Algebraic and symplectic viewpoint on compactifications of two-dimensional cluster varieties of finite type
AAlgebraic and symplectic viewpoint oncompactifications of two-dimensional clustervarieties of finite type
Man-Wai Mandy Cheung and Renato Vianna
Abstract
In this article we explore compactifications of cluster varieties of finitetype in complex dimension two. Cluster varieties can be viewed as the spec of aring generated by theta functions and a compactification of such varieties can begiven by a grading on that ring, which can be described by positive polytopes [17].In the examples we exploit, the cluster variety can be interpreted as the complementof certain divisors in del Pezzo surfaces. In the symplectic viewpoint, they can be de-scribed via almost toric fibrations over R (after completion). Once identifying themas almost toric manifolds, one can symplectically view them inside other del Pezzosurfaces. So we can identify other symplectic compactifications of the same clustervariety, which we expect should also correspond to different algebraic compactifi-cations. Both viewpoints are presented here and several compactifications have theircorresponding polytopes compared. The finiteness of the cluster mutations are ex-plored to provide cycles in the graph describing monotone Lagrangian tori in delPezzo surfaces connected via almost toric mutation [34]. Cluster algebras, introduced by Fomin and Zelevinsky [12], are subalgebras of ratio-nal functions in n variables. The generators of cluster algebras are called the clustervariables. Instead of being given the complete sets of generators and relations asother commutative rings, a cluster algebra is defined from an (initial) seed, which Man-Wai Mandy CheungHarvard University, One Oxford Street, Cambridge, MA 02138, United States of America,e-mail: [email protected]
Renato ViannaInstitute of Mathematics, Federal University of Rio de Janeiro; Av. Athos da Silveira Ramos, 149- Ilha do Fundo, Rio de Janeiro - RJ, 21941-909, Brazil,e-mail: [email protected] a r X i v : . [ m a t h . S G ] A ug Man-Wai Mandy Cheung and Renato Vianna includes a set of the generators and a matrix. An iterative procedure called mutationwould produce new seeds from a given seed and this process gives all the clustervariables. The cluster algebra is then defined to be the ring generated by all clustervariables.Geometrically, the cluster varieties, described by Fock and Goncharov [11], andby Gross, Hacking, Keel in [16], are defined in a similar manner. A seed data nowwould be associated to an algebraic torus. The mutation procedures give the bira-tional transformations used to glue the tori. A cluster variety is then the union of thetori under the gluing.The compactification of the cluster varieties can be given by a Rees construction.Combinatorially, the construction can be described by ‘convex’ polytopes, calledthe positive polytopes [17]. The article [8] showed that the positive polytopes sat-isfy a convexity condition called ‘broken line convexity’. As the seed mutates, thepolytope mutates correspondingly. One can then give a mutation process to the poly-topes. Note that under this type of mutation, there is no change in the compactifica-tion.More generally, one can similarly describe the compactification of the log Calabi-Yau surfaces studied in [15]. In this case, one would construct the dual intersectioncomplex of a given Looijenga pair. The underlying topological space of the complexwill carry an affine manifold structure. The affine structures would correspond toanother type of mutation for the positive polytopes.On the other hand, in the symplectic viewpoint, mutations were exploited infour dimensional symplectic geometry [33, 34], inspired by the pioneering workof Galkin-Usnhish [13] (further developed in [1]), and being grounded on the de-velopment of almost toric fibrations (ATFs) by Symington [32]. Upon identifyingan almost toric fibration of a open variety, we can symplectically identify it as asymplectic submanifold of some closed symplectic manifold. We will refer to it asa (symplectic) compactification. In the examples of this paper, we can identify thesymplectic form as the K¨ahler form of del Pezzo surfaces. We expect that symplec-tic compactifications can be translated to algebraic compactifications under certainnuances discussed in Section 3.This paper is an attempt to understand the two notions. The motivation of bothsides come from the Strominger-Yau-Zaslow conjecture – the conjecture suggeststhere are special Lagrangian fibrations for the Calabi-Yau manifold and its mirrorspace over the base B . The construction of the log-CY variety from the symplecticside is via the almost toric fibration, Meanwhile, in the algebro-geometric side, theconstruction can be described in terms of the wall crossing structures called thescattering diagrams.We begin with the algebro-geometric perspective in Section 2. In this section,we will discuss cluster varieties, positive polytopes, compactifications, and the mu-tations of the polytopes. Then in Section 3.1, we give a perspective on how clustervarieties and scattering diagrams arise from considering wall crossing correctionsas one attempt to build a mirror in terms of the SYZ picture. In complex dimensiontwo, the wall-crossing happens when we consider singular Lagrangian fibrationsknown as almost toric fibrations (ATFs). In particular, we illustrate the idea in terms ompactifications of cluster varieties 3 of the A cluster variety – compactified as the del Pezzo surface of degree 5 in Sec-tion 3.1.3. Afterward, we explore compactifications of cluster varieties using thealmost toric viewpoint in Section 3.2.The symplectic geometry approach to compactification via almost toric fibrationsmakes no reference to the complex structure, while the algebro-geometric approachdoes not fix a symplectic form. Nonetheless, because a similar set of data can en-code the scattering diagram as well as an ATF, we seem to always be able to relatecompactifications, encoded by the same polytope in both pictures. We aim to showthe correspondence between the symplectic compactification of the cluster varietiesto the algebro-geometric version in our upcoming papers. Acknowledgements
This project initialized from discussions during the conference ”Tropical Ge-ometry And Mirror Symmetry” in the MATRIX Institute. The authors would like to thank theMATRIX institute for their hospitality.The authors would like to thank Denis Auroux, and the referee for helpful feedback on thefirst version of the paper. The first author would like to thank Tim Magee, and Yu-shen Lin forhelpful discussions. The first author is supported by NSF grant DMS-1854512. The second authoris supported by Brazil’s National Council of scientific and technological development CNPq, viathe research fellowships 405379/2018-8 and 306439/2018-2, and by the Serrapilheira Institutegrant Serra-R-1811-25965.
We will first recall some notation used in the definition of a cluster varieties. A fixeddata consists of a lattice N with a skew-symmetric bilinear form {· , ·} : N × N → Q ,an index set I with | I | = rank N , positive integers d i for i ∈ I , a sublattice N ◦ ⊆ N of finite index with some integral properties, the dual lattice M = Hom ( N , Z ) andthe corresponding M ◦ = Hom ( N ◦ , Z ) . One can refer to [16] for the full definition offixed data. Consider N R = N ⊗ R and M R = M ⊗ R .Given this fixed data, a seed data for this fixed data is s : = ( e i ∈ N | i ∈ I ) , where { e i } is a basis for N . The basis for M ◦ would then be f i = d i e ∗ i . One can thenassociate the seed tori A s = T N ◦ = Spec k [ M ◦ ] , X s = T M = Spec k [ N ] . We will denote the coordinates as X i = z e i and A i = z f i and they are called the clustervariables . Similar to the definition of cluster algebras, there is a procedure, called mutation , to produce a new seed data µ ( s ) from a given seed s . The mutation for-mula is stated in [16, Equation 2.3] which we will skip here. The essence is thatwe will obtain new seed tori A µ ( s ) , X µ ( s ) from the mutated seed. Between the tori,there are birational maps µ X : X s (cid:57)(cid:57)(cid:75) X µ ( s ) , µ A : A s (cid:57)(cid:57)(cid:75) A µ ( s ) which are stated in Man-Wai Mandy Cheung and Renato Vianna [16, Equations 2.5, 2.6]. Note that those birational maps are basically the mutationsof cluster variables as in Fomin and Zelevinsky [12].Let A be an union of tori glued by A -mutation µ A . A smooth scheme V isa cluster variety of type A if there is a birational map µ : V (cid:57)(cid:57)(cid:75) A which is anisomorphism outside codimension two subsets of the domain and range. The clustervariety of type X is defined analogously.The A and X cluster varieties can be fit into the formalism of the cluster vari-eties with principal coefficients A prin . The scheme A prin is defined similarly to the A by ‘doubling’ the fixed data, i.e. considering (cid:101) N = N ⊕ M ◦ as fixed data as in [16,Construction 2.11]. Then there are two natural inclusions. The first one is (cid:101) p ∗ : N → (cid:101) M ◦ = M ◦ ⊕ N , n (cid:55)→ ( p ∗ ( n ) , n ) , where p ∗ ( n ) = { n , ·} ∈ M ◦ in the case of no frozen variable. Then for any seed s ,note that A prin , s = T (cid:101) N ◦ , and X s = T M , the there is the exact sequence of tori1 → T N ◦ → A prin , s (cid:101) p −→ X s → . The map (cid:101) p commutes with the mutation maps and thus we get the morphism (cid:101) p : A prin → X . Further the T N ◦ action on A prin , s extends to A prin which makes (cid:101) p a quotient map. Thus, the X variety can be seen as A prin / T N ◦ .The second inclusion is π ∗ : N → M ◦ , n (cid:55)→ ( , n ) . In this case, the π ∗ map induces a projection π : A prin → T M . Then the usual A variety is π − ( e ) , where e is the identity of T M .We would like to indicate another viewpoint of the cluster varieties here. Themutation maps may be described in terms of elementary transformation of P bun-dles. Thus the cluster varieties can also be seen as the blowups of toric varieties (upto codimension two) as well.Given a seed data, consider the fans Σ s , A : = { } ∪ { R ≥ d i e i | i ∈ I } ⊆ N ◦ , Σ s , X : = { } ∪ {− R ≥ d i v i | i ∈ I } ⊆ M , where v i = p ∗ ( e i ) and the i only runs over the unfrozen variables if the frozen vari-ables exist. Let TV s , A and TV s , X be the respective toric varieties. Denote D i tobe the toric divisor corresponding to the one-dimensional ray in one of these fans.Define the closed subschemes Z A , i : = D i ∩ ¯ V ( + z v i ) ⊆ Σ s , A , Z X , i : = D i ∩ ¯ V (cid:16) ( + z e i ) ind d i v i (cid:17) ⊆ Σ s , X , ompactifications of cluster varieties 5 where ¯ V denote the closure of the variety V , and ind d i v i is the greatest degree ofdivisibility of d i v i in M . Then consider the pairs ( (cid:102) TV s , A , D ) and ( (cid:102) TV s , X , D ) consist-ing of the blowups of TV s , A and TV s , X respectively, with D the proper transformof the toric boundaries. Define X s , A = (cid:102) TV s , A \ D and X s , X = (cid:102) TV s , X \ D . When theseed s mutates to s (cid:48) , the corresponding X s , A , X s (cid:48) , A and X s , X , X s (cid:48) , X are isomorphicoutside a codimension two set. In finite type, where there are only finitely clustervariables, the A and X would then also isomorphic to X s , A and X s , X . Note thatthe whole set up here is building a toric model for the cluster varieties. We willintroduce the notion of toric model for log Calabi Yau surfaces later in Section 2.3. Scattering diagrams
Scattering diagrams live in the tropicalization of the cluster varieties. One can alsosee the diagrams encode the structure of the cluster varieties combinatorially.A wall in M R is a pair ( d , f d ) where d ⊆ M R is a convex rational polyhedral coneof codimension one, contained in n ⊥ for some n ∈ N , and f d = + ∑ k ≥ c k z kp ∗ ( n ) ,where c k ∈ C . A wall ( d , f d ) is called incoming if p ∗ ( n ) ∈ d . Otherwise it is called outgoing . A scattering diagram D is then a collection of walls with certain finitenessproperties. Given a seed, an A prin -cluster scattering diagram can be constructed [17]and canonically determined by this given seed data. The A scattering diagram canbe obtained by the projection (cid:101) M R → M R while the X scattering diagrams can bedefined as slicing the A prin scattering diagrams by considering { ( m , n ) | m = p ∗ ( n ) } .It is worth addressing here that for finite type, each chamber, i.e. the maximalcone, of the scattering diagram can be associated to a torus. The wall functions f d are actually representing the birational maps between the tori. Thus the clustervarieties can be seen as gluing of tori associated to the chamber via the wall crossing.In this article, we will focus on the dimension 2 cluster varieties of finite type.The A scattering diagrams are listed as in Figure 1 while the X scattering diagramsof rank 2 finite type are listed as in Figure 2.Fig. 1: A -scattering diagrams for rank 2 finite type. Man-Wai Mandy Cheung and Renato Vianna
Fig. 2: X -scattering diagrams for rank 2 finite type. Mutation of scattering diagrams
As noted in the previous section, a seed determines canonically a scattering diagram.Two mutation-equivalent seeds would then give two different scattering diagrams.It is natural to consider ‘mutation equivalent’ scattering diagrams. This equivalenceis given by piecewise linear maps on the lattices which are very similar to those inSection 3 and hence we will state here.Consider two seeds s and s (cid:48) which are just one mutation step apart, i.e. s (cid:48) = µ k ( s ) for some k ∈ I . Then the corresponding scattering diagrams D s and D s (cid:48) areequivalent to each other by the transformation T k : M ◦ → M ◦ , T k ( m ) = (cid:26) m + (cid:104) d k e k , m (cid:105) v k , for m ∈ H k , + m , for m ∈ H k , − (1)for m ∈ M ◦ , v k = p ∗ ( e k ) , and H k , + = { m ∈ M R |(cid:104) e k , m (cid:105) ≥ } , H k , − = { m ∈ M R |(cid:104) e k , m (cid:105) ≤ } . Extending T k to the wall functions [17, Theorem 1.24] will leadus to another consistent scattering diagram T k ( D s ) which is shown to be equivalentto D µ k ( s ) .We can similarly define the mutation for the X scattering diagrams from A prin .For the scattering diagram of type A in Figure 2, we can obtain the mutation processfor the X scattering diagram as in Figure 3 and Figure 4. ompactifications of cluster varieties 7 Fig. 3: Mutation of the X scattering diagram of type A starting at the index 1Fig. 4: Mutation of the X scattering diagram of type A starting at the index 2Note that the scattering diagrams are determined by seeds while the mutation ofseeds are given by blow ups and blow downs of toric varieties. Thus the mutation ofscattering diagrams actually represents this procedure of blowups and blowdowns.We are going to discuss a similar construction in Section 3 with a symplectic per-spective. Theta functions and the canonical algebras
Theta functions give the generators of the canonical basis of the cluster algebras.Given the cluster variety V = A , A prin , X , the corresponding character lattice is Man-Wai Mandy Cheung and Renato Vianna L = M ◦ , (cid:101) M ◦ , or N . A theta function ϑ p is associated to each point p ∈ L by a combi-natorial object – broken lines which are piecewise linear paths in L R together withdecorating monomials at each linear segment.The free module generated by theta functions is endowed with an algebra struc-ture from the multiplication between theta functions. Indeed the structure constantsin the multiplications of theta functions can be given in terms of counting brokenlines. The product of two theta functions can be expressed as ϑ p · ϑ q = ∑ r α ( p , q , r ) ϑ r , (2)where the structure constants α ( p , q , r ) can be explicitly defined by counting bro-ken lines with certain boundary conditions [17, Proposition 6.4]. In this finite typecase, the structure constants α define ([17, Corollary 8.18]) the finitely generated C -algebra structure on can ( V ) : = (cid:77) r ∈ L C · ϑ r . We will then define X : = Spec ( can ( V )) . With the multiplication structure of the theta functions, we can now state the defi-nition of a positive set– the property required for a set and its dilations to define agraded ring.For S ⊆ L R = L ⊗ R a closed subset, define the cone of S as C ( S ) = { ( p , r ) | p ∈ rS , r ∈ R ≥ } ⊂ L R × R ≥ . Denote dS ( Z ) = C ( S ) ∩ ( L × { d } ) which is viewed as a subset of L .A closed subset S ⊂ L R is called positive if for any non-negative integers d , d , any p ∈ d S ( Z ) , p ∈ d S ( Z ) , and any r ∈ L with α ( p , p , r ) (cid:54) =
0, then r ∈ ( d + d ) S ( Z ) .In the ongoing example of cluster varieties of type A , we consider the polytopewith vertices ( , ) , ( , ) , ( − , ) , ( , − ) , ( , − ) as indicated in Figure 5. Notethat this polytope is in the X diagram thus there is a flip from Figure 21. Thispolytope is indeed positive. In Section 3.1.3, there is a detail discussion of such apolytope in the A side. A similar calculation in this X case will still hold, thus thiswill correspond to the del Pezzo surface of degree 5 [17].We can apply the mutation sequences in Figure 3 and 4 to the polytope in Figure5. Mutations of the polytopes as in Figure 6 and 7 will be obtained respectively.In the next section, we will describe mutations of the polytopes from a symplecticpoint of view. We observe that the mutation sequences of polytope in Figure 6 and 7are the same as the sequences in Figure 23 and 24 respectively. The cluster mutation ompactifications of cluster varieties 9 Fig. 5: Positive polytope of X cluster variety of type A .Fig. 6: Mutation of polytope starting from index 1of the scattering diagrams comes from a change of seed data, i.e. a change of theinitial variables. Thus the underlying spaces are all isomorphic. Compactifications from positive polytopes
We will roughly go over the geometric meaning behind the positive polytopes in thissection. The motivation can be seen as the construction of projective toric varietiesfrom the convex polytopes.For rank 2 cluster varieties, since the A scattering diagrams are well defined, wecan consider ¯ S the positive polytopes in the A scattering diagrams. For this set ¯ S ,define (cid:101) S = ¯ S + N R which is obviously positive. Thus we can define the graded ring Fig. 7: Mutation of polytope starting from index 2 (cid:101) R (cid:101) S = (cid:77) d ≥ (cid:77) q ∈ d (cid:101) S ( Z ) C ϑ q x d ⊂ can ( A prin )[ x ] , with grading defined by x .Define Y A prin : = Proj ( (cid:101) R S ) → T M . For the A variety, we take Y A as the fiber over e ∈ T M in this map. More generally, for the A t variety, t ∈ T M , we can still take Y A t as the fiber over t ∈ T M . Consider X = Spec ( can ( V )) , for V = A prin , A t , as in theprevious subsection. Define B = Y \ X . Then [17] showed that, X is a Gorensteinscheme with trivial dualizing sheaf, in particular, for V = A prin , A t , X is a K -trivialGorenstein log canonical variety. In this finite rank 2 case, for V = A prin , A , X ⊆ Y is a minimal model, i.e. Y is a projective normal variety, B ⊂ Y is a reduced Weildivisor, K Y + B is trivial, and ( Y , B ) is log canonical.For the case of the X varieties, as indicated in Section 2.1, the X varietiesare quotients of the A prin varieties. Thus we will consider still consider A prin butinstead see the lattice as (cid:101) M ◦ instead of (cid:101) N (which are actually isomorphic). We canrepeat the same procedure as before and then obtain the compactification of X = Spec ( can ( X )) . The scheme X is also a K -trivial Gorenstein log canonical variety. In the last section, we note that the cluster mutations of the scattering diagrams arenot changing the underlying schemes. We are proposing another type of mutationwhich is given by the monodromy on B . We are going to understand the ideas be-hind from the mirror construction suggested by Gross, Hacking, and Keel in [15].In Section 3.1, we will discuss the affine structure and monodromy from the SYZperspective. ompactifications of cluster varieties 11 Consider a pair ( Y , D ) , where Y is a smooth rational projective surface, and D isan anti-canonical cycle of projective lines. We will call such a pair a Looijenga pair.Let X = Y \ D . The tropicalization of ( Y , D ) is a pair ( B , Σ ) , where B is an integrallinear manifold with singularities, and Σ is a decomposition of B into cones. Thepair ( B , Σ ) can be constructed by associating each node p i , i + of D a rank two latticewith basis v i , v i + . Denote the cone generated by v i , v i + as σ i , i + ⊂ M i , i + ⊗ R . Thecones σ i , i + and σ i − , i are glued over the ray ρ i = R ≥ v i to obtain a piecewise linearmanifold B homeomorphic to R and Σ = { σ i , i + } ∪ { ρ i } ∪ { } .The integral affine structure on B = B \ { } can be defined by the charts ψ i : U i = Int ( σ i − , i ∪ σ i , i + ) → M R , where ψ i ( v i − ) = ( , ) , ψ i ( v i ) = ( , ) , and ψ i ( v i + ) = ( − , − D ) , and ψ i is linear on σ i − , i and σ i , i + .Now consider Y the del Pezzo surface of degree 5 and D the anti-canonical cycleof five (-1)-curves. The construction of the charts ψ will then give ψ ( v ) = ( , ) , ψ ( v ) = ( , ) , ψ ( v ) = ( − , ) , ψ ( v ) = ( − , ) , ψ ( v ) = ( , − ) . Note however that having ψ ( v ) = ( − , ) , ψ ( v ) = ( , − ) will lead to ψ ( v ) (cid:32) ( , − ) , ψ ( v ) (cid:32) ( , ) and this is NOT what we began with: ψ ( v ) = ( , ) , and ψ ( v ) = ( , ) . Thus wewould like to identify the cone spanned by ( , ) and ( , ) , and the cone spannedby ( − , ) and ( , ) . This introduces the monodromy ( , ) (cid:55)→ ( , ) , ( , ) (cid:55)→ ( , ) , to B . The affine structure is illustrated in Figure 8. • Fig. 8: The tropicalization ( B , Σ ) of the del Pezzo surface of degree 5 Now we would like to define the canonical scattering diagrams from ( B , Σ ) .Rather than obtaining the diagrams by the algorithmic process with some initialdata in [19] [21], the canonical scattering diagrams are defined via some Gromov-Witten type invariants. We will discuss the two types of diagrams are the ‘same’later in the discussion about how to go from canonical scattering diagrams tocluster scattering diagrams. A wall [18] in B is a pair ( d , f d ) where d ⊂ σ i , i + ,for some i , is a ray generated by av i + bv i + (cid:54) = a , b ∈ Z relatively prime, and f d = + ∑ k ≥ c k X − aki X − bki + ∈ C [[ X − ai X − bi + ]] with some finiteness properties, andwhere c k corresponds to the curve counting invariants. Note that the descriptionof the wall functions f d indicates that all the wall are outgoing in the sense stated inthe last section. Then the scattering diagrams are again the collections of walls. Forexample, the canonical scattering diagram associated to Figure 8 is shown in Figure9. + X − + X − + X X − + X + X Fig. 9: Canonical scattering diagramLet B ( Z ) be the set of points of B with integral coordinates in an integral affinechart and { } . Theta functions ϑ q , q ∈ B ( Z ) , can similarly be defined on the scat-tering diagrams. The set of theta functions again generates an algebra structure [15]in terms of broken lines. In the finite case, we can simply consider A = ⊕ q ∈ B ( Z ) ϑ q .Analogous to the setting in the cluster scattering diagrams, we can use the Reesconstruction to compactify the mirrors [20]. Positive polytopes with respect to theaffine structures can be similarly defined to give a graded algebra. Using [25] orthe argument in [8], the polytopes are broken line convex. In this case, since allthe walls are outgoing, the positive polytopes are simply convex with respect to theaffine structures. ompactifications of cluster varieties 13 Relation to the cluster scattering diagrams
In the case of Y a non-singular toric surface and D = ∂ Y the toric boundary of D , theaffine structure on B extends across the origin. This identifies ( B , Σ ) with M R , Σ Y ,where Σ Y is a fan for Y .Now given a Looijenga pair. Assume there is a toric model p : ( Y , D ) → ( ¯ Y , ¯ D ) which blows up distinct points x i j on D i . A toric model of ( Y , D ) is a birationalmorphism ( Y , D ) → ( ¯ Y , ¯ D ) to a smooth toric surface ¯ Y with its toric boundary ¯ D such that D → ¯ D is an isomorphism. Consider the tropicalisation ( ¯ B , ¯ Σ ) of ( ¯ Y , ¯ D ) .Thus ¯ B ∼ = M R = R and ¯ Σ is the fan for ¯ Y . Then there is a canonical piecewise linearmap ν : B → ¯ B which restricts to an integral affine isomorphism on the maximal cones in σ and¯ Σ . One can then define the scattering diagram D as outlined in Section 2.1 or as in[15, Definition 3.21] for the more general setting. This step can be seen as ‘pushingthe singularities to infinity’ or ‘moving worms’ [21]. By definition, the singularityof the affine structure is at { } as indicated in Figure 8. Then the singularity canbe imagined to be pushed to the infinity of the two incoming walls. The map ν canbe extended to act on the canonical scattering diagram D can . It is shown that [15]¯ D = ν ( D can ) .We have discussed in Section 2.1 that every cluster variety can be describedas blow ups of a toric variety, which give the toric models for the cluster variety.Thus the cluster scattering diagrams can be seen as the diagrams arising from thecanonical scattering diagrams by the map ν (as pushing singularities to infinity). Mutation of polytopes according to the affine structures
One can imagine or with symplectic motivation as in Section 3.1, the ‘pushing sin-gularities to infinities’ procedure is more general than just having singularities at theorigin. For example, we can consider Figure 10 which we only push one of the sin-gularities to infinity, resulting in a scattering diagram with one incoming wall. Themonodromy in Figure 10 is (cid:18) (cid:19) , or ( , ) (cid:55)→ ( , ) , ( , ) (cid:55)→ ( , ) . The polytopein Figure 5 with respect to this affine structure would then be of the form in Figure11.We can apply the sequence of mutations in Figure 3 to the polytope in Figure11 and then obtain a new sequence of mutation polytopes (Figure 12). Putting thepolytope in Figure 5 into the sequence (Figure 12) will get us the sequence Figure26 which is motivated from the symplectic perspective.The singularities can also be located on the walls instead of just at infinity orthe origin. For example, one can obtain the canonical scattering diagram shown inFigure 13. Similar calculation shown in [6] indicates that the scattering diagram isconsistent. + z e + z − e + z e + e + z e Fig. 10: Scattering diagrams with monodromy.Fig. 11: Positive polytope with respect with the underlying affine structureNote that the portions of the walls which go from the singularities to infinity areall outgoing. Thus using the idea in [8, Remark 6.2], we can consider convex sets inthis affine structure. For example, one can construct the polytope as in Figure 14.Applying the mutation sequence as in Figure 3, we obtain the sequence of poly-topes described in Figure 15. Interestingly, this is the same sequence as in Figure 27which is motivated from the symplectic perspective. B and G The other types are similar and thus we only roughly go over the mutations of type B and G . For type B , we will take the same skew-symmetric form with d = d = s = { ( , ) , ( , ) } .Then we will obtain the A and X scattering diagrams as in Figure 1 and 2. If we ompactifications of cluster varieties 15 Fig. 12: Mutation of polytopes with monodromy + z − e + z e + z e + e + z e + z − e (cid:18) −
10 1 (cid:19)(cid:18) (cid:19)
Fig. 13: Scattering diagram with monodromy on the walls.mutate at index 1 first, we can get the mutation of scattering diagrams very similarto the type A case.For type B , we can again take the primitive generators of the walls and thenconsider the polytope as the convex hull of those vertices. By using [8], this polytopeis a positive polytope. The multiplication of the theta functions tells us [7] that thecorresponding space is the del Pezzo surfaces of degree 6. The mutation sequenceof the polytopes is described in Figure 16. Fig. 14: Polytope lives in the affine structure indicated in Figure 13.Fig. 15: Mutation for the polytope in Figure 14Fig. 16: Mutation of polytope for type B ompactifications of cluster varieties 17 One may want to repeat the same trick on the type G . The sad fact is that theif we are taking the convex hull of the primitive generators of the walls, the result-ing polytope would no longer be positive. This is because the polytope is no longerbroken line convex as indicated in [8]. Since we only care about the incoming wallsfor broken line convexity [8], one can see that the top left polytope in Figure 29in the next section is broken line convex. The mutation sequence of the G scatter-ing diagrams are similar to those for type A and B . Without duplicating, one cansee that the mutation indicate in Figure 29 is in fact the cluster mutation of scatter-ing diagrams. Thus the mutation of polytopes follows correspondingly. This againtells us that the mutation sequences of the polytope with the algebro-geometric andsymplectic viewpoints coincide. We begin this section giving a perspective on understanding cluster varieties, aswell as scattering diagrams, as a way of building mirrors under SYZ [31] T -duality.In particular, we explain how scattering diagrams can be related to almost toricfibrations. Later, we explain how one can see compactifications of 2-dimensionalcluster varieties into del Pezzo surfaces from an almost toric fibration perspective. In this section we will sketch how to relate a scattering diagram data (describedin Section 2), for constructing a log-CY variety X , with the base of an almost-toricfibration (ATF), describing a SYZ [31] singular Lagrangian fibration of ( X , ω ) , withrespect to a K¨ahler form ω in X . Informally speaking an almost toric fibrations (ATF) in a symplectic 4-manifold X is a smooth map to a two dimensional base B , whose regular fibres are Lagrangiantori, whose allowed singular fibres are of three kinds: • point (toric - rank 0 elliptic) – locally equivalent to the moment map at the originin C with the standard toric action, ( e i θ , e i θ ) · ( x , y ) = ( e i θ x , e i θ y ) ; • circle (toric - rank 1 elliptic) – locally equivalent to S × { } ⊂ C ∗ × C with thestandard toric action; • nodal (a pinched torus) – with some local model described for the singular point.[See [32, 22] for precise definition, and see Section 3.1.2 for a local model ofthe nodal fibre.] The toric singularities appear on the boundary of the base, while the nodal singular-ities project into the interior. For a precise definition of ATFs see [32].Away from the singular fibres, by the Arnold-Liouville theorem [3], X admitslocally action angle coordinates ( p , p , θ , θ ) and the fibration is locally equivalentto ( p , p , θ , θ ) (cid:55)→ ( p , p ) , in other words, away from singular fibres X equivalentto T ∗ B / Λ ∗ , for some lattice Λ ∗ . Hence, B carries a natural dual lattice Λ ⊂ T B . Thelattice has monodromy as we go around the nodal fibre, which is a shear in thedirection dual to the collapsing cycle of that nodal fibre. Locally, the coordinates ( p , p ) , can be thought as the flux f ∈ H ( T , R ) relative to the Lagrangian fibreassociated with ( , ) . The flux f ( γ ) measures the symplectic area of a cylinderswept by a cycle γ ∈ H ( T , Z ) as we move in a path of Lagrangian fibres connecting ( , ) to ( p , p ) . [See, for instance, [30] for a more complete understanding of fluxin ATFs.]So, in practice, we visualise the base minus a set of cuts (one for each nodal fibre)affinely embedded into R endowed with the standard affine structure. We call themalmost-toric base diagrams (ATBDs) representing the ATF. The same ATF can berepresented by different ATBDs, by changing the set of cuts.Figure 17 shows the base diagram of 3 different ATFs in C ; the right-most dia-grams on Figure 18 are different diagrams representing the same ATF in C \ { xy = } , related by a change of cut; Figures 23–33 contain examples of ATBDs in closed4 manifolds. In these diagrams, the crosses represent the nodal fibres, the dashedlines the cuts, the edges the rank 1 and the dots rank 0 toric singularities. Remark 1.
We expect the above mentioned ATFs to be realisable as a special La-grangian fibration in the complement of a complex divisor projecting to the bound-ary of the ATF, with respect to a holomorphic volume form with poles on thesedivisor. This is true for the fibration presented in Section 3.1.2, but we will avoidtalking about the ”special” condition.Fig. 17: Nodal trade and nodal slide operations in ATFs.There are two ways of modifying ATFs within the same symplectic manifold X ,known as nodal trade and nodal slide [32]. The diagrams in Figure 17 illustrate thechange of the ATBDs after a nodal trade and a nodal slide. In del Pezzo surfaces, themonotone symplectic 4-manifolds, we defined mutation of an ATBD, the process ofsliding one nodal fiber through the monotone fibre, and then redrawing the diagramby changing the direction of the cut used to slide [33, 34]. In the end, the ATBD ompactifications of cluster varieties 19 mutates by slicing it in the direction of the cut and applying the inverse of the cor-responding monodromy, which is a shear in the primitive direction associated to thecut. This transformation is the same polytope mutation as in [1, 2], and completelyanalogous to the mutation of seeds, and scattering diagram we will discuss later. Wecan extend this notion of symplectic mutation to exact almost toric manifold, forinstance, the complement of an anti-canonical divisor in a del Pezzo. In this case,the mutation corresponds to sliding a nodal fibre through the exact torus and thentransferring the associated cut to the opposite side. We briefly recall ATF presented in [4, Section 5], [5, Section 3.1.1]. This ATF ap-peared before in [10] and also in [14, Example 1.2], where it was shown to be aspecial Lagrangian fibration [with respect to certain holomorphic volume form]. Weconsider X ∨ = C \ { xy = } , with ω ∨ = i ( dx ∧ d ¯ x + dy ∧ d ¯ y ) the standard sym-plectic form. Using f : X ∨ → C \ { } , f ( x , y ) = xy , Auroux builds an ATF by par-allel transport of orbits of the S action e i θ · ( x , y ) = ( e i θ x , e − i θ y ) , over circles inthe base of f centred at 1. One then gets Lagrangian torus fibres, parametrised by ( r , λ ) ∈ R > × R , as: T r , λ = { ( x , y ) ∈ C ; r = | xy − | , λ = | x | − | y | } . Note that there is a nodal fibre T , , that contains ( , ) , the fixed point of the S action.This almost toric fibration can be represented by applying a nodal trade to thestandard toric fibration of C , replacing the boundary divisor { xy = } with thesmooth divisor { xy = } , and then deleting this divisor living over the boundary ofthe base, as illustrated by Figure 18. Indeed, replacing the role of 1 by 0 in the abovefibration, i.e., considering parallel transport over circles concentric at 0 (considering r = | xy | ) one obtain precisely the standard toric fibration of C . So, consideringanalogous fibrations by changing 0 to 1 in the definition of r constitutes a nodaltrade, and moreover, varying the value of c ∈ R > in the definition of r = | xy − c | provides different fibrations related by nodal slides. Wall-crossing
Dualising this torus fibration, one gets the mirror variety X Λ of X ∨ , over the Novikovfield Λ = { ∑ ni = a i T σ i ; a i ∈ C , σ i ∈ R , lim i σ i = ∞ } , which is the moduli of almost-toric fibres (special Lagrangians), endowed with unitary Λ ∗ -local systems. We willbe able to relate the valuation val ( u ) = min { σ i ; a i (cid:54) = } of an element u ∈ Λ withthe above mentioned flux, whenever val ( u ) measures the symplectic area of a diskwith boundary in a varying family of the Lagrangian torus fibres. [Notation: Λ = Fig. 18: Nodal trade and an ATF for the complement of a conic. The left diagramsare a shear by ( , − ) of the diagrams in Figure 17. The rightmost diagrams, repre-sent the same ATF, and differ by changing the direction of the cut. { u ∈ Λ ; val ( u ) ≥ } , Λ + = { u ∈ Λ ; val ( u ) > } , Λ ∗ = { a + Λ + ; a ∈ C ∗ } .] We willlater consider the mirror of X ∨ as X = X C over C , by replacing T with e − .So we replace the Lagrangian fibre T , by the dual Λ -torus of unitary lo-cal systems hom ( π ( T ) ; Λ ∗ ) ∼ = ( Λ ∗ ) . Locally identifying each relative class, β ∈ π ( C , T r , λ ) , Auroux defined a function z β : X Λ → Λ , for a local system ∇ in T r , λ , z β ( ∇ ) = T ω ∨ ( β ) ∇ · ∂ β . Choosing a basis { α , β } of π ( C , T r , λ ) , one gets that w : = z α , u : = z β , define local coordinates of X Λ . After that, the idea is to define a su-perpotential function W : X Λ → Λ + , which is locally defined as W ( u , w ) , and whosemonomials encode the relative Gromov-Witten count of Maslov index 2 holomor-phic disks in C with boundary on the torus fibre endowed with the respective lo-cal system determined by ( u , w ) . [The pair ( X Λ , W ) is called the Landau-Ginzburgmodel that is mirror dual to C with respect to the divisor D = { xy − } . We refer thereader to [4, 5] for details on mirror symmetry in the complement of divisors.]The issue is that, in the naive definition of the mirror, the superpotential W is dis-continuous. This is due to the presence of fibres T , λ , λ (cid:54) =
0, which bounds Maslovindex 0 holomorphic disks. Let’s denote the relative class represented by this Maslov0 disks by α for λ <
0, and − α for λ >
0. In [5, Section 3.1.1], it is shown thatfor r <
1, the fibres T r , λ (called Chekanov type) bound one holomorphic disk, in aclass we name β . So W ( u , w ) = u , for these fibres. The fibres T r , λ , for r >
1, (calledClifford type) bound 2 holomorphic disks in relative classes β , β , and hence thesuperpotential is of the form W ( z , z ) = z + z , where z i = z β i .We see in [5, Section 3.1.1] that as r approaches 1, from r >
1, we get α = β − β (hence w = z z − ). Moreover, if we cross the wall at λ <
0, the class β is naturally identified with β , and if we cross the wall at λ >
0, the class β isnaturally identified with β = β + α [which is not so surprising, as the monodromyaround the nodal fibre in the ATF would fix ∂ α and maps ∂ β → ∂ β + ∂ α ]. Sothe superpotential W should be corrected by the term ( + w ± ) , representing thefact that the holomorphic disk on class β , would not only survive past the wall, butthe superpotential would also acquire a holomorphic disk in class β ± α , comingfrom the gluing of the Maslov 2 holomorphic disk on class β with the Maslov 0 ompactifications of cluster varieties 21 holomorphic disk on class ± α . Then, instead of u becoming z as we cross over λ >
0, we should correct it to become u = z ( + w ) = z + z , and instead of u becoming z as we cross over λ <
0, we should correct it to become u = z ( + w − ) = z + z ,and, thus, ensuring the continuity of W .By naming v = z − , so z = v − w , we get the corrected u = v − ( + w ) = v − w ( + w − ) . We see that the corrected (and completed) mirror X Λ , is given by X Λ = { ( u , v , w ) ∈ Λ × ( Λ \ { } ) ; uv = + w } . Figure 19 below describes the mirror SYZ fibrations on X ∨ and X Λ . We list sev-eral remarks about the diagrams in Figure 19 and the mirror X Λ .Fig. 19: SYZ fibrations for the complement of a conic, which is self-mirror, whenconsidering X C . Remark 2.
We see that X Λ is given by gluing two torus charts ( u , w ) ∈ ( Λ \ { } ) ,and ( v , w ) ∈ ( Λ \ { } ) , by a rational map defined in the complement of { w = − } .So, for instance, the case u = ( v , − ) in the ( v , w ) -chart. Remark 3.
Considering the symplectic form ω ∨ in X ∨ , the Lagrangian torus fibra-tion would be viewed in a truncated part of the ( u , w ) -chart, with 0 < val ( u ) ≤ a ( val ( w )) or in the ( v , w ) -chart, with a ( val ( w )) ≤ val ( v − ) < ∞ for val ( w ) <
0, forinstance. These bounds on the valuation would give us X t Λ , a truncated version of themirror X Λ . But in symplectic geometry, we can add to ( X ∨ , ω ∨ ) a contact boundary ∂ X ∨ , and it is most natural to consider a completion procedure called the sym-plectization of X ∨ with respect to this boundary. This endows X ∨ with a different symplectic form ω ∨ S . It is equivalent to consider an infinite inflation of ( C , ω ∨ ) with respect to the divisor D = { xy = } . In this limit we would have val ( u ) → ∞ ,and we would get the completed mirror X Λ . Remark 4.
The expectation regarding the correspondence between the count ofMaslov index 2 disks with boundary on a SYZ fibre and its tropical counterpart wasproven in [24] for a SYZ fibration on the complement of an smooth anti-canonicaldivisor in a del Pezzo surface. More precisely, given a del Pezzo surface Y anda smooth anti-canonical divisor D , there exists a special Lagrangian fibration on Y \ D with respect to the complete Ricci-flat Tian-Yau metric [9]. To understand theLandau-Ginzburg superpotential of Y , there exists a sequence of K¨ahler forms ω i on Y converging to the Tian-Yau metric pointwisely with (cid:82) Y ω i → ∞ [24, Lemma 2.4].Thus, these superpotentials of the special Lagrangian fibres can be defined with re-spect to ω i , i (cid:29) Remark 5.
In the ( r , λ ) projection of Figure 19, the singular fibre in position ( , ) isdepicted by an × , and the wall of fibres with r = ( r , λ ) coordinate does not respect the natural affinestructure on the complement of the singular fibre of B . We instead consider Flux ω ∨ ,the flux with respect to a limiting fibre lying over ( , ) , in the next diagram. Thismap is then continuous, but not differentiable over the dashed ray r ≥ λ = φ ∨ is then anaffine isomorphism to B minus the cut. The affine structure of B minus the node, isdescribed by the gluing of the chart φ ∨ and a chart φ ∨ , going from the third diagramof Figure 18, corresponding to taking the cut associated to 0 < r ≤ λ =
0. Thesymplectic manifold X ∨ can be thought then as a local model for gluing in the nodalfibre to the manifold constructed from gluing the Lagrangian torus fibres associatedto φ ∨ and φ ∨ . This is essentially the same model as the description of X ∨ as aself-plumbing of T ∗ S given in [32, Section 4.2]. Remark 6.
The affine structure on B for the dual mirror fibration X Λ → B is endowedwith the dual affine structure in the complement of the node. We call the map thatadjusts this affine structure in R , Flux ω . In the case we take the SYZ mirror X C over C (by replacing T by e − ), it endows a symplectic form ω as described in [5,Proposition 2.3]. In this case, Flux ω becomes the actual flux with respect to thissymplectic form. Remark 7.
The monodromy around the singular fibre of X ∨ → B , represented bythe bottom left diagram of Figure 19, is given by M ± , for M = (cid:20) −
10 1 (cid:21) , fixingthe cut ( , ) . Then, the monodromy around the node for the rightmost diagramrepresenting X Λ → B is given by ( M T ) ∓ , with ( M T ) − = (cid:20) (cid:21) . We see that this ompactifications of cluster varieties 23 fixes the coordinate w , associated to ( , ) , which we then name ϑ ( , ) , and it sendsthe coordinate v − = ϑ ( , ) to v − w = ϑ ( , ) .Fig. 20: In a complex structure limit, the wall becomes straight. We can move thecut to the invariant direction of the monodromy, that is the same direction as thelimit straight wall. Remark 8.
As described by Mikhalkin [27], we can deform the complex structureon X ∨ to a limit where holomorphic curves would converge to tropical curve onthe base with respect to the so-called complex affine structure, which is dual tothe symplectic affine structure. So, (relative) Gromov-Witten invariants of X ∨ areexpected to be described by tropical curves ( ϑ functions) in the base B , with theaffine structure describing X Λ (or X C ). In particular, the wall becomes straight inthis limit, as illustrated by Figure 20.As we mentioned before, we will now replace the formal variable T by e − in ourconstruction, and consider the mirror as the moduli of Lagrangian fibres endowedwith U ( ) -local systems. So, after completion the mirror becomes X = X C = { ( u , v , w ) ∈ C × C ∗ ; uv = + w } endowed with a completed symplectic form ω as described in [5, Proposition 2.3].Its SYZ dual ATF (dual to the one on X ∨ ) is then described by any of the diagramsin Figure 20. Remark 9.
As we take the completion, the val defined in X Λ approaches − log | . | defined in X C .We see now that the rightmost diagram in Figure 20, can describe an ATF, andonce decorated with wall crossing functions [ ( + w ± ) accordingly] along the wall,it can algebraically determine the space X C . The variety X C is then built out of two ( C ∗ ) charts, with coordinates ( u , w ) and ( v , w ) , glued together in a cluster like tran-sition birational map uv = + w , defined in the complement of { w = − } . A uniquewall, decorated with such wall crossing function, describing X C is the simplest ver-sion of a scattering diagram [19, 15] (see Section 2 for more details). A Cluster Variety: ATF and Scattering Diagram
As described in [17, Example 8.40] (taking the parameters X , X to be 1), wedescribe the affine ( ϑ = A Cluster variety by the ring in five variables ϑ i , i = , . . . , ϑ ϑ = + ϑ ϑ ϑ = + ϑ ϑ ϑ = + ϑ ϑ ϑ = + ϑ ϑ ϑ = + ϑ We see that this variety is obtained by gluing five algebraic tori ( C ∗ ) , with coor-dinates ( ϑ i , ϑ i + ) [indices taken mod 5], according to the above cluster relations.We saw in more details in Section 2 that these relations are encoded by the data of ascattering diagram, as illustrated in the top-right picture of Figure 21.Fig. 21: Scattering diagram and ATF for the A cluster variety.Let’s start with the data of an ATF describing a symplectic manifold X , with 2nodal fibres, whose monodromies are encoded by cuts pointing away from the nodesin the directions ( , ) and ( − , ) , respectively, as illustrated in the bottom-left pic-ture of Figure 21. We now think think of this as endowed with the completed infinitevolume symplectic form, so the base diagram covers the whole R . As indicated inthe previous Section, to build complex charts on this space, we add one wall for eachnode, represented by a line in the invariant direction of the monodromy. [These wallsrepresent dual fibres in the mirror X ∨ , bounding Maslov index 0 disks with respect ompactifications of cluster varieties 25 to a limit complex structure j ∞ .] We call the chamber containing the nodes the mainchamber, and we associate to it a complex torus ( C ∗ ) , with coordinates ( ϑ , ϑ ) ,and associated with the corresponding wall is a gluing function of the form ( + ϑ i ) .One can check that changing coordinates around these 4 walls in a full circle, doesnot give you identity on the ( ϑ , ϑ ) algebraic torus. To correct for that one needs toadd an extra slab , in this case corresponding to a ray in direction ( , − ) , and a cor-responding transition function giving you now 5 chambers, each corresponding toan algebraic torus, as illustrated in the top-right picture of Figure 21. This collectionof walls and slabs is called the scattering diagram [19] [recall the details in Section2]. This scattering diagram describe the relations of the A cluster variety given inthe beginning of this Section.We can compactify this A cluster variety by homogenizing its defining equa-tions, as ϑ ϑ = ϑ + ϑ ϑ , . . . , ϑ ϑ = ϑ + ϑ ϑ . As mentioned in [17, Exam-ple 8.40], this gives a del Pezzo surface of degree 5 in C P . Intersecting the hy-perplane ϑ =
0, we see a chained loop of 5 divisors. Symplectically, it is natural toendow the del Pezzo surface with the monotone symplectic form given by restrictingthe Fubini-Study form of C P . The complement of the five above mentioned divi-sors can be seen as a (Weinstein) subdomain of X , whose completion give X . Indeed,there is an ATF on the degree 5 del Pezzo surface, as illustrated in the bottom-rightdiagram of Figure 21. [We can obtain this ATF by performing a monotone blowup ina corner of [34, diagram ( A ) of Figure 16].] The chained loop of 5 divisors is iden-tified with the boundary of this ATF, and the complement of them is a subdomain of X as illustrated by the bottom-left diagram of Figure 21. We saw in the previous section how to relate the data representing an almost-toricfibration in a open symplectic manifold with a set of initial walls, out of whichGross-Siebert [19] explains how to complete to a scattering diagram that providesthis manifold with complex charts given by gluing algebraic tori ( C ∗ ) along thewalls.As mentioned in Section 2.1, we can construct cluster varieties out of this data,and we will focus on the varieties of finite type A , B , G . These are open exactalmost toric manifolds, built out of the scattering diagram with initial data givenby two orthogonal walls, one of them associated to one node and the other withone, two and three nodes, respectively, as indicated in Figure 22. Recall we call thechart containing all nodes the main chart. One sees that symplectic mutation can beassociated to changing the main chart, as illustrated in Figure 22. In other words,without the prior knowledge, the scattering diagram can be recovered by keepingtrack of the “main charts” as we apply the corresponding mutations, as illustrated inFigure 22. [This is not the case when the scattering diagram has a dense regions ofslabs. For instance, when considering the scattering diagram associated to the mirror Fig. 22: Cluster charts via symplectic mutations on affine cluster varietiesof the complement of an elliptic curve in CP . Note that this case is considered in[24].]In this section, we are interested in understanding compactifications of these clus-ter varieties from the symplectic perspective. Compact symplectic manifolds havefinite volume, hence we will consider as X a subdomain, whose completion is themanifold described by the ATF with base diagram covering the whole R . We willconsider equivalent the subdomains with same completion. ompactifications of cluster varieties 27 All the symplectic compactifications considered here are symplectic del Pez-zos, in the sense that they are endowed with a monotone symplectic form, whichis unique up to scaling and symplectomorphisms [26, 23, 28, 29]. This ensures theexistence of a monotone fibre, that can be detected by the intersection point of thelines in the diagrams that go through the nodes and are in the direction of the cuts.A symplectomorphism class invariant of these monotone fibres (the star-shape) isshown [30] to be given by the interior of the polytope seen in H ( T , R ) ∼ = R ,as we forget the nodes and cuts. So there is a symplectomorphism identifying twomonotone fibres of an ATF, if and only if, the associated polytopes are related underSL ( Z ) . If there exists such ambient symplectomorphism, we say the Lagrangiansare symplectomorphic.In the definition of mutation of ATFs on del Pezzo surface [34], besides mutatingthe polytope by changing the direction of the cut, it is required that we slide the cutthrough the monotone fibre. In that sense, we say that the corresponding monotonefibres are related by mutation. We can then form a graph with vertices representingsymplectomorphism class of a Lagrangian and edges represented these Lagrangiansbeing related by mutation. One aspect we can extract from the cyclic behaviour ofthese finite cluster varieties is the existence of cycles in the above mentioned graph.This behaviour does not appear in mutations of monotone almost toric fibres in C P ,and conjecturally in C P × C P .Fig. 23: Mutations on degree 5 del Pezzo – 1 torusLet us start looking at the example from Section 3.1.3 ([17, Example 8.40], [8,Figure 1]), a compactification of the A cluster variety to the degree 5 del Pezzo, byadding a chain of 5 divisors, whose union represents the anti-canonical class. Thiscompactification and its mutations are illustrated in Figures 23, 24. Note that we getthe same pattern as in Figures 6, 7, where we get back to the same picture after, re-spectively, 4 and 6 cycles, depending on the pattern of mutation. This is misleading,as ATFs, the mutations should not depend on which half-space is fixed, and whichyou decide to shear. In fact, in this example, all diagrams are ( SL ( Z ) ) equiva-lent, which in particular implies that the monotone tori in each pictures are mutually Fig. 24: Mutations on degree 5 del Pezzo – 1 torussymplectomorphic. The 5-cycle pattern of the A cluster appears by looking at themain charts, which we have already illustrated in Figure 22. In particular, this ex-ample does not give us a cycle of monotone Lagrangian tori, since we quotient outthe graph associated to mutations by equivalence.We want to extend a bit our notion of compactification. We will say that a (Wein-stein) domain X compactifies to Y , if we have X ⊂ Y ⊂ Y , with X a sub-domain of Y and Y = Y \ (cid:83) i D i , for symplectic divisors D i . This will be used by us to identify ourdomains of interest, described in Figure 22, appearing as open pieces of ATFs in delPezzo surfaces, where we do not include all the nodes. See for instance, Figures 26,28, 32, 33. In these cases, our domain of interest X is not the complement Y of thesymplectic divisors projecting over the boundary of the ATF, but rather a subdomainof Y . The nodes not contained in the ATF describing X will be considered frozen(not used to mutate), and depicted as a blue × .An alternative way of thinking is to disregard the frozen nodes. The total mani-fold ˜ Y becomes singular, and a non-smooth compactification of X , given by addingthe boundary divisor. The singularities are orbifold T -singularities [2] at each ver-tex, that were previously associated with the frozen nodes. Our original smoothmanifold, that included the frozen nodes, is a smoothing of this orbifold. There is acontinuous way of relating the Lagrangian fibrations on the orbifold with the ATFon the smoothing. We like to interpret it as a two step process, which is locally il-lustrated in Figure 25. The first, we keep the symplectic form on Y , and consideralmost-toric fibrations AT F t , t ∈ [ , ) , so that in the limit t → SLF , on Y , such that over the limit vertices live a possiblysingular Lagrangian. The Lagrangian over each limit vertex can be recognised ineach AT F t , t < Y to ˜ Y ,and a family of singular Lagrangian fibrations SLF s , in the fibers corresponding to ompactifications of cluster varieties 29 Fig. 25: Moving frozen nodes to the boundary, is equivalent to have over the vertexa possibly singular Lagrangian representing a vanishing cycle of a degeneration to atoric orbifold singularity. This vanishing cycle is represented by a pentagon over thevertex, and the orbifold singularity by a star in the above diagrams. Up to equiva-lence, the shaded domain can be viewed either as a subdomain of the complement ofthe boundary divisors in the left-picture, or the complement of the singular divisorsin the orbifold diagram. s ∈ ( , ] , where in the limit s → SLF s areidentified under symplectic parallel transport, so the singular Lagrangian over eachvertex degenerating to an orbifold singularity is precisely the vanishing cycle of thatorbifold singularity. In the examples presented here, the singularities associated withthe frozen nodes will always be of A n type, and hence the corresponding Lagrangiana chain of n − Let’s turn our attention now to Figure 26, where we realize the degree 5 del Pezzosurface Y as a compactification of the A cluster variety X , in a different way. Weperform a nodal trade in a vertex at the bottom of the second diagram in Figure23, and we freeze the top node. Now the boundary divisor represents 4 symplecticspheres, and X is a subdomain of the complement of these divisors. In this case,the mutation cycle induced by the nodal singularities in X does provides us with a5-cycle of distinct monotone Lagrangian tori. Recall that monotone fibres of non- SL ( , Z ) related diagrams are distinct [30]. Remark 10.
Disregarding the frozen node creates a double point singularity. In con-trast with [17, Example 8.40], this is the same as considering X = ϑ =
0. We claim that if one smoothsone node of this chain (represented by our nodal trade), and then delete the resultingchain of 4 divisors in this orbifold, one recovers X , the A cluster variety.Fig. 27: Mutations on degree 8 del Pezzo – 2 toriWe can see that there is a simpler compactification of the A cluster variety X by Y = C P C P , a degree 8 del Pezzo, as illustrated in Figure 27 [which is the sameobtained in Figure 14]. Here, X is the complement of two divisors in classes H and2 H − E , where H is the class of the line, and E is the exceptional class. Note that wedo not get a cycle of monotone Lagrangian tori, though not all tori are equivalent,we only see 2 tori in the whole cycle, which gives us only an edge on the unorientedgraph of mutations of monotone tori, modulo equivalence. ompactifications of cluster varieties 31 Fig. 28: Mutations on degree 4 del Pezzo – 3 toriClearly, performing a blowup on one of the divisors of Y gives us another com-pactification of X . The top left diagram of Figure 28 corresponds to a toric blowupof monotone size [recall that in symplectic geometry, the blowups depend on thesize of a symplectic ball one chooses to delete] in the top left diagram of either Fig-ure 23 or Figure 26. Note in this case that the third and fourth, as well as the secondand fifth, diagrams are equivalent, failing to deliver a cycle on the mutation graphof monotone Lagrangian tori in the degree 4 del Pezzo.Fig. 29: Mutations on degree 6 del Pezzo – 6 tori Fig. 30: This diagram differ from the sixth diagram in Figure 29, by one nodaltrade and one inverse nodal trade. Mutations of the displayed nodes give equivalentpolytopes. Fig. 31: Mutations on degree 5 del Pezzo – 3 toriLet us consider now X the B type cluster variety, with almost toric fibrations asin the series of diagrams in the middle of Figure 22. The first compactification Y we look at is the degree 6 del Pezzo, starting with the ATF depicted in the top-leftdiagram of Figure 29. [This diagram is SL ( Z ) equivalent to [34, Diagram ( A ) ,Figure 16] (up to nodal trades).] In this case X is the complement of three divisorsof Y = C P C P , having symplectic areas 1, 2 and 3. We see that in this case wedo get a cycle of size 6, with one torus corresponding to each cluster chart.It is interesting to notice that the sixth diagram seems to have come from the scat-tering diagram Figure 16. But it is not quite the case, since that scattering diagram ompactifications of cluster varieties 33 has a square function corresponding to the horizontal cut in the sixth diagram Figure29, while a simple function corresponding to the vertical cut. This means that thenatural compactification coming from that scattering diagram would be the same delPezzo, but represented by the diagram of Figure 30 coming from applying a nodaltrade to the corner associated to the horizontal cut in the sixth diagram Figure 29,and an inverse nodal trade on one node at the vertical cut. In particular, X would beseen as the complement of 3 divisors in Y = C P C P , each of symplectic area2. The reader can check that in this case, the mutations associated to X would giveequivalent monotone Lagrangian tori, analogous to the previous case depicted inFigures 23, 24.Clearly we can also compactify the B type cluster variety X to the degree 5 delPezzo, as the complement of four divisors as depicted in Figure 31, by simply ap-plying a nodal trade to a diagram in Figure 23. In Figure 31, we depicted segmentsoutside the diagrams to indicate that they come from applying blowups to the dia-grams in Figure 29. Curiously, it behaves similarly to the case in Figure 28, wherewe have only three non-equivalent Lagrangian tori, not providing a cycle.Fig. 32: Mutations on degree 3 del Pezzo – 8 toriWe now finish by presenting two compactifications of the G cluster variety. Wename it X , and consider it as an almost toric variety corresponding to the bottomseries of diagrams in Figure 22. We start noting that the compactifications described Fig. 33: Mutations on degree 4 del Pezzo – 8 toriin [8], see for instance [8, Figure 18], seems to be giving partially, but not fully,smoothable orbifolds. Here we look to two compactifications to degree 3 and 4del Pezzo surfaces. [Both contain frozen variables, so the reader may prefer thealternative idea of seeing X compactifying to a degeneration of these surfaces.]We start with the top left ATBD in Figure 32, which is equivalent to [34, Dia-gram ( B ) , Figure 19], representing an ATF of the cubic C P C P . In this case, X is a subdomain of the complement of two symplectic divisors. Sliding the frozennodes to the corresponding vertex gives one Lagrangian sphere, in the horizontal cut,and a chain of two Lagrangian spheres in ( , ) -cut. This indicates that disregardingthe frozen nodes corresponds to considering an orbifold with one double-point sin-gularity and one triple-point singularity. We do get one monotone Lagrangian torusfor each of the 8 cluster charts in this case.Another compactification of X is given in Figure 33. The top left diagram ofFigure 33 is equivalent (up to nodal trades) to [34, Diagram ( B ) , Figure 18]. Here, X is viewed as a subdomain of the complement of three symplectic divisors in Y = C P C P . Sliding the frozen node to the vertex provides Lagrangian sphere, orequivalently, disregarding the node gives a double-point orbifold singularity at thevertex. As before, we get one monotone Lagrangian torus for each cluster chart. ompactifications of cluster varieties 35 References
1. Akhtar, M., Coates, T., Galkin, S., Kasprzyk, A.M.: Minkowski polynomials and mutations.SIGMA Symmetry Integrability Geom. Methods Appl. , Paper 094, 17 (2012)2. Akhtar, M.E., Kasprzyk, A.M.: Mutations of fake weighted projective planes. Proc. Edinb.Math. Soc. (2) (2), 271–285 (2016)3. Arnol´d, V.I.: Mathematical methods of classical mechanics, Graduate Texts in Mathematics ,vol. 60. Springer-Verlag, New York (1989). Translated from the 1974 Russian original by K.Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition4. Auroux, D.: Mirror symmetry and T-duality in the complement of an anticanonical divisor. J.G¨okova Geom. Topol. , 51–91 (2007)5. Auroux, D.: Special Lagrangian fibrations, wall-crossing, and mirror symmetry. In: Surveysin differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty yearsof the Journal of Differential Geometry, Surv. Differ. Geom. , vol. 13, pp. 1–47. Int. Press,Somerville, MA (2009). DOI 10.4310/SDG.2008.v13.n1.a16. Cheung, M.W., Lin, Y.S.: Some examples of Family Floer mirror. In preparation7. Cheung, M.W., Magee, T.: Towards Batyrev duality for finite-type cluster varieties. In prepa-ration8. Cheung, M.W., Magee, T., N´ajera-Ch´avez, A.: Compactifications of cluster varieties and con-vexity. arXiv preprint arXiv:1912.13052 (2019)9. Collins, T., Jacob, A., Lin, Y.S.: Special lagrangian submanifolds of log calabi-yau manifolds10. Eliashberg, Y., Polterovich, L.: Unknottedness of Lagrangian surfaces in symplectic4-manifolds. Internat. Math. Res. Notices (11), 295–301 (1993). DOI 10.1155/S1073792893000339. URL http://dx.doi.org/10.1155/S1073792893000339
11. Fock, V., Goncharov, A.B.: Cluster ensembles, quantization and the dilogarithm. Annalesscientifiques de l’cole Normale Suprieure (6), 865–930 (2009)12. Fomin, S., Zelevinsky, A.: Cluster algebras I: Foundations. J. Amer. Math. Soc. , 497–529(2002)13. Galkin, S., Usnich, A.: Laurent phenomenon for landau-ginzburg potential (2010). Availableat http://research.ipmu.jp/ipmu/sysimg/ipmu/417.pdf
14. Gross, M.: Special Lagrangian fibrations. I: Topology. AMS/IP Stud. Adv. Math. , 65–93(2001)15. Gross, M., Hacking, P., Keel, S.: Mirror symmetry for log Calabi-Yau surfaces I. Publicationsmath´ematiques de l’IH ´ES pp. 1–104 (2011)16. Gross, M., Hacking, P., Keel, S.: Birational geometry of cluster algebras. Algebraic Geometry (2), 137–175 (2015)17. Gross, M., Hacking, P., Keel, S., Kontsevich, M.: Canonical bases for cluster algebras. Journalof the American Mathematical Society (2), 497–608 (2018)18. Gross, M., Hacking, P., Keel, S., Siebert, B.: The mirror of the cubic surface. arXiv preprintarXiv:1910.08427 (2019)19. Gross, M., Siebert, B.: From real affine geometry to complex geometry. Annals of mathemat-ics (3), 1301–1428 (2011)20. Gross, M., Siebert, B.: Intrinsic mirror symmetry. arXiv preprint arXiv:1909.07649 (2019)21. Kontsevich, M., Soibelman, Y.: Affine structures and non-Archimedean analytic spaces. In:The unity of mathematics, Progr. Math. , vol. 244, pp. 321–385. Birkh¨auser Boston (2006)22. Leung, N.C., Symington, M.: Almost toric symplectic four-manifolds. J. Symplectic Geom. (2), 143–187 (2010)23. Li, T.J., Liu, A.: Symplectic structure on ruled surfaces and a generalized adjunction formula.Math. Res. Lett. (4), 453–471 (1995). DOI 10.4310/MRL.1995.v2.n4.a6. URL https://doi.org/10.4310/MRL.1995.v2.n4.a6
24. Lin, Y.S.: Enumerative geometry of del pezzo surfaces. arXiv preprint arXiv:2005.08681(2020)25. Mandel, T.: Tropical theta functions and log calabi–yau surfaces. Selecta Mathematica (3),1289–1335 (2016)6 Man-Wai Mandy Cheung and Renato Vianna26. McDuff, D.: The structure of rational and ruled symplectic 4-manifolds. J. Amer. Math. Soc. (3), 679–712 (1990). DOI 10.2307/1990934. URL http://dx.doi.org/10.2307/1990934
27. Mikhalkin, G.: Amoebas of algebraic varieties and tropical geometry. In: Different faces ofgeometry,
Int. Math. Ser. (N. Y.) , vol. 3, pp. 257–300. Kluwer/Plenum, New York (2004)28. Ohta, H., Ono, K.: Notes on symplectic 4-manifolds with b + =
1. II. Internat. J. Math. (6),755–770 (1996). DOI 10.1142/S0129167X96000402. URL http://dx.doi.org/10.1142/S0129167X96000402
29. Ohta, H., Ono, K.: Symplectic 4-manifolds with b + =
1. In: Geometry and physics (Aarhus,1995),
Lecture Notes in Pure and Appl. Math. , vol. 184, pp. 237–244. Dekker, New York(1997)30. Shelukhin, E., Tonkonog, D., Vianna, R.: Geometry of symplectic flux and Lagrangian torusfibrations. arXiv:1804.02044 (2018)31. Strominger, A., Yau, S.T., Zaslow, E.: Mirror symmetry is T -duality. Nuclear Phys. B (1-2), 243–259 (1996). DOI 10.1016/0550-3213(96)00434-8. URL http://dx.doi.org/10.1016/0550-3213(96)00434-8
32. Symington, M.: Four dimensions from two in symplectic topology. In: Topology and geometryof manifolds (Athens, GA, 2001),
Proc. Sympos. Pure Math. , vol. 71, pp. 153–208. Amer.Math. Soc., Providence, RI (2003)33. Vianna, R.: Infinitely many exotic monotone Lagrangian tori in CP . J. Topol. (2), 535–551(2016)34. Vianna, R.: Infinitely many monotone Lagrangian tori in del Pezzo surfaces. Selecta Math.(N.S.)23