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 LOOฤฒENGA 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 ๐ p q . We obtain su๏ฌcient 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 Looฤณenga pairs. Introduction
We outline the symplectic part of the story โ in particular, the applications to sub-leadingasymptotics of ECH capacities โ in ยง1.1 and describe the novel aspects of our algebro-geometricmethods and constructions in ยง1.2.1.1.
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 ๐ , ๐ q and p ๐ , ๐ q of the same dimension, when does there exist a smooth embedding ๐ : ๐ ร ๐ such that ๐ ห ๐ โ ๐ ? Such embeddings are called symplectic embeddings . If there is a symplecticembedding p ๐ , ๐ q ร p ๐ , ๐ q we write p ๐ , ๐ q s รฃ ร p ๐ , ๐ q .For each symplectic embedding problem there is a โconstructiveโ aspect in which the aim is toshow the existence of a symplectic embeddings. Conversely there is an โobstructiveโ aspect thatusually involves ๏ฌnding invariants that are โmonotoneโ 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 ๐ , ๐ q ECH associates a non-decreasing sequence t ๐ ech ๐ p ๐ , ๐ qu ๐ P Z ฤ of (extended) real numbers such that p ๐ , ๐ q s รฃ ร p ๐ , ๐ q รนรฑ ๐ ech ๐ p ๐ , ๐ q ฤ ๐ ech ๐ p ๐ , ๐ q for all ๐ P Z ฤ . ECH capacities have found many applications to notable embedding problems[2, 4, 18] and have been shown to have signi๏ฌcant 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 re๏ฌnethese connections in the context of a wider class of spaces.The algebraic analogues โ algebraic capacities โ 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 ๐ ร 8 [5, 22]. To a pair p ๐, ๐ด q consisting of a projectivealgebraic surface ๐ with an ample (or big and nef) R -divisor ๐ด โ a polarised surface โ the associatedalgebraic capacities form a non-decreasing sequence t ๐ alg ๐ p ๐, ๐ด qu ๐ P Z ฤ 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 โWeyl lawโ for ECH. Theorem 1 ( [7, Thm. 1.1]) . Suppose p ๐ , ๐ q is a Liouville domain such that all ๐ ech ๐ p ๐ , ๐ q are ๏ฌnite.Then lim ๐ ร8 ๐ ech ๐ p ๐ ฮฉ q ๐ โ p ๐ , ๐ q In other words ๐ ech ๐ p ๐ , ๐ q โ a p ๐ , ๐ q ๐ . This was used by Cristofaro-GardinerโHutchings[6] to prove a re๏ฌnement of the Weinstein conjecture. One can hence de๏ฌne error terms ๐ ๐ p ๐ , ๐ q : โ ๐ ech ๐ p ๐ , ๐ q ยด b p ๐ , ๐ q These are manifestly ๐ p? ๐ q but we consider what more there is to say. A series of estimates dueto Sun [20], Cristofaro-GardinerโSavale [10], and Hutchings [15] found bounds of the form ๐ p ๐ ๐ q for ๐ โ { , ๐ โ { , ๐ โ { Conjecture 1.
Let p ๐ , ๐ q be a star-shaped domain in R . (i) (c.f. [15, Ex. 1.6]) ๐ ๐ p ๐ , ๐ q โ ๐ p q . (ii) ( [15, Conj. 1.5]) If p ๐ , ๐ q is generic then ๐ ๐ p ๐ , ๐ q converges with limit lim ๐ ร8 ๐ ๐ p ๐ , ๐ q โ ยด
12 Ru p ๐ , ๐ q where Ru p ๐ , ๐ q is the Ruelle invariant of p ๐ , ๐ q . The Ruelle invariant was de๏ฌned for nice star-shaped domains in R by Hutchings [15, Def. 1.4]following ideas of Ruelle [19], and one can extend this de๏ฌnition to more general star-shapeddomains by continuity. Some of the results of this paper can be viewed as seeking to makeprecise what โgenericโ means.We consider a class of toric symplectic 4-manifolds that will play a central role for us. Suppose ฮฉ ฤ R ฤ is a simply-connected region. De๏ฌne the toric domain ๐ ฮฉ : โ ๐ ยด p ฮฉ q where ๐ : C ร R is the moment map for the ๐ ห ๐ -action on C . If B ฮฉ intersects the coordinateaxes in sets of the form tp ๐ฅ, q : ๐ฅ P r , ๐ su and tp , ๐ฆ q : ๐ฆ P r , ๐ su and the remaining part ofthe boundary โ which we denote by B ` ฮฉ โ is a convex curve (that is, the region ฮฉ is convex),we say that ฮฉ is a convex domain and that ๐ ฮฉ is a convex toric domain . This class includes balls,ellipsoids, polydisks, and many other classic symplectic manifolds. We denote the quantities ๐ and ๐ appearing above by ๐ p ฮฉ q and ๐ p ฮฉ q .In [22, Thm. 4.10] it was shown that when ฮฉ โ ๐ ฮฉ for some lattice polygon ฮฉ and some ๐ P R ฤ , we have ๐ ๐ p ๐ ฮฉ q โ ๐ p q with an explicit calculation of the lim sup and lim inf, which are always di๏ฌerent and so ๐ ๐ p ๐ ฮฉ q does not converge in this situation. We say that such ฮฉ are of scaled-lattice type . The approach toproving this result uses the fact that the interior ๐ ห ฮฉ can be realised as the complement of an ampledivisor ๐ด ฮฉ in the (possibly singular) projective toric surface ๐ ฮฉ associated to ฮฉ [3, ยง2.3], and thatby [21, Thm. 1.5] the ECH capacities ๐ ech ๐ p ฮฉ q are given by the algebraic capacities ๐ alg ๐ p ๐ ฮฉ , ๐ด ฮฉ q associated to the pair p ๐ ฮฉ , ๐ด ฮฉ q . OOฤฒENGA TOWERS AND ASYMPTOTICS OF ECH 3
In [15] Hutchings showed that Conj. 1(ii) holds for ๐ ฮฉ when ฮฉ is โstrictly convexโ : namely,has smooth boundary and all outward normals to B ` ฮฉ live in the strictly positive quadrant of R . The Ruelle invariant in this setting is equal to ๐ p ฮฉ q ` ๐ p ฮฉ q .These two cases will be subsumed in the following theorem. The a๏ฌne length โ a๏ฌ p ๐ฃ q of a vector ๐ฃ P R is the pseudonorm de๏ฌned by 0 if ๐๐ฃ R Z for any ๐ P R ฤ and by 1 if ๐ฃ is a primitivevector in Z . We de๏ฌne the a๏ฌne length of a continuous curve by the (possibly empty) sum ofthe a๏ฌne lengths of the direction vectors de๏ฌning each linear segment of the curve. Theorem 2 (Cor. 3.5) . Let ๐ ฮฉ be a convex toric domain, then ยด ห ๐ p ฮฉ q ` ๐ p ฮฉ q ยด โ a๏ฌ pB ` ฮฉ q ห ฤ lim sup ๐ ร8 ๐ ๐ p ๐ ฮฉ qฤ lim inf ๐ ร8 ๐ ๐ p ๐ ฮฉ q ฤ ยด ห ๐ p ฮฉ q ` ๐ p ฮฉ q ` โ a๏ฌ pB ` ฮฉ q ห where B ` ฮฉ is the part of B ฮฉ not on the coordinate axes. In particular, ๐ ๐ p ๐ ฮฉ q โ ๐ p q . Notice that the upper and lower bounds can easily be translated to involve the Ruelle invariant,and that their midpoint is exactly ยด Ru p ๐ ฮฉ q when ๐ ฮฉ is strictly convex. We immediately obtainthe following corollary generalising [15, Thm. 1.10]. Corollary 1 (Cor. 3.5) . Let ๐ ฮฉ be a convex toric domain. When B ` ฮฉ has no rational-sloped edges wehave that ๐ ๐ p ๐ ฮฉ q is convergent and lim ๐ ร8 ๐ ๐ p ๐ ฮฉ q โ ยด p ๐ p ฮฉ q ` ๐ p ฮฉ qq If a region ฮ ฤ R ฤ is bounded above by the graph of a convex function ๐ : r , ๐ s ร R ฤ we saythat ๐ ฮ 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 ๐ ฮ be a concave toric domain. Then ยด ` ๐ p ฮ q ` ๐ p ฮ q ยด โ a๏ฌ pB ` ฮ q ห ฤ lim sup ๐ ร8 ๐ ๐ p ๐ ฮ qฤ lim inf ๐ ร8 ๐ ๐ p ๐ ฮ q ฤ ยด ` ๐ p ฮ q ` ๐ p ฮ q ` โ a๏ฌ pB ` ฮ q ห and so ๐ ๐ p ๐ ฮ q โ ๐ p q . If B ` ฮ has no rational-sloped edges then lim ๐ ร8 ๐ ๐ p ๐ ฮ q โ ยด p ๐ p ฮ q ` ๐ p ฮ qq We note that one consequence of better understanding the asymptotics of ๐ ๐ p ๐ , ๐ q is to obtain๏ฌner 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 ๐ ฮฉ is โadmissibleโ if either ฮฉ is concave or convex and B ` ฮฉ has no rational-sloped edges, or if ฮฉ is convex and of scaled-lattice type. Corollary 2.
Let ๐ ฮฉ and ๐ ฮฉ be admissible toric domains of the same symplectic volume. Then ๐ ห ฮฉ s รฃ ร ๐ ฮฉ รนรฑ โ a๏ฌ pB ฮฉ q ฤ โ a๏ฌ pB ฮฉ q We conclude in ยง3.8 by discussing two invariants โ the number of rational-sloped edges in B ` ฮฉ and the degree of independence over Q of their a๏ฌne lengths โ that we believe might furthergovern the asymptotics of ๐ ๐ p ๐ ฮฉ q for convex and concave toric domains. Hutchingsโ result also shows that Conj. 1(ii) is true when ฮฉ is โstrictly concaveโ. B. WORMLEIGHTON
Algebraic perspective.
Our approach to the results in ยง1.1 is to identify an algebraic objectwhose โalgebraic capacitiesโ agree with the ECH capacities of a convex toric domain ๐ ฮฉ . We willgive the formal de๏ฌnition in ยง2.5 but, in short, one can think of algebraic capacities as positivityinvariants of a polarised surface p ๐, ๐ด q obtained as solutions to quadratic optimisation problemson the nef cone of ๐ . We denote the ๐ th algebraic capacity of p ๐, ๐ด q by ๐ alg ๐ p ๐, ๐ด q .As discussed above, when ฮฉ is a rational-sloped polygon one can recover the ECH capacitiesof ๐ ฮฉ as the algebraic capacities of the polarised toric surface p ๐ ฮฉ , ๐ด ฮฉ q corresponding to ฮฉ .When ฮฉ is a non-polytopal convex domain, we will use the weight expansion of ฮฉ [2, 17] in ยง2.4 tode๏ฌne a tower of polarised toric surfaces p ๐ , ๐ด q ๐ รร p ๐ , ๐ด q ๐ รร . . . ๐ ๐ รร p ๐ ๐ , ๐ด ๐ q ๐ ๐ ` รร . . . for which there exists a notion of algebraic capacities extending the de๏ฌnition for polarisedsurfaces. We denote such towers of polarised surfaces by calligraphic letters p Y , A q and denotetheir algebraic capacities by ๐ alg ๐ p Y , A q . Proposition 1 (Prop. 3.1) . Let ฮฉ be a convex domain, and let p Y ฮฉ , A ฮฉ q denote the tower of polarisedtoric surfaces associated to ฮฉ . Then ๐ ech ๐ p ๐ ฮฉ q โ ๐ alg ๐ p Y ฮฉ , A ฮฉ q for all ๐ P Z ฤ . 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 ฮฉ .Just as in [22] we ๏ฌnd 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 Looฤณenga pair [11, 16] is a pair p ๐ , ๐ฟ q consisting of a Q -factorial rationalsurface ๐ with a singular nodal curve ๐ฟ P |ยด ๐พ ๐ | . Recall that Q -factorial means that an integermultiple of each Weil divisor on ๐ is Cartier. Note that elsewhere in the literature it is standardto assume that ๐ is smooth, in which case ๐ฟ is either an irreducible rational nodal curve or a cycleof smooth rational curves.We consider Looฤณenga 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 Looฤณenga 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 polarisedLooฤณenga towers in ยง2.3, including a natural notion of divisor (which includes the polarisation A ), an intersection pairing, and a notion of canonical divisor ๐พ Y . Theorem 4 (Thm. 3.3 + Thm. 3.4) . Let p Y , A q be a polarised Looฤณenga tower. Then ๐ alg ๐ p Y , A q โ ? A ๐ and the error terms ๐ alg ๐ p Y , A q : โ ๐ alg ๐ p Y , A q ยด ? A ๐ satisfy ๐พ Y ยจ A ยด ๐พ ` Y ยจ A ฤ lim sup ๐ ร8 ๐ alg ๐ p Y , A qฤ lim inf ๐ ร8 ๐ alg ๐ p Y , A q ฤ ๐พ Y ยจ A where ๐พ ` Y is a divisor on Y canonically associated to p Y , A q . OOฤฒENGA 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 ฮฉ we calculate ๐พ Y ยจ A โ ยด โ a๏ฌ pB ฮฉ q โ ยด ` ๐ p ฮฉ q ` ๐ p ฮฉ q ` โ a๏ฌ pB ` ฮฉ q ห and ยด ๐พ ` Y ยจ A โ โ a๏ฌ pB ` ฮฉ q establishing Thm. 2 and its consequences from ยง1.1. 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 โintrinsicโ 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 รan-Daniel Erdmann-Pham for providing the proof of Lemma 3.6.2.
Towers of Looijenga pairs
Looฤณenga pairs and Looฤณenga towers.
In our context we will de๏ฌne a
Looฤณenga pair to be apair p ๐ , ๐ฟ q consisting of: โ a Q -factorial rational surface ๐ , โ a singular nodal curve ๐ฟ P |ยด ๐พ ๐ | .The basic example of a Looฤณenga pair is a toric surface equipped with the union of its torus-invariant divisors. Note that Looฤณenga pairs are usually assumed to be smooth elsewhere in theliterature.A polarised Looฤณenga pair is a triple p ๐ , ๐ฟ, ๐ด q consisting of a Looฤณenga pair and an ample divisorsupported on a subset of ๐ฟ . This implies that the Looฤณenga pair is โpositiveโ in the languageof [11]. If ๐ด is only big and nef we say that p ๐, ๐ฟ, ๐ด q is a pseudo-polarised Looฤณenga pair .A toric blowup of a Looฤณenga pair p ๐, ๐ฟ q is a blowup ๐ : r ๐ ร ๐ with centre a node of ๐ฟ . Observethat in this case the divisor r ๐ฟ ` ๐ธ โ the strict transform of ๐ฟ plus the exceptional divisor โ is suchthat p r ๐, r ๐ฟ ` ๐ธ q is a Looฤณenga pair. We will consider towers Y : p ๐ , ๐ฟ q ๐ รร p ๐ , ๐ฟ q ๐ รร . . . ๐ ๐ ยด รร p ๐ ๐ ยด , ๐ฟ ๐ ยด q ๐ ๐ รร p ๐ ๐ , ๐ฟ ๐ q ๐ ๐ ` รร . . . of Looฤณenga pairs where each map ๐ ๐ is a toric blowup. We call such structures Looฤณenga towers .We can also ask that each Looฤณenga pair is polarised and that the toric blowups are compatiblewith the polarisations. Namely, we want to consider towers p ๐ , ๐ฟ , ๐ด q ๐ รร p ๐ , ๐ฟ , ๐ด q ๐ รร . . . ๐ ๐ ยด รร p ๐ ๐ ยด , ๐ฟ ๐ ยด , ๐ด ๐ ยด q ๐ ๐ รร p ๐ ๐ , ๐ฟ ๐ , ๐ด ๐ q ๐ ๐ ` รร . . . of polarised Looฤณenga pairs where each map ๐ ๐ is a toric blowup, and the polarisations arerelated by ๐ด ๐ โ ๐ ห ๐ ๐ด ๐ ยด ยด ๐ ๐ ๐ธ ๐ for some ๐ ๐ ฤ
0, where ๐ธ ๐ is the exceptional ๏ฌbre of ๐ ๐ . We call such a structure a polarisedLooฤณenga tower and denote it by a pair p Y , A q where Y is the underlying Looฤณenga tower and A isthe sequence of polarisations p ๐ด ๐ q ๐ P Z ฤ . If the polarisations are relaxed to pseudo-polarisations,we call p Y , A q a pseudo-polarised Looฤณenga tower . We say that Y is smooth if ๐ is smooth. Lemma 2.1.
Let p Y , A q be a pseudo-polarised Looฤณenga tower. Then lim ๐ ร8 ๐ด ๐ exists. This also holds if we omit the anticanonical divisor ๐ฟ ๐ and only consider a tower of pseudo-polarised surfaces related by arbitrary blowups. B. WORMLEIGHTON
Proof.
We see that ๐ด ๐ โ p ๐ ห ๐ ๐ด ๐ ยด ยด ๐ ๐ ๐ธ ๐ q โ ๐ด ๐ ยด ` ๐ ๐ ๐ธ ๐ ฤ ๐ด ๐ ยด and so ๐ด ๐ is a decreasing sequence. It is bounded below since ๐ด ๐ ฤ ๐ by the assumptionthat each p ๐ ๐ , ๐ด ๐ q is pseudo-polarised and is hence convergent. (cid:3) As a result we de๏ฌne A : โ lim ๐ ร8 ๐ด ๐ โ inf t ๐ด ๐ : ๐ P Z ฤ u Polarised Looฤณenga towers from weighted posets.
We generalise the previous construc-tion to use a poset other than Z ฤ to index toric blowups. The towers that come from thisconstruction can thus have many spires. We start by constructing the universal Looฤณenga tower Y univ p ๐,๐ฟ q associated to a Looฤณenga pair p ๐, ๐ฟ q and realise all pseudo-polarised Looฤณenga towerswith p ๐ , ๐ฟ q โ p ๐ , ๐ฟ q in terms of it.Let p ๐ , ๐ฟ q be a Looฤณenga pair. We construct a poset P p ๐,๐ฟ q as follows. Let P be the posetconsisting of all nodes of ๐ท with no order relations. Let ๐ ๐ : ๐ ๐ ร ๐ denote the toric blowupat a node ๐ P ๐ฟ with exceptional divisor ๐ธ ๐ . Set P ๐ โ t ๐ , ๐ u Y P where ๐ , ๐ are the twointersection points of ๐ธ ๐ with the strict transform of ๐ฟ . De๏ฌne P โ ฤ ๐ P P P ๐ and view this as poset by setting ๐ ฤ ๐ if and only if ๐ P ๐ธ ๐ . Note that the elements of P z P correspond to nodes on the Looฤณenga pair p ๐ , ๐ฟ q obtained from p ๐ , ๐ฟ q by blowing up all thenodes of ๐ท . Repeating this process by blowing up each node on ๐ฟ produces a new poset P suchthat P z P is the set of nodes of the Looฤณenga pair p ๐ , ๐ฟ q obtained by blowing up all nodes of p ๐ , ๐ฟ q . Continuing this procedure de๏ฌnes a Looฤณenga pair p ๐ ๐ , ๐ด ๐ q for each ๐ P Z ฤ โ letting p ๐, ๐ฟ q โ p ๐ , ๐ฟ q โ and a poset P ๐ such that P ๐ z P ๐ ยด is the set of nodes of p ๐ ๐ , ๐ฟ ๐ 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 โLooฤณenga towerโto include such towers.We call the Looฤณenga tower arising from this construction the universal Looฤณenga tower associatedto p ๐ , ๐ฟ q and denote it by Y univ p ๐,๐ฟ q . We will later compare this to a construction of Hutchings [15, ยง3].De๏ฌne the poset P p ๐,๐ฟ q โ ฤ ๐ ฤ P ๐ Observe that P p ๐,๐ฟ q is a graded poset with grading de๏ฌned by the ๏ฌltration P ๐ . For ๐ P P p ๐,๐ฟ q ofdegree ๐ we obtain a Looฤณenga pair p ๐ ๐ , ๐ฟ ๐ q obtained as the toric blowup of p ๐ ๐ , ๐ฟ ๐ q at ๐ . De๏ฌnition 2.2.
Let P be a countable poset. We call a function wt : P ร R ฤ a weight function if โ wt is a poset homomorphism where R ฤ is regarded as a poset in the usual way, โ ล ๐ P P wt p ๐ q ฤ 8 .A pair p P , wt q of a poset with a weight function is called a weighted poset . We de๏ฌne the weightsequence wt p P q associated to a weighted poset p P , wt q to be the multiset t wt p ๐ q : ๐ P P u .We also write wt p Y , A q : โ wt p P p ๐ ,๐ฟ q q and refer to this as the weight sequence of p Y , A q . Wesay that an element ๐ of a poset P is a direct descendant of ๐ P P if ๐ ฤ ๐ and there is no ๐ P P such that ๐ ฤ ๐ ฤ ๐ .From the data of a pseudo-polarised Looฤณenga pair p ๐ , ๐ฟ, ๐ด q and a weight function satisfyingsome conditions on P p ๐,๐ฟ q we can produce a pseudo-polarised Looฤณenga tower p Y , A q . Let wt bea weight function on P p ๐,๐ฟ q . To de๏ฌne a polarisation on each p ๐ ๐ , ๐ฟ ๐ q we start by setting ๐ด โ ๐ ห ๐ด ยด รฟ ๐ P P wt p ๐ q ๐ธ ๐ OOฤฒENGA TOWERS AND ASYMPTOTICS OF ECH 7 and then recurse by setting ๐ด ๐ โ ๐ ห ๐ ๐ด ๐ ยด ยด รฟ ๐ P P ๐ z P ๐ ยด wt p ๐ q ๐ธ ๐ If this recipe de๏ฌnes a polarisation (resp. pseudo-polarisation) on each p ๐ ๐ , ๐ฟ ๐ q then we say thatwt is an ample (resp. big and nef) weight function on P p ๐,๐ฟ q . Thus, after choosing a big and nefweight function, to each ๐ P P p ๐,๐ฟ q there is a pseudo-polarised Looฤณenga pair p ๐ ๐ , ๐ฟ ๐ , ๐ด ๐ 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 Looฤณenga tower in which each map ๐ ๐ isa single toric blowup as in ยง2.1 from this data. We choose a bฤณective poset homomorphism โ : P p ๐,๐ฟ q ร Z op ฤ , where Z op ฤ is Z with reverse ordering, and de๏ฌne p ๐ ๐ , ๐ฟ ๐ , ๐ด ๐ q to be the pseudo-polarised Looฤณenga pair obtained by blowing up in the nodes โ ยด t , . . . , ๐ u . One can easilyverify that this is well-de๏ฌned by the requirement that โ is a poset homomorphism.In later sections we will choose โ such that wt p โ ยด p ๐ qq ฤ wt p โ ยด p ๐ ` qq ; in other words, thereis a commutative diagram in the category of posets of the form: P p ๐,๐ฟ q โ / / wt " " โโโโโโโโ Z op ฤ } } โฃ โฃ โฃ โฃ R ฤ Of course โ is not a poset isomorphism in general!We view two pseudo-polarised Looฤณenga pairs p ๐ , ๐ฟ, ๐ด q and p ๐ , ๐ฟ , ๐ด q as โequivalentโ if thereis a Looฤณenga pair p ๐ , ๐ฟ q and two maps ๐ : ๐ ร ๐ and ๐ : ๐ ร ๐ given as compositions oftoric blowups such that ๐ ห ๐ด โ p ๐ q ห ๐ด In this way, one can indeed recover any pseudo-polarised Looฤณenga tower p Y , A q with p ๐ , ๐ฟ q โp ๐, ๐ฟ q up to equivalence from Y univ p ๐,๐ฟ q by assigning a weight of zero to all nodes on toric blowupsof p ๐, ๐ฟ q that are not blown up in Y . We will revisit this notion of equivalence in ยง2.5.2.3. Divisors on Looฤณenga towers.
Throughout this subsection we ๏ฌx a Looฤณenga tower Y โtp ๐ ๐ , ๐ฟ ๐ qu with toric blowup maps ๐ ๐ and exceptional divisors ๐ธ ๐ . 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 . De๏ฌnition 2.3. A K -divisor on Y is a sequence D โ t ๐ท ๐ u where ๐ท ๐ is a K -divisor on ๐ ๐ such that ๐ท ๐ โ ๐ ห ๐ ๐ท ๐ ยด ยด ๐ ๐ ๐ธ ๐ for some ๐ ๐ P K .Clearly one can view a polarisation A on Y as an R -divisor on Y . We call the sequence p ๐ ๐ q ๐ P Z ฤ 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 de๏ฌned in ยง2.2. Y has a canonical divisor ๐พ Y de๏ฌned as the sequence ๐พ Y โ t ๐พ ๐ ๐ u ๐ P Z ฤ When Y is smooth the weight sequence of ๐พ Y is p , , . . . q . We denote the set of K -divisors on Y by Div p Y q K . One can easily modify this de๏ฌnition to produce numerical or linear equivalenceclasses of divisors on Y . We de๏ฌne Div ` p Y q K to be the set of K -divisors on Y whose weightsequences are summable, and Div ๐ p Y q K to be the set of K -divisors on Y whose weight sequencesare bounded. B. WORMLEIGHTON
There is evidently a pairingDiv ๐ p Y q K b Div ` p Y q K ร R , D ยจ D โ ๐ท ยจ ๐ท ` รฟ ๐ ฤ ๐ ๐ ๐ ๐ ๐ธ ๐ where p ๐ ๐ q ๐ P Z ฤ and p ๐ ๐ q ๐ P Z ฤ 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 ๐ ๐ in the sense that we can view Div p ๐ ๐ q K as the subspace ofDiv ` p Y q K ฤ Div ๐ p Y q K consisting of all K -divisors on Y whose weight sequences vanish afterthe ๐ th term. When A is a polarisation on a smooth Looฤณenga tower Y we will make much useof the quantity ยด ๐พ Y ยจ A โ ยด ๐พ ๐ ยจ ๐ด ยด รฟ ๐ P wt p Y , A q ๐ Toric Looฤณenga towers.
We will study a class of polarised Looฤณenga towers arising fromweighted posets that come from convex domains in R ฤ . The Looฤณenga pairs constituting thesetowers are toric surfaces. Key to our construction to is the weight sequence wt p ฮฉ q associated to aconvex domain ฮฉ following [2,17]. It is well-known (e.g. the work of GrossโHackingโKeel [11,12])that the geometry of Looฤณenga 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 ฮ ๐ denote the triangle in R with vertices p , q , p ๐, q , p , ๐ q . De๏ฌnition 2.4.
Let ฮฉ be a concave domain. The weight sequence wt p ฮฉ q of ฮฉ is de๏ฌned recursivelyas follows. โ Set wt pHq โ H and wt p ฮ ๐ q โ p ๐ q . โ Otherwise let ๐ be the largest real number such that ฮ ๐ ฤ ฮฉ . This divides ฮฉ into three(possibly empty) pieces: ฮ ๐ , ฮฉ , ฮฉ . โ If not empty, ฮฉ and ฮฉ are a๏ฌne-equivalent to concave domains ฮฉ and ฮฉ . De๏ฌnewt p ฮฉ q โ p ๐ q Y wt p ฮฉ q Y wt p ฮฉ q regarded as a multiset.Note that wt p ฮฉ q is ๏ฌnite if and only if ฮฉ is a real multiple of a lattice concave domain but willbe in๏ฌnite in general. We de๏ฌne an analogous sequence for convex domains. De๏ฌnition 2.5.
Let ฮฉ be a convex domain. The weight sequence wt p ฮฉ q of ฮฉ is de๏ฌned recursivelyas follows. โ Let ๐ be the smallest real number such that ฮฉ ฤ ฮ ๐ . โ This divides ฮ ๐ into three (possibly empty) pieces: ฮฉ , ฮฉ , ฮฉ . โ If non-empty, ฮฉ and ฮฉ are a๏ฌne-equivalent to concave domains ฮฉ and ฮฉ . De๏ฌnewt p ฮฉ q โ p ๐ q Y wt p ฮฉ q Y wt p ฮฉ q using Def. 2.4. We regard this as a multiset with a distinguished element ๐ from therecursion above that we call the head of wt p ฮฉ q . We set wt ยด p ฮฉ q : โ wt p ฮฉ qzt ๐ u .We depict the decompositions used to recursively de๏ฌne 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 ฮ ๐ away from ฮฉ by dashed lines. De๏ฌnition 2.6.
Let p Y , A q be a pseudo-polarised Looฤณenga tower. We say p Y , A q is toric if p ๐ , ๐ด q is a toric surface polarised by a torus-invariant R -divisor and each blowup map ๐ ๐ is equivariant. OOฤฒENGA TOWERS AND ASYMPTOTICS OF ECH 9
Figure 1.
Weight sequence decompositions โ โโ โโ โโ โโ โโ โ โโ โโ โโ โโ โโ โโ โ โโ โโ โโ โโ โโ โโ โ โโ โโ โโ โโ โโ โโ โ โโ โโ โโ โโ โโ โโ โ โโ ฮฉ ฮฉ ฮฉ ฮฉ (a) (b)We associate a toric pseudo-polarised Looฤณenga tower p Y ฮฉ , A ฮฉ q to a convex domain ฮฉ . Thiswill have the property wt p ฮฉ q โ wt p Y ฮฉ , A ฮฉ q . We write wt p ฮฉ q โ t ๐ u Y wt ยด p ฮฉ q .Consider P with moment image ฮ shown in Fig. 2 where the lower left vertex is the origin.We denote the hyperplanes corresponding to the three edges of ฮ by ๐ป , ๐ป , ๐ป as shown. Figure 2.
Moment polytope of P โโ โโ ๐ป ๐ป ๐ป We start with p ๐, ๐ฟ, ๐ด q โ p P , ๐ป ` ๐ป ` ๐ป , ๐๐ป q . Set P ฮฉ โ P p ๐,๐ฟ q . We will construct a bigand nef weight function on P ฮฉ via the recursion de๏ฌning the weight sequence for ฮฉ , and hencea (toric) pseudo-polarised Looฤณenga tower.Each element ๐ P P ฮฉ by de๏ฌnition corresponds to a node on a toric blowup of p ๐, ๐ฟ q but fromDef. 2.4 and Def. 2.5 ๐ also corresponds to a step in the weight sequence recursion. Recall theconstruction of P p ๐,๐ฟ q โ ลค ๐ ฤ P ๐ . In this notation the elements of P correspond to the threetorus-๏ฌxed points of P . We assign weight zero to the torus-๏ฌxed point ๐ โ ๐ป X ๐ป 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 ๐ , ๐ P P correspond to the concave domains ฮฉ and ฮฉ from Def. 2.5.Set wt p ๐ ๐ q โ ๐ ๐ , where ฮ ๐ ๐ is the largest regular triangle that ๏ฌts inside ฮฉ ๐ for ๐ โ ,
3. Iteratingthis procedure assigns a weight to each element of P ฮฉ as the side length of the largest regulartriangle that ๏ฌts inside the corresponding concave domain.More precisely, we ๏ฌx notation as follows. Let ฮฉ and ฮฉ be as above. Applying the weightsequence recursion to ฮฉ yields two concave domains ฮฉ and ฮฉ and similarly applying it to ฮฉ yields concave domains ฮฉ and ฮฉ . Repeating this process yields the diagram in Fig. 3(a).Notice that this is naturally in bฤณection with the part of the Haase diagram of the poset P ฮฉ excluding the 2-valent tree with maximum ๐ . We denote by ฮ p ๐ q the concave domain (i.e. either ฮฉ or ฮฉ in Def. 2.4) corresponding to ๐ P P ฮฉ zt ๐ P P ฮฉ : ๐ ฤ ๐ u .We de๏ฌne a weight function on P ฮฉ bywt p ๐ q โ ๐ ฤ ๐ ๐ p ๐ q elsewhere ฮ ๐ p ๐ q is the largest regular triangle contained in ฮฉ ๐ . 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 ๐ ๐ , ๐ด ๐ q corresponds to the polytope ฮฉ ๐ obtained after the ๐ th step of theweight sequence recursion; for comparison, see [4, ยง3.2-3.3]. Figure 3.
Weight sequence recursion (a)(b) ฮฉ ฮฉ ฮฉ ฮฉ ฮฉ ฮฉ ฮฉ ฮฉ ... ฮฉ ฮฉ . . .. . .๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ...๐ ๐ . . .. . . Figure 4.
Constructing P ฮฉ โโโโโโ โโโโโโ โโโโโโ โโโโโโ โโโโโโ โโโโโโ โโโ โโโ โโโ โโโโโโ โโโ โโโ โโโ โโโ โโโ โโโ โโโ Example 2.7.
We will work out the construction of P ฮฉ in detail for the convex domain ฮฉ fromFig. 1(b) with weight sequence p
5; 1 , , q . In Fig. 4(a) we show ฮฉ with the ๏ฌrst step of the weightsequence recursion expressing the 4-ball ๐ต p q as a union of ๐ ฮฉ , ๐ต p q and the ellipsoid ๐ธ p , q .In Fig. 4(b) we show the two regions ฮฉ and ฮฉ . The weights for elements of P ฤ P ฮฉ areillustrated below the domains. In Fig. 4(c) the ๏ฌnal stage of the recursion is shown โ of the fourconcave domains coming from ฮฉ and ฮฉ only one is nonempty, and is equal to ฮ โ and thecorresponding weights are listed below. Throughout we omit the tree with maximum ๐ withweights all zero. OOฤฒENGA TOWERS AND ASYMPTOTICS OF ECH 11
This example terminated after ๏ฌnitely many stages because ฮฉ was a rational-sloped polytope.In this case all p ๐ ๐ , ๐ด ๐ q for large enough ๐ are equivalent to the polarised toric surface p ๐ ฮฉ , ๐ด ฮฉ q associated to ฮฉ . That is, for large enough ๐ there is a series of toric blowups ๐ : ๐ ๐ ร ๐ ฮฉ with ๐ด ๐ โ ๐ ห ๐ด ฮฉ .Note that the sequence of polarised toric surfaces p ๐ ๐ , ๐ด ๐ q produced using the structure of P p ๐,๐ฟ q as a graded poset from ยง2.2 recovers the sequence of approximations used by Hutchingsin [15, ยง3] by setting ฮฉ ๐ to be the polytope for ๐ด ๐ .2.5. Algebraic capacities for Looฤณenga towers.
Recall the construction of algebraic capacities fora Q -factorial pseudo-polarised surface p ๐ , ๐ด q : ๐ alg ๐ p ๐ , ๐ด q : โ inf Nef p ๐ q Z t ๐ท ยจ ๐ด : ๐ p ๐ท q ฤ ๐ ` ๐ p O ๐ qu When ๐ is smooth this reduces to ๐ alg ๐ p ๐, ๐ด q : โ inf Nef p ๐ q Z t ๐ท ยจ ๐ด : ๐ผ p ๐ท q ฤ ๐ u where ๐ผ p ๐ท q : โ ๐ท ยจ p ๐ท ยด ๐พ ๐ q . It was shown in [22, Prop. 2.11] for all smooth or toric pseudo-polarised surfaces p ๐, ๐ด q we have that ๐ alg ๐ p ๐ , ๐ด q is obtained by ranging over e๏ฌective Z -divisorsin place of nef Z -divisors. Lemma 2.8.
Suppose p Y , A q โ tp ๐ ๐ , ๐ฟ ๐ , ๐ด ๐ qu Z ฤ is a pseudo-polarised Looฤณenga tower that is smoothor toric. Then lim ๐ ร8 ๐ alg ๐ p ๐ ๐ , ๐ด ๐ q exists and is ๏ฌnite. This result also holds for any tower p Y , A q โ tp ๐ ๐ , ๐ด ๐ qu of pseudo-polarised surfaces relatedby blowups. Notice that in this case ๐ p O ๐ ๐ q โ ๐ p O ๐ ๐ q โ : ๐ p O Y q for all ๐ , ๐ P Z ฤ . Proof.
Let ๐ท ๐ be a nef Z -divisor computing ๐ alg ๐ p ๐ ๐ , ๐ด ๐ q . We have in the smooth or toric casesthat ๐ p ๐ ห ๐ ` ๐ท ๐ q โ ๐ p ๐ท ๐ q ฤ ๐ ` ๐ p O Y q so that ๐ alg ๐ p ๐ ๐ ` , ๐ด ๐ ` q ฤ ๐ ห ๐ ` ๐ท ๐ ยจ ๐ด ๐ ` โ ๐ท ๐ ยจ ๐ด ๐ โ ๐ alg ๐ p ๐ ๐ , ๐ด ๐ q It follows that ๐ alg ๐ p ๐ ๐ , ๐ด ๐ q is a decreasing sequence in ๐ that is bounded below, and is henceconvergent. (cid:3) We thus de๏ฌne ๐ alg ๐ p Y , A q : โ lim ๐ ร8 ๐ alg ๐ p ๐ ๐ , ๐ด ๐ q We will see in the next section that this de๏ฌnition 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 ยง2.1 was motivated by the structure of algebraiccapacities.
Lemma 2.9.
Let p ๐ , ๐ฟ, ๐ด q and p ๐ , ๐ฟ , ๐ด q be smooth or toric Looฤณenga pairs. If p ๐ , ๐ฟ, ๐ด q and p ๐ , ๐ฟ , ๐ด q are equivalent, then ๐ alg ๐ p ๐, ๐ด q โ ๐ alg ๐ p ๐ , ๐ด q for all ๐ P Z ฤ . 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 ๏ฌx a particularuniversal Looฤณenga tower and choose a weight function on it to calculate the algebraic capacitiesof any pseudo-polarised Looฤณenga tower. We will hence also not specify the function โ we havechosen to produce a bona ๏ฌde Looฤณenga tower (indexed by Z ฤ ) from the poset P p ๐,๐ฟ 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 . De๏ฌne Nef p Y q to be the submonoid of Div ` p Y q R consistingof divisors D such that D ยจ ๐ธ ฤ ๐ธ P NE p ๐ ๐ q for each ๐ . Note that Nef p ๐ ๐ q naturally embedsinto Nef p Y q . Set Nef p Y q Z โ 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 Looฤณenga tower, then ๐ alg ๐ p Y , A q โ inf D P Nef p Y q Z t D ยจ A : D ยจ p D ยด ๐พ Y q ฤ ๐ u We write D โ p ๐ท , ๐ , . . . q for a K -divisor on Y where ๐ท is a K -divisor on ๐ and t ๐ ๐ u ๐ P Z ฤ 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 ๐ผ p ๐ท q . Proof.
Since Nef p ๐ ๐ q Z can be viewed as a subset of Nef p Y q Z we obtain ๐ alg ๐ p ๐ ๐ , ๐ด ๐ q ฤ inf D P Nef p Y q Z t D ยจ A : D ยจ p D ยด ๐พ Y q ฤ ๐ u for each ๐ , and so ๐ alg ๐ p Y , A q ฤ inf D P Nef p Y q Z t D ยจ A : D ยจ p D ยด ๐พ Y q ฤ ๐ u For the converse it su๏ฌces that for each ๐ ฤ ๐ P Z ฤ such that ๐ alg ๐ p ๐ ๐ , ๐ด ๐ q ฤ inf D P Nef p Y q Z t D ยจ A : D ยจ p D ยด ๐พ Y q ฤ ๐ u ` ๐ for all ๐ ฤ ๐ . Let D โ p ๐ท , ๐ , . . . q P Nef p Y q Z be such that D ยจ A ฤ inf D P Nef p Y q Z t D ยจ A : D ยจ p D ยด ๐พ Y q ฤ ๐ u ` ๐ As D P Div ` p Y q we must have that ๐ ๐ โ ๐ ฤ ๐ for some ๐ P Z ฤ . There is thus a nef Z -divisor ๐ท ๐ P Nef p ๐ ๐ q for ๐ ฤ ๐ that is mapped to D under the embedding Nef p ๐ ๐ q ร Nef p Y q .Hence, for all ๐ ฤ ๐ , ๐ alg ๐ p ๐ ๐ , ๐ด ๐ q ฤ ๐ท ๐ ยจ ๐ด ๐ โ ๐ท ๐ ยจ A โ D ยจ A ฤ inf D P Nef p Y q Z t D ยจ A : D ยจ p D ยด ๐พ Y q ฤ ๐ u ` ๐ as required. (cid:3) In fact this in๏ฌmum is realised in the toric case via a sympletic argument using Prop. 3.1 below.3.
Sub-leading asymptotics of ECH capacities
Looฤณenga towers and ECH.
To each symplectic 4-manifold p ๐ , ๐ q ECH associates an in-creasing sequence t ๐ ech ๐ p ๐ , ๐ qu ๐ P Z ฤ of (extended) real numbers called the ECH capacities of p ๐ , ๐ q . These obstruct symplectic em-beddings in the sense that p ๐ , ๐ q s รฃ ร p ๐ , ๐ q รนรฑ ๐ ech ๐ p ๐ , ๐ q ฤ ๐ ech ๐ p ๐ , ๐ q for all ๐ Proposition 3.1.
For any convex domain ฮฉ ฤ R the toric polarised Looฤณenga tower p Y ฮฉ , A ฮฉ q has ๐ ech ๐ p ๐ ฮฉ q โ ๐ alg ๐ p Y ฮฉ , A ฮฉ q Proof.
Consider the sequence of polygons t ฮฉ ๐ u ๐ P Z ฤ arising as the polytopes associated to thedivisors ๐ด ๐ . We know that lim ๐ ร8 ๐ ech ๐ p ๐ ฮฉ ๐ q โ ๐ ech ๐ p ๐ ฮฉ q by Hausdor๏ฌ continuity [2, Lem. 2.3].Since ฮฉ ๐ is rational-sloped [21, Thm. 1.5] gives that ๐ ech ๐ p ๐ ฮฉ ๐ q โ ๐ alg ๐ p ๐ ๐ , ๐ด ๐ 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].
OOฤฒENGA TOWERS AND ASYMPTOTICS OF ECH 13
Proposition 3.2 (c.f. [15, Lem. 3.6]) . Let ฮฉ be a convex domain whose weight sequence has head ๐ . Then ๐ ยด รฟ ๐ P wt p ฮฉ q ๐ โ ๐ p ฮฉ q ` ๐ p ฮฉ q ` โ a๏ฌ pB ` ฮฉ q Proof.
Note that3 ๐ ยด รฟ ๐ P wt p ฮฉ q ๐ โ ยด ๐พ Y ฮฉ ยจ A ฮฉ โ lim ๐ ร8 ยด ๐พ ๐ ๐ ยจ ๐ด ๐ โ lim ๐ ร8 ๐ p ฮฉ ๐ q ` ๐ p ฮฉ ๐ q ` โ a๏ฌ pB ` ฮฉ ๐ q and the result follows from continuity of ๐ pยจq and ๐ pยจq 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 โWeyl lawโ controllingthe growth of ๐ alg ๐ p Y , A q . Theorem 3.3.
Let p YA q be a pseudo-polarised Looฤณenga tower. Then lim ๐ ร8 ๐ alg ๐ p Y , A q ๐ โ A We will not prove this directly, but will instead appeal to the analysis of the error terms ๐ alg ๐ p Y , A q : โ ๐ alg ๐ p Y , A q ยด a A ๐ below, where we will show that ๐ alg ๐ p Y , A q โ ๐ p q . Thm. 3.3 follows immediately from Prop. 3.1when p Y , A q is a toric polarised Looฤณenga tower arising from a convex domain.These error terms associated to p Y , A q are analogous to the error terms in ECH ๐ ๐ p ๐ , ๐ q : โ ๐ ech ๐ p ๐ , ๐ q ยด b p ๐ , ๐ q ๐ and agree when p Y , A q comes from a convex domain.3.3. Bounds for error terms.
For a pseudo-polarised Looฤณenga tower p Y , A q we de๏ฌne a divisor ยด ๐พ ` Y on Y by ยด ๐พ ` Y โ ยด ๐พ Y ` ๐พ ๐ ยด ๐พ ` ๐ where ยด ๐พ ` ๐ is the support of ๐ด viewed as a reduced divisor. As a sequence of divisors indexedby ๐ like in ยง2.3, ยด ๐พ ` Y has as its ๐ th term the support of ๐ด ๐ viewed as a reduced divisor. In thissense ยด ๐พ ` Y can be viewed as the โsupportโ of A . When p Y , A q is actually a pseudo-polarised toricsurface corresponding to a rational-sloped polygon ฮฉ , we see that ยด ๐พ ` Y is the preimage of B ` ฮฉ under the moment map, giving ยด ๐พ ` Y ยจ A โ โ a๏ฌ pB ` ฮฉ q . Our next aim is to prove the followingtheorem. Theorem 3.4.
Suppose p Y , A q is a pseudo-polarised Looฤณenga tower such that Y is smooth or toric. Then ๐พ Y ยจ A ยด ๐พ ` Y ยจ A ฤ lim sup ๐ ร8 ๐ alg ๐ p Y , A qฤ lim inf ๐ ร8 ๐ alg ๐ p Y , A q ฤ ๐พ Y ยจ A In particular, ๐ alg ๐ p Y , A q โ ๐ p q . Corollary 3.5.
Let ๐ ฮฉ be a convex toric domain. Then ยด ห ๐ p ฮฉ q ` ๐ p ฮฉ q ยด โ a๏ฌ pB ` ฮฉ q ห ฤ lim sup ๐ ร8 ๐ ๐ p Y , A qฤ lim inf ๐ ร8 ๐ ๐ p Y , A q ฤ ยด ห ๐ p ฮฉ q ` ๐ p ฮฉ q ` โ a๏ฌ pB ` ฮฉ q ห When ฮฉ has no rational-sloped edges we have that ๐ ๐ p ๐ ฮฉ q is convergent and lim ๐ ร8 ๐ ๐ p ๐ ฮฉ q โ ยด p ๐ p ฮฉ q ` ๐ p ฮฉ qq Over the next two subsections we will establish these asymptotic upper and lower bounds.Since โ a๏ฌ pB ` ฮฉ q โ B ` ฮฉ 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 Looฤณenga towers.3.4. Upper bound for error terms.
Observe that any nef Z -divisor ๐ท on a Q -factorial surface ๐ gives an upper bound ๐ alg ๐ p ๐, ๐ด q ฤ ๐ท ยจ ๐ด when 2 ๐ ฤ ๐ผ p ๐ท q . By [22, Prop. 2.11] this also works if ๐ท is an e๏ฌective Z -divisor. Let p Y , A q โtp ๐ ๐ , ๐ฟ ๐ , ๐ด ๐ qu ๐ P Z ฤ be a pseudo-polarised Looฤณenga tower. We obtain an upper bound for ๐ alg ๐ p ๐ ๐ , ๐ด ๐ q in terms of ๐ and ๐ by using Z -divisors of the form r ๐๐ด ๐ s and then consideringhow the resulting bound behaves as ๐ and ๐ become large. We let the components of ๐ด ๐ bedenoted ๐ท , . . . , ๐ท ๐ ; that is, ยด ๐พ ` ๐ ๐ โ ล ๐ ๐ โ ๐ท ๐ .Consider the constraint ๐ผ p r ๐๐ด ๐ s q โ p ๐๐ด ๐ ` ฮ ๐ q ยจ p ๐๐ด ๐ ` ฮ ๐ ยด ๐พ ๐ ๐ q ฤ ๐ where ฮ ๐ โ r ๐๐ด ๐ s ยด ๐๐ด ๐ . That is, ๐ ๐ด ๐ ยด ๐๐ด ๐ ยจ ๐พ ๐ ๐ ` ๐๐ด ๐ ยจ ฮ ๐ ยด ๐ ` ฮ ๐ ยด ฮ ๐ ยจ ๐พ ๐ ๐ ฤ ๐๐ด ๐ ยจ ฮ ๐ ฤ ฮ ๐ is e๏ฌective and so we ignore that term. We bound ฮ ๐ ยจ ฮ ๐ ยด ฮ ๐ ยจ ๐พ ๐ ๐ in terms of the geometry of ๐ ๐ . Notice that ฮ ๐ ฤ รฟ ๐ท ๐ ฤ ๐ท ๐ and ยด ฮ ๐ ยจ ๐พ ๐ ๐ ฤ รฟ ๐ท ๐ ฤยด p ` ๐ท ๐ q giving ฮ ๐ ยด ฮ ยจ ๐พ ๐ ๐ ฤ ยด tpยด q -curves on ๐ ๐ u ` รฟ ๐ท ๐ ฤยด p ` ๐ท ๐ q Hence we see that ๐ผ p r ๐๐ด ๐ s q ฤ ๐ when ๐ ๐ด ๐ ยด ๐๐ด ๐ ยจ ๐พ ๐ ๐ ยด ๐ ยด tpยด q -curves on ๐ ๐ u ` รฟ ๐ท ๐ ฤยด p ` ๐ท ๐ q ฤ ๐ is bounded below by the larger solution of the quadratic obtained by replacing ฤ with โ in the above. Write ยด ๐ด ๐ ยจ ๐พ ๐ ๐ { ๐ด ๐ โ : ๐ ๐ . We thus have ๐ผ p r ๐๐ด ๐ s q ฤ ๐ if ๐ ฤ ยด ๐ ๐ ` gffe ๐๐ด ๐ ` tpยด q -curves on ๐ ๐ u ยด ล ๐ท ๐ ฤยด p ` ๐ท ๐ q ๐ด ๐ ` ๐ ๐ ๐ด ๐ Set ๐น p ๐ q โ tpยด q -curves on ๐ ๐ u ยด รฟ ๐ท ๐ ฤยด p ` ๐ท ๐ q We study how ๐น p ๐ q changes with ๐ by measuring ๐น p ๐ ` qยด ๐น p ๐ q ; i.e. how ๐น changes under a singleblowup in a torus-๏ฌxed point between two torus-invariant curves ๐ถ and ๐ถ . Let t ๐, ๐ u โ t , u .The options are: OOฤฒENGA TOWERS AND ASYMPTOTICS OF ECH 15 โ ๐ถ ฤ ๐ถ ฤ รนรฑ ๐น p ๐ ` q ยด ๐น p ๐ q โ โ ๐ถ ๐ โ ๐ถ ๐ ฤ รนรฑ ๐น p ๐ ` q ยด ๐น p ๐ q โ โ ๐ถ ๐ โ ยด ๐ถ ๐ ฤ รนรฑ ๐น p ๐ ` q ยด ๐น p ๐ q โ โ ๐ถ ๐ ฤ ยด ๐ถ ๐ ฤ รนรฑ ๐น p ๐ ` q ยด ๐น p ๐ q โ โ ๐ถ ๐ โ ยด ๐ถ ๐ โ รนรฑ ๐น p ๐ ` q ยด ๐น p ๐ q โ โ ๐ถ ๐ โโ ๐ถ ๐ โ ยด รนรฑ ๐น p ๐ ` q ยด ๐น p ๐ q โ โ ๐ถ ๐ ฤ ยด ๐ถ ๐ โ รนรฑ ๐น p ๐ ` q ยด ๐น p ๐ q โ โ ๐ถ ๐ ฤ ยด ๐ถ ๐ โ ยด รนรฑ ๐น p ๐ ` q ยด ๐น p ๐ q โ โ ๐ถ ๐ ฤ ยด ๐ถ ๐ ฤ ยด รนรฑ ๐น p ๐ ` q ยด ๐น p ๐ q โ ๐น p ๐ ` q ยด ๐น p ๐ q ฤ
5. In the toric case we have ๐ โ P and so ๐น p ๐ q ฤ ๐ since P has no negative curves. In general we will have ๐น p ๐ q ฤ ๐ ` ๐น p q but, as it makes nosigni๏ฌcant di๏ฌerence to the argument, we will ignore the constant for notational convenience.Therefore we see that ๐ผ p r ๐๐ด ๐ s q ฤ ๐ when ๐ ฤ ยด ๐ ๐ ` d ๐๐ด ๐ ` ๐๐ด ๐ ` ๐ ๐ ๐ด ๐ โ : ๐ ๐,๐ It follows that ๐ alg ๐ p ๐ ๐ , ๐ด ๐ q ฤ r ๐ ๐,๐ ๐ด ๐ s ยจ ๐ด ๐ ฤ ๐ ๐,๐ ๐ด ๐ ยด ๐พ ` ๐ ๐ ยจ ๐ด ๐ โ ยด ๐ ๐ ๐ด ๐ ` d ๐ด ๐ ๐ ` ๐ด ๐ ๐ ` ๐ ๐ p ๐ด ๐ q ยด ๐พ ` ๐ ๐ ยจ ๐ด ๐ This is an explicit bound for ๐ alg ๐ p ๐ ๐ , ๐ด ๐ q valid for all ๐ and ๐ . We require an elementary lemmafrom analysis to study what happens as ๐ and ๐ get large. Lemma 3.6.
Suppose p ๐ ๐ q is a decreasing summable sequence. Let ๐ p ๐ q โ ล ๐ ฤ ๐ ๐ ๐ . Then there exists astrictly increasing sequence p ๐ ๐ q of natural numbers such that ๐ ๐ โ ๐ p? ๐ q and ๐ p ๐ ๐ q โ ๐ p {? ๐ 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 di๏ฌerence to demandthat ๐ ๐ โ ๐ p ๐ q and ๐ p ๐ ๐ q โ ๐ p { ๐ q instead of ๐ ๐ โ ๐ p? ๐ q and ๐ p ๐ ๐ q โ ๐ p {? ๐ q , which we adoptfor notational convenience. Proof.
We ๏ฌrst show that ๐ ๐ โ ๐ p ๐ q . We can choose p ๐ ๐ q to be non-increasing and so we mayinterpret it as the tail probabilities ๐ ๐ โ ๐ p ๐ ฤ ๐ q for some random variable ๐ with values in N .As ๐ ๐ is summable, ๐ has ๏ฌnite expectation: E ๐ โ ล ๐ ๐ p ๐ ฤ ๐ q โ ล ๐ ๐ ๐ ฤ 8 . Now, ๐ ยจ ๐ ๐ โ ๐ ยจ ๐ p ๐ ฤ ๐ q โ ๐ ยจ E ๐ ฤ ๐ โ E ๐ ๐ ฤ ๐ ฤ E ๐ ๐ ฤ ๐ which approaches 0 as ๐ ร 8 by the dominated convergence theorem. It follows that ๐ ๐ ฤ ๐ ๐ { ๐ for some ๐ ๐ P ๐ p q , which again without loss of generality we may choose to be decreasing. Wenow de๏ฌne ๐ ๐ โ inf t ๐ก : ล ๐ ฤ ๐ก ๐ ๐ ฤ ๐ ยด u . Claim 3.7. ๐ ๐ โ ๐ p ๐ q . Computing tails we ๏ฌnd, using the monotonicity of p ๐ ๐ q ,( โ ) รฟ ๐ ฤ ๐ก ๐ ๐ ฤ รฟ ๐ ฤ ๐ก ๐ ๐ { ๐ ฤ ๐ ๐ก รฟ ๐ ฤ ๐ก ๐ ยด โ ๐ ๐ก { ๐ก , De๏ฌne ๐ ๐ โ inf t ๐ : ๐ ๐ { ๐ ฤ ๐ ยด u . We see that ๐ ๐ ฤ ๐ ๐ , so it su๏ฌces that ๐ ๐ โ ๐ p ๐ q . But byde๏ฌnition, p ๐ ๐ ยด q{ ๐ ฤ ๐ ๐ ๐ ยด P ๐ p q and so we have shown the claim.To ๏ฌnish the proof, we know from ( โ ) that ๐ p ๐ก q โ ๐ p ๐ก ยด q , i.e. ๐ p ๐ก q ฤ ๐ ๐ก { ๐ก for some non-increasing ๐ ๐ก P ๐ p q . Consequently, we are looking for a sequence ๐ ๐ โ ๐ p ๐ q such that ๐ ๐ ๐ { ๐ ๐ โ ๐ ` ๐ ยด ห , or equivalently for a sequence ๐ ๐ โ ๐ p q for which ๐ ๐๐ ๐ { ๐ ๐ โ ๐ p q . Here is a constructionof such a sequence ๐ ๐ : โ De๏ฌne ๐ก ๐ โ inf t ๐ก : ๐ ๐ก ฤ ยด ๐ u . โ Set ๐ ๐ โ ล ๐ โ t ๐๐ก ๐ ฤ ๐ ๐ ฤp ๐ ` q ๐ก ๐ ` u ๐ ยด .These ๐ ๐ are certainly ๐ p q , and with ๐ p ๐ q โ sup t ๐ : ๐๐ก ๐ ฤ ๐ u we have ๐ ๐๐ ๐ { ๐ ๐ โ ๐ ๐ ยจ ๐ p ๐ q ยด ยจ ๐ p ๐ q ฤ ๐ ๐ก ๐ p ๐ q ยจ ๐ p ๐ q โ ๐ p ๐ q ยจ ยด ๐ p ๐ q โ ๐ p q as desired. (cid:3) In this context Lemma 3.6 implies that there is a function ๐ p ๐ q that that depends only on p Y , A q and is ๐ p? ๐ q such that ล ๐ ฤ ๐ p ๐ q ๐ ๐ โ ๐ p {? ๐ q . It follows that | ๐ด ๐ ๐ ยด A | โ ๐ p {? ๐ q . Since ๐ alg ๐ p Y , A q ฤ ๐ alg ๐ p ๐ ๐ , ๐ด ๐ q for all ๐ and ๐ we get ๐ alg ๐ p Y , A q ฤ ยด ๐ ๐ p ๐ q ๐ด ๐ p ๐ q ยด ๐พ ` ๐ ๐ p ๐ q ยจ ๐ด ๐ p ๐ q ` d ๐ด ๐ p ๐ q ๐ ` ๐ด ๐ p ๐ q ๐ p ๐ q ` p ๐ ๐ p ๐ q ๐ด ๐ p ๐ q q ยด a A ๐ โ ยด ๐ ๐ p ๐ q ๐ด ๐ p ๐ q ยด ๐พ ` ๐ ๐ p ๐ q ยจ ๐ด ๐ p ๐ q ` d ห A ` ๐ ห ? ๐ หห ๐ ` ห A ` ๐ ห ? ๐ หห p ๐ p ๐ q ` q ` ๐ p q ยด a A ๐ โ ยด ๐ ๐ p ๐ q ๐ด ๐ p ๐ q ยด ๐พ ` ๐ ๐ p ๐ q ยจ ๐ด ๐ p ๐ q ` b A ๐ ` ๐ p? ๐ q ยด a A ๐ By letting ๐ ร 8 and substituting ๐ ๐ ๐ด ๐ โ ยด ๐ด ๐ ยจ ๐พ ๐ ๐ we achieve the following. Proposition 3.8.
Let p Y , A q be a pseudo-polarised Looฤณenga tower. Then, lim sup ๐ ร8 ๐ alg ๐ p Y , A q ฤ ๐พ Y ยจ A ยด ๐พ ` Y ยจ A We convert this into combinatorial language.
Corollary 3.9.
Let ฮฉ be a convex domain. Then, lim sup ๐ ร8 ๐ ๐ p ๐ ฮฉ q ฤ ยด ห ๐ p ฮฉ q ` ๐ p ฮฉ q ยด โ a๏ฌ pB ` ฮฉ q ห In particular, if B ` ฮฉ has no rational-sloped edge then lim sup ๐ ร8 ๐ ๐ p ๐ ฮฉ q ฤ ยด p ๐ p ฮฉ q ` ๐ p ฮฉ qq Lower bound for error terms.
To deduce a lower bound we can in fact generalise to thesetting of a tower of blowups Y โ tp ๐ ๐ , ๐ด ๐ qu ๐ P Z ฤ of polarised surfaces where ยด ๐พ Y is โe๏ฌectiveโโ that is, each ยด ๐พ ๐ ๐ is e๏ฌective. DenoteNS p ๐ q ๐ด ฤ : โ t ๐ท P NS p ๐ q : ๐ท ยจ ๐ด ฤ u De๏ฌne for a pseudo-polarised surface p ๐ , ๐ด q ๐ ` ๐ p ๐, ๐ด q : โ inf NS p ๐ q ๐ด ฤ t ๐ท ยจ ๐ด : ๐ท ยจ p ๐ท ยด ๐พ ๐ q ฤ ๐ u This is a variation on the asymptotic capacity ๐ asy ๐ p ๐ , ๐ด q from [22, ยง4.1] or the estimate using theโapproximate ECH indexโ of [15, ยง5.2]. These invariants will have preferable numerics to studylower bounds for ๐ alg ๐ p ๐, ๐ด q . It is already clear that ๐ ` ๐ p ๐, ๐ด q ฤ ๐ alg ๐ p ๐, ๐ด q for all ๐ . As usual we write ๐ โ ยด ๐พ ๐ ยจ ๐ด { ๐ด . OOฤฒENGA TOWERS AND ASYMPTOTICS OF ECH 17
Lemma 3.10.
Suppose p ๐, ๐ด q is a pseudo-polarised surface such that ๐ is smooth or toric. If ๐ is nottoric, assume that ยด ๐พ ๐ is e๏ฌective. When ๐ ฤ ยด p ๐พ ๐ ยจ ๐ด q ๐ด ยด ๐พ ๐ ยฏ we have ๐ ` ๐ p ๐, ๐ด q โ ๐พ ๐ ยจ ๐ด ` b ๐พ ๐ ๐ด ` ๐ด ๐ Proof.
Without loss of generality, we assume that ๐ is smooth. From the Hodge index theoremwe have an orthogonal basis ๐ด, ๐ , . . . , ๐ ๐ of NS p ๐ q . Set ๐ ๐ โ ยด ๐ ๐ . Let ยด ๐พ ๐ โ ๐ ๐ด ` ล ๐ฟ ๐ ๐ ๐ . Wesee that an optimiser for ๐ ` ๐ p ๐, ๐ด q is ๐ท ๐ โ ๐ ๐ ๐ด ยด รฟ ๐ฟ ๐ ๐ ๐ where ๐ ๐ is the smallest nonnegative real number ๐ such that ๐ p ๐ ` ๐ q ฤ ๐ด ห ๐ ยด รฟ ๐ฟ ๐ ๐ ๐ ยธ Solving for ๐ , we see that the two solutions are ยด ๐ ห d ๐ ยด รฟ ๐ฟ ๐ ๐ ๐ ๐ด ` ๐๐ด We also note that ๐พ ๐ โ ๐ ๐ด ยด รฟ ๐ฟ ๐ ๐ ๐ and so the solutions for ๐ can be rewritten as ยด ๐ ห d ๐พ ๐ ๐ด ` ๐๐ด There is a unique nonnegative solution given by the larger value of ๐ precisely when ๐ ฤ ๐พ ๐ ๐ด ` ๐๐ด or when ๐ ฤ ๐ ๐ด ยด ๐พ ๐ โ ห p ๐พ ๐ ยจ ๐ด q ๐ด ยด ๐พ ๐ ห Substituting in the larger value for ๐ ๐ gives the result. (cid:3) Note that ๐พ ๐ ๐ โ ๐พ ๐ ยด ๐ . Hence, we see that ๐ alg ๐ p ๐ ๐ , ๐ด ๐ q ฤ ๐ ` ๐ p ๐ ๐ , ๐ด ๐ q โ ๐พ ๐ ๐ ยจ ๐ด ๐ ` b ๐พ ๐ ๐ ๐ด ๐ ` ๐ด ๐ ๐ for all ๐ ฤ ยด p ๐พ ๐๐ ยจ ๐ด ๐ q ๐ด ๐ ยด ๐พ ๐ ๐ ยฏ โ ๐ ` ยด p ๐พ ๐๐ ยจ ๐ด ๐ q ๐ด ๐ ยด ๐พ ๐ ยฏ . For notational convenience we notethat p ๐พ ๐ ๐ ยจ ๐ด ๐ q ๐ด ๐ ยด ๐พ ๐ ฤ p ๐พ ๐ ยจ ๐ด q A ยด ๐พ ๐ โ ๐ ๐ ยด ล ๐ ๐ ยด ๐พ ๐ โ : ๐ We choose a sequence ๐ ๐ as in Lemma 3.6 with ๐ ๐ โ ๐ p? ๐ q and ๐ด ๐ ๐ ยด A โ ๐ p {? ๐ q . Forsu๏ฌciently large ๐ we have ๐ ๐ ` ๐ ฤ ๐ . Then, for all such ๐ we have ๐ alg ๐ p ๐ ๐ ๐ , ๐ด ๐ ๐ q ฤ ๐พ ๐ ๐๐ ยจ ๐ด ๐ ๐ ` b ๐พ ๐ ๐๐ ๐ด ๐ ๐ ` ๐ด ๐ ๐ ๐ โ ๐พ ๐ ๐๐ ยจ ๐ด ๐ ๐ ` d p ๐พ ๐ ยด ๐ ๐ q ห A ` ๐ ห ? ๐ หห ` ห A ` ๐ ห ? ๐ หห ๐ โ ๐พ ๐ ๐๐ ยจ ๐ด ๐ ๐ ` d p ๐พ ๐ ` ๐ p? ๐ qq ห A ` ๐ ห ? ๐ หห ` ห A ` ๐ ห ? ๐ หห ๐ โ ๐พ ๐ ๐๐ ยจ ๐ด ๐ ๐ ` b A ๐ ` ๐ p? ๐ q As a result, letting ๐ ร 8 giveslim inf ๐ ร8 ๐ alg ๐ p Y , A q ฤ lim ๐ ร8 ๐พ ๐ ๐๐ ยจ ๐ด ๐ ๐ ` b A ๐ ` ๐ p? ๐ q ยด a A ๐ โ ๐พ Y ยจ A Proposition 3.11.
Let p Y , A q be a pseudo-polarised Looฤณenga tower with Y either smooth or toric. Then lim inf ๐ ร8 ๐ alg ๐ p Y , A q ฤ ๐พ Y ยจ A This implies the following in combinatorial terms.
Corollary 3.12.
Suppose ๐ ฮฉ is a convex toric domain. Then lim inf ๐ ร8 ๐ ๐ p ๐ ฮฉ q ฤ ยด ` ๐ p ฮฉ q ` ๐ p ฮฉ q ` โ a๏ฌ pB ` ฮฉ q ห In particular, if ฮฉ has no rational-sloped edge then lim inf ๐ ร8 ๐ ๐ p ๐ ฮฉ q ฤ ยด p ๐ p ฮฉ q ` ๐ p ฮฉ 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 ฮฉ is a convex toric domain with weight sequence given by wt p ฮฉ q โ p ๐ ; wt p ฮฉ q , wt p ฮฉ qq where ฮฉ , ฮฉ are concave domains as in Def. 2.5. Then ๐ ech ๐ p ๐ ฮฉ q โ inf ๐ ,๐ ฤ t ๐ ech ๐ ` ๐ ` ๐ p ๐ต p ๐ qq ยด ๐ ech ๐ p ๐ ฮฉ q ยด ๐ ๐ p ๐ ฮฉ qu Let ฮ be a concave toric domain. It is clear that there exists a convex domain ฮฉ such that either ฮฉ โ ฮ and ฮฉ โ H , or ฮฉ โ H and ฮฉ โ ฮ . We assume the former without loss of generality. Proposition 3.14.
Let ๐ ฮ be a concave toric domain. Then lim inf ๐ ร8 ๐ ๐ p ๐ ฮ q ฤ ยด p ๐ p ฮ q ` ๐ p ฮ q ` โ a๏ฌ pB ` ฮ qq Proof.
Let ฮฉ be as discussed above and let ๐ be the head of wt p ฮฉ q . Then ๐ ech ๐ p ๐ ฮฉ q โ inf ๐ ฤ t ๐ ech ๐ p ๐ต p ๐ qq ยด ๐ ech ๐ p ๐ ฮ qu This in๏ฌmum is attained for each ๐ ; we denote an optimiser for ๐ by ๐ so that ๐ ech ๐ p ๐ ฮฉ q โ ๐ ech ๐ p ๐ต p ๐ qq ยด ๐ ech ๐ p ๐ ฮ q OOฤฒENGA TOWERS AND ASYMPTOTICS OF ECH 19
Thus ๐ ๐ p ๐ ฮ q is given by ๐ ech ๐ ` ๐ p ๐ต p ๐ qq ยด ๐ ech ๐ p ๐ ฮฉ q ยด b p ๐ ฮ q ๐ โ ๐ ๐ ` ๐ p ๐ต p ๐ qq ยด ๐ ๐ p ๐ ฮฉ q ` b p vol p ๐ ฮฉ q ` vol p ๐ ฮ qqp ๐ ` ๐ q ยด b p ๐ ฮฉ q ๐ ยด b p ๐ ฮ q ๐ From Cor. 3.5 we see that ๐ ๐ p ๐ ฮฉ q and ๐ ๐ p ๐ต p ๐ qq are bounded and so it follows that ๐ ๐ p ๐ ฮ q isbounded below by ยด ๐ ` p ๐ p ฮฉ q ` ๐ p ฮฉ q ` โ a๏ฌ pB ` ฮฉ qq โ ยด p ๐ p ฮ q ` ๐ p ฮ q ` โ a๏ฌ pB ` ฮ qq using the CauchyโSchwartz inequality. (cid:3) Hutchings shows in [15, Cor. 3.9] that ๐ ๐ p ๐ ฮ q is bounded above by ยด ล ๐ P wt p ฮ q ๐ . We henceobtain the following. Theorem 3.15.
Let ๐ ฮ be a concave toric domain. Then ยด p ๐ p ฮ q ` ๐ p ฮ q ยด โ a๏ฌ pB ` ฮ qq ฤ lim sup ๐ ร8 ๐ ๐ p ๐ ฮ qฤ lim inf ๐ ร8 ๐ ๐ p ๐ ฮ q ฤ ยด p ๐ p ฮ q ` ๐ p ฮ q ` โ a๏ฌ pB ` ฮ qq and so ๐ ๐ p ๐ ฮ q โ ๐ p q . If B ` ฮ has no rational-sloped edges then lim ๐ ร8 ๐ ๐ p ๐ ฮ q โ ยด p ๐ p ฮ q ` ๐ p ฮ 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 ` ฮ has no rational-sloped edges. (cid:3) Algebraic analogues of rational-sloped edges.
We discuss the geometric analogue for po-larised Looฤณenga towers of the combinatorial condition on convex domains of having a rational-sloped edge. In particular, this supplies a criterion for convergence for ๐ alg ๐ p Y , A q in this generality.Given a poset P de๏ฌne its extended poset p P to be P Y t8u with ๐ for all ๐ P P . If p Y , A q is a pseudo-polarised Looฤณenga tower we can de๏ฌne an weight function on the extended poset p P p ๐ ,๐ฟ q by setting wt p8q โ ยด ๐พ ` ๐ ยจ ๐ด . De๏ฌne a subposet p P p ๐ ,๐ฟ q p ๐ q as follows: โ ๐ is the unique maximal element of P p ๐ ,๐ฟ q p ๐ q , โ if ๐ P p P p ๐ ,๐ฟ q p ๐ q then exactly one direct descendant of ๐ is in p P p ๐ ,๐ฟ q p ๐ q , namely the directdescendant corresponding to the point of intersection of ๐ธ ๐ and the strict transform of ๐ธ ๐ in ๐ ๐ .This all works similarly for the weighted poset p P ฮฉ associated to a convex domain ฮฉ ; forinstance, the weight of the element is the a๏ฌne length of the possibly empty edge of slope p , ยด q in B ` ฮฉ , and one can interpret each element ๐ of p P ฮฉ p ๐ q with direct ancestor ๐ as thevertex of ฮฉ ๐ incident to the edge that is the moment image of (the strict transform of) ๐ธ ๐ . Lemma 3.16.
Let ๐ ฮฉ be a convex toric domain. Let p P ฮฉ be the extended weighted poset associated to ฮฉ .Then there is a bฤณectionrational-sloped edges in B ` ฮฉ รร ๐ P p P ฮฉ such that wt p ๐ q ยด รฟ ๐ P P ฮฉ p ๐ q wt p ๐ q ฤ Proof.
It follows from the weight sequence recursion and the construction of P ฮฉ p ๐ q that wt p ๐ q ยด ล ๐ P P ฮฉ p ๐ q wt p ๐ q is the a๏ฌne length of the (possibly empty) edge in B ` ฮฉ introduced at the stepcorresponding to ๐ in the recursion. Rational-sloped edges in B ` ฮฉ are exactly such edges thathave nonzero a๏ฌne length, which gives the result. (cid:3) We see that the extension of P ฮฉ was necessary to capture the (possiby empty) edge of slope p , ยด q from the ๏ฌrst step of the recursion. De๏ฌnition 3.17.
We say that a pseudo-polarised Looฤณenga tower p Y , A q is balanced if wt p ๐ q ยด ล ๐ P P p ๐ ,๐ฟ q p ๐ q wt p ๐ q โ ๐ P p P p ๐ ,๐ฟ q .This is the algebraic analogue for pseudo-polarised Looฤณenga 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 Looฤณenga tower that is balanced. Then ๐ alg ๐ p Y , A q is convergent with lim ๐ ร8 ๐ alg ๐ p Y , A q โ ๐พ Y ยจ A โ p ๐พ ๐ ยด ๐พ ` ๐ q ยจ ๐ด Proof.
We already have12 ๐พ Y ยจ A ยด ๐พ ` Y ยจ A ฤ lim sup ๐ ร8 ๐ alg ๐ p Y , A q ฤ lim inf ๐ ร8 ๐ alg ๐ p Y , A q ฤ ๐พ Y ยจ A from Thm. 3.4, and so it su๏ฌces to show that ยด ๐พ ` Y ยจ A โ p Y , A q is balanced. We have ยด ๐พ ` Y ยจ A โ ยด ๐พ Y ยจ A ` ๐พ ๐ ยจ A ยด ๐พ ` ๐ ยจ A โ ยด ๐พ ๐ ยจ ๐ด ยด รฟ ๐ P p P p ๐ ,๐ฟ q wt p ๐ q ` ๐พ ๐ ยจ ๐ด ยด ๐พ ` ๐ ยจ ๐ด โ wt p8q ยด รฟ ๐ P p P p ๐ ,๐ฟ q wt p ๐ q Let S p q โ t8u . Recursively de๏ฌne S p ๐ q to be the set of maxima of p P p ๐ ,๐ฟ q z ฤ ๐ ฤ ๐ ฤ ๐ P S p ๐ q p P p ๐ ,๐ฟ q p ๐ q By construction we have from the above that ยด ๐พ ` Y ยจ A โ รฟ ๐ โ รฟ ๐ P S p ๐ q ยจหห wt p ๐ q ยด รฟ ๐ P p P p ๐ ,๐ฟ q p ๐ q wt p ๐ q หโนโ which is zero by the assumption that p Y , A q is balanced. The second equality in the statementfollows from ยด ๐พ ` Y ยจ A โ (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 whatโgenericโ means in Hutchingsโ conjecture [15, Conj. 1.5].Given convex or concave ฮฉ we let ๐ p ฮฉ q be the Q -vector subspace of R spanned by the a๏ฌnelengths of rational-sloped edges in B ` ฮฉ . We denote the dimension of ๐ p ฮฉ q by ๐ฃ p ฮฉ q .Let ๐ ฮฉ be a convex or concave toric domain. We believe that two ingredients for strongerconvergence criteria are this ๐ฃ p ฮฉ q and the number ๐ p ฮฉ q of rational-sloped edges in B ` ฮฉ .If ๐ p ฮฉ q ฤ 8 then we suspect ๐ ๐ p ๐ ฮฉ q converges if ๐ฃ p ฮฉ q ยญโ
1. If ฮฉ has in๏ฌnitely many rational-sloped edges then it seems likely that ๐ ๐ p ๐ ฮฉ q converges. In each case of convergence we expectthat the limit is( ห ) ยด
12 Ru p ๐ ฮฉ q โ ยด p ๐ p ฮฉ q ` ๐ p ฮฉ qq though it is possible that there are toric domains for which ๐ ๐ converge but that are not genericin the sense that they do not satisfy Hutchingsโ conjecture and have limit di๏ฌerent to ( ห ). In thecase of non-convergence, we expect that ( ห ) is the midpoint of the lim inf and lim sup of ๐ ๐ p ๐ ฮฉ q . OOฤฒENGA TOWERS AND ASYMPTOTICS OF ECH 21
Note that the case ๐ฃ p ฮฉ q โ ฮฉ having no rational-sloped edges โ which is coveredby Cor. 3.5 and Thm. 3.15 โ and ๐ฃ p ฮฉ q ฤ ฮฉ having at least two rational-slopededges whose a๏ฌne lengths are independent over Q .There is a distinction between the case that ฮฉ is a of scaled-lattice type as in [22] โ that is, where ฮฉ โ ๐ ฮฉ for some lattice polygon ฮฉ and some ๐ P R ฤ โ and the complementary case: whereeither ฮฉ is polytopal and has ๐ฃ p ฮฉ q ฤ
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Department of Mathematics & Statistics, Washington University in St. Louis, St. Louis, MO, 63130, USA
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