governed by the Pellnumbers. Cristofaro-Gardiner, Frenkel, and Schlenk have shown that the only infinite stair-case in the ellipsoid embedding function for polydisks P ( , b ) with b ∈ N is when b = P ( , b ) [40]and found the first infinite families of infinite staircases. Usher’s families all have b quadraticirrationalities of a special form.Despite these myriad examples, a general theory of infinite staircases does not currentlyexist, however; a goal of this paper is to lay some possible steps for such a classification.1.1. Accumulation points of infinite staircases.
Our first result is aimed at rootingout the “germs” of infinite staircases. If the function c X ( a ) has an infinite staircase, thenthe singular points must accumulate at some set of points, otherwise this would contradictpacking stability. We show that if an infinite staircase exists, there is in fact a unique suchaccumulation point; moreover, it can be characterized as the solution to an explicit quadraticequation over the integers determined by Ω.We now make this precise. To a convex toric domain we can associate a blowup vector ( b ; b , b , . . . ) , but not in a unique way. To do so, first we need to define a b -triangle to bethe triangle with vertices ( , ) , ( b, ) and ( , b ) or any AGL ( , Z ) transformation of it .We proceed inductively: let b > b -triangle. If Ω equalsthat triangle, we are done. Otherwise, let b > b -triangle minus a b -triangle that is removed at a corner of the b -triangle. If Ω equals thisquadrilateral, we are done. Otherwise, let b > b -triangle that is removed at one of its corners. The removing ofthe b i -triangles is reminiscent of what is done to the moment polytope when performingequivariant symplectic blowups at fixed points, hence the name blowup vector. We notethat when Ω is a lattice polygon, this blowup process is finite.Note that the same convex toric domain has several different possible blowup vectors: forexample, (
2; 1 , , ) and ( ) describe the same region. We can address this by picking b assmall as possible and b i as large as possible at each step; these choices give a blowup vectorthat is also called the negative weight expansion in [6]. Conversely, two different convextoric domains can have the same associated blowup vector, see Figure 1.9 for examples. Wewill see in Remark 2.8 that the relevant feature of a convex toric domain in the context ofthis paper is its blowup vector, and not the actual shape of Ω. Definition 1.7.
We say that a blowup vector ( b ; b , b , . . . ) is finite there are finitely manynon-zero b i ’s. Given a convex toric domain X with finite blowup vector ( b ; b , . . . , b n ) wedefine: per = b − n ∑ i = b i vol = b − n ∑ i = b i Remark 1.8.
The quantities per and vol are, respectively, the affine perimeter and twicethe area of the region in R representing X . They are well-defined as invariants of X . In By AGL ( , Z ) transformation we mean a GL ( , Z ) transformation followed by an affine translation. articular, vol is the symplectic volume of X . Note also that per vol is invariant under scalingof (the region representing) X . (3) (4;2,2)(3;1) (4;2,2) (3;1,1) (3;1,1)(3;1,1,1) (3;1,1,1) (3;1,1,1) (3;1,1,1,1) (3;1,1,1,1)(3;1,1,1)(3;1,1,1,1,1) (3;1,1,1,1,1) (3;1,1,1,1,1,1)(3;1,1,1,1,1) Figure 1.9.
Regions corresponding to convex toric domains, and their blowup vec-tors ( b ; b , . . . , b n ) . The blowup vectors (
3; 1 ) and (
3; 1 , ) correspond to the J = J =
2; cf. Table 1.13 and Remark 1.17. Note that all of thesepolygons are reflexive.
We can now state precisely our theorem about finding accumulation points.
Theorem 1.10.
Let X be a convex toric domain with finite blowup vector. If the ellipsoidembedding function c X ( a ) has an infinite staircase then it accumulates at a , a real solution of the quadratic equation (1.11) a − ( per vol − ) a + = . Furthermore, at a the ellipsoid embedding function touches the volume curve: c X ( a ) = √ a vol . In the face of all the examples in Figure 1.12, the power of Theorem 1.10 is that it providesa means for showing when there is not an infinite staircase. For example in Figure 1.12(c)below, we can see clearly that the ellipsoid embedding function is obstructed, and so a toricdomain with blowup vector (
4; 2 , ) cannot have an infinite staircase. There is a similarobstruction at Figure 1.12(d) if you zoom in sufficiently.1.2. Reflexive polygons and infinite staircases.
Having explained in the previous sec-tion where infinite staircases must accumulate, we now turn our attention to finding them.We begin by showing in Figure 1.12 the types of graphs we can produce of embedding func-tions using Mathematica. These types of plots were essential in our early investigations ofinfinite staircases. This is discussed further in Appendix C. Note that if equation (1.11) has two distinct real solutions, then there is a unique solution greater than1. Thus, when a exists as in the statement of the theorem, it is unique on the domain of c X . .5 2.0 2.5 3.00.500.550.600.650.700.75 (a) Blowup vector (3;1,1,1,1). (b)
Blowup vector (3;1). (c)
Blowup vector (4;2,1). (d)
Blowup vector (6;1).
Figure 1.12.
Plots of ellipsoid embedding functions for domains with differentblowup vectors. The red curves are the volume curves and the vertical lines indicatewhere the accumulation points would necessarily be located, if a staircase existed,per Theorem 1.10. The top two plots have infinite staircases: in (a) we have a J = J = Our next result identifies infinite staircases for the ellipsoid embedding functions of twelveconvex toric domains, including the already known ball, polydisk P ( , ) , and E ( , ) . Ourproof of Theorem 1.14 provides a uniform approach to prove the existence of all twelve inone fell swoop. The graphs of these functions are related to certain recurrence sequences,which are given in Table 1.13. Blowup vector Recurrence relation Seeds K = per vol − J a g ( n + J ) = Kg ( n + J ) − g ( n ) g ( ) , . . . , g ( J − )( ) g ( n + ) = g ( n + ) − g ( n ) , , , + √ (
4; 2 , ) g ( n + ) = g ( n + ) − g ( n ) , , , + √ (
3; 1 , , ) g ( n + ) = g ( n + ) − g ( n ) , , , + √ (
3; 1 , , , ) g ( n + ) = g ( n + ) − g ( n ) , , , + √ (
3; 1 ) g ( n + ) = g ( n + ) − g ( n ) , , , , , + √ (
3; 1 , ) g ( n + ) = g ( n + ) − g ( n ) , , , , , + √ Table 1.13.
The key recurrence relations. heorem 1.14. Let X be a convex toric domain with blowup vector ( b ; b , . . . , b n ) equal to ( ) , (
3; 1 ) , (
3; 1 , ) , (
3; 1 , , ) , (
3; 1 , , , ) , or (
4; 2 , ) . Then the ellipsoid embedding function c X ( a ) has an infinite staircase which alternates be-tween horizontal lines and lines through the origin connecting inner and outer corners ( x in , y in ) , ( x out , y out ) , ( x in , y in ) , ( x out , y out ) , . . . respectively with coordinates: ( x in n , y in n ) = ( g ( n + J ) ( g ( n + ) + g ( n + + J ))( g ( n ) + g ( n + J )) g ( n + ) , g ( n + J ) g ( n ) + g ( n + J ) ) , ( x out n , y out n ) = ( g ( n + J ) g ( n ) , g ( n + J ) g ( n ) + g ( n + J ) ) . Remark 1.15.
The recurrence relations that appear in Table 1.13 do not immediately ap-pear to be the ones previously associated to infinite staircases. But a quick computationshows that for ( ) , this does recover the odd-index Fibonacci numbers McDuff and Schlenkfound in [35]; for (
4; 2 , ) it recovers Pell and half-companion Pell numbers as found byFrenkel and M¨uller [17]; and for (
3; 1 , , ) the sequences of Cristofaro-Gardiner and Klein-man [13]. Writing them in this uniform way simplifies the statement of Theorem 1.14. Remark 1.16.
Combining Theorems 1.2 and 1.14, we conclude that the ellipsoid embeddingfunction c X ( a ) has an infinite staircase for the compact symplectic manifolds C P × C P and C P k C P for k = , , , ,
4. The smooth polygons in Figure 1.9 are Delzant polygons:they are the moment polygons of compact toric symplectic manifolds, namely for C P × C P and C P k C P for k = , , ,
3. The only blowup vector from the list that does not havea smooth Delzant polygon representative is (
3; 1 , , , ) . This manifold is well known notto admit a Hamiltonian circle or 2-torus action [20]. We may identify this manifold asequilateral pentagon space and as such, it is well known to admit a completely integrablesystem from bending flows whose image is shown in the bottom right picture in Figure 1.9. Remark 1.17.
For each convex toric domain, the accumulation point of the infinite staircaseis on the volume curve. However, two fairly distinct behaviors can be observed, related tothe order of the recurrence relation in Table 1.13. In the J = J = J = J = J = e complete the introduction with a conjecture that the list in Theorem 1.14 is in asuitable sense exhaustive. Recall that a convex lattice polygon is reflexive if it has exactlyone interior lattice point; this is equivalent to requiring that its dual polygon is also alattice polygon. Up to AGL ( Z ) , the only domains which have blowup vectors listed inTheorem 1.14 are the ones shown in Figure 1.9. These are well known as twelve of thesixteen reflexive lattice polygons in R ; the other four appear in Figure 6.5, and do not haveinfinite staircases as part of their ellipsoid embedding function. Conjecture 1.18.
If the ellipsoid embedding function of a rational convex toric domain hasan infinite staircase, then its moment polygon is a scaling of a reflexive polygon.
In particular, if Conjecture 1.18 holds, we will see that the only rational convex toricdomains whose ellipsoid embedding function has an infinite staircase are indeed the ones fromTheorem 1.14 or any scaling of those, by ruling out the remaining four reflexive polygons.We will give some evidence supporting Conjecture 1.18 in this paper; as further evidence,Cristofaro-Gardiner’s paper [7] applies Theorem 1.10 to prove Conjecture 1.18 in the specialcase of ellipsoids. In light of Usher’s work [40] about infinite staircases for irrational polydisks P ( , b ) , it is crucial in the conjecture that the toric domain be rational. Organization of the paper.
We begin in Section 2 by reviewing the basic properties ofellipsoid embedding functions, ECH capacities, convex lattice paths, obstructive classes, toricmanifolds, and almost toric fibrations. Next, we explore the relationship between convex toricdomains and compact toric manifolds in Section 3, proving Theorem 1.2. In Section 4, weturn to the proof of Theorem 1.10. We are then able give our unified proof of the existenceof the infinite staircases (Theorem 1.14) in Section 5. We conclude by describing evidencesupporting our Conjecture, in Section 6, that the six examples described here are the onlyexamples among rational convex toric domains.The paper also includes three appendices: the first, Appendix A, draws together some com-binatorial data used to define families of convex lattice paths Λ n needed to find obstructionsfor the proof of Theorem 1.14. The second, Appendix B, describes seeds for the families ofalmost toric fibrations needed to provide embeddings in the proof of Theorem 1.14. Finally,in Appendix C, we recall the very beginning of the project, including a surprise connec-tion to the numbered stops on a Philadelphia subway line. This appendix also contains theMathematica code we used to estimate ellipsoid embedding functions and search for infinitestaircases. Acknowledgements . We are deeply grateful for the support of the Institute for AdvancedStudy, without which we would not have been in the same place at the same time (withlovely lunches) to begin this project, nor continue our conjectural work during an engagingSummer Collaborators visit. We were encouraged and helped along by conversations with:Alex Kantorovich, Dusa McDuff, Emily Maw, Felix Schlenk, Helmut Hofer, Ivan Smith,Margaret Symington, Morgan Weiler, Nicole Magill, Peter Sarnak, Renato Vianna, andRoger Casals.Joyful delays in the completion of the project were caused by the births of Am´alia (ARP),Hannah (TSH), and Emily (DCG).DCG was supported by NSF grant DMS 1711976 and the Minerva Research Foundation. SH was supported by NSF grant DMS 1711317 and the Simons Foundation.AM was supported by FCT/Portugal through project PTDC/MAT-PUR/29447/2017.AP was supported by a Simons Foundation Collaboration Grant for Mathematicians.
Relation to [4]. This article has been posted simultaneously with that of Roger Casals andRenato Vianna [4]. Their Theorem 1.2 coincides with our Proposition 5.9 for the blowupvectors ( ) , (
3; 1 , , ) , (
3; 1 , , , ) , and (
4; 2 , ) . Both collaborations have benefitted fromour exchanges of ideas. Indeed, our initial proof relied solely on ECH capacities, but re-quired additional technical details and guaranteed existence of an infinite staircase withoutcompletely computing the embedding capacity function.When Pires gave a talk on this topic at a 2017 KCL/UCL Geometry Seminar, Casals sharedhis beautiful idea: that mutation sequences of ATFs provided explicit symplectic embeddingsfor the Fibonacci staircase and should do the same whenever the target region Ω is a triangle,that is, corresponding to the blowup vectors ( ) , (
4; 2 , ) and (
3; 1 , , ) . Following thissuggestion, we were then able to implement these ATFs explicitly and uniformly for all ofour target regions, including the non-triangular ones. This greatly simplified our work andallowed us to pin down the embedding capacity function entirely, rather than just providingan existence proof for infinite staircases. Independently and without mutual knowledge,Casals and Vianna went on to explore the embeddings arising from mutation sequences ofATFs, also studying connections to tropical geometry and cluster algebras. They use tropicaltechniques to go from the base diagrams to the existence of embeddings, and we tackle thesame issue by using local normal form results for toric actions on non-compact manifoldsand heavily using ECH machinery, cf. Theorem 1.2 and Remark 3.3.2. Preliminaries and tools
Properties of the ellipsoid embedding function.Proposition 2.1.
Let X be a convex toric domain with finite negative weight expansion.The ellipsoid embedding function c X ( a ) (1) is non-decreasing;(2) has the following scaling property: c X ( t ⋅ a ) ≤ t ⋅ c X ( a ) for t ≥ ;(3) is continuous;(4) is equal to the volume curve for sufficiently large values of a ;(5) is piecewise linear, when not on the volume curve, or at the limit of singular points.Proof. We prove only the first three points here, delaying the proof of the fourth to Section 3,and the fifth to Section 4, because the methods used to prove it are similar to the methodsused to prove the results there. The first three properties actually hold for general symplectic4-manifolds X .(1) Let a < a . For all λ such that E ( , a ) s ↪ λX we have E ( , a ) s ↪ E ( , a ) s ↪ λX , so c X ( a ) ≤ λ . Therefore c X ( a ) ≤ c X ( a ) .(2) Let t ≥
1. For all λ such that E ( , a ) s ↪ λX we have E ( , ta ) s ↪ E ( t, ta ) s ↪ tλX , so c X ( ta ) ≤ tλ . Therefore c X ( ta ) ≤ tc X ( a ) .(3) Let ( a i ) i ∈ N be an increasing sequence converging to a , and define t i ∶ = aa i >
1. Usingproperties (2) and (1) we have c X ( a ) = c X ( t i a i ) ≤ t i c X ( a i ) ≤ t i c X ( a ) . ividing through by t i and letting i → ∞ we conclude that lim i → ∞ c X ( a i ) = c X ( a ) .Now let ( a i ) i ∈ N be a decreasing sequence converging to a , and define t i ∶ = a i a > c X ( a ) ≤ c X ( a i ) = c X ( t i a ) ≤ t i c X ( a ) . Dividing through by t i and letting i → ∞ implies that lim i → ∞ c X ( a i ) = c X ( a ) .Therefore c X is continuous at a . (cid:3) ECH capacities.
Let c ECH ( X ) = ( c ( X ) , c ( X ) , c ( X ) , . . . ) represent the non-decreasingsequence of ECH capacities of the toric domain X , as defined in [24]. The sequence inequality c ECH ( X ) ≤ c ECH ( Y ) means that c k ( X ) ≤ c k ( Y ) for all k ∈ N .The sequence of ECH capacities for an ellipsoid E ( a, b ) is the sequence N ( a, b ) , where for k ≥
0, the term N ( a, b ) k is the ( k + ) st smallest entry in the array ( am + bn ) m,n ∈ N , countedwith repetitions [33]. Equivalently, the terms of the sequence N ( a, b ) are the numbers inTable 2.2 arranged in nondecreasing order:+ 0 a a a . . .0 0 a a a . . . b b a + b a + b a + b . . .2 b b a + b a + b a + b . . .3 b b a + b a + b a + b . . . ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ Table 2.2.
The terms of the sequence N ( a, b ) , before being ordered. Proposition 2.3.
There are at most ( a + )( b + ) − terms in the sequence N ( a, b ) that arelesser than or equal to ab .Proof. For a, b =
1, imagine drawing a line through two equal numbers i + j = i + j = N on (the interior of) Table 2.2. Any number above/on/below that line is respectively small-er/equal/larger than N . For other values of a, b , with i a + j b = i a + j b = N , the sameholds, since we are just looking at the a = b = ba + b = a + ab . There are at most ( a + )( b + ) − ab . (cid:3) We now turn to some algebraic operations on ECH capacities. efinition 2.4. Let ( S k ) k ≥ and ( T k ) k ≥ be the sequences of ECH capacities of two convextoric domains X and Y . We define the sequence sum and sequence subtraction as: ( S T ) k = max m + n = k ( S m + T n )( S − T ) k = inf m ≥ ( S k + m − T m ) . Remark 2.5.
In the definition of sequence subtraction above we require that T ≤ S . Ifadditionally lim k → ∞ S k − T k = ∞ , then inf can be replaced by min. This will happen in all instances in this paper, becausevolume ( X ) > volume ( Y ) . See [6, Remark A.2] and [12, Theorem 1.1] for more details.By [6, Theorem A.1], the sequence of ECH capacities of the convex toric domain X withnegative weight expansion ( b ; b , . . . , b n ) is obtained by the sequence subtraction(2.6) c ECH ( X ) = c ECH ( B ( b )) − c ECH ( n ⨆ i = B ( b i )) = c ECH ( B ( b )) − i c ECH ( B ( b i )) . Since E ( , a ) is a concave toric domain in the sense of [6, § X is a convex toricdomain, the main result [6, Theorem 1.2] implies the following. Proposition 2.7.
There is a symplectic embedding E ( , a ) s ↪ λX if and only if c ECH ( E ( , a )) ≤ c ECH ( λX ) . Remark 2.8.
The existence of a symplectic embedding is equivalent to an inequality of ECHcapacities, which are determined by the blowup vector. Thus, the function c X ( a ) dependsonly on the blowup vector ( b ; b , . . . , b n ) , not on any particular shape of a region in R ≥ withthat blowup vector.Combining this with the definition (1.1) of the ellipsoid embedding function c X ( a ) , wehave(2.9) c X ( a ) = sup k c k ( E ( , a )) c k ( X ) . An equivalent way to compute ECH capacities for convex toric domains uses the combi-natorics of convex lattice paths. The definitions below are based on [6, Definitions A.6, A.7,A.8] and can be found there in more detail.
Definition 2.10. A convex lattice path is a piecewise linear path Λ ∶ [ , c ] → R suchthat all its vertices, including the first ( , x ( Λ )) and last ( y ( Λ ) , ) , are lattice points andthe region enclosed by Λ and the axes is convex. An edge of Λ is a vector ν from one vertexof Λ to the next. The lattice point counting function L ( Λ ) counts the number of latticepoints in the region bounded by a convex lattice path Λ and the axes, including those onthe boundary.Let Ω ⊂ R ≥ be a convex region in the first quadrant. The Ω -length of a convex latticepath Λ is defined as(2.11) (cid:96) Ω ( Λ ) = ∑ ν ∈ Edges ( Λ ) det [ ν p Ω ,ν ] , here for each edge ν we pick an auxiliary point p Ω ,ν on the boundary of Ω such that Ω liesentirely “to the right” of the line through p Ω ,ν and direction ν .Convex lattice paths provide a combinatorial way of computing ECH capacities of a convextoric domain, which we will use to prove Proposition 5.6. Theorem 2.12. [6, Corollary A.5]
Let X be the toric domain corresponding to the region Ω . Then its k th ECH capacity is given by: c k ( X ) = min { (cid:96) Ω ( Λ ) ∶ Λ is a convex lattice path with L ( Λ ) = k + } . Hutchings indicates [26, Ex. 4.16(a)] that the minimum can be taken over those latticepaths Λ that have edges parallel to edges of the region Ω. This simplifies the search forobstructing paths Λ and explains why the lattice paths in Figure A.1 look similar to someof the domains in Figure 1.9.2.3.
Obstructive classes.
To find classes that obstruct the ellipsoid embedding questionfor E ( , a ) , we must introduce the weight expansion ( a , . . . , a n ) of the rational number a ≥
1. The definition is recursive and can be found in [35, Definition 1.2.5] where it is called“weight sequence.” When a is irrational, we may still define the weight expansion in thesame recursive way. It has infinite length.We recall here the essential properties of weight expansions that we use later in Section 4. Lemma 2.13. [35, Lemma 1.2.6]
Let ( a , . . . , a n ) be the weight expansion of a = pq ≥ ,where a is expressed in lowest terms. Then:(1) a n = q ;(2) n ∑ i = a i = a ; and(3) n ∑ i = a i = a + − q . Let ( b ; b , . . . , b N ) be the negative weight expansion of the convex toric domain X . Theweight expansion of a is related to the problem of embedding the ellipsoid E ( , a ) into λX in the following way, following [6, Theorem 2.1]:(2.14) E ( , a ) s ↪ λX ⟺ n ⨆ j = B ( a j ) s ↪ λX ⟺ n ⨆ j = B ( a j ) ⊔ N ⨆ i = B ( λb i ) s ↪ B ( λb ) . Equation (2.14) highlights how the problem of embedding an ellipsoid E ( , a ) into a scalingof a convex toric domain X is similar to the problem of embedding it into a scaling of aball studied in [35]: both boil down to the problem of embedding a disjoint union of ballsinto another ball. Therefore it is no surprise that our proof uses similar tools to those in[35], adapted to this more general case. In particular we use classes ( d ; m ) , which for us aretuples of non-negative integers of the form(2.15) ( d ; m ) = ( d ; ˜ m , . . . , ˜ m N , m , . . . , m n ) hat satisfy the following Diophantine equations (cf. [35, Proposition 1.2.12(i)]): ∑ ˜ m i + ∑ m j = d − ∑ ˜ m i + ∑ m j = d + . (2.17)Fix a convex toric domain X and its negative weight expansion ( b ; b , . . . , b N ) . Each class ( d ; m ) determines a function µ ( d ; m ) as follows. First, pad the tuple ( d ; m ) with zeros on theright in order to make it infinitely long. Then, for a ∈ Q with weight expansion ( a , . . . , a n ) we define:(2.18) µ ( d ; m ) ( a ) ∶ = ∑ m j a j d b − ∑ ˜ m i b i . Formula (2.18) also makes sense for irrational values of a ; as above, these have weightexpansions of infinite length.The following is analogous to [35, Corollary 1.2.3]: Proposition 2.19.
Let ( a , . . . , a n ) be the weight expansion of a ∈ Q and ( b ; b , . . . , b N ) bethe negative weight expansion of X .If the ellipsoid E ( , a ) embeds symplectically into X , then either c X ( a ) = √ a volor there exists a class ( d ; m ) satisfying conditions (2.16) and (2.17) such that (2.20) µ ( d ; m ) ( a ) > √ a vol . In the latter case, c X ( a ) = max ( d ; m ) { µ ( d ; m ) ( a )} . A class ( d ; m ) satisfying (2.20) (in addition to (2.16) and (2.17)) is called an obstructiveclass and the corresponding function µ ( d ; m ) is called an obstruction . Proof.
Hutchings’ survey article [23] gives a nice overview of these ideas. By [35, Theo-rem 1.2.2, Proposition 1.2.12], for an embedding as in the rightmost side of (2.14) to exist,we must have ∑ ˜ m i b i b + ∑ m j a j λb < d for all obstructive classes ( d ; m ) . Rearranging, this is equivalent to the condition that λ > ∑ m j a j db − ∑ ˜ m i b i hence the lemma. (cid:3) The length (cid:96) ( m ) of the class ( d ; m ) = ( d ; ˜ m , . . . , ˜ m N , m , . . . , m n ) is the number ofnonzero m j ’s (not the ˜ m i ’s). For a rational number a ∈ Q , its length (cid:96) ( a ) is the number n of entries in the weight expansion ( a , . . . , a n ) of a . Lemma 2.21.
Let X be a convex toric domain and ( b ; b . . . , b N ) its negative weight expan-sion. Then,(1) If (cid:96) ( a ) < (cid:96) ( m ) then µ ( d ; m ) ( a ) ≤ √ a vol .
2) For all a for which the right hand side is defined, (2.22) µ ( d ; m ) ( a ) ≤ √ avol ⎛⎜⎜⎜⎝ √ b − ∑ b i √ b d d + − ∑ b i ⎞⎟⎟⎟⎠ . Proof.
First assume that (cid:96) ( a ) < (cid:96) ( m ) , and therefore not all m j ’s appear in the sum ∑ m j a j .Then we have µ ( d ; m ) ( a ) = ∑ m j a j d b − ∑ ˜ m i b i ≤ √ ∑ a j ≠ m j √ ∑ a j d b − ∑ ˜ m i b i (by Cauchy-Schwarz) ≤ √ ∑ m j − √ ∑ a j d b − ∑ ˜ m i b i (because at least one m j ∈ Z was excluded) = √ d − ∑ ˜ m i √ ad b − ∑ ˜ m i b i (by (2.17))The light-cone inequality (an analogue of the Cauchy-Schwarz inequality for the Lorentzproduct, [38, Problem 4.5]) guarantees that √ d − ∑ ˜ m i √ b − ∑ b i ≤ d b − ∑ ˜ m i b i , so we obtain the desired inequality: µ ( d ; m ) ( a ) ≤ √ a √ b − ∑ b i = √ a vol . To prove (2.22) for general a we repeat the argument above – minus the line where weused the fact that at least one m j was excluded – and conclude that µ ( d ; m ) ( a ) ≤ √ + d − ∑ ˜ m i √ ad b − ∑ ˜ m i b i . The light-cone inequality from above guarantees that √ + d − ∑ ˜ m i √ b d d + − ∑ b i ≤ d b − ∑ ˜ m i b i , which gives us the desired bound µ ( d ; m ) ( a ) ≤ √ a √ b d d + − ∑ b i . (cid:3) Proposition 2.23.
An obstruction µ ( d ; m ) is continuous and piecewise linear. Furthermore,on each maximal interval I where µ ( d ; m ) ( a ) > √ a vol , the obstruction µ ( d ; m ) has a uniquenon-differentiable point, and it is at a value a such that (cid:96) ( a ) = (cid:96) ( m ) .
3) (3;1) (4;2,2) (4;2,2) (3;1,1) (3;1,1)(3;1,1,1) (3;1,1,1) (3;1,1,1) (3;1,1,1) (3;1,1,1,1) (3;1,1,1,1)(3;1,1,1,1,1) (3;1,1,1,1,1) (3;1,1,1,1,1) (3;1,1,1,1,1,1) xx xxx xxx xx xxxx o ⃝ i ⃝ o ⃝ i ⃝ i ⃝ (3) (3;1) (4;2,2) (3;1,1) (3;1,1,1) (3;1,1,1,1) √ a vol µ ( d ; m ) ( a ) a :: Figure 2.24.
The graph of an obstruction function µ ( d ; m ) ( a ) , together with thevolume curve √ a vol in red. The marked : s represent the unique singular pointsguaranteed in Proposition 2.23. Proof.
Let I be a maximal interval where µ ( d ; m ) ( a ) > √ a vol . Then by Lemma 2.21, (cid:96) ( a ) ≥ (cid:96) ( m ) for all a ∈ I . Assume towards a contradiction that (cid:96) ( a ) > (cid:96) ( m ) for all a in I . Then inparticular 1 ∉ I because (cid:96) ( ) = i th weight in the weight expansion of a , considered as afunction of a , is linear on any open interval that does not contain a point whose weightexpansion has length less than or equal to i . Thus, with (cid:96) ( a ) > (cid:96) ( m ) for all a in I , thefunction µ ( d ; m ) ( a ) would be linear on I . But this is impossible: the volume curve is concaveand the interval I is necessarily bounded above (and below by 1), as the graph of c X ( a ) is equal to the volume curve for sufficiently large a . Thus, there is some point ˜ a with (cid:96) ( ˜ a ) = (cid:96) ( m ) .The uniqueness follows from Lemma 2.21 together with the following basic fact aboutweight expansions, proved in [35, Proof of Lemma 2.1.3]: if b > a are two rational numbersand (cid:96) ( a ) = (cid:96) ( b ) , then there must be some number y ∈ ( a, b ) with (cid:96) ( y ) < (cid:96) ( a ) = (cid:96) ( b ) .We conclude that µ ( d ; m ) ( a ) is piecewise linear on I , with ˜ a the unique singular point. (cid:3) Toric manifolds and almost toric fibrations. A toric symplectic manifold is a symplectic manifold M equipped with an effective Hamiltonian T action satisfyingdim ( T ) = dim ( M ) . Delzant established a one-to-one correspondence between compacttoric symplectic manifolds (up to equivariant symplectomorphism) and Delzant polytopes(up to
AGL n ( Z ) equivalence).A polytope ∆ in R n may be defined as the convex hull of a set of points, or alternativelyas a (bounded) intersection of a finite number of half-spaces in R n . We say ∆ is simple if there are n edges adjacent to each vertex, and it is rational if the edges have rationalslope relative to a choice of lattice Z n ⊂ R n . For a vector with rational slope, the primitivevector with that slope is the shortest positive multiple of the vector that is in the lattice Z n ⊆ R n . A simple polytope is smooth at a vertex if the n primitive edge vectors emanating An action is effective if no positive dimensional subgroup acts trivially. rom the vertex span the lattice Z n ⊆ R n over Z . It is smooth if it is smooth at each vertex.A Delzant polytope is a simple, rational, smooth, convex polytope.To each compact toric symplectic manifold, the polytope we associate to it is its mo-ment polytope. There is a more complicated version of this classification theorem for toricsymplectic manifolds without boundary which are not necessarily compact. In this case, polytopes are replaced by orbit spaces , which are possibly unbounded. Given such an orbitspace, the manifold M is not unique but determined by a choice of cohomology class in H ( M / T ; Z n × R ) . For further details, the reader should consult [1, Chapter VII] and [28, Theorem 1.3].
Remark 2.25.
Note that when M / T is contractible, the above cohomology group is trivialand the corresponding T -space is unique. For example, Euclidean space C n equipped withthe coordinate T n action ( t , . . . , t n ) ⋅ ( z , . . . , z n ) = ( t ⋅ z , . . . , t n ⋅ z n ) is a toric symplectic manifold. The moment map is µ ∶ C n → R n , ( z , . . . , z n ) ↦ ( π ∣ z ∣ , . . . , π ∣ z n ∣ ) , with image the positive orthant in R n . Note that C n / T n is equal to this positive orthant,which is contractible and so by [28, Theorem 1.3], this is the unique toric symplectic manifoldwith this moment map image.More generally, for any relatively open subset Ω ⊂ R n ≥ , the toric domain X Ω = µ − ( Ω ) inherits a linear symplectic form and Hamiltonian torus action from C n . Thus endowed, X Ω is a (non-compact) toric symplectic manifold with X Ω / T = Ω. When Ω is contractible, forexample, the cohomology group H ( X Ω / T ; Z n × R ) = X Ω is the uniquetoric symplectic manifold with moment map image Ω.The moment map on a toric symplectic manifold M is a completely integrable systemwith elliptic singularities. We now focus on four-dimensional manifolds. An almost toricfibration or ATF is a completely integrable system on a four-manifold M with elliptic andfocus-focus singularities. An almost toric manifold is a symplectic manifold equippedwith an almost toric fibration. These were introduced by Symington [39], building on workof Zung [43]. Almost toric fibrations on compact four-manifolds without boundary wereclassified by Leung and Symington in [30] in terms of the base diagram , which includes theimage of the Hamiltonians with decorations to indicate the focus-focus singularities. Evansgives a particularly nice exposition of these ideas [16].For a toric symplectic M , we can identify the singular points of the Hamiltonians in termsof the moment map image. In the four dimensional case, the preimage of each vertex in themoment polygon is a single point for which the moment map has an elliptic singularity ofcorank two. The preimage of a point on the interior of an edge is a circle, for each point ofwhich the moment map has an elliptic singularity of corank one. The preimage of a pointon the interior of the polygon is a 2-torus, of which each point is a regular point. Thus, inFigure 2.26(a), there are three corank two elliptic singularities, three open intervals’ worthof circles of corank two elliptic singularities, and a disc’s worth of tori of regular points. × × (a) (b) (c) (d) Figure 2.26.
Figure (a) is the Delzant polygon for the standard T -action on C P (where the line has symplectic area 3). From (a) to (b), we perform a nodal tradeat the top vertex. From (b) to (c), we perform a nodal slide. From (c) to (d), weperform a mutation on the base diagram. In (d), the light gray portion is just theshadow of portion of the triangle that has changed, it is not part of of the new basediagram, which is outlined in black. Thus, each of these figures represents an almosttoric fibration on C P . Note that the last figure allows us to find an embedding from E ( , ) into C P , which gives E ( , ) s ↪ B ( ) . This embedding is only as explicit asthe diffeomorphisms described here pictorially (which is to say, not explicit!). There are three important operations on the base diagram of an almost toric manifold thatfix the symplectomorphism type of the manifold (cf. [16, 30, 39, 41]). The first is a nodaltrade . Geometrically, this involves excising the neighborhood of a fixed point and gluingin a local model of a focus-focus singularity. This does not change the underlying manifold,but it does change the Hamiltonian functions. The effect on the base diagram is that wemust insert a ray with a mark for the focus-focus singularity thereon. In Figure 2.26, such aray has appeared in (b). The singularities of the Hamiltonian function are still recorded inthe base diagram. Above the marked point on the ray, there is a pinched torus. The pinchpoint is a focus-focus singularity for the new Hamiltonians; the other points on the pinchedtorus are regular. Everything else is as before except for the vertex that anchors the ray.This has been transformed into a circle, for each point of which the new Hamiltonians havean elliptic singularity of corank one.The second operation is a nodal slide . The local model for a focus-focus singularity hasone degree of freedom. A shift in that degree of freedom moves the focus-focus singularityfurther or closer to the preimage of the corner where the ray is anchored. In the base diagram,the marked point moves along the ray. Such a slide is occurring in Figure 2.26 from (b) to(c). The singularities remain as they were.The third operation is a mutation with respect to a nodal ray of the base diagram. Thischanges the shape of the base diagram as follows. The base diagram is sliced in two bythe nodal ray. One piece remains unchanged and the other is acted on by an affine lineartransformation in
ASL ( Z ) that fixes the anchor vertex; • fixes the nodal ray; and • aligns the two edges emanating from the anchor vertex.The operation creates a new (anchor) vertex and nodal ray (in the opposite direction frombefore) in the base diagram. This result is shown in Figure 2.26 from (c) to (d). As before,the preimage of the anchor vertex is a circle, for each point of which the new Hamiltonianshave an elliptic singularity of corank one. The old anchor vertex is now in the interior of anedge, and its preimage remains a circle of corank one elliptic singularities.It is important to note that a mutation is only allowed when the nodal ray hits • the interior of an edge; or • a vertex which is the anchor of a nodal ray in the opposite direction.In the latter case, the marked points accumulate on the nodal ray. See, for example, thesequence of mutations described in Figure B.4 where many nodes have accumulated. Proposition 2.27.
Suppose that a symplectic manifold M is equipped with an almost toricfibration with base diagram ∆ M that consists of a closed region in R ≥ that is bounded bythe axes and a convex (piecewise-linear) curve from ( a, ) to ( , b ) , for a, b ∈ R + . Supposein addition that there is no nodal ray emanating from ( , ) . Then there exists a symplecticembedding of the ellipsoid ( − ε ) E ( a, b ) into M for any < ε < .Proof. The region ∆ M resembles Figure 2.28(a). We slide all nodes so that they are containedin small neighborhoods of the vertices from which their rays emanate. The neighborhoodsshould be sufficiently small so that they are disjoint from the triangle with vertices ( , ) , (( − ε ) ⋅ a, ) , and ( , ( − ε ) ⋅ b ) . The result now resembles Figure 2.28(b). × × × ×× × × ×× ××× ××× × × × × (a) (b) ( a, )( , b ) Figure 2.28.
Figure (a) is a base diagram satisfying the hypotheses of Proposi-tion 2.27. Figure (b) shows the new base diagram after nodal slides. The nodal raysare contained in the small disks indicated at the corresponding vertices.
We now remove the small disks from the base diagram to produce a non-compact regionΩ. We also remove the corresponding neighborhoods from M to produce a non-compactsymplectic manifold M Ω ⊂ M with a pair of Poisson-commuting Hamiltonian functions thathave only elliptic singularities. Thus, M Ω is actually a toric symplectic manifold. BecauseΩ is contractible, following Remark 2.25, M Ω is the unique toric symplectic manifold with his moment map image. The preimage of the origin is a fixed point. Because Ω containsthe dark, closed triangle in Figure 2.28(b) with vertices ( , ) , (( − ε ) ⋅ a, ) , and ( , ( − ε ) ⋅ b ) , the Local Normal Form theorem [28, Theorem B.3] now guarantees that for the fixed pointabove ( , ) , there is an equivariant neighborhood that is symplectomorphic to the closedellipsoid ( − ε ) ⋅ E ( a, b ) . This guarantees that for any ε >
0, there is a symplectic embedding ( − ε ) ⋅ E ( a, b ) s ↪ M Ω ⊂ M (centered at the fixed point), as desired. (cid:3) Passing to closed symplectic manifolds
We will see in this section how ellipsoid embeddings into compact target spaces, including C P blown up 0 to 4 times and C P × C P , are equivalent to ellipsoid embeddings into ap-propriate convex toric domains. We begin with the compact targets that are toric symplecticmanifolds, the context for Theorem 1.2. (3) (3;1) (4;2,2) (4;2,2) (3;1,1) (3;1,1)(3;1,1,1) (3;1,1,1) (3;1,1,1) (3;1,1,1) (3;1,1,1,1) (3;1,1,1,1)(3;1,1,1,1,1) (3;1,1,1,1,1) (3;1,1,1,1,1) (3;1,1,1,1,1,1) xx xxx xxx xx xxxx o ⃝ i ⃝ o ⃝ i ⃝ i ⃝ (3) (3;1) (4;2,2) (3;1,1) (3;1,1,1) (3;1,1,1,1) (a) (b) (c) (d) (e) Figure 3.1.
The regions in R ≥ that are Delzant polygons and whose convex toricdomains admit infinite staircases. These polygons correspond to (a) C P ; (b) C P C P ; (c) C P × C P ; (d) C P C P ; and (e) C P C P . Proof of Theorem 1.2. ( ⟸ ) First suppose we have an embedding E ( d, e ) s ↪ X Ω . The ellip-soid E ( d, e ) is an open ellipsoid, so the image of the symplectic embedding is containedin int ( X Ω ) . Because the Delzant polygon for M coincides with Ω, we have an inclu-sion int ( X Ω ) s ↪ M . Indeed, this is a symplectic embedding, so we may simply compose E ( d, e ) s ↪ int ( X Ω ) s ↪ M to get an embedding (1.3).( ⟹ ) For the other direction, suppose that M is a toric symplectic manifold whose momentmap image is a Delzant polygon (these are shown in Figure 3.1). Assume there is an em-bedding E ( d, e ) s ↪ M . To show that there is an embedding E ( d, e ) s ↪ X Ω , [6, Corollary 1.6]establishes that it is sufficient to produce embeddings of closed ellipsoids ( − ε ) E ( d, e ) s ↪ X Ω for any 0 < ε <
1. Given such an ε , we first choose d ′ , e ′ so that e ′ / d ′ is rational and ( − ε ) E ( d, e ) ⊂ E ( d ′ , e ′ ) ⊂ E ( d, e ) . In particular, because E ( d, e ) s ↪ M , there is also a symplectic embedding of the closed ellipsoid E ( d ′ , e ′ ) → M .A closed toric symplectic four-manifold M is either a product of two symplectic two-spheres, or can be obtained from C P by a series of equivariant blowups, see for example
27, Corollary 2.21]. In Figure 3.1, the square in (c) corresponds to S × S with symplecticform that has area 2 on each S . The polygons in Figure 3.1(a), (b), (d), and (e) arethe polygons for those that are equivariant blowups of C P . We consider these two casesseparately. Case 1: Blowups of C P . Assume first that M is obtained from C P by a series of equivariant symplectic blowups;as in the proof of [27, Corollary 2.21], these equivariant blowups correspond to corner chopson the polygon, resulting finally in Ω. As has been our convention, we may assume that wechoose the blowup vector for Ω with b as small as possible and the b i as large as possible ateach step, resulting in the negative weight expansion ( b ; b , . . . , b n ) (where here 0 ≤ n ≤ E ( d ′ , e ′ ) ⊂ M . Because e ′ / d ′ is rational, this ellipsoid has a finite weight expansion ( a , . . . , a m ) . We may use this weight expansion to blow up along that closed ellipsoid (asin [6, § C P n C P m C P . Specifically, we think of the first n C P factors as corresponding to the n blowups requiredto produce M , and we think of the remaining m factors as those required to blowup E ( d ′ , e ′ ) ;The symplectic form on (3.2) satisfies P D [ ω ] = bL − n ∑ i = b i E i − m ∑ j = a j E j and P D ( − c ( T M )) = − L + n ∑ i = E i + m ∑ j = E j . These two equations are analogues of [23, Equations [6] & [7]] (where we have normalizedthe line to have symplectic area b ).By [23, Proposition 6], having such a blowup symplectic form is equivalent to a symplecticembedding m ⨆ i = B ( a i ) ⊔ n ⨆ i = B ( b i ) s ↪ B ( b ) . This immediately implies that the open balls embed m ⨆ i = B ( a i ) ⊔ n ⨆ i = B ( b i ) s ↪ B ( b ) , which allows us to use [6, Theorem 2.1] to deduce that there is a symplectic embedding E ( d ′ , e ′ ) s ↪ X Ω , and hence the desired embedding ( − ε ) E ( d, e ) s ↪ X Ω exists. ase 2: M = C P × C P . If M is a product of two symplectic two-spheres, we use the trick that after performing asingle (arbitrarily small) blowup, we are back in Case 1. Using the same notation as before,we first find a small embedded B ( δ ) disjoint from the image of E ( d ′ , e ′ ) . Blow up along thisball, let F denote the homology class of the exceptional fiber, and let S and S denote thehomology classes of the spheres. There is a diffeomorphism from the resulting manifold ̂ M to C P C P mapping F ↦ L − E − E , S ↦ L − E , S ↦ L − E . This is described, for example, in [17]. The canonical class gets mapped − c ( T M ) ↦ − L + E + E and there is an embedding E ( d ′ , c ′ ) → ̂ M , and so we can repeat the argument from Case1 above. More precisely, if the spheres have areas b and b , respectively, then under thisdiffeomorphism the symplectic form on ̂ M induces a symplectic form on C P C P that isobtained from C P , normalized so that the line class has area b + b − δ , by blowups of size b − δ and b − δ . The triple ( b + b − δ ; b − δ, b − δ ) is the negative weight expansion for arectangle of side lengths b and b with its top right corner removed, so the argument fromCase 1 gives an embedding of E ( d, e ) into this toric domain, which in turn embeds into thetoric domain associated to a rectangle of side lengths b and b . (cid:3) Remark 3.3.
One of the toric domains, shown in Figure 3.4, is the image of an inte-grable system on a smooth, compact manifold P o(cid:96) ( , , , , ) that is not toric. Indeed, P o(cid:96) ( , , , , ) is known not to admit any Hamiltonian circle action [20, Theorem 3.2],even though P o(cid:96) ( − δ, + δ, , − δ, + δ ) is a toric symplectic manifold for any 0 < δ < P o(cid:96) ( , , , , ) .The integrable system in question is know as the “bending flow” on the polygon space. Thisintegrable system does come from a toric action on an open dense subset of P o(cid:96) ( , , , , ) :we must simply remove two Lagrangian S s that live above the points ( , ) and ( , ) inthe Figure 3.4. These Lagrangian S s are the loci of points where the “bending diagonals”vanish. The dense subset has moment image the polytope in Figure 3.4 with the two pointsremoved. The Local Normal Form theorem [28, Theorem B.3] now guarantees that therelevant int ( X Ω ) is in fact a subset of P o(cid:96) ( , , , , ) . This allows us to conclude that ifan ellipsoid E ( d, e ) s ↪ X Ω , it must also embed in P o(cid:96) ( , , , , ) .On the other hand, to prove that if E ( d, e ) s ↪ P o(cid:96) ( , , , , ) , it also embeds into X Ω ,we use the same argument in the Proof of Theorem 1.2, Case 1, because we may identify P o(cid:96) ( , , , , ) ≅ C P C P . (3) (3;1) (4;2,2) (4;2,2) (3;1,1) (3;1,1)(3;1,1,1) (3;1,1,1) (3;1,1,1) (3;1,1,1) (3;1,1,1,1) (3;1,1,1,1)(3;1,1,1,1,1) (3;1,1,1,1,1) (3;1,1,1,1,1) (3;1,1,1,1,1,1) xx xxx xxx xx xxxx o ⃝ i ⃝ o ⃝ i ⃝ i ⃝ (3) (3;1) (4;2,2) (3;1,1) (3;1,1,1) (3;1,1,1,1) Figure 3.4.
The image of the “bending flow” integrable system on equilateralpentagon space P o(cid:96) ( , , , , ) = C P C P . he second half of the argument in the proof of Theorem 1.2 also guarantees the following. Proposition 3.5.
Let ( b ; b , b , . . . , b n ) be a vector of non-negative integers that both repre-sents a blowup symplectic form on an n -fold blowup of projective space, M = C P b i C P b i ,and also is the negative weight expansion of a convex toric domain X Ω . Then E ( c, d ) s ↪ M ⟹ E ( c, d ) s ↪ X Ω . Remark 3.6.
The argument in the proof of Theorem 1.2 also implies that to producea symplectic embedding of E ( a, b ) into a convex toric domain X Ω with negative weightexpansion ( a + b ; a, b, c , c , . . . ) , it is enough to find an embedding into a closed symplecticmanifold that is obtained from C P a × C P b by symplectic blowups of size ( c , c , . . . ) . Indeed,just as in the proof of Case 2 above, we can find a small embedded B ( δ ) disjoint from theimage of E ( d ′ , c ′ ) , blow up, and then reduce to the case of blow-ups of C P . We now also give the promised proof of the fourth item in Proposition 2.1, which usessome of the same ideas in the of proof of Theorem 1.2 .
Proof of Proposition 2.1(4).
Recall from (2.14) that finding an embedding E ( , a ) s ↪ X Ω isequivalent to finding a ball-packing(3.7) n ⨆ i = B ( a i λ ) ⊔ N ⨆ j = B ( b j ) s ↪ B ( b ) . As in the proof of Theorem 1.2 and by the argument for [6, Corollary 1.6], in order to findan embedding (3.7), it suffices to find, for any 0 < ε <
1, an embedding(3.8) n ⨆ i = B (( − ε ) a i λ ) ⊔ N ⨆ j = B (( − ε ) b j ) s ↪ B ( b ) . We will find this embedding by looking at the closed symplectic manifold ( M, ω ) whichis the N -fold blowup of C P b with blowups of sizes ( − ε ) b j . By the strong packing sta-bility property [3, Theorem 1], there is some number δ associated to M such that the onlyobstruction to embedding any number of (open) balls of parameter less than δ is given bythe volume constraint. Now choose a sufficiently large, so that each a i λ above is smaller than δ , where λ = √ a vol ; we can do this, because each a i is bounded above by 1. Then, strongpacking stability applies to find an embedding of these balls into M ; we can then find anembedding of closed balls B (( − ε ) a i λ ) as well. As in the proof of Theorem 1.2 above, wecan then blow down to get an embedding of the desired form (3.8). (cid:3) Pinpointing the location of the accumulation point
In this Section we prove Theorem 1.10. We will first collect several equations below, withthe idea of highlighting how the accumulation point arises in the problem. We then completethe proof of the theorem, using the key equality (4.4). The result [6, Corollary 1.6] is stated for a single domain, but as was already observed by Gutt-Usher[18, §
3] the proof works just as well for disjoint unions. o prove Theorem 1.10, it is convenient to introduce further notation. Let a ≥ ( a , . . . , a n ) and X be a convex toric domain withnegative weight expansion ( b ; b , . . . , b N ) . Let also λ a = √ a vol . We introduce the vector w = ( λ a b , . . . , λ a b N , a , . . . , a n ) and use it to define the error vector (cid:15)(cid:15)(cid:15) following [35, (2.1.1)] by(4.1) m = dλ a b w + (cid:15)(cid:15)(cid:15), where ( d ; m ) is a class as in equation (2.15) satisfying (2.16) and (2.17). Furthermore, itcan be checked that ( d ; m ) satisfies (2.20) and is thus called an obstructive class if and onlyif the inner product(4.2) (cid:15)(cid:15)(cid:15) ⋅ w > . We can now derive the key equality (4.4) below, which highlights why the accumulationpoint arises in this context. We know that(4.3) − (cid:15)(cid:15)(cid:15) = dλ a b w − m . Let ( d ; m ) be an obstructive class and let s i denote the entries in w . Then combiningequation (4.3) with (2.16) gives − ∑ i (cid:15) i = dλ a b ( ∑ i s i ) − ( d − ) = + dλ a b (( ∑ i s i ) − λ a b ) . Using Lemma 2.13(3) and taking the absolute value of both sides, we can further rewrite theabove as(4.4) (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) − ∑ i (cid:15) i (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) = (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) + dλ a b ( a + + ( ∑ i λ a b i ) − λ a b − q )(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) . This is the genesis of the quadratic equation (1.11). Essentially, we would like to know when (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) a + + ( ∑ i λ a b i ) − λ a b − q (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) > , since this will eventually give us a bound on d , which will bound the number of obstructiveclasses and therefore the complexity of the graph of c X ( a ) . Intuitively, since q can be madearbitrarily large by small perturbation of a , the contribution of the term q is negligible, sothe interesting behavior is determined by (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) a + + ∑ i λ a b i − λ a b (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) . Now, to actually use (4.4) to bound d , we need a bound on ∣ ∑ i (cid:15) i ∣ . We get this by adaptinga strategy from McDuff-Schlenk, cf [35, Lemma 2.1.3]. roof of Theorem 1.10. Step 0: Ordering the class . We now assume here and below thatthe entries of m satisfy ˜ m i ≥ ˜ m j and m i ≥ m j , whenever i ≤ j . In other words, we willonly analyze classes m for which this property holds; we call such an m ordered . Themotivation for doing this is that if we have an arbitrary m , and we permute its entries tomake it ordered, then the left hand side of (2.20) for the permuted m will be at least asmuch as the value for the original m . Hence, in computing µ ( d ; m ) ( a ) , we can restrict toordered m . Step 1: A preliminary estimate.
The purpose of this step is to prove a basic, but veryimportant, estimate on any obstructive class, namely (4.5) below.Let m be an ordered obstructive class, and let p be the unique point from Proposition 2.23where (cid:96) ( p ) = (cid:96) ( m ) . Write p = p / q , where p and q are in lowest terms. Assume that p ≠ p must be 1 / q . Moreover, weknow that E ( , p ) is not a ball. Hence, the smallest weight of p must repeat at least twice.We now claim that we must have(4.5) dqλ a b > / . To see why, first note that by condition (2.17) and equation (4.1), we have d + = m ⋅ m = ( dλ a b w + (cid:15)(cid:15)(cid:15) ) ⋅ ( dλ a b w + (cid:15)(cid:15)(cid:15) ) = d λ a b w ⋅ w + dλ a b w ⋅ (cid:15)(cid:15)(cid:15) + (cid:15)(cid:15)(cid:15) ⋅ (cid:15)(cid:15)(cid:15). We know that w ⋅ w = a + λ a ( b − vol ) . Noting that λ a = a vol , this simplifies to w ⋅ w = λ a b ,and so d + = d + dλ a b w ⋅ (cid:15)(cid:15)(cid:15) + (cid:15)(cid:15)(cid:15) ⋅ (cid:15)(cid:15)(cid:15). Now recalling that (4.2) says w ⋅ (cid:15)(cid:15)(cid:15) >
0, we conclude that(4.6) ∑ i (cid:15) i < . Hence, in particular, each (cid:15) i must be less than 1. Remember now that we have m = ( ˜ m , . . . , ˜ m N , m , . . . , m n ) w = ( λ a b , . . . , λ a b N , a , . . . , a n ) where the entries in each box are in decreasing order, the m i are positive integers, and the a i are the weight expansion for a . In particular, we must have a n − = a n = q where a = pq in lowest terms. Thus, examining (cid:15)(cid:15)(cid:15) = m − dλ a b w , the last two entries are m n − − dλ a bq and m n − dλ a bq . Because (cid:15) i <
1, we must have m n − = m n =
1. If contrary to the assumption (4.5)we had dqλ a b ≤ , then each of these last two terms would be at least , and so we wouldconclude that ∑ i (cid:15) i ≥ / + / > , ontradicting (4.6). Step 2. The key estimate.
We can now prove a strong estimate on d , namely (4.8) below.We do this, using the estimate (4.5), as follows. Recall that − ∑ i (cid:15) i = + dλ a b ( a + + ( ∑ i λ a b i ) − λ a b − q ) . Let L be the length of the weight expansion of p , plus a finite number N of terms corre-sponding to the number of b i . Applying Cauchy-Schwarz to (cid:15)(cid:15)(cid:15) and the vector ( , . . . , ) oflength L , and using (4.6), we know that (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) ∑ i − (cid:15) i (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) < √ L. The triangle inequality guarantees that (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) − − ∑ i (cid:15) i (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) ≤ + (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) − ∑ i (cid:15) i (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) . We therefore get that (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) − − ∑ i (cid:15) i (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) = dλ a b ((cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) p + + ( ∑ i λ a b i ) − λ a b − q (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)) ≤ + √ L. We now want to bound L , using the fact that the length of p is bounded. It is a basic factabout weight expansions, see [35, Lemma 5.1.1], that the length of the weight expansion for p is bounded from above by q where p = pq in lowest terms. To simplify the notation, define(4.7) f ( a ) = a + + ( ∑ i λ a b i ) − λ a b = ( a + ) − ( λ a ⋅ ( b − ∑ i b i )) = a + − √ a ⋅ per vol . We note that f ( a ) = f ( a ) = ( a + + √ a ⋅ per vol ) ; we will use this fact below.We thus get dλ a b ((cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) f ( p ) − q (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)) ≤ + √ N + q. Rearranging (4.5), we have q < dλ a b . Hence, we get(4.8) dλ a b ((cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) f ( p ) − q (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187)) ≤ + √ N + dλ a b . The key point is now that if f ( p ) − / q is nonzero, then clearly there are only a finitenumber of d satisfying (4.8). (If p =
1, then this is still true, even though we assumed p ≠ tep 3. Capacity function at accumulation equals volume. Recall that a point a at which thegraph of c X is not smooth is called a singular point . Assume that there exists an infinitesequence of distinct singular points z , z , . . . . If a is sufficiently large, then c X ( a ) lies on thevolume curve by Proposition 2.1(4); hence, the z i must converge to some finite z ∞ ; we canassume that z ∞ ≠ z i for all i . We will eventually want to conclude that z ∞ = a where a isthe solution to (1.11).We begin with the following two observations, which we will use repeatedly in this stepand the next. We remind the reader, for motivation, that at any point p with c X ( p ) greaterthan the volume bound, the number c X ( p ) is the supremum of the obstructions over allobstructive classes, by Proposition 2.19.(1) Any obstructive class is obstructive on finitely many intervals, on which it is linear.(2) There are only finitely many obstructive classes with d less than any fixed number.The first observation holds because for any obstructive class ( d ; m ) , there are only finitelymany values a with (cid:96) ( a ) = (cid:96) ( m ) , and by Proposition 2.23 any such interval must have sucha point. For the second, we note that a bound on d bounds the individual entries ˜ m k and m k , as well as the total number of nonzero entries, as a result of (2.17). But we are assumingfrom Step 0 that our classes are ordered, so once d is bounded, there are only finitely manypossibilities.Now we show that c X ( z ∞ ) lies on the volume curve. Otherwise, by continuity, there issome neighborhood of z ∞ in which c X ( a ) is some uniformly bounded distance above thevolume curve. However, this cannot occur: in this neighborhood, any obstructive classwhose obstruction gives c X ( a ) must have a uniform bound on d , using (2.22). Hence the twoobservations above would apply to give a contradiction, since a finite number of obstructionssatisfying the conclusions of observation (1) could not generate the infinitely many singularpoints in this neighborhood. Step 4. Accumulation point must be a . With the key estimate (4.8), we can complete theproof of Theorem 1.10. We now assume that z ∞ ≠ a and, noting as above that a is a zeroof the function f from (4.7), we will derive a contradiction. We assume first that the z i areconverging to z ∞ from the left; the argument in the case where the z i are converging fromthe right will be essentially the same. We pass to a subsquence of z i that increase to z ∞ (from the left).Take a sequence of obstructive classes ( d i ; m i ) that are obstructive at points z ′ i withindistance ∣ z i − z ∞ ∣ i + of z i ; we know that such a sequence exists because otherwise c X ( a ) wouldlie on the volume curve on an open neighborhood of z i , and so z i would not be a singularpoint. In addition, choose the ( d i ; m i ) so that infinitely many of these ( d i ; m i ) are distinct.We know that we can do this, because otherwise only finitely many obstructive classes woulddetermine the behavior of c X in open neighborhoods of z i s, and by observation (1) abovethis could not generate infinitely many singular points. We again pass to a subsequence sothat all of the ( d i ; m i ) are distinct.Now for each ( d i , m i ) , let a i be the unique point corresponding to z ′ i with (cid:96) ( a i ) = (cid:96) ( m i ) ,whose existence is guaranteed by Proposition 2.23. We will show the following: it cannot bethe case that infinitely many a i ≥ z ′ i ; and, it cannot be the case that infinitely many a i < z ′ i .This will give the desired contradiction. ase 1: a i ≥ z ′ i We first establish a contradiction in the case where infinitely many of the a i satisfy a i ≥ z ′ i .Under this assumption, again pass to a subsequence so that all a i have this property.We must have a i < z ∞ , since c X ( a ) is obstructed on [ z ′ i , a i ] but c X ( z ∞ ) lies on the volumecurve. Thus, in this case, the a i must also be converging to z ∞ from the left. By ourassumption that z ∞ ≠ a is not a solution of the quadratic equation (1.11), we know that δ ∶ = ∣ f ( z ∞ )∣ >
0, where f is as in (4.7). There are only finitely many rational numbers p / q with 1 / q > δ /
4, so for sufficiently large i , (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) f ( a i ) − q i (cid:187)(cid:187)(cid:187)(cid:187)(cid:187)(cid:187) has a positive lower bound,independent of i , where a i = p i q i . Hence, by (4.8) there is therefore a uniform upper bound on d i across all ( d i , m i ) , hence only finitely many possible ( d i , m i ) , by the second observationabove. However, we are assuming that the ( d i , m i ) are all distinct, providing a contradiction. Case 2: a i < z ′ i We now establish a contradiction in the case where infinitely many of the a i satisfy a i < z ′ i .Under this assumption, pass again to a subsequence so that all a i have this property.Because a i < z ′ i , the function µ ( d i , m i ) ( a ) is linear on the maximal interval [ z ′ i , z ∗ i ) on which ( d i , m i ) is obstructive. These points z ∗ i satisfy z ∗ i ≤ z ∞ , because we showed above that c X ( z ∞ ) lies on the volume curve and we know that z ′ i < z ∞ . As the z i are converging to z ∞ ,it then follows that the z ∗ i must be as well; it therefore follows that the slope of the volumecurve at z ∗ i is converging to the slope of the volume curve at z ∞ . Now the line segment from ( a i , µ ( d i , m i ) ( a i )) to ( z ∗ i , √ z ∗ i vol ) lies above the volume curve. So this line segment must also beabove the tangent line to the volume curve at z ∗ i on the interval ( a i , z ∗ i ) , because the volumecurve is concave. Regarding the points a i , we now split into two subsequences, one wherethe a i are uniformly bounded away from z ∞ and the other where the a i converge to z ∞ . Ona subsequence of the a i which are uniformly bounded away from z ∞ , we must have a uniformbound on d i across the ( d i , m i ) , by (2.22). This follows because the length of the interval ( a i , z ∗ i ) is also uniformly bounded from below, so (cid:187)(cid:187)(cid:187)(cid:187)(cid:187) µ ( d i , m i ) ( a i ) − √ a i vol (cid:187)(cid:187)(cid:187)(cid:187)(cid:187) is uniformly boundedfrom below as well, as a consequence of the upper bound on the slope of the line segmentdescribed above. On the other hand, on any subsequence of the a i converging to z ∞ , we mustalso have a uniform bound on d i , by the same argument as in the case where a i ≥ z i . Thus,in both cases, the uniform bounds on the d i mean we have only finitely many obstructiveclasses by observation (2) above; but we are assuming that the ( d i , m i ) are distinct, whichis a contradiction.When the z i are converging to z ∞ from the right, we can argue completely analogously:when a i < z ′ i , a i is sandwiched between z ∞ and z i , and when a i > z ′ i , we can repeat theargument from Case 2 above. (cid:3) Remark 4.9.
It would be interesting to understand whether an analogue of Theorem 1.10still holds, without the assumption of finitely many b i ; this could be useful for understandingembeddings into an irrational ellipsoid, for example. Most of the above argument shouldgo through, except that now the number N in (4.8) would be infinite. It is neverthelessplausible that there is a way around this. emark 4.10. We could alternatively think about Theorem 1.10 from the point of view ofthe ECH capacities reviewed in § E (√ vol a , √ a vol ) into X . If we assume that a is irrational, and setthe perimeter of the domain and the target equal to each other, we get the equation(4.11) √ vol a + √ a vol = per , which can be rearranged to (1.11).There is in turn a heuristic for why (4.11) is natural to consider in view of the questionof finding infinite staircases for this problem from the point of view of ECH capacities.The justification for normalizing the volumes to be equal is as follows: by packing stability(Proposition 2.1(4)), an infinite staircase must accumulate at some point s . It’s not hardto show in addition that the embedding function at s must lie on the volume curve, as inStep 3 of the Proof of Theorem 1.10 above.Now, it has been shown [12, Theorem 1.1] that for the manifolds we consider here, asymp-totically ECH capacities recover the volume; moreover, the subleading asymptotics haverecently been studied, see for example [15, Theorem 3], and in the present situation thesenext order asymptotics are well-understood as well. These can be interpreted as recoveringthe perimeter (see [15, Proposition 16]). These asymptotics dominate when we normalizethe leading asymptotics, which are the volume.With all of this understood, here is the promised heuristic: if the subleading asymptoticsof the domain are larger than the subleading asymptotics of the target (which happens when s irrational is smaller than the solution to (4.11)), then no volume preserving embeddingcan exist. On the other hand, if the subleading asymptotics of the domain are smaller thanthe subleading asymptotics of the target (which happens when s irrational is larger thanthe solution to (4.11)), then only finitely many ECH capacities can give an obstruction, andthese are not enough to generate an infinite staircase. Thus, the only possibility is that theaccumulation point is actually given by the relevant solution to (4.11).Note, however, that this is quite different than the proof we give above for Theorem 1.10.It is easy to make the heuristic above rigorous concerning the case where the subleadingasymptotics of the domain are smaller than the subleading asymptotics of the target; but tomake the other case rigorous, one would want a uniform bound on the maximal number ofobstructive ECH capacities close to s ; it might be possible to get this, but it is potentiallydelicate. Another issue is that if a is rational instead of irrational, then the perimeter of thedomain is different than what is said above, so (4.11). This is why we give a rather differentargument, inspired by the work of McDuff and Schlenk in [35].We now also give the promised proof of the fifth item in Proposition 2.1, which uses someof the same ideas as in the proof of Theorem 1.10. Proof of Proposition 2.1 (5) . Let ˜ a be a point which is not a limit of singular points. Then,there is some open interval I = ( m, n ) containing ˜ a on which the only possible singular pointof c X ( a ) is ˜ a itself. If c X ( a ) is equal to the volume obstruction on I , then the conclusion ofthe proposition holds near ˜ a . Thus we can assume there is some point y in I on which c X ( y ) is strictly greater than the volume obstruction; without loss of generality, assume that y < ˜ a . s in Step 3 of the proof of Theorem 1.10 above, there is now some subinterval I ′ ⊂ I ,containing y , on which c X ( a ) is the supremum of finitely many obstructive classes, each ofwhich is piecewise linear on I ′ , with at most one singular point. It follows that c X is piecewiselinear on I ′ ; since ˜ a is the only possible singular point of c X ( a ) on I , it follows that in fact c X ( a ) is linear on ( m, ˜ a ] . We now apply the same argument to the interval ( ˜ a, n ) . Namely, if c X ( a ) is the volume on ( ˜ a, n ) , then the conclusion of the proposition holds near ˜ a , so we aredone. Otherwise, we can assume there is some point y ′ in ( ˜ a, n ) such that c X ( y ′ ) is strictlygreater the volume obstruction. Then, as in the y < ˜ a case, c X ( a ) is linear on [ ˜ a, n ) , asdesired. (cid:3) The existence of the Fano staircases
To prove Theorem 1.14, we begin by showing that the purported x -coordinates intertwine: x out n < x in n < x out n + . We then take a limit as n → ∞ , verifying that the x -coordinates x out n tendto a (and therefore also x in n tend to a as well) and y out n = y in n tend to √ a vol . Next we showthat y out n ≤ c X ( x out n ) and that c X ( x in n ) ≤ y in n . For the first inequality, we find an obstruction,and for the second, we produce an explicit embedding. Finally, we use the fact that c X ( a ) is continuous, non-decreasing, and has the scaling property to conclude that the graph ofthe function must consist of line segments alternately joining points of the two sequences ( x in n , y in n ) and ( x out n , y out n ) , and that these line segments alternate: some are horizontal and theothers, when extended to be lines, pass through the origin. This is illustrated in Figure 5.1. (3) (3;1) (4;2,2) (4;2,2) (3;1,1) (3;1,1)(3;1,1,1) (3;1,1,1) (3;1,1,1) (3;1,1,1) (3;1,1,1,1) (3;1,1,1,1)(3;1,1,1,1,1) (3;1,1,1,1,1) (3;1,1,1,1,1) (3;1,1,1,1,1,1) xx xxx xxx xx xxxx o ⃝ i ⃝ o ⃝ i ⃝ i ⃝ Figure 5.1.
The inner corners are marked i ○ and the outer corners are marked o ○ .The exed out lines represent the inequalities y out n ≤ c X ( x out n ) and c X ( x in n ) ≤ y in n . Theproperties of the embedding capacity function then imply that its graph consists ofline segments between the corners. Before we begin, it will be convenient to catalogue certain combinatorial identities thathold for our sequences. These will be essential for the inductive proofs that follow.
Lemma 5.2.
Let X be a convex toric domain with blowup vector ( b ; b , . . . , b n ) equal to ( ) , (
3; 1 ) , (
3; 1 , ) , (
3; 1 , , ) , (
3; 1 , , , ) , or (
4; 2 , ) . When J = , that is, for the sequences with recurrence relation g ( n + ) = Kg ( n + ) − g ( n ) , the following identities hold: ( ♣ ) g ( n ) + g ( n + ) = β n + g ( n + ) ♦ ) g ( n ) + g ( n + ) − Kg ( n ) g ( n + ) = − αβ n + , ( ♥ ) g ( n ) g ( n + ) = g ( n + ) g ( n + ) + α where K = vol − , the sequence seeds, α , and β n are: Blowup vector K Seeds α β n ( ) , , , (
4; 2 , ) , , , ⎧⎪⎪⎪⎨⎪⎪⎪⎩ , n odd , n even (
3; 1 , , ) , , , ⎧⎪⎪⎪⎨⎪⎪⎪⎩ , n odd , n even (
3; 1 , , , ) , , , ⎧⎪⎪⎪⎨⎪⎪⎪⎩ , n odd , n even When J = , that is, for the sequences with recurrence relation g ( n + ) = Kg ( n + ) − g ( n ) ,the following identities hold: ( ♣ for (3;1)) g ( n ) + g ( n + ) = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ g ( n + ) + g ( n + ) , n ≡ g ( n + ) + g ( n + ) , n ≡ g ( n + ) + g ( n + ) , n ≡ ♣ for (3;1,1)) g ( n ) + g ( n + ) = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ g ( n + ) + g ( n + ) , n ≡ g ( n + ) + g ( n + ) , n ≡ g ( n + ) + g ( n + ) , n ≡ ♦ ) g ( n ) + g ( n + ) − Kg ( n ) g ( n + ) = − β n + , ( ♥ . g ( n ) g ( n + ) = g ( n + ) g ( n + ) + δ n ( ♥ . g ( n ) g ( n + ) = g ( n + ) g ( n + ) + µ n where K = vol − , the sequence seeds, β n , δ n and µ n are: Blowup vector K Seeds β n δ n µ n (
3; 1 ) , , , , , ⎧⎪⎪⎪⎨⎪⎪⎪⎩ , n ≡ , n ≡ , ⎧⎪⎪⎪⎨⎪⎪⎪⎩ , n ≡ , , n ≡ (
3; 1 , ) , , , , , ⎧⎪⎪⎪⎨⎪⎪⎪⎩ , n ≡ , n ≡ , ⎧⎪⎪⎪⎨⎪⎪⎪⎩ , n ≡ , , n ≡ Proof.
These identites can be proved by induction for each congruence class of n . For the J = β n β n + = vol = K + (cid:3) e now use the identities in Lemma 5.2 to establish the relationships among the the x -and y -coordinates of purported corners of the ellipsoid embedding functions. Proposition 5.3.
The recurrence relations above define inner and outer corners respectivelywith coordinates: ( x in n , y in n ) = ( g ( n + J ) ( g ( n + ) + g ( n + + J ))( g ( n ) + g ( n + J )) g ( n + ) , g ( n + J ) g ( n ) + g ( n + J ) ) , ( x out n , y out n ) = ( g ( n + J ) g ( n ) , g ( n + J ) g ( n ) + g ( n + J ) ) . These coordinates satisfy:(1) x out n < x in n < x out n + ;(2) lim n → ∞ x out n = lim n → ∞ x in n = a ; and(3) lim n → ∞ y out n = lim n → ∞ y in n = √ a vol .Proof. For (1): Both inequalities boil down to showing that g ( n + ) g ( n + J ) < g ( n ) g ( n + J + ) , which follows immediately from the identities ( ♥ ) in Lemma 5.2.For (2): In view of (1), it suffices to show that lim n → ∞ x out n = a . The linear recurrence relation g ( n + J ) = Kg ( n + J ) − g ( n ) is of order 2 J but can be replaced by J linear recurrence relations of order 2, one for eachof the J subsequences of g ( n ) with n ≡ j (mod J ), for j = , , . . . , J −
1. Each of thesesubsequences has the recurrence relation(5.4) g j ( n + ) = Kg j ( n + ) − g j ( n ) , where g j ( n ) = g ( J n + j ) .We can get a closed form for g j ( n ) by solving the polynomial equation λ = Kλ −
1. Let λ , λ be the roots of this equation, we note that in each of the cases we are considering wehave λ > > λ >
0. Then for appropriate coefficients D j , E j depending on the seed of thesequences,(5.5) g j ( n ) = D j λ n + E j λ n . For each j = , , . . . , J − n → ∞ g j ( n + ) g j ( n ) = lim n → ∞ D j λ n + + E j λ n + D j λ n + E j λ n = λ . Noting that a is exactly λ , the larger solution of λ − Kλ + =
0, we conclude as desiredthat lim n → ∞ x n = λ = a . Finally, for (3): In view of the fact that y out n = y in n , it suffices to show thatlim n → ∞ y out n = √ a vol . ndeed we have lim n → ∞ y out n = lim n → ∞ ( x out n + ) = a + = √ vol a , the last equality uses the facts that a − Ka + = = K +
2. This completes theproof. (cid:3)
Next, we show that at the outer corners, the ellipsoid embedding function is indeed ob-structed in all of our six examples.
Proposition 5.6.
Let X be a convex toric domain whose blowup vector ( b ; b , . . . , b n ) is ( ) , (
3; 1 ) , (
3; 1 , ) , (
3; 1 , , ) , (
3; 1 , , , ) , or (
4; 2 , ) . For each ( x out n , y out n ) = ( g ( n + J ) g ( n ) , g ( n + J ) g ( n ) + g ( n + J ) ) , we have y out n ≤ c X ( x out n ) .Proof. Recall that N ( a, b ) k denotes the k th ECH capacity of E ( a, b ) and c k ( X ) denotes the k th ECH capacity of X . Let k n ∶ = ( g ( n ) + )( g ( n + J ) + ) −
1. If we prove that(5.7) g ( n + J ) ≤ N ( , x out n ) k n and g ( n ) + g ( n + J ) ≥ c k n ( X ) , then we have the desired inequality: y out n = g ( n + J ) g ( n ) + g ( n + J ) ≤ N ( , x out n ) k n c k n ( X ) ≤ sup k N ( , x out n ) k c k ( X ) = c X ( x out n ) . The first part of (5.7) can be rewritten as g ( n ) g ( n + J ) ≤ N ( g ( n ) , g ( n + J )) k n , and by Proposition 2.3 it is indeed true that there are at most k n terms of the sequence N ( g ( n ) , g ( n + J )) strictly smaller than g ( n ) g ( n + J ) .Next, we tackle the second part of (5.7): g ( n ) + g ( n + J ) ≥ c k n ( X ) . By Theorem 2.12, it suffices to find a convex lattice path Λ n that encloses k n + g ( n ) + g ( n + J ) :(5.8) L ( Λ n ) = ( g ( n ) + )( g ( n + J ) + ) (cid:96) Ω ( Λ n ) = g ( n ) + g ( n + J ) . We do this separately for each of the six cases under consideration in Appendix A.The convex lattice paths Λ n for each case (and sub-case) can be found in Figure A.1, whilethe formulæ for s n and t n are provided in (A.2). Using the identities ( ♦ ) in Lemma 5.2 andinduction we conclude that s n and t n are indeed integers.Formulæ for the number of lattice points L ( Λ n ) enclosed by the path and its Ω-length (cid:96) Ω ( Λ n ) are provided in Table A.5 for each case and sub-case. As discussed after Table A.5,it is a computational check that these give the correct numbers that satisfy (5.8). (cid:3) Next, we show that at the inner corners, there are explicit ellipsoid embeddings realizingthe purported value of the ellipsoid embedding function. We do this by exploring recursivefamilies of ATFs, following a suggestion of Casals. The idea that the recurrence sequencesinvolved in the coordinates of the corners of the infinite staircases may be related to theMarkov-type equations that show up when performing ATF mutations was first mentioned o us by Smith and is studied in detail by Maw for symplectic del Pezzo surfaces [31]. Thisprocedure is explained nicely in Evans’ lecture notes [16, Example 5.2.4]. We use a seriesof mutations first described by Vianna [41, §
3] on the compact manifolds corresponding toour blowup vectors with J =
2. The J = Proposition 5.9.
Let X be a convex toric domain whose blowup vector ( b ; b , . . . , b n ) is ( ) , (
3; 1 ) , (
3; 1 , ) , (
3; 1 , , ) , (
3; 1 , , , ) , or (
4; 2 , ) . For each ( x in n , y in n ) = ( g ( n + J )( g ( n + ) + g ( n + + J ))( g ( n ) + g ( n + J )) g ( n + ) , g ( n + J ) g ( n ) + g ( n + J ) ) , there is a symplectic embedding (5.10) E ( , x in n ) s ↪ y in n X, which forces c X ( x in n ) ≤ y in n .Proof. We use Theorem 1.2 and Proposition 2.27 to prove that there is an embedding(5.11) E ( g ( n ) + g ( n + J ) g ( n + J ) , g ( n + ) + g ( n + + J ) g ( n + ) ) s ↪ X, which is equivalent to (5.10). By definition of c X ( a ) , this implies the desired inequality.The proof consists of applying successive mutations to base diagrams, beginning with aDelzant polygon. This allows us to use Proposition 2.27 to find ellipsoids embedded incompact manifolds. Theorem 1.2 then allows us to deduce that those ellipsoids must also beembedded in the corresponding convex toric domain. Since two convex toric domains withthe same blowup vectors have identical ellipsoid embedding functions (see Remark 2.8), itsuffices to exhibit the embeddings for one convex toric domain per blowup vector. We musttake particular care with the blowup vector (
3; 1 , , , ) , making use of Remark 3.3.We begin by producing ATFs on the compact manifolds M corresponding to our blowupvectors. The manifolds are C P ; C P C P ; C P C P ; C P C P ; C P C P ; and C P × C P . Except for C P C P , these manifolds may be endowed with toric actions. The corre-sponding Delzant polygons are displayed in Figure 3.1. Our first step is to apply mutationsto the Delzant polygons to produce a base diagram that is a triangle with two nodal rayswhen J = J =
3. For C P C P , we useVianna’s trick [41, § C P C P given in Figure B.4(e). This ATF has a smooth toric cornerat the origin where we may perform a toric blowup of symplectic size 1, resulting in an ATFon C P C P . These initial maneuvers are described in Appendix B and the results areshown in Figure 5.12. × × × × × × × × ×× × × × × × × × × × × ×× C P C P C P C P C P C P C P C P C P C P × C P Figure 5.12.
The base diagrams for ATFs on our manifolds. These are a trianglewith two nodal rays when J = J = We now want to show that for any 0 < ε < ( − ε ) ⋅ E ( g ( n ) + g ( n + J ) g ( n + J ) , g ( n + ) + g ( n + + J ) g ( n + ) ) s ↪ M, where M is the compact manifold from our list. We achieve this by showing that the basediagram obtained at each additional mutation contains the triangle with vertices ( , ) , ( g ( n ) + g ( n + J ) g ( n + J ) , ) , ( , g ( n + ) + g ( n + + J ) g ( n + ) ) . We proceed by induction. In Table 5.14, we record theadditional data we will need for our recursive mutation procedure. lowup vector J σ n ( ) (
3; 1 , , ) { , n odd3 , n even (
3; 1 , , , ) { , n odd5 , n even (
3; 1 ) (
3; 1 , ) { , n ≡ , n ≡ , (
4; 2 , ) { , n odd2 , n even Table 5.14.
Additional data, by blowup vector.
We now treat separately the cases where J = J =
3, starting with J =
2. Inthis case, starting with base diagrams in Figure 5.12 and continuing to apply mutations, allfurther base diagrams will be triangles. The induction hypothesis is that the triangle ∆ n hasside lengths a n , b n , c n , nodal rays v n and u n and hypotenuse direction vector w n as shown inFigure 5.15(a) , and that the matrix that takes ∆ n to ∆ n + is M n : v n = ( g ( n + ) − g ( n + ) ) , u n = ( − g ( n ) g ( n + ) ) , w n = ( σ n + g ( n + ) − σ n + g ( n + ) ) ,a n = g ( n + ) + g ( n + ) g ( n + ) , b n = g ( n ) + g ( n + ) g ( n + ) , and M n = ⎛⎜⎝ − σ n + g ( n + ) − σ n + g ( n + ) σ n + g ( n + ) + σ n + g ( n + ) ⎞⎟⎠ , where σ n is as in Table 5.14. × × ×× × ××× ×× × × × × v n u n w n a n b n c n (a) × × × ×× × ××× ×× × × × × u n v n w n s n r n a n b n c n d n (b) Figure 5.15.
The general base diagrams (a) ∆ n for the cases when J =
2; and (b) □ n for the cases when J = The base case is immediate from Figure 5.12. For the induction step, we must check firstthat the matrix M n is indeed performing the mutation from ∆ n to ∆ n + , that is:(1) M n v n = v n ,(2) M n w n = ( ) , and(3) det ( M n ) = n + :(4) w n + = M n ( − ) ,(5) v n + = M n u n ,(6) u n + = − v n ,(7) a n + = a n + c n ,(8) b n + = a n st entry of v n nd entry of v n , and(9) c n + = b n − b n + .The proof of these uses the identities in Lemma 5.2. Finally, we note that at each step,the base diagram ∆ n is exactly the triangle with vertices ( , ) , ( g ( n ) + g ( n + ) g ( n + ) , ) = ( b n , ) and ( , g ( n + ) + g ( n + ) g ( n + ) ) = ( , a n ) , which is what we wanted to prove.Next we tackle the J = □ n : u n = ( − g ( n ) g ( n + ) ) , w n = ( g ( n + ) − g ( n + ) ) , s n = ( σ n + g ( n + ) − σ n + g ( n + ) g ( n + ) ) , r n = ( σ n g ( n ) g ( n + ) − − σ n + g ( n + ) ) ,a n = g ( n + ) + g ( n + ) g ( n + ) , b n = g ( n ) + g ( n + ) g ( n + ) , nd M n = ⎛⎜⎝ − σ n + g ( n + ) g ( n + ) − σ n + g ( n + ) σ n + g ( n + ) + σ n + g ( n + ) g ( n + )⎞⎟⎠ , where σ n is again as in Table 5.14.Performing a mutation on □ n uses the matrix M n and yields □ n + . The matrix M n satisfies(1) M n w n = w n (2) M n r n = ( ) (3) det ( M n ) = □ n + is obtained via(4) r n + = M n s n (5) s n + = M n ( − ) (6) v n + = M n u n and w n + = M n v n , or simply w n + = M n M n − u n − (7) u n + = − w n (8) a n + = a n + d n (9) d n + = c n (10) b n + = − a n st entry of w n nd entry of w n (11) c n + = b n − b n + .The proof of these relations uses the identities in Lemma 5.2. Finally, we note that thetriangle with vertices ( , ) , ( g ( n ) + g ( n + ) g ( n + ) , ) = ( b n , ) and ( , g ( n + ) + g ( n + ) g ( n + ) ) = ( , a n ) fits inthe base diagram □ n for each n , which is what we wanted to prove.The ATFs described above and Proposition 2.27 allow us to conclude that we have thedesired embeddings (5.13) with target the compact manifold M . We now argue that there arealso such embeddings with target a convex toric domain. Suppose that X is the convex toricdomain with the same blowup vector as M . By Theorem 1.2, we then have an embedding ( − ε ) ⋅ E ( g ( n ) + g ( n + J ) g ( n + J ) , g ( n + ) + g ( n + + J ) g ( n + ) ) s ↪ X for every 0 < ε <
1. Now note that ( − ε ) ⋅ E ( a, b ) ⊂ ( − ε ) ⋅ E ( a, b ) , so we may conclude that we have symplectic embeddings of the closed ellipsoids ( − ε ) ⋅ E ( g ( n ) + g ( n + J ) g ( n + J ) , g ( n + ) + g ( n + + J ) g ( n + ) ) s ↪ X for every 0 < ε <
1. We may now apply [6, Cor. 1.6] to deduce that there is a symplecticembedding of the form (5.11), as desired. In fact, we may also apply Theorem 1.2 one moretime to deduce that E ( g ( n ) + g ( n + J ) g ( n + J ) , g ( n + ) + g ( n + + J ) g ( n + ) ) s ↪ M. Thus we have shown that there are ellipsoid embeddings representing the purported interiorcorners. (cid:3)
We now have all of the ingredients in place to complete the proof that the Fano infinitestaircases exist. roof of Theorem 1.14. We know from Propositions 5.6 and 5.9 that y out n ≤ c X ( x out n ) and c X ( x in n ) ≤ y in n .Now we use Proposition 5.3. Since x out n < x in n and y out n = y in n , because c X ( a ) is continuousand non-decreasing, it must be constant and equal to y out n between x out n and x in n . Furthermore,since x in n < x out n + and the points ( , ) , ( x in n , y in n ) and ( x out n + , y out n + ) are colinear, the scalingproperty of c X ( a ) implies that between each x in n and x out n + , the graph of c X ( a ) consists of astraight line segment (which extends through the origin). We thus have an infinite staircasein each of the cases studied.Finally, by continuity and because x out n → a , x in n → a , and y out n = y in n → √ a vol as n → ∞ ,we know that the infinite staircase accumulates from the left at ( a , c X ( a ) = √ a vol ) , whichcompletes the proof of the theorem. (cid:3) Remark 5.16.
One must take care to interpret the base diagrams in Figure 5.12 correctly.These represent almost toric fibrations on smooth manifolds, not moment map images oftoric orbifolds.6.
Conjecture: why these may be the “only” infinite staircases
Each of the convex toric domains in Theorem 1.14 has a finite blowup vector with allinteger entries. A toric domain with this property is called rational . In this section wedescribe some evidence towards Conjecture 1.18, which speculates that the only rationalconvex toric domains that admit an infinite staircase are those whose moment polygon, upto scaling, is
AGL ( Z ) -equivalent to one in Figure 1.9.In light of Usher’s work [40] about infinite staircases for irrational polydisks P ( , b ) , it iscrucial in the conjecture that the toric domain be rational. Let X Ω be a rational convex toricdomain with negative weight expansion ( b ; b , b , . . . , b n ) . The ellipsoid embedding functionof the scaling of a convex toric domain is a scaling of the ellipsoid embedding function of theoriginal domain. Thus, we may assume that the negative weight expansion of X Ω is integral.By Theorem 1.10, if the ellipsoid embedding function of X Ω has an infinite staircase, then c X Ω ( a ) = √ a vol . This implies that E ( , a ) s ↪ √ a vol X Ω , which by Proposition 2.7 and conformality of ECH capacities is equivalent to an inequalityof sequences of ECH capacities:(6.1) c ECH ( E (√ vol a , √ a vol )) ≤ c ECH ( X Ω ) . To rewrite this inequality we introduce the cap function of a convex toric domain X , for T ∈ N :(6.2) cap X ( T ) ∶ = { k ∶ c k ( X ) ≤ T } . With u = √ vol a and v = √ a vol, the inequality (6.1) is equivalent to:(6.3) cap E ( u,v ) ( T ) ≥ cap X Ω ( T ) , for all T ∈ N . e first look at the right hand side of inequality (6.3). In [42, Lemma 5.9], Wormleightonproves that if X Ω belongs to a certain class of convex toric domains, then its cap function iseventually equal to a quasipolynomial:cap X Ω ( T ) = T + per2vol T + Γ r , where r ∈ { , . . . , vol − } is a congruence class of T (mod vol) and each Γ r is a constant.Furthermore, [42, Lemma 5.6] states that if one of the b i ’s in the negative weight expansionof X Ω is equal to 1 then X Ω is in that class, and [42, Conjecture 5.7] hypothesizes that if gcd ( b, b , . . . , b n ) = E ( u, v ) , thecap function equals the Ehrhart function of the triangle ∆ u,v , with vertices ( , ) , ( / u, ) , and ( , / v ) : cap E ( u,v ) ( T ) = { k ∶ c k ( E ( u, v )) ≤ T }) = { k ∶ N ( u, v ) k ≤ T } = ( T ∆ u,v ∩ Z ) = ehr ∆ u,v ( T ) . In the particular case of the ellipsoid in (6.3), since a is irrational and vol is not, both u = √ vol a and v = √ a vol are irrational, and their ratio is irrational as well. We thus satisfythe conditions of [14, Lemma 2], which allows us to write:cap E ( u,v ) ( T ) = ehr ∆ ( u,v ) = T + per2vol T + d ( T ) , where d ( T ) is asymptotically o ( T ) . We use here the fact that (1.11) implies(6.4) 1 u + v = pervol . Based on experimental evidence and [19], we conjecture that unless the term d ( T ) isperiodic, its asymptotic behavior is actually O ( ± log ( T )) , and therefore d ( T ) is unboundedabove and below. This unboundedness would then imply that (6.3) holds exactly whenehr ∆ ( u,v ) is a quasipolynomial. Following [14, Theorem 1(i)] and using (4.11) and (6.4), theEhrhart function ehr ∆ ( u,v ) is a quasipolynomial if and only if both pervol and per vol are in N .Consider now the scaled convex toric domain ̃ X = pervol X Ω , whose corresponding region ̃ Ωis a scaling of the original Ω by the same factor pervol . The region ̃ Ω is still a lattice polygon,with area = ̃ vol2 = per vol and = ̃ per = per vol , and by Pick’s Theoremarea = + − ̃ Ω has exactly one interior lattice point: that is, it is a reflexive polygon. p to AGL ( Z ) equivalence, there are exactly sixteen reflexive polygons: the twelve inFigure 1.9 plus the four in Figure 6.5, with reduced blow-up vectors (
3; 1 , , , , ) and (
3; 1 , , , , , ) . Solving the quadratic equation (1.11) in these two cases, we obtain a = a ∈ C respectively, and plotting the corresponding embedding capacity functions weconfirm that they do not have infinite staircases. (3) (4;2,2)(3;1) (4;2,2) (3;1,1) (3;1,1)(3;1,1,1) (3;1,1,1) (3;1,1,1) (3;1,1,1,1) (3;1,1,1,1)(3;1,1,1)(3;1,1,1,1,1) (3;1,1,1,1,1) (3;1,1,1,1,1,1)(3;1,1,1,1,1) Figure 6.5.
The remaining four reflexive polygons. ppendix A. Lattice Paths
In this appendix, we compile the combinatorial data we need for Proposition 5.6. } s n (3) } (4;2,2) n odd } } t n (4;2,2) n even } t n - } t n } s n (3;1,1,1,1) } t n } (3;1,1) } t n } s n (mod 3) n ≡ t n } (3;1,1) } t n } s n (mod 3) n ≡ t n - } s n } } } t n t n t n (3;1,1,1) } t n } n odd t n } t n s n } (3;1,1,1) } t n } t n t n + s n } } (mod 4) n ≡ t n } t n } s n } (3;1,1,1) } (mod 4) n ≡ t n - s n } s n } t n t n Figure A.1.
The lattices paths Λ n . All triangles drawn with size s n or t n on oneof their sides are right-angle equilateral triangles (in the “length” that counts latticepoints). The sequences s n and t n that determine the paths Λ n have formulæ in terms of theirblowup vectors. In all six cases, the blowup vector is of the form ( B ; b, . . . , b ) . In terms ofthose B and b , we have(A.2) s n = B ⋅ ( g ( n ) + g ( n + J )) + c n vol and t n = b ⋅ ( g ( n ) + g ( n + J )) + d n vol , where c n and d n are given in Table A.4. We also have, in terms of B , b , and k the numberof b s,(A.3) (cid:96) Ω ( Λ n ) = B ⋅ s n + kb ⋅ t n + e n . The e n are given explicitly in Table A.4 and implicitly in Table A.5. B ; b, . . . , b ) vol c n d n e n ( ) ∄ d n (
4; 2 , ) n even0 n odd 2 n even0 n odd (
3; 1 ) n ≡ − n ≡ n ≡ − n ≡ n ≡ − n ≡ n ≡ − n ≡ n ≡ − n ≡ n ≡ − n ≡ (
3; 1 , ) n ≡ − n ≡ n ≡ − n ≡ n ≡ − n ≡ n ≡ − n ≡ n ≡ n ≡ n ≡ − n ≡ n ≡ n ≡ , (
3; 1 , , ) n ≡ n ≡ n ≡ − n ≡ − n ≡ n ≡ n ≡ − n ≡ − n ≡ n ≡ , n ≡ (
3; 1 , , , ) n ≡ n ≡ − n ≡ n ≡ n ≡ n ≡ − n ≡ n ≡ Table A.4.
The constants c n and d n used in the formulæ in (A.2), by blowupvector; and the constants e n for (A.3). B ; b, . . . , b ) L ( Λ n ) (cid:96) Ω ( Λ n )( ) ( s n + )( s n + ) s n (
4; 2 , ) ( s n + )( s n + ) − t n ( t n + ) − t n ( t n − ) n even ( s n + )( s n + ) − ⋅ t n ( t n + ) n odd 2 ( s n − ( t n − )) + ( s n − t n ) n even2 ( s n − t n ) + ( s n − t n ) n odd (
3; 1 ) ( s n + )( s n + ) − t n ( t n + ) ( t n ) + ( s n − t n )(
3; 1 , ) ( s n + )( s n + ) − ⋅ t n ( t n + ) n ≡ ( s n + )( s n + ) − t n ( t n + ) − t n ( t n − ) n ≡ , t n + ( s n − t n ) + t n n ≡ t n + ( s n − t n + ) + ( t n − ) n ≡ , (
3; 1 , , ) ( s n + )( s n + ) − t n ( t n + ) − ( t n + )( t n + ) n ≡ ( s n + )( s n + ) − t n ( t n + ) n ≡ , ( s n + )( s n + ) − t n ( t n + ) − ( t n − ) t n n ≡ t n + ( s n − t n − ) + ( t n + ) n ≡ t n + ( s n − t n ) + t n n ≡ , t n + ( s n − t n + ) + ( t n − ) n ≡ (
3; 1 , , , ) ( s n + )( s n + ) − t n ( t n + ) t n + ( s n − t n ) + t n Table A.5.
The quantities L ( Λ n ) and (cid:96) Ω ( Λ n ) , by blowup vector. The first, L ( Λ n ) , counts lattice points enclosed byΛ n . The second, (cid:96) Ω ( Λ n ) , is a notion of length of the path, defined in (2.11); the constant term in each expression is e n in (A.3). To prove Proposition 5.6, these quantities L ( Λ n ) and (cid:96) Ω ( Λ n ) must satisfy (5.8). Using the definitions of s n and t n , as well asthe properties ♡ and ♣ of Lemma 5.2, one can argue directly that L ( Λ n ) = ( g ( n ) + )( g ( n + J ) + ) . We also have (cid:96) Ω ( Λ ) = B ⋅ s n + kb ⋅ t n + e n = ( B − kb )( g ( n ) + g ( n + J )) + Bc n − bkd n vol + e n = g ( n ) + g ( n + J ) + Bc n − bkd n + vol ⋅ e n vol . Checking that Bc n − kbd n + vol ⋅ e n = n then guarantees (cid:96) Ω ( Λ ) = g ( n ) + g ( n + J ) , as desired. ppendix B. ATFs
In this appendix, we describe the initial ATF maneuvering required to produce ATFs onthe manifolds C P ; C P C P ; C P C P ; C P C P ; C P C P ; and C P × C P that have base diagram a triangle with two nodal rays when J = J = C P , J = ×× × × ×× × × × × ×× × × × × × × ×× × × × × × × × × (a) (b) Figure B.1.
In (a), we see the Delzant polygon for C P . From (a) to (b), we haveapplied two nodal trades to add two singular fibers, creating a new almost toricfibration on C P . For C P C P , J = ×× × × ×× × × × × ×× × × × × × × ×× × × × × × × × × × × × (a) (b) Figure B.2.
In (a), we see the Delzant polygon for C P C P . From (a) to (b), wehave applied three nodal trades to add three singular fibers, creating a new almosttoric fibration on C P C P . or C P C P , J = ×× × × ×× × × × × ×× × × × × × × ×× × × × × × × × × (a) (b) (c) Figure B.3.
In (a), we see the Delzant polygon for C P C P . From (a) to (b),we have applied four nodal trades to add four singular fibers, creating a new almosttoric fibration on C P C P . Finally, from (b) to (c), we apply a mutation, withresulting base diagram a quadrilateral with three nodal rays, as desired. In (c), twoof the nodal rays have a single singular fiber and the third has two singular fibers.This is yet a third almost toric fibration on C P C P . or C P C P , J = × × ××× × × × × × ×× × × × ×× × × × ×× × × × × ×× × × × × ×× × × × × ×× × × × ×× × × × × ×× × × × × ×× × × × (a) (b) (c)(d) (e) Figure B.4.
In (a), we see the Delzant polygon for C P C P . From (a) to (b),we have applied five nodal trades to add five singular fibers, creating a new almosttoric fibration on C P C P . From (b) to (c), we apply a mutation, with resultingbase diagram a pentagon with four nodal rays. From (c) to (d), we apply anothermutation, with resulting base diagram a quadrilateral with three nodal rays. Finally,from (d) to (e), we perform a third mutation, with resulting base diagram the desiredtriangle with two nodal rays. In (e), one of the nodal rays has two singular fibersand the other has three singular fibers. or C P C P , J = § C P C P given in Figure B.4(e). This ATF has a smooth toric corner at the originwhere we may perform a toric blowup of symplectic size 1. In terms of the base diagram,this corresponds to chopping off a 1 × C P C P , shown in Figure B.5(b). There isthen a sequence of ATF moves that achieves a triangle with two nodal rays. See Figure B.5. × × ××× × × × × × ×× × × × ×× × × × ×× × × × × ×× × × × × ×× × × × × ×× × × × ×× × × × × ×× × × × × ×× × × × (a) (b)(c) (d) (e) Figure B.5.
In (a), we see the base diagram for an ATF on C P C P . From (a)to (b), we have applied a toric blowup of size 1 at the origin, resulting in an almosttoric fibration on C P C P . From (b) to (c), we apply one nodal trade. From (c)to (d), we apply mutation, with resulting base diagram a quadrilateral with threenodal rays. Finally, from (d) to (e), we perform a second mutation, with resultingbase diagram the desired triangle with two nodal rays. In (e), one of the nodal rayshas one singular fiber and the other has five singular fibers. or C P × C P , J = ×× × × ×× × × × × ×× × × × × × × ×× × × × × × × × × (a) (b) (c) Figure B.6.
In (a), we see the Delzant polygon for C P × C P . From (a) to (b), wehave applied three nodal trades to add three singular fibers, creating a new almosttoric fibration on C P × C P . Finally, from (b) to (c), we apply a mutation, withresulting base diagram a triangle with two nodal rays, as desired. In (c), one of thenodal rays has a single singular fiber and the other has two singular fibers. ppendix C. Behind the scenes
In this section, we give an account of how we found the six blowup vectors that appearin Theorem 1.14. At the beginning of this project, the ellipsoid embedding functions for theball, polydisk, and ellipsoid E ( , ) were known to have infinite staircases. These correspondto the blowup vectors ( ) , (
4; 2 , ) , and (
3; 1 , , ) .By Theorem 1.10, we knew where the accumulation point would occur for any domain, if aninfinite staircase were to exist. We wrote Mathematica code that generates an approximationof the graph of c X ( a ) for a given X , and started by trying a number different integer blowupvectors. By chance, we first tried (
3; 1 ) and found an infinite staircase. The vector (
3; 1 , ) admitted one too, and we were off, trying to prove that there was always an infinite staircase.The actual answer, of course, has turned out to be more subtle. The code we used in ourearly searches is included below and the notebook is available online at [11].The idea behind the code is that the ellipsoid embedding function can be computed asthe supremum of ratios of ECH capacities, as in equation (2.9). We compute a large (butfinite!) number of ECH capacities of the domain X using the sequence subtraction operationof Definition 2.4. We also compute a large but finite number of ECH capacities of E ( , a i ) ,for equally spaced values a i within a given range. Next, for each a i we find the maximum ofthe ratios of the computed ECH capacities, obtaining a list of points ( a i , ˜ c X ( a i )) . Our codethen approximates the graph by connecting the dots. Figure 1.12 illustrates four examples.The graph of ˜ c X ( a ) is an approximation of the graph of the embedding function c X ( a ) intwo senses: we are only using a finite number of points in the domain, and the the computedvalues ˜ c X ( a i ) are not completely accurate because we have restricted to a finite list of ECHcapacities. Nonetheless, the approximation does allow us to visually rule out certain domainsfrom the possibility of having an infinite staircase. For example, in the bottom left graphin Figure 1.12, there clearly exists an obstruction where the infinite staircase would have toaccumulate, so that blowup vector must not admit an infinite staircase. In cases where itis more ambiguous, we change the range of points a i , ask the code to compute more ECHcapacities, and hence zoom in on the graph to probe further. For example, zooming in onthe bottom right example in Figure 1.12 shows that in fact there exists an obstruction atthe potential accumulation point.Whenever this zooming in process suggested that there indeed exists an infinite staircasefor that domain, the next step was to find the coordinates of the inner and outer cornersof the staircase. Recall that in the ball case these are ratios of certain Fibonacci numbers,so we were interested in obtaining a recurrence sequence from the numerators and denomi-nators of these coordinates. We used the Mathematica function Rationalize to approximatethe values ˜ c X ( a i ) by fractions with small denominators and then fed the integer sequencesobtained into OEIS, the Online Encyclopedia of Integer Sequences, sometimes unearthingunexpected connections . Eventually we switched to using the function FindLinearRecur-rence on Mathematica to find the linear recurrence for the sequences found. In one instance, the integer sequence that came up on the OEIS search engine was sequence A007826:numbered stops on the Market-Frankford rapid transit (SEPTA) railway line in Philadelphia, PA USA. Thisconstitutes possibly the first ever application of symplectic geometry to mass transit. he Mathematica Code. List of first ( (⌊ kab ⌋ + )( k + ) − ) ECH capacities of the ellipsoid E ( a, b ) , we usually set k = E C H e l l i p s o i d [ a , b , k ] :=
Module [ { l = Floor [ ( k + 1)
Floor [ 1 + k a/b ] / 2 ] − } , Take [ Sort [ Flatten [ Table [ N [m a + n b ] , { m, 0 , k } , { n , 0 , k } ] ] ] , l ] ] ; List of first ( ( k − ) ) ECH capacities of the ball E ( , ) , usually we set k = ECHball [ k ] :=
Take [ Sort [ Flatten [ N [ Array [ Array [ k − Floor [ ( k − Sequence subtraction operation: a u x l i s t [ l i s t 1 , l i s t 2 , i ] :=
Block [ { a = l i s t 1 , b = l i s t 2 , l } , l = Min [ Length [ a ] ,
Length [ b ] ] ;
Array [ a [ [ i + − b [ [ − i ] ] ;minus [ l i s t 1 , l i s t 2 ] := Block [ { a = l i s t 1 , b = l i s t 2 , l } , l = Min [ Length [ a ] ,
Length [ b ] ] ;
Array [ Min [ a u x l i s t [ a , b , − Ellipsoid embedding function c X ( a ) where ECHlist is (the beginning of) the sequence ofECH capacities of X : c [ a , ECHlist ] := Module [ { p = Length [ ECHlist ] , k , l , nn } ,k = Floor [ (
Sqrt [ a ˆ2 + 6 a + 1 + 8 a p ] − − a ) / 2 ] ;nn = E C H e l l i p s o i d [ 1 , a , k ] ;l = Min [ p ,
Length [ nn ] ] ;
Max [ Array [ nn [[ − Creates a list of points ( a i , c X ( a i )) where ECHlist is (the beginning of) the sequence of ECHcapacities of X : c l i s t [ amin , amax , a s t e p , ECHlist ] := Block [ { l = Floor [ ( amax − amin )/ a s t e p ] } , Array [ { amin + ( −
1) astep , c [ amin + ( −
1) astep , ECHlist ] } &,l ] ] ; Plots the volume curve, plus a vertical line at the location of the potential accumulationpoint: c o n s t r a i n t [ amin , amax , v o l , a c c ] :=
Plot [ Sqrt [ t / v o l ] , { t , amin , amax } , PlotStyle − > Red , Epilog − > { I n f i n i t e L i n e [ { acc , 0 } , { } ] } ] ; Example C.1.
Let X have blowup vector (
4; 2 , ) . Make changes as necessary. Accpointgives the location of the potential accumulation point, so that one can choose the range ofpoints to plot. The output of this example is in Figure 1.12. ECHcap =
Block [ { seq = ECHball [ 1 0 0 ] } , minus [ minus [ 4 ∗ seq , 2 ∗ seq ] , seq ] ] ;per = 3 ∗ − − ( ∗ per = 3 b − sum b i ∗ ) v o l = 4ˆ2 − − ( ∗ v o l = b ˆ2 − sum b i ˆ2 ∗ ) a c c p o i n t = x / . NSolve [ xˆ2 + (2 − per ˆ2/ v o l ) x + 1 == 0 , x ] [ [ 2 ] ]5 . 1 7 0 2 2 50amin = 1 ;aamax = 6 ;a a s t e p = 0 . 0 1 ; Show [ { c o n s t r a i n t [ aamin , aamax , vol , a c c p o i n t ] , ListPlot [ c l i s t [ aamin , aamax , aastep , ECHcap ] , Joined − > True ] } ] References [1] M. Audin,
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Institute for Advanced Study
E-mail address : [email protected] Cornell University
E-mail address : [email protected] Universidade Federal Fluminense and IST, Universidade de Lisboa
E-mail address : [email protected] University of Edinburgh
E-mail address : [email protected]@ed.ac.uk