ECH embedding obstructions for rational surfaces
EECH EMBEDDING OBSTRUCTIONS FOR RATIONAL SURFACES
J. CHAIDEZ AND B. WORMLEIGHTONAbstract. Let p Y , A q be a smooth rational surface or a possibly singular toric surface with ampledivisor A . We show that a family of ECH-based, algebro-geometric invariants c alg k p Y , A q proposedin [34] obstruct symplectic embeddings into Y . Precisely, if p X , ω X q is a 4-dimensional star-shapeddomain and ω Y is a symplectic form Poincaré dual to r A s then p X , ω X q embeds into p Y , ω Y q symplectically ùñ c ECH k p X , ω X q ď c alg k p Y , A q We give three applications to toric embedding problems: (1) these obstructions are sharp for embed-dings of concave toric domains into toric surfaces; (2) the Gromov width and several generalizationsare monotonic with respect to inclusion of moment polygons of smooth (and many singular) toricsurfaces; and (3) the Gromov width of such a toric surface is bounded by the lattice width of itsmoment polygon, addressing a conjecture of Averkov–Hofscheier–Nill in [2].
1. IntroductionA symplectic embedding of symplectic manifolds p X , ω q Ñ p X , ω q of the same dimension isa smooth embedding ϕ : X Ñ X that intertwines the symplectic form, i.e. ϕ ˚ ω “ ω . The studyof symplectic embeddings has been a major topic in symplectic geometry ever since Gromovproved his eponymous non-squeezing theorem, stating that B n p r q symplectically embeds into B p R q ˆ (cid:67) n ´ ðñ r ď R Symplectic capacities provide the primary tool for obstructing symplectic embeddings. Roughlyspeaking, a symplectic capacity c is a numerical invariant associated to a symplectic manifold(usually in a restricted class, e.g. exact) such that c p X q ď c p X q whenever X symplecticallyembeds into X . The most famous example is the Gromov width of X , defined by(1.1) c G p X q : “ sup t π r : B p r q symplectically embeds into X u Capacities like c G have been used to great effect to provide complete solutions to many symplecticembedding problems.One family of capacities that have been applied with particular success in dimension 4 are the ECH capacities c ECH k (one for each integer k ě
1) introduced by Hutchings in [17]. These capacitiesare defined using embedded contact homology (or ECH for short), a version of Floer homologyfor contact 3-manifolds with a deep connection to Seiberg-Witten theory. They also providesharp embedding obstructions for several 4-dimensional symplectic embedding problems, suchas ellipsoids into ellipsoids [21] and (more generally) of concave toric domains into convex toricdomains [8]. This paper is about symplectic embedding obstructions derived using ECH.1.1.
ECH capacities via algebraic geometry.
Our present story begins with the work of Worm-leighton (the second author of this paper) in [34], which we now review in some detail.Recall that a toric domain X Ω is the inverse image µ ´ p Ω q of a compact subset Ω Ă r , withopen interior under the standard moment map on (cid:67) . µ : (cid:67) Ñ (cid:82) p z , z q ÞÑ p π | z | , π | z | q The region Ω is called the moment image . A toric domain X Ω is convex if Ω “ K X r , where K Ă (cid:82) is a convex set and 0 P K . Likewise, X Ω is concave if Ω “ C X r , where (cid:82) z C is a r X i v : . [ m a t h . S G ] A ug J. CHAIDEZ AND B. WORMLEIGHTON convex and 0 P C . Finally, a rational toric domain is a convex toric domain where Ω is the convexhull of finitely many rational points in r , .The ECH capacities of toric domains have been studied extensively (c.f. [6, 8, 15, 16]). Forrational toric domains, the ECH capacities can be combinatorially computed using the momentpolytope Ω , and these computations bear a remarkable resemblance to calculations arising in thealgebraic geometry of (cid:81) -line bundles over toric surfaces. This observation was first leveraged (forellipsoids) in the work of Cristofaro-Gardiner–Kleinman [9]. In [34], Wormleighton formalizedit as a theorem.To state this theorem we observe that, given a moment polytope Ω , there is in addition to X Ω , an associated projective algebraic surface Y Ω described by the inner normal fan of Ω . Thissurface can be singular, and may alternately be viewed as a toric, symplectic orbifold withmoment polytope Ω . It comes equipped with a canonical ample (cid:82) -divisor A Ω on Y Ω . Theorem 1.1 ( [34, Thm. 1.5]) . Let X Ω be a rational toric domain and p Y Ω , A Ω q be the correspondingpolarized toric surface. Then (1.2) c ECH k p X Ω q “ inf D P nef p Y Ω q (cid:81) t D ¨ A Ω : h p D q ě k ` u Here the infimum is over all nef (cid:81) -divisors in Y Ω . For the more symplectically minded reader, anef divisor may be thought of as a homology class that is represented by a disconnected J -curve,and which has non-negative intersection with any other J -curve. For example, in (cid:80) this is everynon-negative multiple of the hyperplane class r (cid:80) s , while in (cid:80) ˆ (cid:80) this is every non-negativecombination of r (cid:80) ˆ pt s and r pt ˆ (cid:80) s .Theorem 1.1 allows one to leverage the computational tools developed for toric geometry toperform calculations, and implies a number of nice results about the asymptotics of the ECHcapacities as k Ñ 8 . See [34] for more results.1.2.
Geometric explanation.
The proof of Theorem 1.1 in [34] is largely combinatorial, andamounts to checking that the two quantities agree using previously known explicit formulas.Thus, it is natural to wonder if there is some deeper geometric phenomenon at play. We nowsketch a heuristic argument suggesting that this is indeed the case.To start, given a moment polytope Ω , we observe that the surface with divisor p Y Ω , A Ω q and domain X Ω are related. Indeed, the interior X ˝ Ω of X Ω and the complement Y Ω z A Ω areequivariantly symplectomorphic and one can write down a “collapsing map” π : B X Ω Ñ A Ω whose fibers are generically circles. If Y Ω is smooth, we can (roughly speaking) write(1.3) Y Ω “ X Ω Y Z N Ω where N Ω is a very thin neighborhood of A Ω and Z is the boundary of N Ω . Thus we have thefollowing picture. Figure 1. The relationship between Y Ω and X Ω . CH EMBEDDING OBSTRUCTIONS FOR RATIONAL SURFACES 3
Now we return to a discussion of capacities. Dissecting the construction of c ECH k , we findthat the 1st ECH capacity of X Ω is (again, roughly speaking) computed as the minimum area ofcertain disconnected holomorphic curves u in p Z “ (cid:82) ˆ Z satisfying some conditions. First, eachcomponent C of u is embedded, cylindrical at ˘8 and comes with an integer weight n C P (cid:90) ` .Second, u must pass through a point p P p Z (fixed for all u ). The k th ECH capacity of X Ω is givenby sequences u i of k such curves with matching ends at ˘8 .One way that sequences u i of this form arise naturally is by neck stretching Y Ω along thehypersurface Z . Namely, a disconnected curve D Ă Y Ω with embedded components that isequipped with integer weights on its components and that passes through k generic points in Y Ω will (if it survives the stretching process) produce a sequence u i as above.Figure 2. Neck stretching divisors to acquire ECH curves.The curve D is essentially an effective, integral Weil divisor. If D passes through k points, thenwe expect the moduli of divisors M D in the class of D to satisfy dim p M D q ě k . Furthermore,the area of D in Y Ω is given by A Ω ¨ D since A Ω is Poincare dual to the Kahler form on Y Ω .The above discussion leads us to expect an inequality of the following form, which stronglyresembles one direction of the equality (1.2). c ECH k p X Ω q ď min t A Ω ¨ D | effective divisors D with dim p M D q ě k + more (?)) u Note that, in the above discussion, we did not reference the fact that X Ω and Y Ω arose via toricgeometry or that X Ω “ Y Ω z A Ω . In fact, the entire argument seems sensible if p Y , A q is an arbitraryprojective surface with ample divisor and X Ă Y is an embedded exact symplectic sub-domain. Remark 1.2.
A more precise perspective on the curve D in Y Ω is that it arises in the modulispace count used to define the Gromov-Taubes invariant of a symplectic 4-manifold [22, 31]. Thisneck stretching phenomenon is, morally speaking, the reason that ECH is the Floer theorycategorifying the Gromov-Taubes invariants.In practice, this fact is formalized using the isomorphism of ECH with a variant of Seiberg-Witten-Floer homology [32], and the equivalent of the Gromov-Taubes invariants with theSeiberg-Witten invariants [33]. In order to make the discussion of this section (§1.2) rigorous, wewill make use of these equivalences via a result of Hutchings (see Theorem 2.8 in §2.3).1.3.
Main results.
We are now ready to state the main theorem of this paper, which formalizesthe discussion of §1.2. First we recall the notion of algebraic capacity from [34–36].
Definition 1.3 (Definition 3.2) . The k th algebraic capacity c alg k p Y , A q of a rational projective surface Y with ample (cid:82) -divisor A is c alg k p Y , A q : “ inf D P Nef p Y q (cid:90) t D ¨ A : χ p D q ě k ` χ p O Y qu Here Nef p Y q (cid:90) denotes the set of nef (cid:90) -divisors on Y . J. CHAIDEZ AND B. WORMLEIGHTON
Recall that a star-shaped domain X Ă (cid:67) is a codimension 0 sub-manifold with boundarypossessing a point p P X with the property that any other point q P X is connected to p by a linesegment in X . We do not require X to have smooth boundary. Theorem 1.4. (Theorem 3.5) Let X Ñ Y be a symplectic embedding of a star-shaped domain X into asmooth rational projective surface p Y , ω A q with a ample (cid:82) -divisor A with r ω A s “ PD r A s . Then ( ‹ ) c ECH k p X q ď c alg k p Y , A q Remark 1.5.
Methods of algebraic geometry have been applied extensively to symplectic em-bedding problems for rational and toric surfaces, and our result is just one more perspectiveon this story. We refer the reader to the work of McDuff [23], McDuff-Polterovich [24], Anjos-Lalonde-Pinsonnault [1], Casals-Vianna [5] and Christofaro-Gardiner-Holm-Mandini-Pires [10]for just a few examples. Likewise, rationality is a key assumption in many embedding results(even those that use purely symplectic methods). See, for example, the work of Buse-Hind [3]and Opshtein [28]. Note that our references here are not at all exhaustive.
Remark 1.6.
The formula ( ‹ ) provides a new computational tool for studying the ECH capacitiesof star shaped domains living within divisor complements. Indeed, the nef cones of surfaces arevery well studied and many structural results exist which may be brought to bear while studying c ECH via Theorem 3.5. Furthermore, the nef cone is often polyhedral, and thus methods fromconvex optimization can be utilised to compute c alg . We hope to explore the combinatoral andcomputational implications of ( ‹ ) in future work.Although we were originally motivated to prove Theorem 1.4 in order to study non-toric sur-faces, many interesting implications appear even in the toric setting. In particular, [34, Thm. 1.5]implies that the inequality in Theorem 1.4 is an equality for certain divisor complements, andthis is key to our applications. We will now discuss the three results on symplectic embeddingsinto smooth toric surfaces that we will prove.For our first application, we prove that these obstructions are sharp for embeddings of concavetoric domains into toric surfaces. Theorem 1.7. (Theorem 4.13) Let X ∆ be a concave toric domain with interior X ˝ ∆ Ă X ∆ , and let p Y Ω , A Ω q be a smooth toric surface. Then X ˝ ∆ symplectically embeds into Y Ω ðñ c ECH k p X ∆ q ď c alg k p Y Ω , A Ω q This result uses a similar result of Christofaro-Gardiner in [8], for embeddings of concave domainsinto convex domains. Theorem 1.7 essentially shows that the extra freedom provided by gluingthe divisor A Ω into X ˝ Ω makes no difference for embeddings of concave domains.For our next application, we prove the following result that includes a folk conjecture aboutthe Gromov width. Let Ξ be the moment polygon of a concave toric domain and define the Ξ -width by c Ξ p X q : “ sup t r : X r Ξ symplectically embeds in X u When Ξ is the triangle with vertices p , q , p , q , p , q the Ξ -width c Ξ is just the Gromov width c G . Theorem 1.8. (Corollary 4.14 + Corollary 4.15) Let Ξ be the moment polygon of a concave toric domain.Suppose Ω Ă ∆ is an inclusion of moment polytopes of smooth toric projective surfaces. Then c Ξ p Y Ω q ď c Ξ p Y ∆ q In particular, the Gromov widths satisfy c G p Y Ω q ď c G p Y ∆ q CH EMBEDDING OBSTRUCTIONS FOR RATIONAL SURFACES 5
In fact, we prove Theorem 1.8 (and Theorem 1.7) for all projective surfaces (even singular ones)that possess a smooth fixed point. Note that any smooth symplectic toric 4-manifold is a smoothprojective toric surface (c.f. [19]) so Theorem 1.8 may be stated in those terms as well.
Remark 1.9.
There have been previous results (c.f. [10, Thm. 1.2]) indicating that a ball (or moregenerally, ellipsoid) embeds into a toric domain if and only if it embeds into the correspondingtoric surface. These results are related to Theorem 1.8, and can actually be used to recover somecases. See §4.4 for more discussion.Finally, we prove an estimate of the Gromov width of a toric surface in terms of the latticewidth of its moment polygon. This result is [2, Conjecture 3.12].
Definition 1.10.
The lattice width w p Ω q of a moment polytope is defined by w p Ω q : “ min l P (cid:90) n z ´ max p , q P Ω x l , p ´ q y ¯ Theorem 1.11. (Corollary 4.19) Let Ω be a moment polygon with a smooth vertex. Then c G p Y Ω q ď w p Ω q In particular, this holds when Ω is Delzant or, equivalently, when the toric surface Y Ω is smooth. Theorem 1.11 follows from Theorem 1.8 and a rigorous version of a heuristic argument from [2].
Remark 1.12.
The assumption that the moment polytope has a smooth vertex in Theorems 1.7,1.8 and 1.11 is an technical assumption that may be removable with different methods.1.4.
Future directions.
There are a number of interesting research directions along the linesof [34] and this paper that are worth exploring. We will comment on these now.First, Theorem 1.1 in [34] gives an equality for the ECH capacities, and it is natural to ask whenTheorem 3.5 can be upgraded to an equality as well. Here is a guess along those lines.
Conjecture 1.13 (ECH of divisor complements) . Let p Y , ω A q be a rational projective surface with anample (cid:82) -divisor A such that sing p Y q Ď supp p A q and suppose Y z supp p A q is deformation-equivalent toa ball. Then, c ECH k p Y z supp p A qq “ c alg k p Y , A q Note that Y z A can still be viewed as the interior of a star shaped domain with corners X .Proving Conjecture 1.13 would require either a clever argument for packing X or a very delicateunderstanding of the ECH and Reeb dynamics of smoothings of X .Beyond the ECH capacities, there are finer obstructions defined (by Hutchings in [16]) forembeddings of convex toric domains into other convex toric domains. These invariants are stillpoorly understood. The hope is that they could help solve some of the more obstinate embeddingproblems, such as the problem of embedding polydisks into ellipsoids. Question 1.14.
Let ∆ and Ω be rational moment polytopes. Is there a framework for treating theobstructions of [16] to embeddings X ∆ Ñ X Ω in terms of the algebraic geometry of Y ∆ and Y Ω ?Finally, our proof of Theorem 1.8 for the Gromov width requires only a family of capacitiesthat provide sharp obstructions for embeddings of the ball into convex toric domains, and anextension of these invariants to closed toric surfaces satisfying a set of axioms (see Proposition4.9). It is interesting to ask if the proof of Theorem 1.8 can be ported to higher dimensions usinganother family of holomorphic curve based capacities, such as the S -equivariant symplectichomology capacities of Gutt-Hutchings [13] or the rational SFT capacities of Siegel [30]. J. CHAIDEZ AND B. WORMLEIGHTON
Outline.
This concludes §1 , the introduction. The rest of the paper is organized as follows.In §2 , we cover preliminaries in Seiberg-Witten theory (2.1) and embedded contact homology(2.2). We then prove an important estimate of the ECH capacities of a star-shaped domain interms of a minimum area over Seiberg-Witten non-zero classes. We should note that this is wherethe “neck stretching” part of the argument is made formal.In §3 , we discuss the algebraic capacities in earnest (§3.1). We then prove Theorem 1.4 usingthe results of §2 and methods from algebraic geometry (§3.2).In §4 , we discuss the applications to toric surfaces. We start with a review of toric surfaces(4.1) and toric domains (4.2). We then show that the algebraic capacities of a (possibly singular)surface satisfy a set of nice axioms (4.3). Finally, we apply the axioms to prove Theorems 1.7-1.11. Acknowledgements.
We would like to thank Michael Hutchings for sharing [14] with us andsuggesting that its contents were relevant to the arguments in §1.2. We would also like tothank David Eisenbud and Sam Payne for helpful conversations. JC was supported by the NSFGraduate Research Fellowship under Grant No. 1752814.2. ECH capacities and Seiberg–Witten theoryIn this section, we review some aspects of Seiberg–Witten theory (§2.1) and embedded contacthomology (§2.2). Our goal is to prove an estimate for the ECH capacities in terms of the Seiberg–Witten invariants in §2.3.2.1.
Seiberg–Witten invariants.
The Seiberg–Witten invariants are a family of integer-valuedinvariants for a closed 4-manifold X with b ` p X q ě s . These invariantsare constructed using moduli spaces arising from gauge theory. We direct the reader to [26]and [20] for a detailed review.When X is symplectic, a canonical spin-c structure s X exists and we can view the Seiberg–Witten invariants as a function(2.1) SW X : H p X ; (cid:90) q Ñ (cid:90) A ÞÑ SW X p A q : “ SW X p s X ` A q We note that in the b ` p X q “ ` versionSW ` X of the Seiberg–Witten invariants. These invariants satisfy a number of useful axioms. Proposition 2.1 (SW Axioms) . The Seiberg–Witten invariants of symplectic -manifolds with b ` ě have the following properties. Fix a symplectic -manifold X and a cohomology class C P H p X ; (cid:90) q .(a) (Index) The index I p C q of X and the spin-c structure s “ s ˝ ` C , given by I p C q “ c p L s q ´ χ p X q ´ σ p X q “ c p X q ¨ C ` C satisfies I p C q ě if SW X p C q ‰ .(b) (Blow Up) Let π : ˜ X Ñ X be the blow up of X with exceptional divisor E Ă ˜ X . Fix a cohomologyclass ˜ C of the form ˜ C “ π ˚ C ` r ¨ r E s P H p ˜ C ; (cid:90) q with I p ˜ C , ˜ A q ě Then the Seiberg–Witten invariants of A and ˜ A are related by SW ˜ X p ˜ C q “ SW X p C q .(c) (Projective Plane) The Seiberg–Witten invariants of (cid:80) are given by SW (cid:80) p d ¨ r (cid:80) sq “ " if d ě else CH EMBEDDING OBSTRUCTIONS FOR RATIONAL SURFACES 7
Instead of the Seiberg-Witten function SW Y , we will often refer to the set of Seiberg-Wittennon-zero classes, denoted bySW p Y q : “ t A P H p Y ; (cid:90) q : SW Y p A q “ u Ă H p Y ; (cid:90) q This set of homology classes will be particularly referred to in §3.2.
Example 2.2.
As an important example (used in §3.2), we compute the set of SW non-zero classesSW p Y q when X is a minimal rational surface; Y “ (cid:80) , (cid:80) ˆ (cid:80) or a Hirzebruch surface (cid:70) r .For (cid:80) , Proposition 2.1(c) states that SW p (cid:80) q is precisely the cone generated by the line class H “ r (cid:80) s , meaning that SW p (cid:80) q “ Cone p H q “ t dH : d ě u For (cid:80) ˆ (cid:80) , we note that a blowup of (cid:80) ˆ (cid:80) can be realised as the blowup of (cid:80) in two points.Using Proposition 2.1(b) we find thatSW p (cid:80) ˆ (cid:80) q “ Cone p H , H q where H , H are the classes of two lines each in different rulings of (cid:80) ˆ (cid:80) . For the Hirzebruchsurfaces, we note that (cid:70) is the blowup of (cid:80) in one point and so we find that SW p (cid:70) q is given bythe lattice points inside the polyhedron shown in Fig. 3.Figure 3. Seiberg–Witten nonzero divisors for (cid:70) ‚ ‚ ‚ ‚ D D ` D F SW p (cid:70) q Here we use the basis F , D for H p (cid:70) , (cid:90) q where F is a fibre class and D is the section at ,and denote the zero section by D . Finally, for r ě p (cid:70) r q to SW p (cid:70) r ` q bya blowup/blowdown procedure – often called ‘elementary transformations’ – allowing us toapply Proposition 2.1(b). We find thatSW p (cid:70) r q “ Cone p F , D ´ rF q Here F and D are analogous to the classes used for (cid:70) .2.2. Embedded Contact Homology.
Here we review embedded contact homology as a sym-plectic field theory, as presented in [14] (also see [15]).
Definition 2.3. A contact -manifold p Y , ξ q is a 3-manifold Y with a 2-plane bundle ξ Ă TY thatis the kernel ξ “ ker p α q of a contact form. A contact form α is a 1-form satisfying α ^ d α ą Reeb vector-field R of α is the unique vector-field satisfying α p R q “ d α p R , ¨q “
0, and a
Reeb orbit is a closed orbit (modulo reparametrization) of R .The embedded contact homology , or ECH for short, of a closed contact 3-manifold p Y , ξ q is a (cid:90) { (cid:90) { p Y , ξ q “ à r Γ sP H p Y ; (cid:90) q ECH p Y , ξ ; r Γ sq J. CHAIDEZ AND B. WORMLEIGHTON
The ECH group comes equipped with a degree ´ U-map and a distinguished empty set class. U : ECH p Y , ξ ; r Γ sq Ñ ECH p Y , ξ ; r Γ sq rHs P ECH p Y , ξ ; r sq The (cid:90) { p Y , ξ ; r sq can be canonically enhanced to a (cid:90) { m -grading where rHs has grading 0 and m is defined by m : “ min tx c p ξ q ; r Σ sy : r Σ s P H p Y ; (cid:90) qu The simplest example of ECH groups are those of the 3-sphere.
Proposition 2.4. (c.f. [15]) The embedded contact homology
ECH p S , ξ q of the -sphere is given by ECH p S , ξ q “ (cid:90) { r U ´ s as a (cid:90) { r U s -module, where | U ´ | “ and U acts in the obvious way. Given a choice of contact form α for p Y , ξ q , one can enhance the ECH groups of Y to a familyof filtered ECH groups ECH L p Y , α ; r Γ sq parametrized by L P r , equipped with natural maps(2.2) ι KL : ECH L p Y , α ; r Γ sq Ñ ECH K p Y , α ; r Γ sq and ι L : ECH L p Y , α ; r Γ sq Ñ ECH p Y , ξ ; r Γ sq Each filtered ECH group comes equipped with a U-map and empty set class, and these structuresare compatible with the maps (2.2). U L : ECH L p Y , α ; r Γ sq Ñ ECH L p Y , α ; r Γ sq rHs L P ECH L p Y , ξ ; r sq Furthermore, the inclusions ι KL respect composition and the ordinary ECH is the colimit of thefiltered ECH groups via the maps ι L .We can give a simple definition of the ECH capacities in terms of the formal structure of ECHdescribed above. Definition 2.5.
The k-th ECH capacity c k p Y , α q of a closed contact 3-manifold is defined by c k p Y , α q “ min ! L : rHs “ U k ˝ ι L p σ q for σ P ECH L p Y , α ; r sq ) The k -th ECH capacity c k p X , λ q of a Liouville domain p X , λ q is the k -th ECH capacity of itsboundary pB X , λ | B X q as a contact manifold.The ECH capacities are (non-normalized) capacities on the category of Liouville domains. Proposition 2.6.
The ECH capacities c k p¨q satisfy the following axioms.(a) (Inclusion) If X Ñ X is a symplectic embedding of Liouville domains, then c k p X , λ q ď c k p X , λ q .(b) (Scaling) If p X , λ q is a Liouville domain then c k p X , C ¨ λ q “ C ¨ c k p X , λ q for any constant C ą . The ECH groups are the homology of an ECH chain group ECC p Y , α, J q depending on achoice of non-degenerate1 contact form α and a complex structure J on the symplectization of Y satisfying certain conditions. The chain group is freely generated over (cid:90) { orbit sets Γ “ tp γ i , m i qu ki “ γ i is an embedded Reeb orbit and m i P (cid:90) ` satisfying the condition that m i “ γ i is hyperbolic. Given an element x of ECC p Y , α, J q and an orbit set Γ , we denote the Γ -coefficient of x by x x , Γ y . The differential B : ECC p Y , α, J q Ñ ECC p Y , α, J q is defined by a holomorphic curve count. More precisely, if Γ ` “ tp γ i , m i qu k and Γ ´ “ tp η i , n i qu l are admissible orbit sets, then the Γ ´ -coefficient of B Γ ` is given by xB Γ ` , Γ ´ y “ M p Y , J q{ (cid:82)
1A non-degenerate contact form is one where the differential of the Poincare return map along any orbit has no1-eigenvalues.
CH EMBEDDING OBSTRUCTIONS FOR RATIONAL SURFACES 9
Here M p Y , J q{ (cid:82) is a count of (possibly disconnected) holomorphic curves in the symplectiza-tion of Y that have ECH index 1 with positive ends at Γ ` and negative ends at Γ ´ . The ECH index I p C q of a homology class in C P H p Y , Γ ` Y Γ ´ q is defined by(2.3) I p C q “ x c τ p ξ q , C y ` Q τ p C , C q ` k ÿ i “ m i ÿ j “ CZ τ p γ ij q ´ l ÿ i “ n i ÿ j “ CZ τ p η ij q Here c τ p ξ q is the relative 1st Chern class, Q τ p C , C q is the relative intersection form and CZ τ isthe Conley-Zehnder index (all relative to a trivialization τ of the contact structure).Embedded contact homology has a vaguely TQFT-like structure, whereby certain types ofcobordisms between contact manifolds induce maps on the (filtered) ECH groups. Definition 2.7. A strong symplectic cobordism X between contact manifolds Y ˘ with contact form,denoted by p X , ω q : p Y ` , α ` q Ñ p Y ´ , α ´ q is a symplectic manifold p X , ω q with oriented boundary B X “ Y ` ´ Y ´ such that ω | Y ˘ “ ˘ d α ˘ .The area class r ω, α ˘ s P H p X , B X q of p X , ω q is the class of the relative de Rham cycle p ω, α ` ´ α ´ q P Ω p X q ‘ Ω p Y ` Y ´ Y ´ q We use r Σ s : r Γ ` s Ñ r Γ ´ s to denote a relative class in H p X , B X q whose image under theboundary map B : H p X , B X q Ñ H pB X q is given by r Γ ` s ‘ r Γ ´ s P H p Y ` q ‘ H p Y ´ q » H pB X q For convenience, we use ρ r Σ s Ñ (cid:82) denote the pairing of r Σ s with the area class r ω, α ˘ s . Explicitly,we have the formula ρ r Σ s “ ż Σ ω ´ ż B ` Σ α ` ` ż B ´ Σ α ´ With the above notation, we can state the following result of Hutchings regarding the functorialityof ECH with respect to strong symplectic cobordisms.
Theorem 2.8 (Hutchings [14]) . A strong symplectic cobordism X : Y ` Ñ Y ´ and let r Σ s : r Γ ` s Ñ r Γ ´ s be a class in H p X , B X q . Then there is a canonical, ungraded map (2.4) ECH L p X ; r Σ sq : ECH L p Y ` , α ` ; r Γ ` sq Ñ ECH L ` ρ r Σ s p Y ´ , α ´ ; r Γ ´ sq (2.5) ECH p X ; r Σ sq : ECH p Y ` , ξ ` ; r Γ ` sq Ñ ECH p Y ´ , ξ ´ ; r Γ ´ sq These maps are compatible with composition, and satisfy the following axioms.(a) (Curve Counting) There exists a chain map Φ inducing ECH p X ; r Σ sq , of the form Φ L : ECC L p Y ` , α ` ; r Γ ` sq Ñ ECC L ` ρ r Σ s p Y ´ , α ´ ; r Γ ´ sq that “counts curves” in the following sense. If Γ ˘ are orbit sets in Y ˘ , and x Φ A p Γ ` q , Γ ´ y “ ,then there is a holomorphic current2 of ECH index asymptotic at ˘8 to Γ ˘ .(b) (Filtration) The maps commute with the inclusion maps ι KL and ι L , e.g. ECH L p Y ` , α ` ; r Γ ` sq ECH L ` ρ r Σ s p Y ´ , α ´ ; r Γ ´ sq ECH K p Y ` , α ` ; r Γ ` sq ECH K ` ρ r Σ s p Y ´ , α ´ ; r Γ ´ sq ECH L p X ; r Σ sq ECH K p X ; r Σ sq (c) (U-Map) The maps commute with the U -maps, e.g. U L ` ρ r Σ s ˝ ECH L p X ; r Σ sq “ ECH L p X ; r Σ sq ˝ U L (d) (Seiberg–Witten) Let p P , ξ q be a contact -manifold. Consider a pair of strong symplectic cobor-disms and their composition, denoted by N : H Ñ
P X : P Ñ H and Y “ N Y Z X : H Ñ H
Fix homology classes r A s P H p N , Z q and r B s P H p X , Z q with Br A s “ Br B s . Then ECH p X , r B sq ˝ U k ˝ ECH p N , r A sq “ ÿ r C sP S SW Y pr C sq Here S Ă H p X q is shorthand for the set of homology classes satisfying r C s X N “ r A s r C s X X “ r B s and I pr C sq “ k From ECH to SW.
We now conclude the section by applying the formal structure of theECH groups in §2.2 to estimate for the ECH capacities of a star-shaped domain embedded intoclosed symplectic manifolds.
Proposition 2.9.
Let p X , λ q Ă (cid:82) be a star-shaped domain with restricted Liouville form λ and let p Y , ω q be a closed symplectic -manifold. Fix an embedding ι : p X , d λ q Ñ p Y , ω q Then the ECH capacities of X satisfy (2.6) c k p X q ď inf r Σ sP SW p Y q tx ω, r Σ sy : I pr Σ sq ě k u Remark 2.10.
This result is based on the proofs in [14, §2.2].
Proof.
Let p Z , α q be the contact boundary of p X , λ q and let r Σ s P H p Y q be any (cid:90) -homology classsatisfying the constraints laid out in (2.6).SW Y pr Σ sq “ I pr Σ sq ě k It suffices to demonstrate the following inequality for any such r Σ s . c k p X q ď A : “ x ω, r Σ sy Since c k p X q ď c j p X q for j “ I pr Σ sq{
2, we can assume that k “ j “ I pr Σ sq{
2. Furthermore, it isequivalent to show that for all (cid:15) ą η P ECH A ` (cid:15) p Z , ξ ; r sq with U k ι A η “ rHs P ECH p Z , ξ ; r sq To find an η that satisfies (2.7), we consider the splitting of Y into X (or rather, the image ι p X q )and N “ Y z X . If we denote the contact boundary of X by p Z , ξ q , we can interpret this as pair ofstrong symplectic cobordisms N : H Ñ
Z X : Z Ñ H
Since X is diffeomorphic to a 4-ball, the pair of maps H p Y q ´X X ÝÝÝÑ H p X , B X q and H p Y q ´X P ÝÝÝÑ H p P , B P q are, respectively, the 0 map and an isomorphism. Let r S s “ r Σ s X X be the intersection of r Σ s with X . Note that we have A “ xr ω s , r Σ sy “ ρ r S s ` ρ r s “ ρ r S s Now we let (cid:15) ą η by η “ ECH A p P ; r S sqrHs P ECH A ` (cid:15) p Z , ξ ; r sq where rHs P ECH (cid:15) pH ; r sq » (cid:90) { rHs We would like to show that U k ι A ` (cid:15) η “ rHs . To start, pick a chain map lifting the ECHcobordism map as in Thm. 2.8(a). That is, Φ : (cid:90) { Ñ ECC (cid:15) ` A p Z , α ; r sq with r Φ pHqs “ η CH EMBEDDING OBSTRUCTIONS FOR RATIONAL SURFACES 11 If Γ ´ is any orbit set such that x Φ pHq , Γ ´ y “
1, then by Theorem 2.8(a) we know that there is aholomorphic current C of ECH index 0 with empty positive boundary and negative boundary Γ ´ . If we let C Ă Z be a surface with positive boundary Γ ´ , so that | Γ ´ | “ I p C q , then by theadditivity of the ECH index we have2 k “ I pr Σ sq “ I p C q ` I p C q “ I p C q “ | Γ ´ | Thus we know that η is homogenous of grading 2 k , and so U k ˝ ι A ` (cid:15) p η q is grading 0. In particular,by Proposition 2.4, we have U k ι A ` (cid:15) η P ECH p Z , ξ ; r sq » ECH p S ; r sq “ (cid:90) { rHs On the other hand, by Theorem 2.8(b) and (d), we know thatECH p X ; r sq ˝ U k ˝ ι A ` (cid:15) η “ ECH A p X ; r sq ˝ U k ˝ ECH p X ; r S sqrHs “ c Here c P (cid:90) { r C s with r C s X X “ r s and r C s X P “ r S s of SW Y pr C sq mod 2.Since r Σ s is the unique such class and SW Y pr Σ sq “ c “
1. Thus, U k ι A ` (cid:15) η is non-zero and we must have U k ι A ` (cid:15) η “ rHs This proves that for every (cid:15) , there is a class η P ECH A ` (cid:15) p Z , ξ ; r sq satisfying (2.7), and thusconcludes the proof. (cid:3) Remark 2.11.
The proof of Proposition 2.9 generalizes immediately to Liouville domains p X , λ q that satisfy the following criteria.(a) The map H pB X q ι ˚ ÝÑ H p X q is 0.(b) The contact manifold pB X , ξ q has torsion chern class, i.e. c p ξ q “ P H pB X ; (cid:81) q .(c) The empty set rHs is the unique class of ECH grading 0 in the image of the U -map.The conclusion of Proposition 2.9 must be appropriately modified so that (2.6) is a minimumover all classes r Σ s such that r Σ s X X “
0. In practice, the most difficult criterion to verify is (c).This holds, for instance, when rHs is the unique ECH index 0 class. It is also believed to holdfor circle bundles over a 2-sphere (c.f. the unpublished thesis of Ferris [11] and the forthcomingwork of Nelson–Weiler [27]).3. Algebraic capacities and birational geometryWe now construct of the algebraic capacities (§3.1) and prove Theorem 3.5 (§3.2).
Conventions 3.1.
In this section, all surfaces will be projective normal algebraic surfaces overthe complex numbers, not necessarily smooth, unless otherwise specified.Let (cid:75)
P t (cid:90) , (cid:81) , (cid:82) u . We work in the Néron–Severi group NS p Y q Ď H p Y , (cid:90) q of Weil (cid:90) -divisorsregarded up to algebraic equivalence. We denote NS p Y q (cid:75) : “ NS p Y qb (cid:90) (cid:75) . We say that a (cid:90) -divisor D on a surface Y is (cid:81) -Cartier if some integer multiple of D is Cartier; that is, the sheaf O p D q isa line bundle. Y is said to be (cid:81) -factorial if every Weil (cid:90) -divisor on Y is (cid:81) -Cartier. Every toricsurface is (cid:81) -factorial. A (cid:81) -Cartier (cid:82) -divisor D on Y is nef if D ¨ C ě C Ď Y .Denote by Nef p Y q (cid:75) the classes in NS p Y q (cid:75) corresponding to nef divisors.3.1. Construction of algebraic capacities.
Let Y be a (cid:81) -factorial projective surface and let A bean ample (cid:82) -divisor on Y . We recall the optimisation problems of [34–36] that are designed toemulate ECH capacities in the context of algebraic geometry. Definition 3.2 ( [34, §4.5] or [36, Def. 2.2]) . The k th algebraic capacity of p Y , A q are given by(3.1) c alg k p Y , A q : “ inf D P Nef p Y q (cid:90) t D ¨ A : χ p D q ě k ` χ p O Y qu Remark 3.3.
Note that it follows from Kleiman’s criterion for nef-ness that this infimum in (3.1)is always achieved.The index of a (cid:90) -divisor D on Y is given by I p D q : “ D ¨ p D ´ K Y q . When Y is smooth or has atworst canonical singularities [29] we have Noether’s formula(3.2) χ p D q “ χ p O Y q ` I p D q Furthermore, if ω A is the Kahler class induced by A via the embedding into (cid:80) H p k O p A qq for k " A is ample) we may write(3.3) D ¨ A “ x ω A , D y “ ż D ω A In these cases, we can alternatively write the algebraic capacities as(3.4) c alg k p Y , A q “ inf D P Nef p Y q (cid:90) tx ω A , D y : I p D q ě k u which is very similar to the upper bound for c ECH k in Proposition 2.9.3.2. Relating ECH capacities and algebraic capacities.
We seek to prove the following result.
Theorem 3.4.
Suppose Y is a smooth rational surface, and let A be an ample (cid:82) -divisor on Y . Then inf D P SW p Y q t D ¨ A : I p D q ě k u “ inf D P Nef p Y q (cid:90) t D ¨ A : I p D q ě k u “ : c alg k p Y , A q By combining Proposition 2.9, the formula (3.4) and Theorem 3.4, we immediately acquire themain result, which we state again for completeness.
Theorem 3.5.
Suppose that X Ñ Y is a symplectic embedding of a star-shaped domain X into a smoothrational projective surface Y with a ample (cid:82) -divisor A and symplectic form ω A satisfying r ω A s “ PD r A s .Then ( ‹ ) c ECH k p X q ď c alg k p Y , A q Remark 3.6.
We only require an upper bound of the Seiberg-Witten quantity by c alg k for the pur-poses of this paper. However, Theorem 3.4 is satisfying because it demonstrates that the algebraiccapacities are (as obstructions) just as sensitive as the Seiberg-Witten theoretic quantities.We treat the case of smooth rational surfaces using the Minimal Model Program. We startwith a result most easily proved in symplectic geometry. For a proof, see [12, §2]. Proposition 3.7.
Let Y be a smooth surface. Then SW p Y q Ď NE p Y q . For a nef (cid:82) -divisor D on a (cid:81) -factorial variety Y and a point p P Y recall the (local) Seshadriconstant , which is defined by ε p D , p q : “ sup t ε ě π ˚ D ´ ε E is nef u where π : r Y Ñ Y is the blowup of Y at p with exceptional divisor E Ď r Y . Lemma 3.8.
For Y a smooth surface, a point p P Y , and π : r Y Ñ Y the blowup at p with exceptionaldivisor E , Nef p r Y q (cid:82) “ ď D P Nef p Y q (cid:82) π ˚ D ´ r , ε p D , p qs E CH EMBEDDING OBSTRUCTIONS FOR RATIONAL SURFACES 13
Proof.
Suppose r D is a nef divisor on r Y . It suffices to verify that r D “ r D ` p r D ¨ E q E is nef, since r D is orthogonal to E and is hence of the form π ˚ D for some D P Nef p Y q (cid:82) . Now E is effective andso p r D ¨ E q E is effective, from which it follows that for any other prime divisor r D i “ E r D ¨ r D i ě r D ¨ r D i ě r D is nef. (cid:3) Proposition 3.9.
Let Y be a smooth rational surface. Then (3.5) t D P Nef p Y q (cid:90) : I p D q ě u Ď SW p Y q Proof.
We induct on the number b of blow ups required to acquire Y from a minimal surface. If Y is minimal, (3.5) can be verified directly from the calculation of SW p Y q in Example 2.2.Now suppose Y is acquired by blowing up of a minimal surface b ą E Ď Y bea p´ q -sphere and let π : Y Ñ Y be the corresponding contraction. By Lemma 3.8 it suffices toshow that π ˚ D ´ mE P SW p Y q whenever I p π ˚ D ´ mE q ě D P Nef p Y q (cid:90) and all integers m with 0 ď m ď ε p D , p q . Since ¯ Y is a blow up of a minimal surface b ´ t D P Nef p Y q : I p D q ě u Ď SW p Y q Now observe the following index inequality for the proper transform I p π ˚ D ´ mE q “ I p π ˚ D q ´ m p m ` q “ I p D q ´ m p m ` q ď I p D q Thus I p π ˚ D ´ mE q ě I p D q ě
0. Hence, all eligible D are Seiberg–Witten nonzero andthus it follows from the blowup formula Proposition 2.1(b) that π ˚ D ´ mE is Seiberg–Wittennonzero when D is nef and m is as above such that I p π ˚ D ´ mE q ě (cid:3) Remark 3.10.
One can prove Proposition 3.9 combinatorially for smooth toric surfaces. A similarproof strategy yields that if Y is a smooth rational surface with ´ K Y effective then we haveNef p Y q (cid:90) Ď SW p Y q .The inclusion in Proposition 3.9 automatically implies an inequality in one direction.inf D P SW p Y q t D ¨ A : I p D q ě k u ď inf D P Nef p Y q (cid:90) t D ¨ A : I p D q ě k u For the converse inequality, we will show for that each Seiberg–Witten nonzero divisor thereis a nef divisor that is ‘preferable’ from the perspective of the optimisation problems above. Forthis purpose, we adopt the following terminology.
Definition 3.11.
Let Y be a (cid:81) -factorial surface. We say that a Weil (cid:81) -divisor D is(a) index-preferable to another Weil (cid:81) -divisor D if I p D q ě I p D q and(b) area-preferable D if D ¨ A ď D ¨ A for all ample (cid:82) -divisors A on Y .A Weil (cid:81) -divisor D that is both area- and index-preferable will simply be called preferable . Notethat D is area-preferable to D if and only if D ´ D is effective.To construct preferable divisors in general we will use the isoparametric transform IP Y of [4].This takes an effective divisor D to a new divisor IP Y p D q given by(3.6) IP Y p D q : “ D ´ ÿ D ¨ D i ă S D ¨ D i D i W D i Here the sum is over prime divisors D i with D ¨ D i ă Y p D q “ D if D is nef.We denote by IP nY p D q the result of iterating IP nY n times. In [4], the following result is proven. Theorem 3.12 ( [4, Thm. 1.1 + 1.2]) . For any effective divisor D on a smooth surface Y we have h p D q “ h p IP Y p D qq Then for all sufficiently large n " , we have IP nY p D q “ IP Y p D q for some nef IP Y p D q P Nef p Y q (cid:90) . We will need to know what IP Y does to area and index. For area, the answer is quite simple. Lemma 3.13.
Let D be effective and A be ample. Then A ¨ IP Y p D q ď A ¨ D .Proof. If D i is a prime divisor with D i ¨ D ă D is effective, then D i ă
0. Thus the coefficientsof the sum in (3.6) are positive. Since A is ample, A ¨ D i ă
0. These two facts imply the result. (cid:3)
The answer for the index is more complicated. For this, we need the following lemma.
Lemma 3.14.
Let Y be a smooth surface with D an effective divisor on Y . Suppose C , . . . , C n is acollection of curves intersecting D negatively. Then either one of the C i is a p´ q -curve or I p D q ě I p D q where D “ D ´ n ÿ i “ S D ¨ C i C i W C i In particular, I p IP Y p D qq ě I p D q if no p´ q -curve intersects D negatively.Proof. Suppose n “ C . If C “ ´ C “ ´ r for r ě
2. Let D ¨ C “ ´ (cid:96) so that D “ D ´ R (cid:96) r V C “ : D ´ mC Let π : Y Ñ Y be the contraction of C to the singular surface Y . We can compute I p D q “ p D ´ mC q ¨ p D ´ mC ´ K Y q“ I p D q ´ mD ¨ C ` p´ mC q ¨ p´ mC ´ K Y q“ I p D q ` m (cid:96) ` p´ mC q ¨ p´ mC ´ π ˚ K Y ´ ´ rr C q“ I p D q ` m (cid:96) ´ m r ´ p ´ r q m Now observe that 1 ą m ´ (cid:96) r ě (cid:96) ` r ą rm . Furthermore, r ě m ě m (cid:96) ´ mr p m ` ´ rr q ą m (cid:96) ´ p (cid:96) ` r qp m ` ´ rr q“ m (cid:96) ` (cid:96) ¨ r ´ r ´ mr ` r ´ ě m (cid:96) ´ r p m ´ q ´ ą p m ´ q (cid:96) ´ ě ´ I p D q ą I p D q ´
2. However I p¨q is even and so we must have I p D q ě I p D q .Now induct on the number of curves. Suppose the formula holds for a set of n curves meetingan effective divisor negatively. Suppose curves C , . . . , C n , C intersect D negatively. If any of thecurves is a p´ q -curve then we are done. Assume not. Notate D ¨ C “ ´ (cid:96), C “ ´ r , R D ¨ CC V “ m and F “ n ´ ÿ i “ m i C i so that D “ D ´ F ´ mC . Compute I p D ´ F ´ mC q ““ I p D ´ F q ` mF ¨ C ´ mD ¨ C ` I p´ mC qě I p D ´ F q ` m (cid:96) ´ mr p m ` ´ rr qą I p D ´ F q ` p m ´ q (cid:96) ´ ě I p D ´ F q ´ CH EMBEDDING OBSTRUCTIONS FOR RATIONAL SURFACES 15 where we used that F ¨ C ě F is effective and supported away from C . By inductiveassumption I p D ´ F q ě I p D q and so we have I p D q ą I p D q ´
2. Since I p¨q is even we can concludethat I p D q ě I p D q as desired. (cid:3) Proof of Thm. 3.4.
In view of Proposition 3.7, we simply need to show that for any divisor inSW p Y q , there exists a preferable nef divisor. In other words, we must construct a map N Y : SW p Y q Ñ Nef p Y q (cid:90) taking a Seiberg–Witten nonzero divisor to a preferable nef (cid:90) -divisor. We now construct thesemaps by induction on the number of blow ups necessary to make Y from a minimal surface.For minimal rational surfaces the existence of an N Y is clear. In the cases of (cid:80) and (cid:80) ˆ (cid:80) ,we have SW p Y q “ Nef p Y q (cid:90) . Hirzebruch surfaces, on the other hand, have no p´ q -curves. Thuswe can set N Y p D q “ IP nY p D q for n "
0. Lemmas 3.14 and 3.13 imply that the result is preferable.Now assume that such a function exists for all rational surfaces expressible as b ´ Y be a surface expressed as b blowups of a minimal rationalsurface, and for any p´ q -curve E Ď Y denote the contraction by π E : Y Ñ Y E . We define N Y p D q by the following procedure.(a) If D ¨ C ě C Ď Y then D is nef and we define N Y p D q “ D .(b) If D ¨ E ď p´ q -curve E , write D “ π ˚ E D ` mE for some D P SW p Y E q andfor some m ě
0. The inductive hypothesis implies that there exists a nef (cid:90) -divisor D preferable to D . Define N p D q “ π ˚ E D .(c) If D ¨ E ą p´ q -curves E on Y but D ¨ C ă p´ r q -curve C with r ě Y p D q instead of D and define N Y p D q as the result.This procedure terminates: if IP nY p D q eventually intersects a p´ q -curve negatively then (b)outputs a nef divisor. If IP nY p D q does not intersect a p´ q -curve nonpositively for any n then aftera finite number of steps we reach IP Y p D q P Nef p Y q (cid:90) by Theorem 3.12, which is returned by (a).Note that the application of Theorem 3.12 is valid by Proposition 3.7.We claim that N Y p D q is nef and preferable to D . Indeed, all three steps (a)-(c) only improvethe area and index constraints. This claim is trivial for (a) and follows from Lemmas 3.13 and3.14 for (c). (b) produces a preferable nef (cid:90) -divisor since π ˚ E D is preferable to D “ π ˚ E D ` mE from direct calculation (noting that m ě π ˚ E D is nef and preferable to π ˚ E D since D is preferable to D . (cid:3)
4. Toric SurfacesWe now apply Theorem 3.5 to the study of embeddings into projective toric surfaces. We beginwith a review of toric surfaces (§4.1) and toric domains (§4.2). We then demonstrate that thealgebraic capacities on toric surfaces are uniquely characterized by a set of axioms (§4.3). Finally,we discuss the main applications: obstructing embeddings of concave toric domains into toricsurfaces, and monotonicity of the Gromov width under inclusion of moment polygons (§4.4).4.1.
Toric varieties.
We start with a brief review of toric varieties. Our main reference is [7].
Definition 4.1. A (projective normal) toric variety Y of dimension n over (cid:67) is a projective normalvariety with a p (cid:67) ˆ q n -action acting faithfully and transitively on a Zarisiki open subset of Y .Every toric variety Y can be described (uniquely, up to isomorphism) by either a fan Σ Ă (cid:82) n [7,Def 3.1.2 and Cor 3.1.8] or a moment polytope Ω Ă (cid:82) n [7, Def 2.3.14]. A fan Σ for Y can be recoveredfrom a moment polytope Ω for Y by passing to the inner normal fan Σ p Ω q of Ω [7, Prop 3.1.6]. Wewill focus on the polytope perspective, since it will be more important in this paper. Definition 4.2. A moment polytope Ω Ă (cid:82) n is a convex polytope with rational vertices and openinterior. We denote the corresponding toric variety by Y Ω .Note that given a scalar S ą (cid:81) or an affine map T : (cid:90) Ñ (cid:90) , we can scale Ω to S Ω or apply T to acquire T Ω . There are naturally induced isomorphisms of varieties Y S Ω » Y Ω and Y T Ω » Y Ω . Definition 4.3. A smooth vertex v P Ω of a moment polytope is a vertex such that there exists aneighborhood U Ă (cid:82) n ě of 0, a neighborhood V Ă Ω of v , a scaling S and a (cid:90) -affine isomorphism T such that ST p U q “ V and ST p v q “
0. Otherwise a vertex is singular .On a projective toric variety, each face F Ă Ω determines a (cid:81) -Cartier divisor D F . Every torusinvariant divisor is in the span of these divisors D F , and every divisor class is represented by atorus-invariant divisor [7, 4.1.3]. Furthermore, every moment polytope Ω for a toric variety Y Ω is associated to a natural divisor A Ω given as a combination of these face divisors. Definition 4.4.
The associated divisor A Ω of the moment polytope Ω is defined as A Ω “ ÿ F a F D F Here for each face F Ă Ω , we define u F P (cid:90) n and a F P (cid:81) by the following conditions. x u F , x y “ ´ a F for x P F u F is primitive in (cid:90) n , inward to Ω and normal to F Note that the equation x u F , x y “ ´ a F defines a hyperplane that we denote by Π F . Lemma 4.5.
The associated divisor A Ω of a moment polytope Ω has the following properties.(a) (Ample) A Ω is an ample divisor, and so defines an projective embedding to projective space. (4.1) | kD Ω | : Y Ω Ñ (cid:80) H p Y Ω , kA Ω q for k " (b) (Translation/Scaling) Let T P GL n p (cid:90) q , V P (cid:90) n and S P (cid:81) . Then D T Ω “ D Ω D Ω ` V “ D Ω ` P V D S Ω “ S ¨ D Ω Here P V is a principle divisor depending on V .Proof. For (a), see [7, Prop 6.1.10]. For (b), see [7, §4.2, Ex 4.2.5(a)] for the translation property.The scaling and linear map properties follow from Definition 4.4. (cid:3)
More generally, any (cid:84) n -equivariant (cid:81) -divisor D “ ř a F D F is associated to a half-space ar-rangement consisting of the half-spaces H F and a (possibly empty) polytope P F given by H F “ t x P (cid:82) : x u F , x y ě ´ a F u P F “ X F Π F The dimension of the space of sections h p D q is given by the number of lattice points | P F X (cid:90) n | [7,§7.1, p. 322]. A divisor is ample if and only if B H F X P D is an open subset of B H F for each F , andnef if B H F X P D is non-empty for each F .We are primarily interested in toric surfaces , i.e. projective toric varieties of complex dimension2. In this case, the embedding (4.1) gives Y the structure of a symplectic orbifold by restrictionof the Kahler form on (cid:80) N . Every toric surface is an orbifold [7, Thm. 3.1.19] since every two-dimensional fan is simplicial (dually, every polygon is simple).4.2. Toric domains.
We next review the theory of toric domains. Let ω std denote the standardsymplectic form on (cid:67) n and let µ denote the standard moment map µ : (cid:67) n Ñ (cid:82) n ě p z , . . . , z n q ÞÑ p π | z | , . . . , π | z n | q CH EMBEDDING OBSTRUCTIONS FOR RATIONAL SURFACES 17
Definition 4.6. A toric domain p X Ω , ω q is the inverse image µ ´ p Ω q of a closed subset Ω Ă r , with open interior, equipped with the symplectic form ω std | X Ω and moment map µ | X Ω .A toric domain X Ω is convex if Ω “ C X r , n where C Ă (cid:67) n is a convex and contains 0 in itsinterior, concave if the compliment (cid:82) ` z Ω is convex in (cid:67) n and free if Ω is convex and contained in (cid:82) n ` Ă (cid:82) n ě (i.e. disjoint from the coordinate axes). Finally, Ω is rational if it is a moment polytopein the sense of Definition 4.2 (i.e. a polytope with rational vertices).A fundamental fact in this paper is that a convex rational domain X can be compactified totoric surfaces Y by collapsing the boundary B Y so that it becomes the associated ample divisor A of Y . More precisely, we have the following result. Lemma 4.7.
Let Ω be a rational, convex domain polytope with toric variety p Y Ω , A Ω q and toric surface X Ω . Then there is a (cid:84) n -equivariant symplectomorphism Y Ω z supp p A Ω q » X ˝ Ω Proof.
Let µ : Y Ω Ñ (cid:82) n ě and ν : X Ω Ñ (cid:82) n ě denote the moment maps of Y Ω and X Ω . Define Ω ˝ tobe the complement Ω zpB Ω X (cid:82) n ` q . First note that Ω ˝ is the moment image of both Y Ω z supp p A Ω q under µ and X ˝ Ω under ν . For X ˝ Ω this is clear, and true for any convex domain.For Y Ω , write the associated ample divisor as A Ω “ ř F a F ¨ D F . By examination of Definition4.4, we see that a F “ F is on a plane passing through 0. Since Ω “ K X (cid:82) n ě forsome convex K , we know that Ω intersects each coordinate hyperplane H i “ t x P (cid:82) n | x i “ u along a single face F i and every other face F i is not contained in a plane containing the origin(essentially by convexity). Thus a F i “ i and a F ‰ F . Thus Y Ω z supp p A Ω q “ µ ´ p Ω ˝ q .Figure 4. Moment polytopes for Y Ω and Y Ω z D Ω Now that we have shown that X ˝ Ω and Y Ω z supp p A Ω q have the same moment images, we justapply an open version of Delzant’s theorem, e.g. the result of Kershon-Lerman [18, Thm 1.3].Note that, in that result, there is a homological obstruction o to the equivalence of two spaceswith the same moment image o P H p X ˝ Ω ; R q “ H p Y Ω z supp p A Ω q ; R q for some abelian group R . This obstruction necessarily vanishes since X ˝ Ω is contractible. (cid:3) Note that (essentially by definition) a moment polytope Ω Ă (cid:82) n is equivalent to a convex,rational polytope (cid:82) n ě by scalings and GL n p (cid:90) q -affine maps if and only if Ω has a smooth vertex. Example 4.8.
Considering ellipsoids X Ω “ E p a , b q and the corresponding toric varieties (cid:80) p , a , b q ,we recover the (well-)known compactifications (cid:80) z H “ B p q ˝ and (cid:80) p , a , b qz H “ E p a , b q ˝ where H “ O p q is a hyperplane section in each variety respectively.4.3. Axioms of c alg for toric surfaces. This section is devoted to proving that the algebraiccapacities of toric surfaces satisfy a set of important formal properties.
Theorem 4.9.
Let Y Ω be a projective toric surface with moment polytope Ω and associated ample divisor A Ω . Then the k th algebraic capacity satisfies the following axioms. (a) (Scaling/Affine Maps) If S ą is a constant and T : (cid:90) Ñ (cid:90) is an affine isomorphism, then c alg k p Y S Ω , A S Ω q “ S ¨ c alg k p Y Ω , A Ω q and c alg k p Y T Ω , A T Ω q “ c alg k p Y Ω , A Ω q (b) (Inclusion) If Ω Ă ∆ is an inclusion of moment polytopes, then c alg k p Y Ω , A Ω q ď c alg k p Y ∆ , A ∆ q (c) (Blow Up) If π : Y r Ω Ñ Y Ω is a birational toric morphism with one exceptional fiber E andassociated ample divisor A r Ω “ π ˚ A Ω ´ (cid:15) E for (cid:15) ą small, then c alg k p Y r Ω , A r Ω q ď c alg k p Y Ω , A Ω q (d) (Embeddings) If X Ă (cid:82) be a star-shaped domain that symplecically embeds into Y Ω , then c ECH k p X q ď c alg k p Y Ω , A Ω q (e) (Domains) If Ω is a (convex or free) domain polytope and X Ω is the associated toric domain, then c ECH k p X Ω q “ c alg k p Y Ω , A Ω q Furthermore, axioms (a)-(e) uniquely characterize the algebraic capacities c alg k on toric surfaces.Proof. We will need some of these properties to prove the others, so we must proceed in aparticular order. We first prove (a), (c) and (e) which are mutually independent. We then applythese properties to acquire (b) and apply Theorem 3.5 to acquire (d). (a) - Scaling/Affine Maps.
First, note that a toric domain transforms as Y S Ω “ Y Ω and the divisortransforms as A S Ω “ S ¨ A Ω . So the scaling axiom follows from Definition 3.2.Next, we must show invariance if T is either linear or a translation. If T P GL p (cid:90) q is linear,then T is an automorphism on the Lie algebra (cid:82) » t of (cid:84) induced by a group automorphismof (cid:84) . Thus p Y Ω , A Ω q and p Y T Ω , A T Ω q are identical after pulling back by this automorphism, andthe algebraic capacities must agree. If T is a translation then Y Ω “ Y T Ω and A Ω “ A T Ω ` R where R is a principle divisor determined by T . On the other hand, A Ω ¨ D for a divisor D depends onlyon the divisor class of A Ω , and so invariance follows from Definition 3.2. (c) - Blow Up. Let D be a nef (cid:81) -divisor on Y that achieves the optimum defining c alg k p Y , A q , i.e. c alg k p Y , A q “ D ¨ A and χ p D q ě k ` π ˚ D of D on r Y , which is nef. This has χ p π ˚ D q “ χ p D q ě k ` c alg k p Y r Ω , A r Ω q ď π ˚ D ¨ A r Ω “ π ˚ D ¨ p π ˚ A Ω ´ (cid:15) E q “ D ¨ A Ω “ c alg k p Y Ω , A Ω q (e) - Domains. This is simply a restatement of Theorem 4.15 and Theorem 4.18 of [34], whichstate that if Ω is is a convex domain polytope or a convex free polytope, then(4.2) c ECH k p X Ω q “ inf D P nef p Y Ω q (cid:81) t D ¨ A Ω : h p D q ě k ` u This result is phrased in terms of (cid:81) -divisors, and also uses global sections instead of the Eulercharacteristic. However, since Y Ω is toric we have Demazure vanishing. Lemma 4.10 ( [7, Thm. 9.3.5.]) . Suppose Y is a toric surface and D is a nef (cid:81) -divisor. Then h p p D q “ p ą h p D q “ χ p D q . Moreover, we have the following Lemma (see [36, Lem. 2.1]). Lemma 4.11.
Let D be a nef (cid:81) -divisor on Y Ω . Then there exists a nef (cid:90) -divisor with h p D q “ h p D q A Ω ¨ D ď A Ω ¨ D CH EMBEDDING OBSTRUCTIONS FOR RATIONAL SURFACES 19
Proof.
Without loss of generality assume D is a torus-invariant divisor and let D “ ř a F D F .Consider the round-down of D , defined by t D u : “ ÿ t a F u D F which is a (cid:90) -divisor with P D X (cid:90) n “ P t D u X (cid:90) n . The difference D ´ t D u is effective and so t D u ¨ A ď D ¨ A . Unfortunately, t D u may not be nef.To fix this, we modify t D u to a nef divisor D by translating some of the hyperplanes H F “t x |x u F , x y ě ´ t a F u u (see §4.1) for t D u inwards if necessary. (Here we are using the nef criteriondiscussed in §4.1.) This is equivalent to subtracting some integer multiple of the prime divisor D F and hence only reduces the area. We must also translate each hyperplane only until it meetsa lattice point in P t D u for t D u , so that h p D q “ h p t D u q . Note that every lattice point in (cid:90) n is inone of the translates of H F , for each F , so we can always perform this translation process whileensuring that P D X (cid:90) n “ P D X (cid:90) n “ P t D u X (cid:90) n . In particular, h p D q “ h p D q . (cid:3) Lemmas 4.10 and 4.11 together imply that the following two infima are equal.inf D P nef p Y Ω q (cid:81) t D ¨ A Ω : h p D q ě k ` u “ inf D P nef p Y Ω q (cid:90) t D ¨ A Ω : χ p D q ě k ` χ p O Y qu In view of (4.2) and Definition 3.2, we conclude that c ECH k p X Ω q “ c alg k p Y Ω , A Ω q . (b) - Inclusion. Let Ω Ă ∆ be an inclusion of moment polytopes. By the application of an affinetransformation T : (cid:90) Ñ (cid:90) to both Ω and ∆ , we may assume that Ω and ∆ are in p , Ă (cid:82) ,and thus are convex free polytopes. By (e) and the fact that X Ω Ă X ∆ , we have c alg k p Y Ω , A Ω q “ c ECH k p X Ω q ď c ECH k p X ∆ q “ c alg k p Y ∆ , A ∆ q (d) - Embeddings. Let X Ñ Y Ω be a symplectic embedding of a star-shaped domain. If Y Ω hasno singularities (i.e. no singular fixed points), this is simply Theorem 3.5. Otherwise, since X is a smooth and compact, its image misses the singular fixed points. Thus we can take a toricresolution π : Y r Ω Ñ Y Ω , where r Ω is acqurired from Ω by cutting off small triangles from thesingular corners. For sufficiently small cuts, Y r Ω inherits an embedding X Ñ Y r Ω and thus wehave c ECH k p X q ď c alg k p Y r Ω , A r Ω q ď c alg k p Y Ω , A Ω q Here we apply either the blow up axiom (c) or the inclusion axiom (b).
Uniqueness.
Finally, to argue that these axioms uniquely determine c alg k , let d alg k be anotherfamily of numerical invariants satisfying axioms (a)-(e). The blow up and inclusion axioms implythat c alg k and d alg k agree if and only if they agree on all polytopes Ω such that Y Ω is non-singular.Any such polytope is equivalent to a domain polytope by scaling and affind transformation, soby (a) we merely need to check those polytopes. Then (e) implies that the invariants must agreefor those polytopes. (cid:3) Remark 4.12.
Theorem 3.5 and the blow up property (c) can be used together to give an indendentproof of the upper bound of the ECH capacities by the algebraic capacities in Theorem 4.15 of [34].However, we are not aware of a proof that establishes a lower bound which is not essentiallyequivalent to the one provided in [34]. A fundamentally different proof could potentially shedlight on an approach to Conjecture 1.13.4.4.
Embeddings to toric surfaces.
We now prove the main applications of the paper, whichare easy consequences of the axioms in Theorem 4.9. We start by showing that the algebraiccapacities are complete obstructions for embeddings of the interiors of concave toric domainsinto a toric surfaces, in terms of c ECH and c alg . Theorem 4.13.
Let X ∆ be a concave toric domain and let p Y Ω , A Ω q be a projective toric surface with asmooth fixed point. Then X ˝ ∆ symplectically embeds into Y Ω ðñ c ECH k p X ∆ q ď c alg k p Y Ω , A Ω q Proof.
Suppose that X ˝ ∆ Ñ Y Ω is a symplectic embedding, and let X i be an exhaustion of X ˝ ∆ bystar-shaped domains. Then c ECH k p X ∆ q “ lim i Ñ8 c ECH k p X i q ď c alg k p Y Ω , A Ω q On the other hand, suppose that c ECH k p X ∆ q ď c alg k p Y Ω , A Ω q . Since Y Ω has a torus fixed point,we can scale by an S ą T : (cid:90) Ñ (cid:90) so that TS p Ω q is a convex domainpolygon for convex toric domain X TS p Ω q . Applying axioms (a) and (e) of Theorem 4.9, we acquire c ECH k p X S ∆ q ď c alg k p Y TS p Ω q , A TS p Ω q q “ c ECH k p X TS p Ω q q Now we apply a well-known result [8, Thm. 1.2] of Cristofaro-Gardiner stating that a concavetoric domain X S ∆ embeds into a convex toric domain X TS p Ω q if and only if the ECH capacities of X S ∆ are bounded by those of X TS p Ω q . Thus we acquire a symplectic embedding X ˝ S ∆ Ñ X ˝ TS p Ω q Ă Y TS p Ω q » Y S Ω Since scaling the moment image merely scales the symplectic form accordingly, we thus acquirea symplectic embedding X ˝ ∆ Ñ Y Ω . (cid:3) Corollary 4.14.
Let Ω Ă ∆ be an inclusion of moment polygons, each of which has a smooth vertex. Thenthe Gromov widths satisfy c G p Y Ω q ď c G p Y ∆ q In particular, c G is monotonic with respect to inclusions of the moment polytope for smooth toric surfaces.Proof. Let B p r q Ñ Y Ω be a symplectic embedding of a closed ball of symplectic radius r . Then bythe embedding axiom and inclusion axiom in Theorem 4.9, we have c ECH k p B p r qq ď c alg k p Y Ω , A Ω q ď c alg k p Y ∆ , A ∆ q Thus by Theorem 4.13, we have an embedding B ˝ p r q Ñ Y ∆ of the open ball of symplectic radius r ,so r ď c G p Y ∆ q . Taking the sup over all such embeddings B p r q Ñ Y Ω yields c G p Y Ω q ď c G p Y ∆ q . (cid:3) In fact, we can prove a more general result than Corollary 4.14. Namely, given a momentimage Ξ for a concave toric domain and a symplectic manifold Y , define the Ξ -width c Ξ p Y q by c Ξ p Y q : “ sup t r : X r Ξ embeds symplectically into Y u Then by the same argument as in Corollary 4.14, we have the following result.
Corollary 4.15.
Let Ω Ă ∆ be an inclusion of moment polygons, each of which has a smooth vertex. Then c Ξ p Y Ω q ď c Ξ p Y ∆ q Remark 4.16.
It seems that one can also execute the proof of Corollary 4.15 using only the factthat a ball B p r q embeds into X Ω if and only if it embeds into Y Ω (see [10, Thm 1.2]) and theinclusion axiom (b) of Theorem 4.9. However, this would not cover any singular surfaces, andfurthermore the stronger Corollary 4.15 requires the results of this paper.A consequence of Theorem 4.13 is that the Ξ -width of a convex toric domain X Ω where Ω hasrational slopes agrees with the Ξ -width of the toric surface Y Ω . Corollary 4.17.
Suppose Ω is a convex domain with rational slopes. Then c Ξ p X Ω q “ c Ξ p Y Ω q CH EMBEDDING OBSTRUCTIONS FOR RATIONAL SURFACES 21
Gromov width and lattice width.
We use Corollary 4.14 to provide a combinatorial upperbound for the Gromov width of a toric surface as conjectured in [2]. We recall the definition ofthe lattice width.
Definition 4.18.
The lattice width w p Ω q of a moment polytope is defined by w p Ω q : “ min l P (cid:90) n z ´ max p , q P Ω x l , p ´ q y ¯ Corollary 4.19.
Let Ω be a moment polygon with a smooth vertex. Then c G p X Ω q ď w p Ω q .Proof. We implement the heuristic argument in [2, Rmk 3.13] rigorously. Let l P (cid:90) z w p Ω q “ sup p , q P Ω |x l , p ´ q y| We can choose an element A P GL p (cid:90) q such that p A T q ´ p l q “ e “ p , q is the x -basis vector. Thisimplies that x e , A p p ´ q qy “ xp A ´ q T l , A p p ´ q qy “ x l , p ´ q y “ w p Ω q “ w p A Ω q Thus the lattice width of A Ω is achieved in the direction of e . We can thus fit A Ω in a rectangle R of width a “ w p Ω q and very large height a " a . Since A Ω Ă R , we apply Corollary 4.19 toacquire the inequality c G p Y Ω q “ c G p Y A Ω q ď c G p Y R q On the other hand, Y R » (cid:80) p a qˆ (cid:80) p a q and since a " a , we have that c G p Y R q “ a “ w p Ω q . (cid:3) References [1] Anjos, S., Lalonde, F. Pinsonnault, M.
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