Towers of Looijenga pairs and asymptotics of ECH capacities
aa r X i v : . [ m a t h . S G ] J a n TOWERS OF LOOΔ²ENGA PAIRS AND ASYMPTOTICS OF ECH CAPACITIES
B. WORMLEIGHTONAbstract. ECH capacities are rich obstructions to symplectic embeddings in 4-dimensions thathave also been seen to arise in the context of algebraic positivity for (possibly singular) projectivesurfaces. We extend this connection to relate general convex toric domains on the symplectic sidewith towers of polarised toric surfaces on the algebraic side, and then use this perspective to showthat the sub-leading asymptotics of ECH capacities for all convex and concave toric domains are π p q . We obtain suο¬cient criteria for when the sub-leading asymptotics converge in this context,generalising results of Hutchings and of the author, and derive new obstructions to embeddingsbetween toric domains of the same volume. We also propose two invariants to more preciselydescribe when convergence occurs in the toric case. Our methods are largely non-toric in nature,and apply more widely to towers of polarised LooΔ³enga pairs. Introduction
We outline the symplectic part of the story β in particular, the applications to sub-leadingasymptotics of ECH capacities β in Β§1.1 and describe the novel aspects of our algebro-geometricmethods and constructions in Β§1.2.1.1.
Symplectic perspective.
A great deal of symplectic geometry has been stimulated by sym-plectic embedding problems. These are problems of the form: given two symplectic manifolds p π , π q and p π , π q of the same dimension, when does there exist a smooth embedding π : π Γ π such that π Λ π β π ? Such embeddings are called symplectic embeddings . If there is a symplecticembedding p π , π q Γ p π , π q we write p π , π q s Γ£ Γ p π , π q .For each symplectic embedding problem there is a βconstructiveβ aspect in which the aim is toshow the existence of a symplectic embeddings. Conversely there is an βobstructiveβ aspect thatusually involves ο¬nding invariants that are βmonotoneβ under symplectic embeddings and henceobstruct their existence. We will focus on the latter here. ECH capacities were introduced by Hutching [13] to obstruct embeddings between symplectic4-manifolds. To a symplectic 4-manifold p π , π q ECH associates a non-decreasing sequence t π ech π p π , π qu π P Z Δ of (extended) real numbers such that p π , π q s Γ£ Γ p π , π q ΓΉΓ± π ech π p π , π q Δ π ech π p π , π q for all π P Z Δ . ECH capacities have found many applications to notable embedding problems[2, 4, 18] and have been shown to have signiο¬cant connections with the geometry of divisors onalgebraic surfaces [1, 21, 22] and the lattice combinatorics of polytopes [8, 9, 21]. For a generalintroduction to ECH, see [14]. One of the main contributions of this paper is to extend and reο¬nethese connections in the context of a wider class of spaces.The algebraic analogues β algebraic capacities β to ECH capacities were introduced by the authorin [21, 23] and have since been applied to embedding problems into closed surfaces [1] and tothe asymptotics of ECH capacities as π Γ 8 [5, 22]. To a pair p π, π΄ q consisting of a projectivealgebraic surface π with an ample (or big and nef) R -divisor π΄ β a polarised surface β the associatedalgebraic capacities form a non-decreasing sequence t π alg π p π, π΄ qu π P Z Δ of real numbers with certain appealing properties; some of which we will describe shortly.As it will occupy much of this paper, we outline the current asymptotic understanding of ECHcapacities and the motivation for further study. We start with the βWeyl lawβ for ECH. Theorem 1 ( [7, Thm. 1.1]) . Suppose p π , π q is a Liouville domain such that all π ech π p π , π q are ο¬nite.Then lim π Γ8 π ech π p π Ξ© q π β p π , π q In other words π ech π p π , π q β a p π , π q π . This was used by Cristofaro-GardinerβHutchings[6] to prove a reο¬nement of the Weinstein conjecture. One can hence deο¬ne error terms π π p π , π q : β π ech π p π , π q Β΄ b p π , π q These are manifestly π p? π q but we consider what more there is to say. A series of estimates dueto Sun [20], Cristofaro-GardinerβSavale [10], and Hutchings [15] found bounds of the form π p π π q for π β { , π β { , π β { Conjecture 1.
Let p π , π q be a star-shaped domain in R . (i) (c.f. [15, Ex. 1.6]) π π p π , π q β π p q . (ii) ( [15, Conj. 1.5]) If p π , π q is generic then π π p π , π q converges with limit lim π Γ8 π π p π , π q β Β΄
12 Ru p π , π q where Ru p π , π q is the Ruelle invariant of p π , π q . The Ruelle invariant was deο¬ned for nice star-shaped domains in R by Hutchings [15, Def. 1.4]following ideas of Ruelle [19], and one can extend this deο¬nition to more general star-shapeddomains by continuity. Some of the results of this paper can be viewed as seeking to makeprecise what βgenericβ means.We consider a class of toric symplectic 4-manifolds that will play a central role for us. Suppose Ξ© Δ R Δ is a simply-connected region. Deο¬ne the toric domain π Ξ© : β π Β΄ p Ξ© q where π : C Γ R is the moment map for the π Λ π -action on C . If B Ξ© intersects the coordinateaxes in sets of the form tp π₯, q : π₯ P r , π su and tp , π¦ q : π¦ P r , π su and the remaining part ofthe boundary β which we denote by B ` Ξ© β is a convex curve (that is, the region Ξ© is convex),we say that Ξ© is a convex domain and that π Ξ© is a convex toric domain . This class includes balls,ellipsoids, polydisks, and many other classic symplectic manifolds. We denote the quantities π and π appearing above by π p Ξ© q and π p Ξ© q .In [22, Thm. 4.10] it was shown that when Ξ© β π Ξ© for some lattice polygon Ξ© and some π P R Δ , we have π π p π Ξ© q β π p q with an explicit calculation of the lim sup and lim inf, which are always diο¬erent and so π π p π Ξ© q does not converge in this situation. We say that such Ξ© are of scaled-lattice type . The approach toproving this result uses the fact that the interior π Λ Ξ© can be realised as the complement of an ampledivisor π΄ Ξ© in the (possibly singular) projective toric surface π Ξ© associated to Ξ© [3, Β§2.3], and thatby [21, Thm. 1.5] the ECH capacities π ech π p Ξ© q are given by the algebraic capacities π alg π p π Ξ© , π΄ Ξ© q associated to the pair p π Ξ© , π΄ Ξ© q . OOΔ²ENGA TOWERS AND ASYMPTOTICS OF ECH 3
In [15] Hutchings showed that Conj. 1(ii) holds for π Ξ© when Ξ© is βstrictly convexβ : namely,has smooth boundary and all outward normals to B ` Ξ© live in the strictly positive quadrant of R . The Ruelle invariant in this setting is equal to π p Ξ© q ` π p Ξ© q .These two cases will be subsumed in the following theorem. The aο¬ne length β aο¬ p π£ q of a vector π£ P R is the pseudonorm deο¬ned by 0 if ππ£ R Z for any π P R Δ and by 1 if π£ is a primitivevector in Z . We deο¬ne the aο¬ne length of a continuous curve by the (possibly empty) sum ofthe aο¬ne lengths of the direction vectors deο¬ning each linear segment of the curve. Theorem 2 (Cor. 3.5) . Let π Ξ© be a convex toric domain, then Β΄ Λ π p Ξ© q ` π p Ξ© q Β΄ β aο¬ pB ` Ξ© q Λ Δ lim sup π Γ8 π π p π Ξ© qΔ lim inf π Γ8 π π p π Ξ© q Δ Β΄ Λ π p Ξ© q ` π p Ξ© q ` β aο¬ pB ` Ξ© q Λ where B ` Ξ© is the part of B Ξ© not on the coordinate axes. In particular, π π p π Ξ© q β π p q . Notice that the upper and lower bounds can easily be translated to involve the Ruelle invariant,and that their midpoint is exactly Β΄ Ru p π Ξ© q when π Ξ© is strictly convex. We immediately obtainthe following corollary generalising [15, Thm. 1.10]. Corollary 1 (Cor. 3.5) . Let π Ξ© be a convex toric domain. When B ` Ξ© has no rational-sloped edges wehave that π π p π Ξ© q is convergent and lim π Γ8 π π p π Ξ© q β Β΄ p π p Ξ© q ` π p Ξ© qq If a region Ξ Δ R Δ is bounded above by the graph of a convex function π : r , π s Γ R Δ we saythat π Ξ is a concave toric domain . Via formal properties of ECH capacities we obtain an analogousresult for concave toric domains. Theorem 3 (Thm. 3.15) . Let π Ξ be a concave toric domain. Then Β΄ ` π p Ξ q ` π p Ξ q Β΄ β aο¬ pB ` Ξ q Λ Δ lim sup π Γ8 π π p π Ξ qΔ lim inf π Γ8 π π p π Ξ q Δ Β΄ ` π p Ξ q ` π p Ξ q ` β aο¬ pB ` Ξ q Λ and so π π p π Ξ q β π p q . If B ` Ξ has no rational-sloped edges then lim π Γ8 π π p π Ξ q β Β΄ p π p Ξ q ` π p Ξ qq We note that one consequence of better understanding the asymptotics of π π p π , π q is to obtainο¬ner embedding obstructions between symplectic 4-manifolds of the same volume (far fromvacuous, as discussed in [15, Rmk. 1.14]). As a corollary to Thm. 2 and Thm. 3 we obtain thefollowing embedding obstruction subsuming [15, Cor. 1.13] and [22, Cor. 5.15]. For the purposesof this result we say that a toric domain π Ξ© is βadmissibleβ if either Ξ© is concave or convex and B ` Ξ© has no rational-sloped edges, or if Ξ© is convex and of scaled-lattice type. Corollary 2.
Let π Ξ© and π Ξ© be admissible toric domains of the same symplectic volume. Then π Λ Ξ© s Γ£ Γ π Ξ© ΓΉΓ± β aο¬ pB Ξ© q Δ β aο¬ pB Ξ© q We conclude in Β§3.8 by discussing two invariants β the number of rational-sloped edges in B ` Ξ© and the degree of independence over Q of their aο¬ne lengths β that we believe might furthergovern the asymptotics of π π p π Ξ© q for convex and concave toric domains. Hutchingsβ result also shows that Conj. 1(ii) is true when Ξ© is βstrictly concaveβ. B. WORMLEIGHTON
Algebraic perspective.
Our approach to the results in Β§1.1 is to identify an algebraic objectwhose βalgebraic capacitiesβ agree with the ECH capacities of a convex toric domain π Ξ© . We willgive the formal deο¬nition in Β§2.5 but, in short, one can think of algebraic capacities as positivityinvariants of a polarised surface p π, π΄ q obtained as solutions to quadratic optimisation problemson the nef cone of π . We denote the π th algebraic capacity of p π, π΄ q by π alg π p π, π΄ q .As discussed above, when Ξ© is a rational-sloped polygon one can recover the ECH capacitiesof π Ξ© as the algebraic capacities of the polarised toric surface p π Ξ© , π΄ Ξ© q corresponding to Ξ© .When Ξ© is a non-polytopal convex domain, we will use the weight expansion of Ξ© [2, 17] in Β§2.4 todeο¬ne a tower of polarised toric surfaces p π , π΄ q π ΓΓ p π , π΄ q π ΓΓ . . . π π ΓΓ p π π , π΄ π q π π ` ΓΓ . . . for which there exists a notion of algebraic capacities extending the deο¬nition for polarisedsurfaces. We denote such towers of polarised surfaces by calligraphic letters p Y , A q and denotetheir algebraic capacities by π alg π p Y , A q . Proposition 1 (Prop. 3.1) . Let Ξ© be a convex domain, and let p Y Ξ© , A Ξ© q denote the tower of polarisedtoric surfaces associated to Ξ© . Then π ech π p π Ξ© q β π alg π p Y Ξ© , A Ξ© q for all π P Z Δ . Remark 1.
A point of independent interest here is that the tower of polarised toric surfaces weproduce can be viewed naturally as the object in toric algebraic geometry corresponding to theconvex non-polytopal region Ξ© .Just as in [22] we ο¬nd that our results on the asymptotics of algebraic capacities for towers ofpolarised toric surfaces do not require much of the toric structure and in fact apply to a muchlarger class of algebro-geometric objects.We will say that a LooΔ³enga pair [11, 16] is a pair p π , πΏ q consisting of a Q -factorial rationalsurface π with a singular nodal curve πΏ P |Β΄ πΎ π | . Recall that Q -factorial means that an integermultiple of each Weil divisor on π is Cartier. Note that elsewhere in the literature it is standardto assume that π is smooth, in which case πΏ is either an irreducible rational nodal curve or a cycleof smooth rational curves.We consider LooΔ³enga pairs with a polarisation supported on the anticanonical divisor, andtowers of such objects in which the choices of polarisation and anticanonical divisor are respectedappropriately. We call such objects polarised LooΔ³enga towers and write them as pairs p Y , A q . Thetowers of polarised toric surfaces we consider are examples of these, though there are also manyinteresting non-toric examples. We develop the necessary birational geometry of polarisedLooΔ³enga towers in Β§2.3, including a natural notion of divisor (which includes the polarisation A ), an intersection pairing, and a notion of canonical divisor πΎ Y . Theorem 4 (Thm. 3.3 + Thm. 3.4) . Let p Y , A q be a polarised LooΔ³enga tower. Then π alg π p Y , A q β ? A π and the error terms π alg π p Y , A q : β π alg π p Y , A q Β΄ ? A π satisfy πΎ Y Β¨ A Β΄ πΎ ` Y Β¨ A Δ lim sup π Γ8 π alg π p Y , A qΔ lim inf π Γ8 π alg π p Y , A q Δ πΎ Y Β¨ A where πΎ ` Y is a divisor on Y canonically associated to p Y , A q . OOΔ²ENGA TOWERS AND ASYMPTOTICS OF ECH 5
In the case that p Y , A q is a tower of polarised toric surfaces arising from a convex domain Ξ© we calculate πΎ Y Β¨ A β Β΄ β aο¬ pB Ξ© q β Β΄ ` π p Ξ© q ` π p Ξ© q ` β aο¬ pB ` Ξ© q Λ and Β΄ πΎ ` Y Β¨ A β β aο¬ pB ` Ξ© q establishing Thm. 2 and its consequences from Β§1.1. We prove a convergence criterion similarto Cor. 1 in Prop. 3.18. Using intersection theory on Y we also formulate algebraic capacitiesintrinsically in terms of divisors on Y in Prop. 2.10. We hope that this βintrinsicβ geometry of Y will shed more insight on the asymptotics of algebraic capacities and, hence, of ECH capacities. Acknowledgements.
I am grateful for many encouraging and helpful conversation with DanCristofaro-Gardiner, Michael Hutchings, Julian Chaidez, Vinicius Ramos, Tara Holm, and AnaRita Pires. I am especially grateful to Michael Hutchings for discussing the content of [15] withme, and to Vinicius Ramos for hosting me at IMPA where the idea for this project was seeded. Iam very thankful to Γan-Daniel Erdmann-Pham for providing the proof of Lemma 3.6.2.
Towers of Looijenga pairs
LooΔ³enga pairs and LooΔ³enga towers.
In our context we will deο¬ne a
LooΔ³enga pair to be apair p π , πΏ q consisting of: β a Q -factorial rational surface π , β a singular nodal curve πΏ P |Β΄ πΎ π | .The basic example of a LooΔ³enga pair is a toric surface equipped with the union of its torus-invariant divisors. Note that LooΔ³enga pairs are usually assumed to be smooth elsewhere in theliterature.A polarised LooΔ³enga pair is a triple p π , πΏ, π΄ q consisting of a LooΔ³enga pair and an ample divisorsupported on a subset of πΏ . This implies that the LooΔ³enga pair is βpositiveβ in the languageof [11]. If π΄ is only big and nef we say that p π, πΏ, π΄ q is a pseudo-polarised LooΔ³enga pair .A toric blowup of a LooΔ³enga pair p π, πΏ q is a blowup π : r π Γ π with centre a node of πΏ . Observethat in this case the divisor r πΏ ` πΈ β the strict transform of πΏ plus the exceptional divisor β is suchthat p r π, r πΏ ` πΈ q is a LooΔ³enga pair. We will consider towers Y : p π , πΏ q π ΓΓ p π , πΏ q π ΓΓ . . . π π Β΄ ΓΓ p π π Β΄ , πΏ π Β΄ q π π ΓΓ p π π , πΏ π q π π ` ΓΓ . . . of LooΔ³enga pairs where each map π π is a toric blowup. We call such structures LooΔ³enga towers .We can also ask that each LooΔ³enga pair is polarised and that the toric blowups are compatiblewith the polarisations. Namely, we want to consider towers p π , πΏ , π΄ q π ΓΓ p π , πΏ , π΄ q π ΓΓ . . . π π Β΄ ΓΓ p π π Β΄ , πΏ π Β΄ , π΄ π Β΄ q π π ΓΓ p π π , πΏ π , π΄ π q π π ` ΓΓ . . . of polarised LooΔ³enga pairs where each map π π is a toric blowup, and the polarisations arerelated by π΄ π β π Λ π π΄ π Β΄ Β΄ π π πΈ π for some π π Δ
0, where πΈ π is the exceptional ο¬bre of π π . We call such a structure a polarisedLooΔ³enga tower and denote it by a pair p Y , A q where Y is the underlying LooΔ³enga tower and A isthe sequence of polarisations p π΄ π q π P Z Δ . If the polarisations are relaxed to pseudo-polarisations,we call p Y , A q a pseudo-polarised LooΔ³enga tower . We say that Y is smooth if π is smooth. Lemma 2.1.
Let p Y , A q be a pseudo-polarised LooΔ³enga tower. Then lim π Γ8 π΄ π exists. This also holds if we omit the anticanonical divisor πΏ π and only consider a tower of pseudo-polarised surfaces related by arbitrary blowups. B. WORMLEIGHTON
Proof.
We see that π΄ π β p π Λ π π΄ π Β΄ Β΄ π π πΈ π q β π΄ π Β΄ ` π π πΈ π Δ π΄ π Β΄ and so π΄ π is a decreasing sequence. It is bounded below since π΄ π Δ π by the assumptionthat each p π π , π΄ π q is pseudo-polarised and is hence convergent. (cid:3) As a result we deο¬ne A : β lim π Γ8 π΄ π β inf t π΄ π : π P Z Δ u Polarised LooΔ³enga towers from weighted posets.
We generalise the previous construc-tion to use a poset other than Z Δ to index toric blowups. The towers that come from thisconstruction can thus have many spires. We start by constructing the universal LooΔ³enga tower Y univ p π,πΏ q associated to a LooΔ³enga pair p π, πΏ q and realise all pseudo-polarised LooΔ³enga towerswith p π , πΏ q β p π , πΏ q in terms of it.Let p π , πΏ q be a LooΔ³enga pair. We construct a poset P p π,πΏ q as follows. Let P be the posetconsisting of all nodes of π· with no order relations. Let π π : π π Γ π denote the toric blowupat a node π P πΏ with exceptional divisor πΈ π . Set P π β t π , π u Y P where π , π are the twointersection points of πΈ π with the strict transform of πΏ . Deο¬ne P β Δ π P P P π and view this as poset by setting π Δ π if and only if π P πΈ π . Note that the elements of P z P correspond to nodes on the LooΔ³enga pair p π , πΏ q obtained from p π , πΏ q by blowing up all thenodes of π· . Repeating this process by blowing up each node on πΏ produces a new poset P suchthat P z P is the set of nodes of the LooΔ³enga pair p π , πΏ q obtained by blowing up all nodes of p π , πΏ q . Continuing this procedure deο¬nes a LooΔ³enga pair p π π , π΄ π q for each π P Z Δ β letting p π, πΏ q β p π , πΏ q β and a poset P π such that P π z P π Β΄ is the set of nodes of p π π , πΏ π q . These pairscoalesce to form a slightly more general kind of tower where blowups with multiple centres arepermitted at each stage. For the remainder of this section we will use the term βLooΔ³enga towerβto include such towers.We call the LooΔ³enga tower arising from this construction the universal LooΔ³enga tower associatedto p π , πΏ q and denote it by Y univ p π,πΏ q . We will later compare this to a construction of Hutchings [15, Β§3].Deο¬ne the poset P p π,πΏ q β Δ π Δ P π Observe that P p π,πΏ q is a graded poset with grading deο¬ned by the ο¬ltration P π . For π P P p π,πΏ q ofdegree π we obtain a LooΔ³enga pair p π π , πΏ π q obtained as the toric blowup of p π π , πΏ π q at π . Deο¬nition 2.2.
Let P be a countable poset. We call a function wt : P Γ R Δ a weight function if β wt is a poset homomorphism where R Δ is regarded as a poset in the usual way, β Ε π P P wt p π q Δ 8 .A pair p P , wt q of a poset with a weight function is called a weighted poset . We deο¬ne the weightsequence wt p P q associated to a weighted poset p P , wt q to be the multiset t wt p π q : π P P u .We also write wt p Y , A q : β wt p P p π ,πΏ q q and refer to this as the weight sequence of p Y , A q . Wesay that an element π of a poset P is a direct descendant of π P P if π Δ π and there is no π P P such that π Δ π Δ π .From the data of a pseudo-polarised LooΔ³enga pair p π , πΏ, π΄ q and a weight function satisfyingsome conditions on P p π,πΏ q we can produce a pseudo-polarised LooΔ³enga tower p Y , A q . Let wt bea weight function on P p π,πΏ q . To deο¬ne a polarisation on each p π π , πΏ π q we start by setting π΄ β π Λ π΄ Β΄ ΓΏ π P P wt p π q πΈ π OOΔ²ENGA TOWERS AND ASYMPTOTICS OF ECH 7 and then recurse by setting π΄ π β π Λ π π΄ π Β΄ Β΄ ΓΏ π P P π z P π Β΄ wt p π q πΈ π If this recipe deο¬nes a polarisation (resp. pseudo-polarisation) on each p π π , πΏ π q then we say thatwt is an ample (resp. big and nef) weight function on P p π,πΏ q . Thus, after choosing a big and nefweight function, to each π P P p π,πΏ q there is a pseudo-polarised LooΔ³enga pair p π π , πΏ π , π΄ π q . Itfollows by direct computation that wt is big and nef implies wt is a poset homomorphism.We can non-canonically create a pseudo-polarised LooΔ³enga tower in which each map π π isa single toric blowup as in Β§2.1 from this data. We choose a bΔ³ective poset homomorphism β : P p π,πΏ q Γ Z op Δ , where Z op Δ is Z with reverse ordering, and deο¬ne p π π , πΏ π , π΄ π q to be the pseudo-polarised LooΔ³enga pair obtained by blowing up in the nodes β Β΄ t , . . . , π u . One can easilyverify that this is well-deο¬ned by the requirement that β is a poset homomorphism.In later sections we will choose β such that wt p β Β΄ p π qq Δ wt p β Β΄ p π ` qq ; in other words, thereis a commutative diagram in the category of posets of the form: P p π,πΏ q β / / wt " " ββββββββ Z op Δ } } β£ β£ β£ β£ R Δ Of course β is not a poset isomorphism in general!We view two pseudo-polarised LooΔ³enga pairs p π , πΏ, π΄ q and p π , πΏ , π΄ q as βequivalentβ if thereis a LooΔ³enga pair p π , πΏ q and two maps π : π Γ π and π : π Γ π given as compositions oftoric blowups such that π Λ π΄ β p π q Λ π΄ In this way, one can indeed recover any pseudo-polarised LooΔ³enga tower p Y , A q with p π , πΏ q βp π, πΏ q up to equivalence from Y univ p π,πΏ q by assigning a weight of zero to all nodes on toric blowupsof p π, πΏ q that are not blown up in Y . We will revisit this notion of equivalence in Β§2.5.2.3. Divisors on LooΔ³enga towers.
Throughout this subsection we ο¬x a LooΔ³enga tower Y βtp π π , πΏ π qu with toric blowup maps π π and exceptional divisors πΈ π . We will introduce the notionof divisors on Y , and study classes of divisors that will be relevant to our applications. Let K P t Z , Q , R u . Deο¬nition 2.3. A K -divisor on Y is a sequence D β t π· π u where π· π is a K -divisor on π π such that π· π β π Λ π π· π Β΄ Β΄ π π πΈ π for some π π P K .Clearly one can view a polarisation A on Y as an R -divisor on Y . We call the sequence p π π q π P Z Δ the weight sequence of D . The weight sequence of A regarded as a divisor is by construction theweight sequence of p Y , A q as deο¬ned in Β§2.2. Y has a canonical divisor πΎ Y deο¬ned as the sequence πΎ Y β t πΎ π π u π P Z Δ When Y is smooth the weight sequence of πΎ Y is p , , . . . q . We denote the set of K -divisors on Y by Div p Y q K . One can easily modify this deο¬nition to produce numerical or linear equivalenceclasses of divisors on Y . We deο¬ne Div ` p Y q K to be the set of K -divisors on Y whose weightsequences are summable, and Div π p Y q K to be the set of K -divisors on Y whose weight sequencesare bounded. B. WORMLEIGHTON
There is evidently a pairingDiv π p Y q K b Div ` p Y q K Γ R , D Β¨ D β π· Β¨ π· ` ΓΏ π Δ π π π π πΈ π where p π π q π P Z Δ and p π π q π P Z Δ are the weight sequences of D and D respectively. We choosethe codomain to be R to avoid issues of integrality when Y is not smooth. This pairing extendsthe intersection product for each π π in the sense that we can view Div p π π q K as the subspace ofDiv ` p Y q K Δ Div π p Y q K consisting of all K -divisors on Y whose weight sequences vanish afterthe π th term. When A is a polarisation on a smooth LooΔ³enga tower Y we will make much useof the quantity Β΄ πΎ Y Β¨ A β Β΄ πΎ π Β¨ π΄ Β΄ ΓΏ π P wt p Y , A q π Toric LooΔ³enga towers.
We will study a class of polarised LooΔ³enga towers arising fromweighted posets that come from convex domains in R Δ . The LooΔ³enga pairs constituting thesetowers are toric surfaces. Key to our construction to is the weight sequence wt p Ξ© q associated to aconvex domain Ξ© following [2,17]. It is well-known (e.g. the work of GrossβHackingβKeel [11,12])that the geometry of LooΔ³enga pairs is close to the geometry of toric varieties and so this is a richexample to consider algebraically, as well as being the main source of applications to symplecticgeometry.We start by recalling the weight sequence associated to a concave or convex domain in R . Let Ξ π denote the triangle in R with vertices p , q , p π, q , p , π q . Deο¬nition 2.4.
Let Ξ© be a concave domain. The weight sequence wt p Ξ© q of Ξ© is deο¬ned recursivelyas follows. β Set wt pHq β H and wt p Ξ π q β p π q . β Otherwise let π be the largest real number such that Ξ π Δ Ξ© . This divides Ξ© into three(possibly empty) pieces: Ξ π , Ξ© , Ξ© . β If not empty, Ξ© and Ξ© are aο¬ne-equivalent to concave domains Ξ© and Ξ© . Deο¬newt p Ξ© q β p π q Y wt p Ξ© q Y wt p Ξ© q regarded as a multiset.Note that wt p Ξ© q is ο¬nite if and only if Ξ© is a real multiple of a lattice concave domain but willbe inο¬nite in general. We deο¬ne an analogous sequence for convex domains. Deο¬nition 2.5.
Let Ξ© be a convex domain. The weight sequence wt p Ξ© q of Ξ© is deο¬ned recursivelyas follows. β Let π be the smallest real number such that Ξ© Δ Ξ π . β This divides Ξ π into three (possibly empty) pieces: Ξ© , Ξ© , Ξ© . β If non-empty, Ξ© and Ξ© are aο¬ne-equivalent to concave domains Ξ© and Ξ© . Deο¬newt p Ξ© q β p π q Y wt p Ξ© q Y wt p Ξ© q using Def. 2.4. We regard this as a multiset with a distinguished element π from therecursion above that we call the head of wt p Ξ© q . We set wt Β΄ p Ξ© q : β wt p Ξ© qzt π u .We depict the decompositions used to recursively deο¬ne the weight sequence in Fig. 1, withthe concave case shown in Fig. 1(a) and the convex case in Fig. 1(b). In both cases we denote theparts of B Ξ π away from Ξ© by dashed lines. Deο¬nition 2.6.
Let p Y , A q be a pseudo-polarised LooΔ³enga tower. We say p Y , A q is toric if p π , π΄ q is a toric surface polarised by a torus-invariant R -divisor and each blowup map π π is equivariant. OOΔ²ENGA TOWERS AND ASYMPTOTICS OF ECH 9
Figure 1.
Weight sequence decompositions β ββ ββ ββ ββ ββ β ββ ββ ββ ββ ββ ββ β ββ ββ ββ ββ ββ ββ β ββ ββ ββ ββ ββ ββ β ββ ββ ββ ββ ββ ββ β ββ Ξ© Ξ© Ξ© Ξ© (a) (b)We associate a toric pseudo-polarised LooΔ³enga tower p Y Ξ© , A Ξ© q to a convex domain Ξ© . Thiswill have the property wt p Ξ© q β wt p Y Ξ© , A Ξ© q . We write wt p Ξ© q β t π u Y wt Β΄ p Ξ© q .Consider P with moment image Ξ shown in Fig. 2 where the lower left vertex is the origin.We denote the hyperplanes corresponding to the three edges of Ξ by π» , π» , π» as shown. Figure 2.
Moment polytope of P ββ ββ π» π» π» We start with p π, πΏ, π΄ q β p P , π» ` π» ` π» , ππ» q . Set P Ξ© β P p π,πΏ q . We will construct a bigand nef weight function on P Ξ© via the recursion deο¬ning the weight sequence for Ξ© , and hencea (toric) pseudo-polarised LooΔ³enga tower.Each element π P P Ξ© by deο¬nition corresponds to a node on a toric blowup of p π, πΏ q but fromDef. 2.4 and Def. 2.5 π also corresponds to a step in the weight sequence recursion. Recall theconstruction of P p π,πΏ q β Ε€ π Δ P π . In this notation the elements of P correspond to the threetorus-ο¬xed points of P . We assign weight zero to the torus-ο¬xed point π β π» X π» whosemoment image is the origin and to all its descendants, capturing the fact that there will be noblowups performed with that centre.The two other points π , π P P correspond to the concave domains Ξ© and Ξ© from Def. 2.5.Set wt p π π q β π π , where Ξ π π is the largest regular triangle that ο¬ts inside Ξ© π for π β ,
3. Iteratingthis procedure assigns a weight to each element of P Ξ© as the side length of the largest regulartriangle that ο¬ts inside the corresponding concave domain.More precisely, we ο¬x notation as follows. Let Ξ© and Ξ© be as above. Applying the weightsequence recursion to Ξ© yields two concave domains Ξ© and Ξ© and similarly applying it to Ξ© yields concave domains Ξ© and Ξ© . Repeating this process yields the diagram in Fig. 3(a).Notice that this is naturally in bΔ³ection with the part of the Haase diagram of the poset P Ξ© excluding the 2-valent tree with maximum π . We denote by Ξ p π q the concave domain (i.e. either Ξ© or Ξ© in Def. 2.4) corresponding to π P P Ξ© zt π P P Ξ© : π Δ π u .We deο¬ne a weight function on P Ξ© bywt p π q β π Δ π π p π q elsewhere Ξ π p π q is the largest regular triangle contained in Ξ© π . This is shown in Fig. 3(b) with thesame indexing as in Fig. 3(a). Observe that this weight function is big and nef since the associated polarised toric surface p π π , π΄ π q corresponds to the polytope Ξ© π obtained after the π th step of theweight sequence recursion; for comparison, see [4, Β§3.2-3.3]. Figure 3.
Weight sequence recursion (a)(b) Ξ© Ξ© Ξ© Ξ© Ξ© Ξ© Ξ© Ξ© ... Ξ© Ξ© . . .. . .π π π π π π π π ...π π . . .. . . Figure 4.
Constructing P Ξ© ββββββ ββββββ ββββββ ββββββ ββββββ ββββββ βββ βββ βββ ββββββ βββ βββ βββ βββ βββ βββ βββ Example 2.7.
We will work out the construction of P Ξ© in detail for the convex domain Ξ© fromFig. 1(b) with weight sequence p
5; 1 , , q . In Fig. 4(a) we show Ξ© with the ο¬rst step of the weightsequence recursion expressing the 4-ball π΅ p q as a union of π Ξ© , π΅ p q and the ellipsoid πΈ p , q .In Fig. 4(b) we show the two regions Ξ© and Ξ© . The weights for elements of P Δ P Ξ© areillustrated below the domains. In Fig. 4(c) the ο¬nal stage of the recursion is shown β of the fourconcave domains coming from Ξ© and Ξ© only one is nonempty, and is equal to Ξ β and thecorresponding weights are listed below. Throughout we omit the tree with maximum π withweights all zero. OOΔ²ENGA TOWERS AND ASYMPTOTICS OF ECH 11
This example terminated after ο¬nitely many stages because Ξ© was a rational-sloped polytope.In this case all p π π , π΄ π q for large enough π are equivalent to the polarised toric surface p π Ξ© , π΄ Ξ© q associated to Ξ© . That is, for large enough π there is a series of toric blowups π : π π Γ π Ξ© with π΄ π β π Λ π΄ Ξ© .Note that the sequence of polarised toric surfaces p π π , π΄ π q produced using the structure of P p π,πΏ q as a graded poset from Β§2.2 recovers the sequence of approximations used by Hutchingsin [15, Β§3] by setting Ξ© π to be the polytope for π΄ π .2.5. Algebraic capacities for LooΔ³enga towers.
Recall the construction of algebraic capacities fora Q -factorial pseudo-polarised surface p π , π΄ q : π alg π p π , π΄ q : β inf Nef p π q Z t π· Β¨ π΄ : π p π· q Δ π ` π p O π qu When π is smooth this reduces to π alg π p π, π΄ q : β inf Nef p π q Z t π· Β¨ π΄ : πΌ p π· q Δ π u where πΌ p π· q : β π· Β¨ p π· Β΄ πΎ π q . It was shown in [22, Prop. 2.11] for all smooth or toric pseudo-polarised surfaces p π, π΄ q we have that π alg π p π , π΄ q is obtained by ranging over eο¬ective Z -divisorsin place of nef Z -divisors. Lemma 2.8.
Suppose p Y , A q β tp π π , πΏ π , π΄ π qu Z Δ is a pseudo-polarised LooΔ³enga tower that is smoothor toric. Then lim π Γ8 π alg π p π π , π΄ π q exists and is ο¬nite. This result also holds for any tower p Y , A q β tp π π , π΄ π qu of pseudo-polarised surfaces relatedby blowups. Notice that in this case π p O π π q β π p O π π q β : π p O Y q for all π , π P Z Δ . Proof.
Let π· π be a nef Z -divisor computing π alg π p π π , π΄ π q . We have in the smooth or toric casesthat π p π Λ π ` π· π q β π p π· π q Δ π ` π p O Y q so that π alg π p π π ` , π΄ π ` q Δ π Λ π ` π· π Β¨ π΄ π ` β π· π Β¨ π΄ π β π alg π p π π , π΄ π q It follows that π alg π p π π , π΄ π q is a decreasing sequence in π that is bounded below, and is henceconvergent. (cid:3) We thus deο¬ne π alg π p Y , A q : β lim π Γ8 π alg π p π π , π΄ π q We will see in the next section that this deο¬nition extends the relationship between algebraiccapacities of polarised algebraic surfaces and ECH capacities of related symplectic 4-manifolds.Next we note how our notion of equivalence from Β§2.1 was motivated by the structure of algebraiccapacities.
Lemma 2.9.
Let p π , πΏ, π΄ q and p π , πΏ , π΄ q be smooth or toric LooΔ³enga pairs. If p π , πΏ, π΄ q and p π , πΏ , π΄ q are equivalent, then π alg π p π, π΄ q β π alg π p π , π΄ q for all π P Z Δ . This is essentially the content of [22, Prop. 3.4 + Prop. 3.5]. It is clear that the anticanonicaldivisors play no role in this result. The value of Lem. 2.9 is in allowing us to ο¬x a particularuniversal LooΔ³enga tower and choose a weight function on it to calculate the algebraic capacitiesof any pseudo-polarised LooΔ³enga tower. We will hence also not specify the function β we havechosen to produce a bona ο¬de LooΔ³enga tower (indexed by Z Δ ) from the poset P p π,πΏ q . We end this subsection by showing that one can capture the algebraic capacities of p Y , A q intrinsically in terms of divisors on Y . Deο¬ne Nef p Y q to be the submonoid of Div ` p Y q R consistingof divisors D such that D Β¨ πΈ Δ πΈ P NE p π π q for each π . Note that Nef p π π q naturally embedsinto Nef p Y q . Set Nef p Y q Z β Nef p Y q X Div p Y q Z . Proposition 2.10. If p Y , A q is a smooth or toric pseudo-polarised LooΔ³enga tower, then π alg π p Y , A q β inf D P Nef p Y q Z t D Β¨ A : D Β¨ p D Β΄ πΎ Y q Δ π u We write D β p π· , π , . . . q for a K -divisor on Y where π· is a K -divisor on π and t π π u π P Z Δ is the weight sequence of D . The assumption that Y is smooth or toric allows us by Lem. 2.9 toreduce to the smooth case where the constraint is given in terms of πΌ p π· q . Proof.
Since Nef p π π q Z can be viewed as a subset of Nef p Y q Z we obtain π alg π p π π , π΄ π q Δ inf D P Nef p Y q Z t D Β¨ A : D Β¨ p D Β΄ πΎ Y q Δ π u for each π , and so π alg π p Y , A q Δ inf D P Nef p Y q Z t D Β¨ A : D Β¨ p D Β΄ πΎ Y q Δ π u For the converse it suο¬ces that for each π Δ π P Z Δ such that π alg π p π π , π΄ π q Δ inf D P Nef p Y q Z t D Β¨ A : D Β¨ p D Β΄ πΎ Y q Δ π u ` π for all π Δ π . Let D β p π· , π , . . . q P Nef p Y q Z be such that D Β¨ A Δ inf D P Nef p Y q Z t D Β¨ A : D Β¨ p D Β΄ πΎ Y q Δ π u ` π As D P Div ` p Y q we must have that π π β π Δ π for some π P Z Δ . There is thus a nef Z -divisor π· π P Nef p π π q for π Δ π that is mapped to D under the embedding Nef p π π q Γ Nef p Y q .Hence, for all π Δ π , π alg π p π π , π΄ π q Δ π· π Β¨ π΄ π β π· π Β¨ A β D Β¨ A Δ inf D P Nef p Y q Z t D Β¨ A : D Β¨ p D Β΄ πΎ Y q Δ π u ` π as required. (cid:3) In fact this inο¬mum is realised in the toric case via a sympletic argument using Prop. 3.1 below.3.
Sub-leading asymptotics of ECH capacities
LooΔ³enga towers and ECH.
To each symplectic 4-manifold p π , π q ECH associates an in-creasing sequence t π ech π p π , π qu π P Z Δ of (extended) real numbers called the ECH capacities of p π , π q . These obstruct symplectic em-beddings in the sense that p π , π q s Γ£ Γ p π , π q ΓΉΓ± π ech π p π , π q Δ π ech π p π , π q for all π Proposition 3.1.
For any convex domain Ξ© Δ R the toric polarised LooΔ³enga tower p Y Ξ© , A Ξ© q has π ech π p π Ξ© q β π alg π p Y Ξ© , A Ξ© q Proof.
Consider the sequence of polygons t Ξ© π u π P Z Δ arising as the polytopes associated to thedivisors π΄ π . We know that lim π Γ8 π ech π p π Ξ© π q β π ech π p π Ξ© q by Hausdorο¬ continuity [2, Lem. 2.3].Since Ξ© π is rational-sloped [21, Thm. 1.5] gives that π ech π p π Ξ© π q β π alg π p π π , π΄ π q and so the resultfollows from Lem. 2.8. (cid:3) Using the same sequence of approximations we prove a result similar to [15, Lem. 3.6].
OOΔ²ENGA TOWERS AND ASYMPTOTICS OF ECH 13
Proposition 3.2 (c.f. [15, Lem. 3.6]) . Let Ξ© be a convex domain whose weight sequence has head π . Then π Β΄ ΓΏ π P wt p Ξ© q π β π p Ξ© q ` π p Ξ© q ` β aο¬ pB ` Ξ© q Proof.
Note that3 π Β΄ ΓΏ π P wt p Ξ© q π β Β΄ πΎ Y Ξ© Β¨ A Ξ© β lim π Γ8 Β΄ πΎ π π Β¨ π΄ π β lim π Γ8 π p Ξ© π q ` π p Ξ© π q ` β aο¬ pB ` Ξ© π q and the result follows from continuity of π pΒ¨q and π pΒ¨q and analysis similar to [15, Lem. 3.6]. (cid:3) Asymptotics for algebraic capacities.
Just like for ECH capacities of symplectic 4-manifoldsand algebraic capacities of pseudo-polarised algebraic surfaces we have a βWeyl lawβ controllingthe growth of π alg π p Y , A q . Theorem 3.3.
Let p YA q be a pseudo-polarised LooΔ³enga tower. Then lim π Γ8 π alg π p Y , A q π β A We will not prove this directly, but will instead appeal to the analysis of the error terms π alg π p Y , A q : β π alg π p Y , A q Β΄ a A π below, where we will show that π alg π p Y , A q β π p q . Thm. 3.3 follows immediately from Prop. 3.1when p Y , A q is a toric polarised LooΔ³enga tower arising from a convex domain.These error terms associated to p Y , A q are analogous to the error terms in ECH π π p π , π q : β π ech π p π , π q Β΄ b p π , π q π and agree when p Y , A q comes from a convex domain.3.3. Bounds for error terms.
For a pseudo-polarised LooΔ³enga tower p Y , A q we deο¬ne a divisor Β΄ πΎ ` Y on Y by Β΄ πΎ ` Y β Β΄ πΎ Y ` πΎ π Β΄ πΎ ` π where Β΄ πΎ ` π is the support of π΄ viewed as a reduced divisor. As a sequence of divisors indexedby π like in Β§2.3, Β΄ πΎ ` Y has as its π th term the support of π΄ π viewed as a reduced divisor. In thissense Β΄ πΎ ` Y can be viewed as the βsupportβ of A . When p Y , A q is actually a pseudo-polarised toricsurface corresponding to a rational-sloped polygon Ξ© , we see that Β΄ πΎ ` Y is the preimage of B ` Ξ© under the moment map, giving Β΄ πΎ ` Y Β¨ A β β aο¬ pB ` Ξ© q . Our next aim is to prove the followingtheorem. Theorem 3.4.
Suppose p Y , A q is a pseudo-polarised LooΔ³enga tower such that Y is smooth or toric. Then πΎ Y Β¨ A Β΄ πΎ ` Y Β¨ A Δ lim sup π Γ8 π alg π p Y , A qΔ lim inf π Γ8 π alg π p Y , A q Δ πΎ Y Β¨ A In particular, π alg π p Y , A q β π p q . Corollary 3.5.
Let π Ξ© be a convex toric domain. Then Β΄ Λ π p Ξ© q ` π p Ξ© q Β΄ β aο¬ pB ` Ξ© q Λ Δ lim sup π Γ8 π π p Y , A qΔ lim inf π Γ8 π π p Y , A q Δ Β΄ Λ π p Ξ© q ` π p Ξ© q ` β aο¬ pB ` Ξ© q Λ When Ξ© has no rational-sloped edges we have that π π p π Ξ© q is convergent and lim π Γ8 π π p π Ξ© q β Β΄ p π p Ξ© q ` π p Ξ© qq Over the next two subsections we will establish these asymptotic upper and lower bounds.Since β aο¬ pB ` Ξ© q β B ` Ξ© has no rational-sloped edges the criterion for convergence followsimmediately.We will assume that Y is smooth, passing to the singular toric case by [22, Prop. 4.19] thateasily extends to the case of toric LooΔ³enga towers.3.4. Upper bound for error terms.
Observe that any nef Z -divisor π· on a Q -factorial surface π gives an upper bound π alg π p π, π΄ q Δ π· Β¨ π΄ when 2 π Δ πΌ p π· q . By [22, Prop. 2.11] this also works if π· is an eο¬ective Z -divisor. Let p Y , A q βtp π π , πΏ π , π΄ π qu π P Z Δ be a pseudo-polarised LooΔ³enga tower. We obtain an upper bound for π alg π p π π , π΄ π q in terms of π and π by using Z -divisors of the form r ππ΄ π s and then consideringhow the resulting bound behaves as π and π become large. We let the components of π΄ π bedenoted π· , . . . , π· π ; that is, Β΄ πΎ ` π π β Ε π π β π· π .Consider the constraint πΌ p r ππ΄ π s q β p ππ΄ π ` Ξ π q Β¨ p ππ΄ π ` Ξ π Β΄ πΎ π π q Δ π where Ξ π β r ππ΄ π s Β΄ ππ΄ π . That is, π π΄ π Β΄ ππ΄ π Β¨ πΎ π π ` ππ΄ π Β¨ Ξ π Β΄ π ` Ξ π Β΄ Ξ π Β¨ πΎ π π Δ ππ΄ π Β¨ Ξ π Δ Ξ π is eο¬ective and so we ignore that term. We bound Ξ π Β¨ Ξ π Β΄ Ξ π Β¨ πΎ π π in terms of the geometry of π π . Notice that Ξ π Δ ΓΏ π· π Δ π· π and Β΄ Ξ π Β¨ πΎ π π Δ ΓΏ π· π ΔΒ΄ p ` π· π q giving Ξ π Β΄ Ξ Β¨ πΎ π π Δ Β΄ tpΒ΄ q -curves on π π u ` ΓΏ π· π ΔΒ΄ p ` π· π q Hence we see that πΌ p r ππ΄ π s q Δ π when π π΄ π Β΄ ππ΄ π Β¨ πΎ π π Β΄ π Β΄ tpΒ΄ q -curves on π π u ` ΓΏ π· π ΔΒ΄ p ` π· π q Δ π is bounded below by the larger solution of the quadratic obtained by replacing Δ with β in the above. Write Β΄ π΄ π Β¨ πΎ π π { π΄ π β : π π . We thus have πΌ p r ππ΄ π s q Δ π if π Δ Β΄ π π ` gffe ππ΄ π ` tpΒ΄ q -curves on π π u Β΄ Ε π· π ΔΒ΄ p ` π· π q π΄ π ` π π π΄ π Set πΉ p π q β tpΒ΄ q -curves on π π u Β΄ ΓΏ π· π ΔΒ΄ p ` π· π q We study how πΉ p π q changes with π by measuring πΉ p π ` qΒ΄ πΉ p π q ; i.e. how πΉ changes under a singleblowup in a torus-ο¬xed point between two torus-invariant curves πΆ and πΆ . Let t π, π u β t , u .The options are: OOΔ²ENGA TOWERS AND ASYMPTOTICS OF ECH 15 β πΆ Δ πΆ Δ ΓΉΓ± πΉ p π ` q Β΄ πΉ p π q β β πΆ π β πΆ π Δ ΓΉΓ± πΉ p π ` q Β΄ πΉ p π q β β πΆ π β Β΄ πΆ π Δ ΓΉΓ± πΉ p π ` q Β΄ πΉ p π q β β πΆ π Δ Β΄ πΆ π Δ ΓΉΓ± πΉ p π ` q Β΄ πΉ p π q β β πΆ π β Β΄ πΆ π β ΓΉΓ± πΉ p π ` q Β΄ πΉ p π q β β πΆ π ββ πΆ π β Β΄ ΓΉΓ± πΉ p π ` q Β΄ πΉ p π q β β πΆ π Δ Β΄ πΆ π β ΓΉΓ± πΉ p π ` q Β΄ πΉ p π q β β πΆ π Δ Β΄ πΆ π β Β΄ ΓΉΓ± πΉ p π ` q Β΄ πΉ p π q β β πΆ π Δ Β΄ πΆ π Δ Β΄ ΓΉΓ± πΉ p π ` q Β΄ πΉ p π q β πΉ p π ` q Β΄ πΉ p π q Δ
5. In the toric case we have π β P and so πΉ p π q Δ π since P has no negative curves. In general we will have πΉ p π q Δ π ` πΉ p q but, as it makes nosigniο¬cant diο¬erence to the argument, we will ignore the constant for notational convenience.Therefore we see that πΌ p r ππ΄ π s q Δ π when π Δ Β΄ π π ` d ππ΄ π ` ππ΄ π ` π π π΄ π β : π π,π It follows that π alg π p π π , π΄ π q Δ r π π,π π΄ π s Β¨ π΄ π Δ π π,π π΄ π Β΄ πΎ ` π π Β¨ π΄ π β Β΄ π π π΄ π ` d π΄ π π ` π΄ π π ` π π p π΄ π q Β΄ πΎ ` π π Β¨ π΄ π This is an explicit bound for π alg π p π π , π΄ π q valid for all π and π . We require an elementary lemmafrom analysis to study what happens as π and π get large. Lemma 3.6.
Suppose p π π q is a decreasing summable sequence. Let π p π q β Ε π Δ π π π . Then there exists astrictly increasing sequence p π π q of natural numbers such that π π β π p? π q and π p π π q β π p {? π q . We use some basic techniques from probability theory to prove this result, though a ratherlonger but completely elementary proof also exists. Notice that it makes no diο¬erence to demandthat π π β π p π q and π p π π q β π p { π q instead of π π β π p? π q and π p π π q β π p {? π q , which we adoptfor notational convenience. Proof.
We ο¬rst show that π π β π p π q . We can choose p π π q to be non-increasing and so we mayinterpret it as the tail probabilities π π β π p π Δ π q for some random variable π with values in N .As π π is summable, π has ο¬nite expectation: E π β Ε π π p π Δ π q β Ε π π π Δ 8 . Now, π Β¨ π π β π Β¨ π p π Δ π q β π Β¨ E π Δ π β E π π Δ π Δ E π π Δ π which approaches 0 as π Γ 8 by the dominated convergence theorem. It follows that π π Δ π π { π for some π π P π p q , which again without loss of generality we may choose to be decreasing. Wenow deο¬ne π π β inf t π‘ : Ε π Δ π‘ π π Δ π Β΄ u . Claim 3.7. π π β π p π q . Computing tails we ο¬nd, using the monotonicity of p π π q ,( β ) ΓΏ π Δ π‘ π π Δ ΓΏ π Δ π‘ π π { π Δ π π‘ ΓΏ π Δ π‘ π Β΄ β π π‘ { π‘ , Deο¬ne π π β inf t π : π π { π Δ π Β΄ u . We see that π π Δ π π , so it suο¬ces that π π β π p π q . But bydeο¬nition, p π π Β΄ q{ π Δ π π π Β΄ P π p q and so we have shown the claim.To ο¬nish the proof, we know from ( β ) that π p π‘ q β π p π‘ Β΄ q , i.e. π p π‘ q Δ π π‘ { π‘ for some non-increasing π π‘ P π p q . Consequently, we are looking for a sequence π π β π p π q such that π π π { π π β π ` π Β΄ Λ , or equivalently for a sequence π π β π p q for which π ππ π { π π β π p q . Here is a constructionof such a sequence π π : β Deο¬ne π‘ π β inf t π‘ : π π‘ Δ Β΄ π u . β Set π π β Ε π β t ππ‘ π Δ π π Δp π ` q π‘ π ` u π Β΄ .These π π are certainly π p q , and with π p π q β sup t π : ππ‘ π Δ π u we have π ππ π { π π β π π Β¨ π p π q Β΄ Β¨ π p π q Δ π π‘ π p π q Β¨ π p π q β π p π q Β¨ Β΄ π p π q β π p q as desired. (cid:3) In this context Lemma 3.6 implies that there is a function π p π q that that depends only on p Y , A q and is π p? π q such that Ε π Δ π p π q π π β π p {? π q . It follows that | π΄ π π Β΄ A | β π p {? π q . Since π alg π p Y , A q Δ π alg π p π π , π΄ π q for all π and π we get π alg π p Y , A q Δ Β΄ π π p π q π΄ π p π q Β΄ πΎ ` π π p π q Β¨ π΄ π p π q ` d π΄ π p π q π ` π΄ π p π q π p π q ` p π π p π q π΄ π p π q q Β΄ a A π β Β΄ π π p π q π΄ π p π q Β΄ πΎ ` π π p π q Β¨ π΄ π p π q ` d Λ A ` π Λ ? π ΛΛ π ` Λ A ` π Λ ? π ΛΛ p π p π q ` q ` π p q Β΄ a A π β Β΄ π π p π q π΄ π p π q Β΄ πΎ ` π π p π q Β¨ π΄ π p π q ` b A π ` π p? π q Β΄ a A π By letting π Γ 8 and substituting π π π΄ π β Β΄ π΄ π Β¨ πΎ π π we achieve the following. Proposition 3.8.
Let p Y , A q be a pseudo-polarised LooΔ³enga tower. Then, lim sup π Γ8 π alg π p Y , A q Δ πΎ Y Β¨ A Β΄ πΎ ` Y Β¨ A We convert this into combinatorial language.
Corollary 3.9.
Let Ξ© be a convex domain. Then, lim sup π Γ8 π π p π Ξ© q Δ Β΄ Λ π p Ξ© q ` π p Ξ© q Β΄ β aο¬ pB ` Ξ© q Λ In particular, if B ` Ξ© has no rational-sloped edge then lim sup π Γ8 π π p π Ξ© q Δ Β΄ p π p Ξ© q ` π p Ξ© qq Lower bound for error terms.
To deduce a lower bound we can in fact generalise to thesetting of a tower of blowups Y β tp π π , π΄ π qu π P Z Δ of polarised surfaces where Β΄ πΎ Y is βeο¬ectiveββ that is, each Β΄ πΎ π π is eο¬ective. DenoteNS p π q π΄ Δ : β t π· P NS p π q : π· Β¨ π΄ Δ u Deο¬ne for a pseudo-polarised surface p π , π΄ q π ` π p π, π΄ q : β inf NS p π q π΄ Δ t π· Β¨ π΄ : π· Β¨ p π· Β΄ πΎ π q Δ π u This is a variation on the asymptotic capacity π asy π p π , π΄ q from [22, Β§4.1] or the estimate using theβapproximate ECH indexβ of [15, Β§5.2]. These invariants will have preferable numerics to studylower bounds for π alg π p π, π΄ q . It is already clear that π ` π p π, π΄ q Δ π alg π p π, π΄ q for all π . As usual we write π β Β΄ πΎ π Β¨ π΄ { π΄ . OOΔ²ENGA TOWERS AND ASYMPTOTICS OF ECH 17
Lemma 3.10.
Suppose p π, π΄ q is a pseudo-polarised surface such that π is smooth or toric. If π is nottoric, assume that Β΄ πΎ π is eο¬ective. When π Δ Β΄ p πΎ π Β¨ π΄ q π΄ Β΄ πΎ π Β― we have π ` π p π, π΄ q β πΎ π Β¨ π΄ ` b πΎ π π΄ ` π΄ π Proof.
Without loss of generality, we assume that π is smooth. From the Hodge index theoremwe have an orthogonal basis π΄, π , . . . , π π of NS p π q . Set π π β Β΄ π π . Let Β΄ πΎ π β π π΄ ` Ε πΏ π π π . Wesee that an optimiser for π ` π p π, π΄ q is π· π β π π π΄ Β΄ ΓΏ πΏ π π π where π π is the smallest nonnegative real number π such that π p π ` π q Δ π΄ Λ π Β΄ ΓΏ πΏ π π π ΒΈ Solving for π , we see that the two solutions are Β΄ π Λ d π Β΄ ΓΏ πΏ π π π π΄ ` ππ΄ We also note that πΎ π β π π΄ Β΄ ΓΏ πΏ π π π and so the solutions for π can be rewritten as Β΄ π Λ d πΎ π π΄ ` ππ΄ There is a unique nonnegative solution given by the larger value of π precisely when π Δ πΎ π π΄ ` ππ΄ or when π Δ π π΄ Β΄ πΎ π β Λ p πΎ π Β¨ π΄ q π΄ Β΄ πΎ π Λ Substituting in the larger value for π π gives the result. (cid:3) Note that πΎ π π β πΎ π Β΄ π . Hence, we see that π alg π p π π , π΄ π q Δ π ` π p π π , π΄ π q β πΎ π π Β¨ π΄ π ` b πΎ π π π΄ π ` π΄ π π for all π Δ Β΄ p πΎ ππ Β¨ π΄ π q π΄ π Β΄ πΎ π π Β― β π ` Β΄ p πΎ ππ Β¨ π΄ π q π΄ π Β΄ πΎ π Β― . For notational convenience we notethat p πΎ π π Β¨ π΄ π q π΄ π Β΄ πΎ π Δ p πΎ π Β¨ π΄ q A Β΄ πΎ π β π π Β΄ Ε π π Β΄ πΎ π β : π We choose a sequence π π as in Lemma 3.6 with π π β π p? π q and π΄ π π Β΄ A β π p {? π q . Forsuο¬ciently large π we have π π ` π Δ π . Then, for all such π we have π alg π p π π π , π΄ π π q Δ πΎ π ππ Β¨ π΄ π π ` b πΎ π ππ π΄ π π ` π΄ π π π β πΎ π ππ Β¨ π΄ π π ` d p πΎ π Β΄ π π q Λ A ` π Λ ? π ΛΛ ` Λ A ` π Λ ? π ΛΛ π β πΎ π ππ Β¨ π΄ π π ` d p πΎ π ` π p? π qq Λ A ` π Λ ? π ΛΛ ` Λ A ` π Λ ? π ΛΛ π β πΎ π ππ Β¨ π΄ π π ` b A π ` π p? π q As a result, letting π Γ 8 giveslim inf π Γ8 π alg π p Y , A q Δ lim π Γ8 πΎ π ππ Β¨ π΄ π π ` b A π ` π p? π q Β΄ a A π β πΎ Y Β¨ A Proposition 3.11.
Let p Y , A q be a pseudo-polarised LooΔ³enga tower with Y either smooth or toric. Then lim inf π Γ8 π alg π p Y , A q Δ πΎ Y Β¨ A This implies the following in combinatorial terms.
Corollary 3.12.
Suppose π Ξ© is a convex toric domain. Then lim inf π Γ8 π π p π Ξ© q Δ Β΄ ` π p Ξ© q ` π p Ξ© q ` β aο¬ pB ` Ξ© q Λ In particular, if Ξ© has no rational-sloped edge then lim inf π Γ8 π π p π Ξ© q Δ Β΄ p π p Ξ© q ` π p Ξ© qq This completes the proof of Thm. 3.4.3.6.
Concave toric domains.
We deduce the analogue of Cor. 3.5 for concave domains by usinga formal property of toric ECH. The formal property in question is described by the following.
Proposition 3.13 ( [4, Thm. A.1]) . Suppose Ξ© is a convex toric domain with weight sequence given by wt p Ξ© q β p π ; wt p Ξ© q , wt p Ξ© qq where Ξ© , Ξ© are concave domains as in Def. 2.5. Then π ech π p π Ξ© q β inf π ,π Δ t π ech π ` π ` π p π΅ p π qq Β΄ π ech π p π Ξ© q Β΄ π π p π Ξ© qu Let Ξ be a concave toric domain. It is clear that there exists a convex domain Ξ© such that either Ξ© β Ξ and Ξ© β H , or Ξ© β H and Ξ© β Ξ . We assume the former without loss of generality. Proposition 3.14.
Let π Ξ be a concave toric domain. Then lim inf π Γ8 π π p π Ξ q Δ Β΄ p π p Ξ q ` π p Ξ q ` β aο¬ pB ` Ξ qq Proof.
Let Ξ© be as discussed above and let π be the head of wt p Ξ© q . Then π ech π p π Ξ© q β inf π Δ t π ech π p π΅ p π qq Β΄ π ech π p π Ξ qu This inο¬mum is attained for each π ; we denote an optimiser for π by π so that π ech π p π Ξ© q β π ech π p π΅ p π qq Β΄ π ech π p π Ξ q OOΔ²ENGA TOWERS AND ASYMPTOTICS OF ECH 19
Thus π π p π Ξ q is given by π ech π ` π p π΅ p π qq Β΄ π ech π p π Ξ© q Β΄ b p π Ξ q π β π π ` π p π΅ p π qq Β΄ π π p π Ξ© q ` b p vol p π Ξ© q ` vol p π Ξ qqp π ` π q Β΄ b p π Ξ© q π Β΄ b p π Ξ q π From Cor. 3.5 we see that π π p π Ξ© q and π π p π΅ p π qq are bounded and so it follows that π π p π Ξ q isbounded below by Β΄ π ` p π p Ξ© q ` π p Ξ© q ` β aο¬ pB ` Ξ© qq β Β΄ p π p Ξ q ` π p Ξ q ` β aο¬ pB ` Ξ qq using the CauchyβSchwartz inequality. (cid:3) Hutchings shows in [15, Cor. 3.9] that π π p π Ξ q is bounded above by Β΄ Ε π P wt p Ξ q π . We henceobtain the following. Theorem 3.15.
Let π Ξ be a concave toric domain. Then Β΄ p π p Ξ q ` π p Ξ q Β΄ β aο¬ pB ` Ξ qq Δ lim sup π Γ8 π π p π Ξ qΔ lim inf π Γ8 π π p π Ξ q Δ Β΄ p π p Ξ q ` π p Ξ q ` β aο¬ pB ` Ξ qq and so π π p π Ξ q β π p q . If B ` Ξ has no rational-sloped edges then lim π Γ8 π π p π Ξ q β Β΄ p π p Ξ q ` π p Ξ qq Proof.
The bounds follow immediately from Thm. 3.14 and [15, Cor. 3.9] in combination with [15,Lem. 3.6]. From here convergence is clear when B ` Ξ has no rational-sloped edges. (cid:3) Algebraic analogues of rational-sloped edges.
We discuss the geometric analogue for po-larised LooΔ³enga towers of the combinatorial condition on convex domains of having a rational-sloped edge. In particular, this supplies a criterion for convergence for π alg π p Y , A q in this generality.Given a poset P deο¬ne its extended poset p P to be P Y t8u with π for all π P P . If p Y , A q is a pseudo-polarised LooΔ³enga tower we can deο¬ne an weight function on the extended poset p P p π ,πΏ q by setting wt p8q β Β΄ πΎ ` π Β¨ π΄ . Deο¬ne a subposet p P p π ,πΏ q p π q as follows: β π is the unique maximal element of P p π ,πΏ q p π q , β if π P p P p π ,πΏ q p π q then exactly one direct descendant of π is in p P p π ,πΏ q p π q , namely the directdescendant corresponding to the point of intersection of πΈ π and the strict transform of πΈ π in π π .This all works similarly for the weighted poset p P Ξ© associated to a convex domain Ξ© ; forinstance, the weight of the element is the aο¬ne length of the possibly empty edge of slope p , Β΄ q in B ` Ξ© , and one can interpret each element π of p P Ξ© p π q with direct ancestor π as thevertex of Ξ© π incident to the edge that is the moment image of (the strict transform of) πΈ π . Lemma 3.16.
Let π Ξ© be a convex toric domain. Let p P Ξ© be the extended weighted poset associated to Ξ© .Then there is a bΔ³ectionrational-sloped edges in B ` Ξ© ΓΓ π P p P Ξ© such that wt p π q Β΄ ΓΏ π P P Ξ© p π q wt p π q Δ Proof.
It follows from the weight sequence recursion and the construction of P Ξ© p π q that wt p π q Β΄ Ε π P P Ξ© p π q wt p π q is the aο¬ne length of the (possibly empty) edge in B ` Ξ© introduced at the stepcorresponding to π in the recursion. Rational-sloped edges in B ` Ξ© are exactly such edges thathave nonzero aο¬ne length, which gives the result. (cid:3) We see that the extension of P Ξ© was necessary to capture the (possiby empty) edge of slope p , Β΄ q from the ο¬rst step of the recursion. Deο¬nition 3.17.
We say that a pseudo-polarised LooΔ³enga tower p Y , A q is balanced if wt p π q Β΄ Ε π P P p π ,πΏ q p π q wt p π q β π P p P p π ,πΏ q .This is the algebraic analogue for pseudo-polarised LooΔ³enga towers of having no rational-sloped edges in the case of convex domains. Proposition 3.18.
Suppose p Y , A q is a pseudo-polarised LooΔ³enga tower that is balanced. Then π alg π p Y , A q is convergent with lim π Γ8 π alg π p Y , A q β πΎ Y Β¨ A β p πΎ π Β΄ πΎ ` π q Β¨ π΄ Proof.
We already have12 πΎ Y Β¨ A Β΄ πΎ ` Y Β¨ A Δ lim sup π Γ8 π alg π p Y , A q Δ lim inf π Γ8 π alg π p Y , A q Δ πΎ Y Β¨ A from Thm. 3.4, and so it suο¬ces to show that Β΄ πΎ ` Y Β¨ A β p Y , A q is balanced. We have Β΄ πΎ ` Y Β¨ A β Β΄ πΎ Y Β¨ A ` πΎ π Β¨ A Β΄ πΎ ` π Β¨ A β Β΄ πΎ π Β¨ π΄ Β΄ ΓΏ π P p P p π ,πΏ q wt p π q ` πΎ π Β¨ π΄ Β΄ πΎ ` π Β¨ π΄ β wt p8q Β΄ ΓΏ π P p P p π ,πΏ q wt p π q Let S p q β t8u . Recursively deο¬ne S p π q to be the set of maxima of p P p π ,πΏ q z Δ π Δ π Δ π P S p π q p P p π ,πΏ q p π q By construction we have from the above that Β΄ πΎ ` Y Β¨ A β ΓΏ π β ΓΏ π P S p π q Β¨ΛΛ wt p π q Β΄ ΓΏ π P p P p π ,πΏ q p π q wt p π q ΛβΉβ which is zero by the assumption that p Y , A q is balanced. The second equality in the statementfollows from Β΄ πΎ ` Y Β¨ A β (cid:3) Outlook.
We conclude with a selection of ideas and observations that we hope will lead tostronger criteria for convergence or, if one is even expressible, a complete description of whatβgenericβ means in Hutchingsβ conjecture [15, Conj. 1.5].Given convex or concave Ξ© we let π p Ξ© q be the Q -vector subspace of R spanned by the aο¬nelengths of rational-sloped edges in B ` Ξ© . We denote the dimension of π p Ξ© q by π£ p Ξ© q .Let π Ξ© be a convex or concave toric domain. We believe that two ingredients for strongerconvergence criteria are this π£ p Ξ© q and the number π p Ξ© q of rational-sloped edges in B ` Ξ© .If π p Ξ© q Δ 8 then we suspect π π p π Ξ© q converges if π£ p Ξ© q Ββ
1. If Ξ© has inο¬nitely many rational-sloped edges then it seems likely that π π p π Ξ© q converges. In each case of convergence we expectthat the limit is( Λ ) Β΄
12 Ru p π Ξ© q β Β΄ p π p Ξ© q ` π p Ξ© qq though it is possible that there are toric domains for which π π converge but that are not genericin the sense that they do not satisfy Hutchingsβ conjecture and have limit diο¬erent to ( Λ ). In thecase of non-convergence, we expect that ( Λ ) is the midpoint of the lim inf and lim sup of π π p π Ξ© q . OOΔ²ENGA TOWERS AND ASYMPTOTICS OF ECH 21
Note that the case π£ p Ξ© q β Ξ© having no rational-sloped edges β which is coveredby Cor. 3.5 and Thm. 3.15 β and π£ p Ξ© q Δ Ξ© having at least two rational-slopededges whose aο¬ne lengths are independent over Q .There is a distinction between the case that Ξ© is a of scaled-lattice type as in [22] β that is, where Ξ© β π Ξ© for some lattice polygon Ξ© and some π P R Δ β and the complementary case: whereeither Ξ© is polytopal and has π£ p Ξ© q Δ
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Department of Mathematics & Statistics, Washington University in St. Louis, St. Louis, MO, 63130, USA
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