Geometry of symplectic flux and Lagrangian torus fibrations
GGEOMETRY OF SYMPLECTIC FLUXAND LAGRANGIAN TORUS FIBRATIONS
EGOR SHELUKHIN, DMITRY TONKONOG, AND RENATO VIANNA
Abstract.
Symplectic flux measures the areas of cylinders swept in the processof a Lagrangian isotopy. We study flux via a numerical invariant of a Lagrangiansubmanifold that we define using its Fukaya algebra. The main geometric featureof the invariant is its concavity over isotopies with linear flux.We derive constraints on flux, Weinstein neighbourhood embeddings andholomorphic disk potentials for Gelfand-Cetlin fibres of Fano varieties in termsof their polytopes. We show that Calabi-Yau SYZ fibres have unobstructedFloer theory under a general assumption. We also describe the space of fibresof almost toric fibrations on the complex projective plane up to Hamiltonianisotopy, and provide other applications.
Contents
1. Overview 12. Enumerative geometry in a convex neighbourhood 63. Fukaya algebra basics 94. The invariant Ψ 175. Bounds on flux 226. Computations of shape 267. Space of Lagrangian tori in C P Overview
This paper studies quantitative features of symplectic manifolds, namely the be-haviour of symplectic flux and bounds on Weinstein neighbourhoods of Lagrangiansubmanifolds, using Floer theory. Among other examples, we will apply our tech-nique to Lagrangian torus fibrations in two important contexts:
SYZ fibrations on Calabi-Yau varieties, and broadly defined
Gelfand-Cetlin fibrations on Fanovarieties.Our technique is heavily influenced by the ideas of Fukaya and the Family Floerhomology approach to mirror symmetry. However, it is hard to point at a preciseconnection because a discussion of the latter theory for Fano varieties (or in othercases when the mirror should support a non-trivial Landau-Ginzburg potential) hasnot appeared in the literature yet. Intuitively, the numerical invariant Ψ introduced
ES was supported by an NSERC Discovery Grant and by the Fonds de recherche du Qu´ebec- Nature et technologies. DT was partially supported by the Simons Foundation grant a r X i v : . [ m a t h . S G ] M a y EGOR SHELUKHIN, DMITRY TONKONOG, AND RENATO VIANNA in this paper should be thought of as the tropicalisation of the Landau-Ginzburgpotential defined on the rigid analytic mirror to the given variety.1.1.
Flux and shape.
We begin by reviewing the classical symplectic invariants ofinterest. Let (
X, ω ) be a symplectic manifold and { L t } t ∈ [0 , a Lagrangian isotopy,i.e. a family of Lagrangian submanifolds L t ⊂ X which vary smoothly with t ∈ [0 , L = L . The flux of L t ,Flux( { L t } t ∈ [0 , ) ∈ H ( L ; R ) , is defined in the following way. Fix an element a ∈ H ( L ; R ), realise it by a real1-cycle α ⊂ L , and consider its trace under the isotopy, that is, a 2-chain C a swept by α in the process of isotopy. The 2-chain C a has boundary on L ∪ L .One defines Flux( { L t } ) · a = (cid:82) C α ω ∈ R . Above, the dot means Poincar´e pairing. It is easy to see that Flux( { L t } ) dependsonly on a ∈ H ( L ; R ), and is linear in a . Therefore it can be considered as anelement of H ( L ; R ).Let L ⊂ ( X, ω ) be a Lagrangian submanifold. The shape of X relative to L isthe set of all possible fluxes of Lagrangian isotopies beginning from L : Sh L ( X ) = { Flux( { L t } t ∈ [0 , ) : L t ⊂ X a Lag. isotopy, L = L } ⊂ H ( L ; R ) . At a first sight this is a very natural invariant of L , but we found out thatit frequently behaves wildly for compact symplectic manifolds. For example, theshape of C P relative to the standard monotone Clifford torus is unbounded; infact, that torus has an unbounded product neighbourhood which symplecticallyembeds into C P , viewed in the almost toric fibration shown in Figure 1. SeeSection 6 for details. Figure 1.
On the right, the “wild” unbounded non-convex productneighbourhood L × Q ⊂ T ∗ L = L × R of the Clifford torus L in C P . The domain Q ⊂ R is shown on the left and is viewd also asa subset of Sh L ( X ).To remedy this and obtain a better behaved invariant of Lagrangian submani-folds using flux, it is natural to introduce the following notion. We call a Lagrangianisotopy { L t } t ∈ [0;1] a star-isotopy ifFlux( { L t } t ∈ [0; t ] ) = t · Flux( { L t } t ∈ [0;1] ) , for each 0 ≤ t ≤ . EOMETRY OF FLUX 3
In other words, flux must develop linearly in time along a fixed ray in H ( L ; R ).The star-shape of X relative to L is defined to be Sh (cid:63)L ( X ) = { Flux( { L t } t ∈ [0 , ) : L t ⊂ X a Lag. star-isotopy, L = L } ⊂ H ( L ; R ) . We shall soon see that this invariant captures the geometry of X in a more robustway. One reason is that star-shape is invariant under Hamiltonian isotopies, whileshape is invariant under all Lagrangian isotopies.A historical note is due. Symplectic shape was introduced by Sikorav [38], cf. [22],in the context of exact symplectic manifolds. The paper [24] studied an invariant def L : H ( L, R ) → (0 , + ∞ ] which is equivalent to star-shape. We refer to thatpaper for further context surrounding flux in symplectic topology. Example . Suppose D ⊂ R n is an open domain, X = T n × D ⊂ T ∗ T n ∼ = T n × R n is a product neighbourhood of the n -torus with the standard symplectic form, and0 ∈ D . Let L = T n × { } be the 0-section in this neighbourhood. The Benci-Sikorav theorem [38] says that Sh L ( X ) = D . If D is star-shaped with respect tothe origin, then Sh (cid:63)L ( X ) = D , see [24, Theorem 1.3]. Remark . Suppose L ⊂ X is a Lagrangian torus. Then Sh L ( X ) gives an obviousbound on product Weinstein neighbourhoods of L embeddable into X . Namely, ifthere is a symplectic embedding of T n × D into X taking the 0-section to L , then D ⊂ Sh L ( X ). If D is star-shaped with respect to the origin, then also D ⊂ Sh (cid:63)L ( X ).1.2. The invariant Ψ and its concavity. We are going to study the geometryof flux, including star-shapes, with the help of a numerical invariant that associatesa number Ψ( L ) ∈ (0 , + ∞ ] (possibly + ∞ ) to any orientable and spin Lagrangiansubmanifold L ⊂ X . Fix a compatible almost complex structure J ; the definitionof Ψ( L ) will be given in terms of the Fukaya A ∞ algebra of L .Roughly speaking, Ψ( L ) is the lowest symplectic area ω ( β ) of a class β ∈ H ( X, L ; Z ) such that holomorphic disks in class β exist and, moreover, contributenon-trivially to some symmetrised A ∞ structure map on odd degree elements of H ∗ ( L ). The latter means, again roughly, that there exists a number k ≥ c , . . . , c k ∈ H odd ( L ; R ) such that holomorphic disks in class β whose boundaries are incident c , . . . , c k form a 0-dimensional moduli space, thusposing an enumerative problem. The count for this problem should be non-zero.The definition of Ψ( L ) appears in Section 4, and the background on A ∞ algebrasis revised in Section 3. Quite differently from the above sketch, we take the primarydefinition to be the following:Ψ( L ) = inf { val m ( e b ) : b ∈ H odd ( L ; Λ rel + ) } . Here is a quick outline of the notation: Λ rel + is the maximal ideal in the Novikovring Λ rel ; val : H ∗ ( L ; Λ rel + ) → R > is the valuation; m ( e b ) = m (1) + m ( b ) + m ( b, b ) + . . . is the expression called the Maurer-Cartan prepotential of b ; and the m i are thestructure maps of the (curved) Fukaya A ∞ algebra of L .An important technical detail, reflected in the formula for Ψ( L ), is that we defineΨ( L ) using a classically minimal model of the Fukaya algebra of L , i.e. one overthe singular cohomology vector space H ∗ ( L ; Λ rel ). Such models always exist, by aversion of homological perturbation lemma. EGOR SHELUKHIN, DMITRY TONKONOG, AND RENATO VIANNA
Using the fact that the Fukaya algebra does not depend on the choice of J andHamiltonian isotopies of L up to weak homotopy equivalence, we show in Section 4that Ψ( L ) is well-defined and invariant under Hamiltonian isotopies of L .The above definition is convenient for proving the invariance of Ψ( L ), but notquite so for computations and for understanding its geometric properties. To thisend, we give a more explicit formula which was hinted above, see Theorem 4.8:Ψ( L ) = min ω ( β ) : ∃ c , . . . , c k ∈ H odd ( L ; R ) s.t. (cid:88) σ ∈ S k m k,β ( c σ (1) , . . . , c σ ( k ) ) (cid:54) = 0 . Here m k,β is the A ∞ operation coming from disks in class β ∈ H ( X, L ; Z ). Belowis the main result linking Ψ to the geometry of flux. Theorem A (=Theorem 4.9) . Let { L t } t ∈ [0 , be a Lagrangian star-isotopy. Thenthe function Ψ( L t ) : [0 , → (0 , + ∞ ] is continuous and concave in t .Proof idea. The idea lies in Fukaya’s trick, explained in Section 2. It says thatthere exist compatible almost complex structures J t such that the structure maps m tk,β t for the Lagrangians L t are locally constant in a neighbourhood of a chosenmoment of the isotopy. But the areas of classes β t change linearly in t during a star-isotopy. So Ψ( L t ) is locally computed as the minimum of several linear functions ;hence it is concave.Now suppose that X admits a singular Lagrangian torus fibration X → B over abase B ; it is immaterial how complicated the singularities are, or what their natureis. By the Arnold-Liouville theorem, the locus B ◦ ⊂ B supporting regular fibrescarries a natural integral affine structure. Consider the map Ψ : B ◦ → (0 , + ∞ ]defined by p (cid:55)→ Ψ( L p ), where L p ⊂ X is the smooth Lagrangian torus fibre over p ∈ B ◦ . The previous theorem implies that this function is concave on all affineline segments in B ◦ . This is a very strong property that allows to compute Ψfor wide classes of fibrations on Calabi-Yau and Fano varieties, with interestingconsequences.1.3. Fano varieties.
Fano varieties are discussed in Section 6. We introduce aclass of singular Lagrangian torus fibrations called
Gelfand-Cetlin fibrations (Sec-tion 6). Roughly speaking, they are continuous maps X → P onto a convex latticepolytope P ⊂ R n which look like usual smooth toric fibrations away from the unionof codimension two faces of P . This includes actual toric fibrations and classicalGelfand-Cetlin systems on flag varieties (from which we derived the name). Itis not unreasonable to conjecture that all Fano varieties admit a Gelfand-Cetlinfibration. Theorem B (=Theorem 6.2) . Let X be a Fano variety, µ : X → P ⊂ R n aGelfand-Cetlin fibration, and L ⊂ X its monotone Lagrangian fibre.Let P ∨ L ⊂ H ( L ; R ) be the interior of the dual of the Newton polytope associatedwith the Landau-Ginzburg potential of L (Section 5.5). Let c be the monotonicityconstant of X , and assume P is translated so that the origin corresponds to thefibre L . Then the following three subsets of H ( L ; R ) ∼ = R n coincide: c · P ∨ L = P = Sh (cid:63)L ( X ) . EOMETRY OF FLUX 5
Note that there are obvious star-isotopies given by moving L within the fibres ofthe fibration, achieving any flux within P . The equality P = Sh (cid:63)L ( X ) says thatthere are no other star-isotopies of L in X (which are not necessarily fibrewise)achieving different flux than that.The enumerative geometry part of the theorem, about the Newton polytope ofthe Landau-Ginzburg potential, is interesting in view of the program for classifyingFano varieties via maximally mutable Laurent polynomials, or via correspondingNewton polytopes which are supposed to have certain very special combinatorialproperties [16, 17, 18]. Toric Fano varieties correspond via this bijection to theirtoric polytopes. One wonders about the symplectic meaning of a polytope corre-sponding this way to a non-toric Fano variety X . The answer suggested by theabove theorem is that it should be the polytope of some Gelfand-Cetlin fibrationon X ; the equality 2 c · P ∨ L = P supports this expectation.On the way, we compute the Ψ-invariant of all fibres of a Gelfand-Cetlin system. Proposition C.
Consider a Gelfand-Cetlin fibration over a polytope, with point p corresponding to a monotone fibre L . Consider the function over the interior of thepolytope whose value at a point is given by the value of Ψ of the corresponding fibre.Then this is a PL function whose graph is a cone over the polytope (see Figure 2).The vertex of the cone is located over p , and its height equals the area of the Maslovindex 2 disks with boundary on L . Figure 2.
The graph of Ψ over the moment triangle for C P . Thefunction Ψ is non-linear over the three gray segments.1.4. Calabi-Yau varieties.
The Calabi-Yau case is discussed in Section 8. Inthat section we axiomatise the properties of SYZ fibrations that are expected toexist on all compact Calabi-Yau manifolds: roughly speaking, the base B is atopological n -manifold without boundary, and the discriminant locus in B is anarbitrarily thin neighbourhood of a codimension two subset in B . Under theseassumptions, we settle the following rather long-standing expectation relevant forSYZ mirror symmetry and Family Floer homology. We note that J. Solomon[40] has recently obtained a similar result under a different axiomatization of SYZfibrations, involving an anti-symplectic involution acting by the inverse map on thetorus fibers, considered as groups. Theorem D (=Theorem 8.3) . Let Y be a Calabi-Yau manifold carrying a suitableSYZ fibration (Definition 8.1). Assume that the boundary divisor axiom holds forthe Fukaya algebras of its smooth fibres (Definition 8.4). Then the Fukaya algebraof any smooth fibre F p is fully algebraically unobstructed, that is, the set MC ( F p ) of graded bounding cochains of F p modulo gauge equivalence is maximal: H ( F p , Λ + ) = MC ( F p ) . EGOR SHELUKHIN, DMITRY TONKONOG, AND RENATO VIANNA
Proof idea.
Since B is a manifold without boundary, B ◦ contains line segmentsin almost all directions whose length is arbitrarily large, provided that we makethe discriminant locus sufficiently thin. Because Ψ : B ◦ → (0 , + ∞ ] is concave andpositive on those long segments, we argue that it must be constant. By a quickargument involving the boundary divisor axiom, it follows that Ψ is identicallyequal to + ∞ . This means the stated unobstructedness result.1.5. Other results.
In Section 6 we determine shapes and star-shapes of certaintori in C n , and study the wild behaviour of (non-star) shape in C P . In Section 7we determine the non-Hausdorff moduli space of all (not necessarily monotone)Lagrangian tori in C P arising as fibres of almost toric fibrations, modulo Hamil-tonian isotopy. In Section 2, as a warm-up, we discuss bounds on flux within aconvex neighbourhood L ⊂ U ⊂ X ; this gives us an opportunity to recall theFukaya trick and establish a non-bubbling lemma that is useful for the argumentsin Section 6.1.6. Technical remark.
Our main invariant, Ψ( L ), is defined using the Fukayaalgebra of L . We remind that whenever the symplectic form on X has rationalcohomology class, the Fukaya algebra of L is defined via classical transversalitymethods using the technique of stabilising divisors [14, 8, 9].In general, the definition of the Fukaya algebra requires the choice of a virtualperturbation scheme. Our results are not sensitive to the details of how it isimplemented. They rely on the general algebraic properties of Fukaya algebrasreminded in Section 3. We shall use [29] as the common reference for these basicproperties; in the setting with stabilising divisors, they were established in [8, 9]. Acknowledgements.
We thank Denis Auroux for many valuable conversations.This work was initiated during the ”Symplectic topology, sheaves and mirror sym-metry” summer school at Institut de Math´ematiques de Jussieu, 2016. We ac-knowledge the hospitality of the Institute of Advanced Study, Princeton, and IBSCenter for Geometry and Physics, Pohang, where part of the work was carried out.ES was partially supported by NSF grant No. DMS-1128155 at the IAS, and byan NSERC Discovery Grant, and by the Fonds de recherche du Qu´ebec - Natureet technologies, at the University of Montr´eal.DT was partially supported by the Simons Foundation grant
Enumerative geometry in a convex neighbourhood
This section is mainly a warm-up. Suppose L ⊂ X is a monotone Lagrangiansubmanifold. Using standard Symplectic Field Theory stretching techniques andwithout using Fukaya-categorical invariants, we are going to obtain bounds onthe shape of Lioville neighbourhoods L ⊂ U ⊂ X of L that are symplectically EOMETRY OF FLUX 7 embeddable into X . Along the way we recall Fukaya’s trick and establish a usefulno-bubbling result, Lemma 2.3.Let J be a tame almost complex structure. For a class β ∈ H ( X, L ; Z ) ofMaslov index 2, let M β ( J ) be the 0-dimensional moduli space of unparametrised J -holomorphic disks ( D, ∂D ) ⊂ ( X, L ) with boundary on L , whose boundary passesthrough a specified point pt ∈ L , and whose relative homology class equals β . Wewill be assuming that the above disks are regular, whenever M β ( J ) is computed.Their count M β ( J ) ∈ Z is invariant under choices of J and Hamiltonian isotopiesof L , by the monotonicity assumption. Definition 2.1.
We call an open subset U ⊂ ( T ∗ M, ω std ) a
Liouville neighbourhood(of the zero-section) if U contains the zero-section, and there exists a Liouville 1-form θ on U such that dθ = ω std , and the zero-section is θ -exact.The next theorem establishes a shape bound on a Liouville neighbourhood U admitting a symplectic embedding φ : U → X which takes the zero-section to L . Theorem 2.2.
Let L ⊂ X be a monotone Lagrangian submanifold and J a tamealmost complex structure. Let U ⊂ X be an open subdomain containing L andsymplectomorphic to a Liouville neighbourhood of L ⊂ U (cid:44) → T ∗ L . For a Maslovindex 2 class β , if M β ( J ) (cid:54) = 0 , then the shape Sh L ( U ) belongs to the following affine half-space: Sh L ( U ) ⊂ B β = { f ∈ H ( X ; R ) : 2 c + f · ∂β > } , where · is the Poincar´e pairing and c is the monotonicity constant, i.e. c = ω ( β ) . Fukaya’s trick.
Fukaya’s trick is a useful observation which has been used asan ingredient to set up Family Floer homology [27, 1]. This trick will enable us toapply Gromov and SFT compactness theorems to holomorphic curves with bound-ary on a moving Lagrangian submanifold, when this isotopy is not Hamiltonian.Let L t ⊂ X be a Lagrangian isotopy, t ∈ [0 , f t : X → X, f t ( L ) = L t . Denote ω t = f ∗ t ω . Let J (cid:48) t be a generic family of almost complex structures suchthat J (cid:48) t tames ω t . When counting holomorphic disks (or other holomorphic curves)with boundary on L t , we will do so using almost complex structures of the form(2.2) J t = ( f t ) ∗ J (cid:48) t where J (cid:48) t is as above. The idea is that f − t takes J t -holomorphic curves with bound-ary on L t to J (cid:48) t -holomorphic curves with boundary on L . In this reformulation, theLagrangian boundary condition L becomes constant, which brings us to the stan-dard setup for various aspects of holomorphic curve analysis, such as compactnesstheorems.Although there may not exist a single symplectic form taming all J (cid:48) t , for each t there exists a δ > t ∈ [ t − δ ; t + δ ], J (cid:48) t tames ω t . For thepurposes of holomorphic curve analysis, this property is as good as being tamedby a single symplectic form. Here is a summary of our notation, where the rightcolumn and the left column differ by applying f t : Almost complex str. J (cid:48) t f t −→ J t Tamed by ω t ω Lag. boundary cond. L L t EGOR SHELUKHIN, DMITRY TONKONOG, AND RENATO VIANNA
This should be compared with L t actually being a Hamiltonian isotopy; in this casewe could have taken ω t ≡ ω and J (cid:48) t tamed by the fixed symplectic form ω ; this caseis standard in the literature.2.2. Neck-stretching.
Recall the setup of Theorem 2.2: L ⊂ X is a monotoneLagrangian submanifold, and L ⊂ U ⊂ X where U is symplectomorphic to aLiouville neighbourhood in the sense of Definition 2.1, which identifies L with thezero-section. Lemma 2.3.
For each
E > and any family { L b } b ∈ B of Lagrangian submanifolds L b ⊂ U ⊂ X parametrised by a compact set B which are Lagrangian isotopic to L , there exists an almost complex structure J on X such that each L b bounds no J -holomorphic disks of Maslov index ≤ and area ≤ E in X .Proof. We will show that almost complex structures which are sufficiently neck-stretched around ∂U have the desired property. Pick a tame J on X ; neck-stretching around ∂U produces a family of tame almost complex structures J n , n → + ∞ , see e.g. [23, 4]. We claim that the statement of Lemma 2.3 holds withrespect J n for a sufficiently large n .Suppose, on the contrary, that L b n bounds a J n -holomorphic disk of Maslovindex µ n ≤ ≤ E all sufficiently large n . Passing to a subsequence ifnecessary, we can assume that b n → b ∈ B . Apply the SFT compactness theorem,which is a version [4, Theorem 10.6] for curves with Lagrangian boundary condition L b n . Using Fukaya’s trick, one easily reduces the desired compactness statementto one about the fixed Lagrangian submanifold L b .The outcome of SFT compactness is a broken holomorphic building; see Figure 3for an example of how the building may look like, ignoring C for the moment. Werefer to [4] for the notion of holomorphic buildings and only record the followingbasic properties:— all curves in the holomorphic building have positive ω -area;— topologically, the curves glue to a disk with boundary on L b and µ ≤ S , which has boundaryon L b ; this curve lies inside U and may have several punctures asymptotic toReeb orbits { γ i } ⊂ ∂U .The Reeb orbits mentioned above are considered with respect to the contact form θ | ∂U , where θ is the Liouville form on U provided by Definition 2.1; this meansthat dθ = ω and L is θ -exact. (Recall that the choice of θ near ∂U is built intothe neck-stretching construction, and we assume that we have used this particular θ for it.)Next, consider a topological cylinder C ⊂ U with boundary ∂C ⊂ L b ∪ L ,and such that the L b -component of the boundary of C matches the one of S , seeFigure 3. Note that S ∪ C , considered as a 2-chain in U , has boundary of the form: ∂ ( S ∪ C ) = (cid:80) i γ i − l where l ⊂ L is a 1-cycle. Let us compute the area: ω ( S ∪ C ) = (cid:80) i (cid:82) γ i θ − (cid:82) l θ = (cid:80) i (cid:82) γ i θ > . Indeed, (cid:82) l θ = 0 because L is θ -exact, and (cid:82) γ i θ > γ i . EOMETRY OF FLUX 9
Figure 3.
A broken disk with boundary on L b , consisting of S andthe upper disks. The attaching cylinder C is not part of the brokendisk.Let us now construct a topological disk ( D (cid:48) , ∂D (cid:48) ) ⊂ ( X, L ) by the followingprocedure: first, glue together all pieces of the ω -tamed holomorphic building con-structed above, including S , then additionally glue C on to the result. Clearly,we get a topological disk with boundary on L , and moreover µ ( D (cid:48) ) ≤
0. Finally, ω ( D (cid:48) ) > ω ( C ∪ S ) > ω -area. These two properties of D (cid:48) contradict the fact that L is monotoneinside X . (cid:3) Conclusion of proof.
Proof of Theorem 2.2.
Take J from Lemma 2.3 for energy level E > ω ( β ) . Pick apoint p t ∈ L t . By an application of Gromov compactness, and standard transversal-ity techniques, there exist almost complex structures J t for 0 ≤ t ≤
1, sufficientlyclose J, such that J t still admit no holomorphic disks of Maslov index ≤ ≤ E with boundary on L t , for all t, and that the moduli spaces M β ( J i ) ,i = 0 , L i and p i , are regu-lar, as well as the parametric moduli space (cid:116) ≤ t ≤ M β ( J t ) where now an element of M β ( J t ) passes through p t . As explained in Subsection 2.1, these holomorphic diskscan be understood as J (cid:48) t -holomorphic disks on the fixed Lagrangian submanifold L = L (and the curve p t could be chosen to correspond to a fixed p = p ∈ L ).So Gromov compactness, again, applies to show that the count M β ( J i ) is inde-pendent of i unless a bubbling occurs for some t . However, any such bubbling willproduce a J t -holomorphic disk of Maslov index ≤ L t , whichis impossible by construction. We conclude that M β ( J i ) (cid:54) = 0, so for i = 0 , L i bounds a J i -holomorphic disk D i . As 2-chains, these disks differ by a cylinderswept by a cycle in class − ∂β . By the definition of flux: ω ( D ) − ω ( D ) = − f · ∂β. Finally, we have ω ( D ) = 2 c by monotonicity, and ω ( D ) > D isholomorphic; therefore f · ∂β ≥ − c . (cid:3) Fukaya algebra basics
Fukaya algebras.
Fix a ground field K of characteristic zero. We take K = R throughout, although all arguments are not specific to this. Let X be a symplecticmanifold and L ⊂ X a Lagrangian submanifold. We shall use the following versionof the Novikov ring with formal parameters T and q : Λ rel = (cid:40) ∞ (cid:88) i =0 a i T ω ( β i ) q β i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a i ∈ R , β i ∈ H ( X, L ; Z ) , ω ( β i ) ∈ R ≥ , lim i →∞ ω ( β i ) = + ∞ , ω ( β i ) = 0 ⇐⇒ β i = 0 (cid:41) We also use the ideal:Λ rel + = (cid:40) ∞ (cid:88) i =0 a i T ω ( β i ) q β i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a i ∈ R , β i ∈ H ( X, L ; Z ) , ω ( β i ) ∈ R > , lim i →∞ ω ( β i ) = + ∞ , ω ( β i ) = 0 ⇐⇒ β i = 0 (cid:41) This Novikov ring is bigger than the conventional Novikov ring Λ used in Floertheory, which only involves the T -variable. In the context of the Fukaya A ∞ algebraof a Lagrangian submanifold, the exponents of the q -variable are, by definition,placeholders for relative homology classes of holomorphic disks contributing to thestructure maps. Abstractly, the theory of gapped A ∞ algebras used below worksin the same way for Λ rel as it does for Λ .There are valuation mapsval : Λ rel → R ≥ , val : Λ rel + → R > defined by val (cid:0)(cid:80) i a i T ω ( β i ) q β i (cid:1) = min { ω ( β i ) : a i (cid:54) = 0 } . Fix an orientation and a spin structure on L ⊂ X . Let C ∗ ( L ; Λ rel ) be a cochaincomplex on L with coefficients in Λ rel ; to us, it is immaterial which cochain model isused provided the Fukaya algebra can be defined over it, recall Section 1.6. We usethe natural grading on C ∗ ( L ; R ) and the following gradings of the formal variables: | T | = 0, | q β | = µ ( β ). Only the reduction of the grading to Z / L are even (by theorientability of L ), the reduced grading simply comes from the reduced grading on C ∗ ( L ; R )Fix a tame almost complex structure J on X , and a suitable perturbation scheme s turning the relevant moduli spaces of J -holomorphic disks with boundary on L into transversely cut out manifolds, see e.g. [29, Proposition 3.5.2]. Holomorphiccurve theory shows that the vector space C ∗ = C ∗ ( L ; Λ rel ) has the structure ofa gapped curved A ∞ algebra structure, called the Fukaya algebra of L [29, Theo-rem 3.1.5]. We denote it by(3.1) CF ∗ ( L ; Λ rel ; J, s ) or ( CF ∗ ( L ; Λ rel ) , m )where m = { m k } k ≥ are the A ∞ structure operations.Abstractly, let C ∗ be a graded vector space over Λ rel . We remind that a gappedcurved A ∞ structure ( C ∗ , m ) is determined by a sequence of maps m k : ( C ∗ ) ⊗ k → C ∗ , k ≥ , of degree 2 − k , where m : R → C is called the curvature and is determined by m (1) ∈ C . EOMETRY OF FLUX 11
The curvature term is required to have non-zero valuation, that is:(3.2) m (1) = 0 mod Λ rel + . Next, the operations satisfy the curved A ∞ relations. If we denotedeg x = | x | − | x | is the grading of x ∈ C ∗ , see [29, (3.2.2)], then the A ∞ relations read [29,(3.2.22)](3.3) (cid:88) i,j ( − (cid:122) i m k − j +1 ( x , . . . , x i , m j ( x i +1 , . . . , x i + j ) , x i + j +1 , . . . , x k ) = 0where ( − (cid:122) i = deg x + . . . + deg x i + i (This convention differs from [37] by reversing the order in which the inputs arewritten down.) The inner appearance of m j may be the curvature term m (1)involving no x i -inputs. For example, the first two relations read: m ( m (1)) = 0 ,m ( m (1) , x ) + ( − deg x +1 m ( x, m (1)) + m ( m ( x )) = 0 . Finally, the condition of being gapped means that the valuations of the m k oper-ations “do not accumulate” anywhere except at infinity, which e.g. guarantees theconvergence of the left hand side of (3.3) over the Novikov field (the adic conver-gence ). We refer to [29] for a precise definition of gappedness. The fact that theFukaya algebra is gapped follows from Gromov compactness.The A ∞ relations can be packaged into a single equation by passing to the barcomplex . First, we extend the operations m k toˆ m k : ( C ∗ ) ⊗ i → ( C ∗ ) ⊗ ( i − k +1) viaˆ m k ( x ⊗ . . . ⊗ x i ) = i − k (cid:88) l =0 ( − (cid:122) l x ⊗ . . . , ⊗ x l ⊗ m k ( x l +1 , . . . x l + j ) ⊗ x l + j − ⊗ . . . ⊗ x k . This in particular means that the operations are trivial whenever k > i , and theexpression for k = 0 reads i (cid:88) l =0 x ⊗ . . . , ⊗ x l ⊗ m (1) ⊗ x l +1 ⊗ . . . ⊗ x k . We introduce the bar complex(3.4) B ( C ∗ ) = ∞ (cid:77) i =0 ( C ∗ +1 ) ⊗ i and define(3.5) ˆ m = ∞ (cid:88) k =0 ˆ m k : B ( C ∗ ) → B ( C ∗ ) . (This operation is denoted by ˆ d in [29].) The A ∞ relations are equivalent to thesingle relation(3.6) ˆ m ◦ ˆ m = 0 . Breakdown into classes.
Let ( C ∗ , m ) be a gapped curved A ∞ algebra overΛ rel . We can decompose the A ∞ operations into classes β ∈ H ( X, L ; Z ) as follows: m k ( x , . . . , x k ) = (cid:88) β ∈ H ( X,L ; Z ) T ω ( β ) q β m k,β ( x , . . . , x k ) . The operations m k,β are defined over the ground field R :(3.7) m k,β : C ∗ ( L ; R ) ⊗ k → C ∗ ( L ; R ) , and then extended linearly over Λ rel ; compare [29, (3.5.7)]. The degree of (3.7)is 2 − k + 2 µ ( β ). The gapped condition guarantees that the above sum convergesadically: there is a finite number of classes β of area bounded by a given constantthat have non-trivial appearance in (3.7).Geometrically, if ( C ∗ , m ) = C ∗ ( L ; Λ rel ; J, s ) is the Fukaya A ∞ algebra of a La-grangian submanifold, the m k,β are, by definition, the operations derived from themoduli spaces of holomorphic disks in class β ∈ H ( X, L ; Z ), see again [29].3.3. The classical part of an algebra.
Let ( C ∗ , m ) be a gapped curved A ∞ algebra over Λ rel . Let C ∗ = C ∗ ⊗ Λ rel R be the reduction of the vector space C ∗ to the ground field R . Together with this,one can reduce the structure maps m k modulo Λ rel + . This is equivalent to setting T = 0 or q = 0 (equivalently: T = q = 0 simultaneously), and gives an A ∞ structure defined over the ground field R : m k : ( C ∗ ) ⊗ k → C ∗ , see [29, Definition 3.2.20]. These operations are the same as the m k,β from (3.7)with β = 0, by the gapped property. This A ∞ structure is no longer curved,meaning m = 0, by (3.2). It is called the classical part of ( C ∗ , m ), and denotedby ( C ∗ , m ) . Now suppose that ( C ∗ , m ) = C ∗ ( L ; Λ rel ; J, s ) is the Fukaya algebra of a La-grangian submanifold. Then on chain level, C ∗ = C ∗ ( L ; R ). In this case ( C ∗ , m )is called the topological A ∞ algebra of L . The following is proven in [29, Theo-rem 3.5.11 and Theorem X]. Theorem 3.1.
The topological A ∞ algebra of L is quasi-isomorphic to the de Rhamdg algebra of L . (cid:3) The definition of quasi-isomorphism will be reminded later in this section. Theterm quasi-isomorphism follows Seidel’s terminology [37]; the same notion is termeda weak homotopy equivalence (between non-curved A ∞ algebras) in [29, Defini-tion 3.2.10].Recall that L is called topologically formal if its de Rham dg algebra is quasi-isomorphic to the cohomology algebra H ∗ ( L ; R ) with the trivial differential. Thetheorem below is due to Deligne, Griffiths, Morgan and Sullivan [21]. Theorem 3.2.
If a compact manifold L admits a K¨ahler structure, then it istopologically formal. (cid:3) Example . The n -torus is topologically formal. EOMETRY OF FLUX 13
Weak homotopy equivalences.
Suppose ( C ∗ , m ) and ( C (cid:48)∗ , m (cid:48) ) are twogapped curved A ∞ algebras over Λ rel . We remind the notion of a gapped curved A ∞ morphism between them: f : ( C ∗ , m ) → ( C (cid:48)∗ , m (cid:48) ) . It is composed of maps f k : ( C ∗ ) ⊗ k → C (cid:48)∗ , k ≥ , of degree 1 − k with the following properties. The term f : R → ( C (cid:48) ) is requiredto have non-zero valuation, that is:(3.8) f (1) = 0 mod Λ rel + . Next, the maps { f k } satisfy the equations for being a curved A ∞ functor, seee.g. [37, (1.6)]: (cid:80) i,j ( − (cid:122) i f k − j +1 ( x , . . . , x i , m j ( x i +1 , . . . , x i + j ) , x j +1 , . . . , x k )= (cid:80) r (cid:80) s ,...,s r m (cid:48) r ( f s ( x , . . . , x s ) , . . . , f s r ( x s r , . . . , x k )) . For example, the first equation reads:(3.9) f ( m (1)) = m (cid:48) (1) + m (cid:48) ( f (1)) + m (cid:48) ( f (1) , f (1)) + . . . where the right hand side converges by (3.8). Finally, the maps f k must be gapped:roughly speaking, this again means that their valuations have no finite accumula-tion points so that the above sums converge. We refer to [29] for details.As earlier, we can package the A ∞ functor equations into a single equationpassing to the bar complex. To this end, introduce a single map between the barcomplexes(3.10) ˆ f : B ( C ∗ ( L ; Λ rel )) → B ( C ∗ ( L ; Λ rel ))by its action on homogeneous elements:(3.11) ˆ f ( x ⊗ . . . ⊗ x k ) = (cid:88) s + ... + s r = k f s ( x , . . . , x s ) ⊗ . . . ⊗ f s r ( x s r , . . . , x k ) , and extend it linearly. In particular,ˆ f (1) = 1 + f (1) + f (1) ⊗ f (1) + . . . The A ∞ functor equations are equivalent to:(3.12) ˆ f ◦ ˆ m = ˆ m (cid:48) ◦ ˆ f , where ˆ m , ˆ m (cid:48) are as in (3.5).We proceed to the notion of weak homotopy equivalence. Note that reducing f modulo Λ rel + gives a non-curved A ∞ morphism¯ f : ( C ∗ , m ) → ( C (cid:48)∗ , m (cid:48) )between the non-curved A ∞ algebras: it satisfies ¯ f = 0. The first non-trivial A ∞ relation says that ¯ f is a chain map with respect to the differentials m and m (cid:48) . Definition 3.3.
Let f be a gapped curved A ∞ morphism as above. We say that¯ f is a quasi-isomorphism if ¯ f induces an isomorphism on the level of homology H ( C ∗ , m ) → H ( C (cid:48)∗ , m (cid:48) ). We say that f is a weak homotopy equivalence if ¯ f is aquasi-isomorphism. It is well known that weak homotopy equivalences of A ∞ algebras have weakinverses. See e.g. [37] in the non-curved case, and [29, Theorem 4.2.45] in thegapped curved case. We record a weaker version of this result. Theorem 3.4.
Let ( C ∗ , m ) and ( C (cid:48)∗ , m (cid:48) ) be two gapped curved A ∞ algebras over Λ rel . If there exists a weak homotopy equivalence ( C ∗ , m ) → ( C (cid:48)∗ , m (cid:48) ) , then therealso exists a weak homotopy equivalence ( C (cid:48)∗ , m (cid:48) ) → ( C ∗ , m ) . (cid:3) The theorem above can be strengthened by asserting that the two morphisms inquestion are weak inverses of each other; we shall not need this addition.As a separate but more elementary property, being weakly homotopy equivalentis a transitive relation, because there is an explicit formula for the compositionof two A ∞ morphisms. Combining it with Theorem 3.4, we conclude that weakhomotopy equivalence is indeed an equivalence relation.Finally, we recall the fundamental invariance property of the Fukaya algebra ofa Lagrangian submanifold. Theorem 3.5.
Let L ⊂ X be a Lagrangian submanifold. Given two choices J, s and J (cid:48) , s (cid:48) , there is is weak homotopy equivalence between the Fukaya A ∞ algebras CF ∗ ( L ; Λ rel ; J, s ) and CF ∗ ( L ; Λ rel ; J (cid:48) , s (cid:48) ) . (cid:3) Classically minimal algebras.
The definition of the Ψ-invariant given inthe next section will use classically minimal models of A ∞ algebras. The book [29]uses a different term: it calls them canonical algebras. Definition 3.6.
Let ( C ∗ , m ) be a gapped curved A ∞ algebra over Λ rel . It is called classically minimal if m = 0.The following is a version of the homological perturbation lemma, see e.g. [29,Theorem 5.4.2]. Theorem 3.7.
Let ( C ∗ , m ) be a gapped curved A ∞ algebras over Λ rel . Then it isweakly homotopy equivalent to a classically minimal one.Moreover, suppose that the classical part ( C ∗ , m ) is formal, i.e. quasi-isomorphicto an algebra ( H ∗ , µ ) with an associative product µ and all other structure maps trivial. Then ( C ∗ , m ) isweakly homotopy equivalent to a gapped curved A ∞ structure ( H ∗ ⊗ R Λ rel , m (cid:48) ) where m (cid:48) = µ, and m (cid:48) k = 0 for all k (cid:54) = 2 . The following obvious lemma will play a key role in the invariance property ofthe Ψ-invariant which we will define soon.
Lemma 3.8.
Let f : ( C ∗ , m ) → ( C (cid:48)∗ , m (cid:48) ) be a weak homotopy equivalence betweentwo classically minimal gapped curved A ∞ algebras over Λ rel . Then ¯ f : C ∗ → C (cid:48)∗ is an isomorphism.Proof. By definition, ¯ f is a quasi-isomorphism between the vector spaces C ∗ and C (cid:48)∗ with trivial differential; hence it is an isomorphism. (cid:3) We remind that, by definition,(3.13) f = ¯ f mod Λ rel + . EOMETRY OF FLUX 15
Remark . Suppose J, s and J (cid:48) , s (cid:48) are two choices of a tame almost complex struc-tures with a perturbation scheme. Denote by m k , m (cid:48) k the corresponding Fukaya A ∞ structures on C ∗ = C ∗ ( L ; Λ rel ) extending the same topological A ∞ algebra struc-ture on L . Then there is a weak homotopy equivalence f : ( C ∗ , m ) → ( C (cid:48)∗ , m (cid:48) )satisfying ¯ f = Idregardless of whether the given topological A ∞ structure is minimal. See [29],compare [36, Section 5].3.6. Around the Maurer-Cartan equation.
Let ( C ∗ , m ) be a gapped curved A ∞ algebra over Λ rel . Consider an element(3.14) a ∈ C ∗ , a = 0 mod Λ rel + . The prepotential of a is(3.15) m ( e a ) = m (1) + m ( a ) + m ( a, a ) + . . . ∈ C ∗ . The condition that a = 0 mod Λ rel + guarantees that the above sum converges.Moreover, (3.2) implies that m ( e a ) = 0 mod Λ rel + .One says that a is a Maurer-Cartan element or a bounding cochain if m ( e a ) = 0.Maurer-Cartan elements are of central importance for Floer theory as they allowto deform the initial A ∞ algebra into a non-curved one, so that one can computeits homology. A Lagrangian L whose Fukaya algebra has a Maurer-Cartan elementis called weakly unobstructed .Our interest in the Maurer-Cartan equation is somewhat orthogonal to this. Weshall look at the prepotential itself, and will not be interested in Maurer-Cartanelements per se.Denote(3.16) e a = 1 + a + a ⊗ a + . . . ∈ B ( C ∗ ) , where B ( C ∗ ) is the bar complex (3.4). For ˆ m as in (3.5), it holds that(3.17) ˆ m ( e a ) = (cid:88) i,j ≥ a ⊗ i ⊗ m ( e a ) ⊗ a j ∈ B ( C ∗ ) . In particular the Maurer-Cartan equation on a is equivalent to the equationˆ m ( e a ) = 0 , see [29, Definition 3.6.4]. More relevantly for our purposes, observe the followingidentity:(3.18) val( m ( e a )) = val( ˆ m ( e a )) . This is because the lowest-valuation summand in (3.17) is obviously the one with i = j = 0. Here we have extended the valuation from C ∗ to B ( C ∗ ) in the naturalway: for a general element x = x i ⊗ . . . ⊗ x i r + x i ⊗ . . . ⊗ x i r + . . . ∈ B ( C ∗ ( L ; Λ rel )) , x i jk ∈ C ∗ , we set(3.19) val( x ) = min { val( x i ) . . . val( x i r ) , val( x i ) . . . val( x i r ) , . . . } . This minimum exists for gapped A ∞ algebras. We proceed to the functorialityproperties of the expression ˆ m ( e a ). Consider a gapped A ∞ morphism f : ( C ∗ , m ) → ( C (cid:48)∗ , m (cid:48) ) . Take an element a as in (3.14) and denote after [29, (3.6.37)]:(3.20) f ∗ a = f (1) + f ( a ) + f ( a, a ) + . . . ∈ C (cid:48)∗ . Note that by (3.8), it holds that f ∗ ( a ) = 0 mod Λ rel + . Next, one has the identities(3.21) ˆ f ( e a ) = e f ∗ a and(3.22) ˆ m (cid:48) ( ˆ f ( e a )) = ˆ f ( ˆ m ( e a )) , see [29, Proof of Lemma 3.6.36]. In particular, f ∗ takes Maurer-Cartan elements toMaurer-Cartan elements; however instead of this property, we shall need the keyproposition below. Proposition 3.9.
Suppose ( C ∗ , m ) and ( C (cid:48)∗ , m (cid:48) ) are classically minimal (see Def-inition 3.6) gapped curved A ∞ algebras over Λ rel , and f : ( C ∗ , m ) → ( C (cid:48)∗ , m (cid:48) ) is aweak homotopy equivalence. For any element a as in (3.14), it holds that val( m (cid:48) ( e f ∗ a )) = val( m ( e a )) . Remark . Observe that we are using m, m (cid:48) not ˆ m, ˆ m (cid:48) here, so that m (cid:48) ( e f ∗ a ) and m ( e a ) are elements of C (cid:48)∗ rather than the bar complex, see (3.15). But in view of(3.18), we could have used the hat-versions instead. Proof.
Using (3.18), (3.21) and (3.22) we obtainval( m (cid:48) ( e f ∗ a )) = val( ˆ m (cid:48) ( e f ∗ a )) = val( ˆ m (cid:48) ( ˆ f ( e a ))) = val( ˆ f ( ˆ m ( e a ))) . Therefore we need to show that(3.23) val( ˆ f ( ˆ m ( e a ))) = val( m ( e a )) . Denote λ = val( m ( e a )), so that m ( e a ) has the form m ( e a ) = T λ y + o ( T λ )for some y ∈ C ∗ , i.e. val( y ) = 0. In view of (3.17), it also holds thatˆ m ( e a ) = T λ y + o ( T λ ) ∈ B ( C ∗ )where y is considered as a length-1 element of the bar complex; compare (3.18).Because the application of ˆ f (3.11) to length-1 elements of the bar complex reducesto the application of f , we have:(3.24) ˆ f ( T λ y + o ( T λ )) = T λ f ( y ) + o ( T λ ) = T λ ¯ f ( y ) + o ( T λ ) . Let us now use the hypothesis that C ∗ , C (cid:48)∗ are classically minimal; by Lemma 3.8, itimplies that ¯ f is an isomorphism. In particular ¯ f ( y ) (cid:54) = 0, therefore val( ¯ f ( y )) = 0.We have shown that val( ˆ f ( ˆ m ( e a ))) = λ , which amounts to (3.23). (cid:3) Remark . The above proof breaks down if ¯ f has kernel. Indeed, suppose ¯ f ( y ) =0; this means that val( ¯ f ( y )) equals + ∞ rather than 0, and the valuation of theright hand side of (3.24) is strictly greater than λ . Hence, classical minimality isan essential condition for Proposition 3.9. EOMETRY OF FLUX 17
Remark . Although the above argument is inspecific to this, recall that we areworking with Λ rel + which has an additional variable q . In this ring, a term T λ y canbe written in full form as T λ = c β q β T ω ( β ) ¯ y where ¯ y ∈ C ∗ , c β ∈ R and λ = ω ( β ).Now in the above proof, take a ∈ C ∗ which is zero modulo Λ rel + , and let c β q β T ω ( β ) ¯ y be one of the lowest-valuation terms of m ( e a ): val( m ( e a )) = ω ( β ). Then m (cid:48) ( e f ∗ a )has the corresponding lowest-valuation term c β q β T ω ( β ) ¯ f (¯ y ). This fact will be usedto prove Proposition 5.1 below.4. The invariant
Ψ4.1.
Definition and invariance.
We are ready to define the Ψ-invariant of acurved, gapped A ∞ algebra over a Novikov ring; see the previous section for theterminology. The Novikov ring could be Λ or Λ rel , and we choose the secondoption. For simplicity, we reduce all gradings modulo 2, and denote by C odd thesubspace of odd-degree elements in a graded vector space C ∗ . We denote [0 , + ∞ ] =[0 , + ∞ ) ∪ { + ∞} . Definition 4.1.
Let ( C ∗ , m ) be a classically minimal (see Definition 3.6) curvedgapped A ∞ algebra over the Novikov ring Λ rel . The Ψ -invariant of this algebra isdefined as follows:Ψ( C ∗ , m ) = inf { val( m ( e b )) : b ∈ C odd , b = 0 mod Λ rel + } ∈ [0 , + ∞ ]Recall that the expression m ( e b ) was introduced in (3.15). Now suppose ( C ∗ , m ) isa curved gapped A ∞ algebra over Λ rel + which is not necessarily classically minimal.Let ( C (cid:48)∗ , m (cid:48) ) be a weakly homotopy equivalent classically minimal A ∞ algebra,which exists by Theorem 3.7. We defineΨ( C ∗ , m ) = Ψ( C (cid:48)∗ , m (cid:48) ) . Two remarks are due.
Remark . The definition requires to look at odd degree elements b . This isopposed to the discussion in Subsection 3.6 where we imposed no degree require-ments on the element a . Indeed one could give a definition of the Ψ-invariant usingelements b of any degree; this would be a well-defined invariant which, however,frequently vanishes. Suppose the classical part ( C ∗ , m ) is a minimal topologicalalgebra of a smooth manifold L ; then m is graded-commutative. If we take b = T (cid:15) y, y ∈ C ∗ with (cid:15) sufficiently small, the expansion of m ( e b ) according to (3.15) will containthe lowest-energy term T (cid:15) m ( y, y ) . If y has odd degree, this term vanishes by the graded-commutativity. However,if y has even degree, its square may happen to be non-zero, in which case letting (cid:15) → C ∗ , m ) is the (formal)topological A ∞ algebra of an n -torus (which is the most important case as far asour applications to symplectic geometry are concerned), since the cohomology ofthe torus is generated as a ring by odd-degree elements. But with the general casein mind, and guided by the analogy with Maurer-Cartan theory, we decided thatDefinition 4.1 is most natural if we only allow odd-degree elements b . See alsoRemark 4.6. Remark . The condition that b = 0 mod Λ rel + is important due to guarantee theconvergence of m ( e b ); compare with (3.14) from Subsection 3.6. Theorem 4.2.
Let ( C ∗ , m ) be a curved gapped A ∞ algebra over Λ rel . Then Ψ( C ∗ , m ) is well-defined and is an invariant of the weak homotopy equivalence class of ( C ∗ , m ) .Proof. Any two classically minimal A ∞ algebras which are both weakly homotopyequivalent to ( C ∗ , m ) are weakly homotopy equivalent to each other by Theo-rem 3.4. Therefore the Ψ-invariants of these two classically minimal algebras areequal by Proposition 3.9. Hence Ψ( C ∗ , m ) does not depend on the choice of theclassically minimal model used to compute it. The same argument proves theinvariance under weak homotopy equivalences of ( C ∗ , m ). (cid:3) Definition 4.3.
Let L ⊂ X be a Lagrangian submanifold. We define the Ψ-invariant of L , Ψ( L ) ∈ [0 , + ∞ ] , to be the Ψ-invariant of its Fukaya algebra CF ∗ ( L ; Λ rel ; J, s ) for some choice of atame almost complex structure J and a perturbation scheme s . In Corollary 5.4we will show that Ψ( L ) is strictly positive. Theorem 4.4.
The invariant Ψ( L ) is well-defined and is invariant under Hamil-tonian isotopies of L .Proof. The invariance under the choices of J and s follows from Theorem 4.2 andTheorem 3.5. The invariance under Hamiltonian isotopies is automatic by pullingback J under the given Hamiltonian diffeomorphism. (cid:3) Expanding the Maurer-Cartan equation.
From now on, let ( C ∗ , m ) be a classically minimal gapped curved A ∞ algebra over Λ rel . Choose a basis b , . . . , b N of C odd ; it induces a basis of C odd denoted by the same symbols. Recall that C isa minimal A ∞ algebra over R . In practice, we shall use A ∞ algebras based onvector spaces C ∗ = H ∗ ( L ; R ) , C ∗ = H ∗ ( L ; Λ rel ) , but the discussion applies generally.We introduce the following notation. A k -type is a function ν : [ N ] = { , . . . , N } → Z ≥ such that (cid:80) i ∈ [ N ] ν ( i ) = k. A map f : [ k ] → [ N ] is said to belong to a k -type ν if ν ( i ) = f − ( { i } ) for all i ∈ [ N ]. When f belongs to the k -type ν , we write f ∈ ν .It is clear that an arbitrary pair of functions f, f (cid:48) ∈ ν belonging to the sametype differs by a permutation in S k , i.e. f (cid:48) is the composition[ k ] σ −→ [ k ] f −→ [ N ]for σ ∈ S k .The language of k -types is useful for expanding polynomial expressions in non-commuting variables. Specifically, consider an arbitrary linear combination of basicclasses: b = (cid:88) i ∈ [ N ] l i b i ∈ C odd , b i ∈ C odd , l i ∈ Λ rel + . EOMETRY OF FLUX 19
Denote l ( ν ) = ( l ) ν (1) · . . . · ( l N ) ν ( N ) . We also put = l = 1 for any l ∈ Λ rel + .Expanding m ( e b ) by definition we obtain:(4.1) m ( e b ) = (cid:88) k ≥ (cid:88) ν ∈ k -types l ( ν ) (cid:88) α ∈ H ( X,L ) T ω ( α ) q α (cid:88) f ∈ ν m k,α ( b f (1) , . . . , b f ( k ) ) . Given α ∈ H ( X, L ; Z ) , k ≥ k -type ν , define(4.2) sm k,α ( ν ) := (cid:88) f ∈ ν m k,α ( b f (1) , . . . , b f ( k ) ) . Now let ( i ν , . . . , i νk ) = ( f (1) , . . . , f ( k )) be the sequence of values some fixed f ∈ ν ;we make one choice of f for each k -type ν and each k ≥
0. Then, by the aboveobservation about the S k -symmetry,(4.3) sm k,α ( ν ) = (cid:88) σ ∈ S k m k,α ( b i νσ (1) , . . . , b i νσ ( k ) ) . The meaning of sm k,α ( ν ) is that it is the sum of symmetrised A ∞ operations,evaluated on a collection of basic vectors b i such that the repetitions among thoseinputs are governed by the type ν . Remark . Tautologically, any sequence of numbers ( i , . . . , i k ) where i j ∈ [ N ] isthe sequence of values of a function f ∈ ν for some k -type ν . Therefore (cid:88) σ ∈ S k m k,α ( b i σ (1) , . . . , b i σ ( k ) ) = sm k,α ( ν )for some ν .4.3. Irrationality.
Recall the valuation val : Λ rel → R ≥ and val : Λ rel + → R > .It induces a valuation on C odd denoted by the same symbol. Recall the crucialproperty val( x + y ) ≥ min { val( x ) , val( y ) } , which turns into the equality whenever the lowest-valuation terms of x do notcancel with those of y .Denote Ω := { ω ( β ) : β ∈ H ( X, L ; Q ) } ⊂ R . This is a finite-dimensional vector subspace of R over Q . Recall that R itself isinfinite-dimensional over Q , in fact uncountably so.A k -type ν defines a linear map ν : R N → R / Ω by( λ i ) i ∈ [ N ] (cid:55)→ λ ( ν ) = (cid:88) ν ( i ) λ i . Definition 4.5.
Consider a vector λ = ( λ i ) i ∈ [ N ] ∈ R N . We call it:(1) generic if the map { k -types } → R / Ω given by ν (cid:55)→ λ ( ν ) is injective.(2) Q -independent if ( λ i ) i ∈ [ N ] induces an independent collection of N vectors of in R / Ω considered as a vector space over Q .We call an element b = (cid:88) i ∈ [ N ] l i b i ∈ C odd , b i ∈ C odd , l i ∈ Λ rel + , generic (respectively Q -independent) if ( λ i ) i ∈ [ N ] = (val( l i )) i ∈ [ N ] is generic (respec-tively Q -independent). Remark . It is clear that Q -independence implies genericity. We shall furtheruse Q -independence, but genericity would also suffice for most statements. Remark . Since R / Ω is (uncountably) infinite-dimensional, the set of Q -independentelements of R N is dense. Lemma 4.6. If b ∈ C odd is Q -independent, the expansion (4.1): m ( e b ) = (cid:88) k ≥ (cid:88) ν ∈ k -types l ( ν ) (cid:88) α ∈ H +2 ( X,L ; J ) T ω ( α ) q α sm k,α ( ν ) , has the following property. All non-zero summands in s ( b, ν ) = l ( ν ) (cid:88) α ∈ H ( X,L ; Z ) T ω ( α ) q α sm k,α ( ν ) corresponding to different types ν have different valuations, and s ( b, ν ) is zero ifand only if sm k,α ( ν ) = 0 for each α ∈ H ( X, L ; Z ) . Proof of Lemma 4.6.
Denote the summands in s ( b, ν ) by s α ( b, ν ) = l ( ν ) T ω ( α ) q α sm k,α ( ν ) . Consider the vector λ = ( λ i ) i ∈ [ N ] = (val( l i )). First, one hasval( s α ( b, ν )) = ω ( α ) + λ ( ν )if sm k,α ( b, ν ) (cid:54) = 0, and + ∞ otherwise. Since λ is Q -independent, it holds thatval( s α ( b, ν )) (cid:54) = val( s β ( b, ν (cid:48) ))for ν (cid:54) = ν (cid:48) , and each α, β ∈ H ( X, L ; Z ). Moreover, it holds that s α ( b, ν ) (cid:54) = s β ( b, ν )whenever α (cid:54) = β, and s α ( b, ν ) (cid:54) = 0, s β ( b, ν ) (cid:54) = 0 because the factors q α , q β distin-guish them. (cid:3) Proposition 4.7. If ( C ∗ , m ) is classically minimal, it holds that Ψ( C ∗ , m ) = inf { val( m ( e b )) : b ∈ C odd , b is Q -independent } . Proof.
Let Ψ (cid:48) be the right hand side of the claimed equality. It is clear that Ψ ≤ Ψ (cid:48) .To show the converse, consider an element b = (cid:80) l i b i such that val( m ( e b )) ≤ Ψ + (cid:15) .We can find ( (cid:15) ni ) i ∈ [ N ] such that b (cid:15) n = (cid:80) T (cid:15) ni l i b i is Q -independent, and (cid:15) ni → n → ∞ . Then for some k and a k -type ν , and α ∈ H ( X, L )val( m ( e b )) = val (cid:16) l ( ν ) T ω ( α ) q α sm k,α ( ν ) (cid:17) = ω ( α ) + λ ( ν ) . By Lemma 4.6, taking the summand corresponding to these α and ν, val( m ( e b (cid:15)n )) ≤ ω ( α ) + λ ( ν ) + (cid:88) ν ( i ) (cid:15) ni . Hence for n sufficiently large,val( m ( e b (cid:15)n )) ≤ val( m ( e b )) + (cid:15) ≤ Ψ + 2 (cid:15).
Therefore Ψ (cid:48) ≤ Ψ + 2 (cid:15), which implies Ψ (cid:48) ≤ Ψ since (cid:15) is arbitrary. (cid:3)
EOMETRY OF FLUX 21
A computation of Ψ . The next theorem is very useful for computing Ψ inpractice.
Theorem 4.8. If ( C ∗ , m ) is classically minimal, it holds that Ψ( C ∗ , m ) = min ω ( α ) : ∃ c , . . . , c k ∈ C odd s.t. (cid:88) σ ∈ S k m k,α ( c σ (1) , . . . , c σ ( k ) ) (cid:54) = 0 . Recall that the operations m k,α are from (3.7); they land in C ∗ . The set fromthe statement includes the case when k = 0, m ,α (1) (cid:54) = 0. Proof.
Expanding the brackets by multi-linearity reduces the statement to the casewhen all c i belong to a basis ( b i ) i ∈ [ N ] of C odd . So it is enough to show thatΨ = min { ω ( α ) : ∃ k and a k -type ν such that sm k,α ( ν ) (cid:54) = 0 } . Denote the right hand side by Ψ (cid:48)(cid:48) . This minimum exists by the gapped condition(or by Gromov compactness, in the case C ∗ is the Fukaya algebra).Returning to the definition of Ψ, recall that by Proposition 4.7, we may restrictto elements b = (cid:80) l i b i that are Q -independent. For such an element, Lemma 4.6says thatval( m ( e b )) = min { ω ( α ) + λ ( ν ) : ∃ k and a k -type ν such that s α ( b, ν ) (cid:54) = 0 } , where the minimum exists since l i ∈ Λ + the A ∞ algebra is gapped. From this it isimmediate that Ψ ≥ Ψ (cid:48)(cid:48) . To show the other inequality, let ν be a k -type and α ∈ H ( X, L ; Z ) a class suchthat sm k,α ( ν ) (cid:54) = 0 and Ψ (cid:48)(cid:48) = ω ( α ). Consider the ring automorphism r n : Λ rel → Λ rel given by T (cid:55)→ T /n . Clearly, it preserves Λ rel + and satisfiesval( r n ( l )) = 1 n val( l ) . It induces a map r n : C odd → C odd , which preserves Q -independence. Then,evaluating at the ν -summand of the expansion (4.1), we haveval( m ( e r n ( b ) )) ≤ ω ( α ) + 1 n λ ( ν ) . The right hand side converges to Ψ (cid:48)(cid:48) as n → ∞ , so Ψ ≤ Ψ (cid:48)(cid:48) . (cid:3) Remark . Because Ψ is determined by the symmetrised A ∞ operations, whichis evident from both Definition 4.1 and Theorem 4.8, the Ψ-invariant can alterna-tively be defined in the setting of L ∞ algebras, cf. [19, 20, 12] and [29, A.3]. Acomparison between Definition 4.1 and Theorem 4.8 in the L ∞ context follows thesame arguments as above.4.5. Concavity.
Below is the main geometric property of Ψ, that it is concaveunder Lagrangian star-isotopies. This is a very powerful property that makes Ψamenable to explicit computation, for example, for fibres of various (singular) torusfibrations.
Theorem 4.9.
Let X be a symplectic manifold and { L t ⊂ X } t ∈ [0 , a Lagrangianstar-isotopy, i.e. a Lagrangian isotopy whose flux develops linearly. Then the func-tion Ψ( L t ) : [0 , → [0 , + ∞ ] is continuous and concave in t . Remark . If Ψ( L t ) achieves value + ∞ for some t , concavity means that Ψ( L t ) ≡ + ∞ for all t . Proof.
By definition of star-isotopy, there is a vector f ∈ H ( L ) such thatFlux( { L t } t ∈ [0 ,t ] ) = t · f , t ∈ [0 , . Denote L = L .Consider an integer k ≥ β ∈ H ( X, L ; Z ), c , . . . , c k ∈ C odd = H odd ( L ; R ). By continuity, the canonically define classes β t ∈ H ( X, L t ; Z )and c t , . . . , c tk ∈ H odd ( L t ; R ). Let { m tk } k ≥ be structure maps of the Fukaya algebraof L t . By Fukaya’s trick, there exists a choice of almost complex structures suchthat for all sufficiently small t , m tk,β t ( c t , . . . , c tk ) does not depend on t, for any k , β and c i as above. Recall that these operations, by definition (3.7), donot remember the area of β t which can change in t . Now let I = { β i } ⊂ H ( X, L ; Z )be the set of all classes β such that ω ( β ) = Ψ( L ) and β satisfies the property fromthe right hand side of the formula from Theorem 4.8: that is, holomorphic disks inclass β witness the value Ψ( L ). By Gromov compactness, this set is finite.By Theorem 4.8 and the Fukaya trick, it is precisely some of the disks among { β ti } i ∈ I that witness the value Ψ( L t ). Namely:Ψ( L t ) = min { ω ( β ti ) } i ∈ I . By definition of flux, for all t : ω ( β ti ) = ω ( β i ) + t ( f · ∂β i ) = Ψ( L ) + t ( f · ∂β i )where f · ∂β i ∈ R is the pairing. Observe that this function is linear in t . So Ψ( L t ),for small enough t , is the minimum of several linear functions. It follows that Ψis continuous and concave at t = 0. The argument may be repeated at any otherpoint t . (cid:3) Bounds on flux
Indicator function.
For the first two subsections, we continue working in thegeneral setting of Sections 3, 4. Let ( C ∗ , m ) be a classically minimal (Definition 3.6)curved gapped A ∞ algebra over the Novikov ring Λ rel . Define the indicator function η L : H ( X, L ; Z ) → { , } as follows. We put η L ( β ) = 1 if and only if ω ( β ) = Ψ( L ) and ∃ k ≥ , ∃ c , . . . , c k ∈ C odd such that (cid:88) σ ∈ S k m k,β ( c σ (1) , . . . , c σ ( k ) ) (cid:54) = 0 . It means that holomorphic disks in class β witness the minimum from Theorem 4.8,computing Ψ. In other words, holomorphic disks in class β are the lowest areaholomorhic disks that contribute non-trivially to some symmetrised A ∞ structuremap on odd-degree elements. The indicator function depends on ( C ∗ , m ), althoughthis is not reflected in the notation. Proposition 5.1.
The indicator function η L is invariant under weak homotopyequivalences ( C ∗ , m ) → ( C (cid:48)∗ , m (cid:48) ) between classically minimal A ∞ algebras. EOMETRY OF FLUX 23
Proof.
This is analogous to the proof of Proposition 3.9. Specifically, see Re-mark 3.4. (cid:3)
Remark . One can use η L to distinguish non-monotone Lagrangian submani-folds up to Hamiltonian isotopy. In particular, this implies [46, Conjecture 1.5]distinguishing certain Lagrangian tori in ( C P ) m .5.2. Curvature term.
The following lemma will be useful in Section 6.
Lemma 5.2.
Suppose β ∈ H ( X, L ; Z ) is a class satisfying η L ( β ) = 1 . Then m ,β (1) ∈ C even is invariant under weak homotopy equivalences f : ( C ∗ , m ) → ( C (cid:48)∗ , m (cid:48) ) between classically minimal algebras. Namely, one has m (cid:48) ,β (1) = f ( m ,β (1)) where f : C ∗ → C (cid:48)∗ is an isomorphism.Proof. Let us look at the A ∞ functor equation (3.9): f ( m (1)) = m (cid:48) (1) + m (cid:48) ( f (1)) + m (cid:48) ( f (1) , f (1)) + . . . Recall that f (1) has positive valuation, by definition. It now follows from Theo-rem 4.8 and the invariance of Ψ that the above equation taken modulo the ideal t Ψ( L ) · Λ + is: (cid:88) β : η L ( β )=1 f ( m ,β (1)) q β T ω ( β ) = (cid:88) β : η L ( β )=1 m (cid:48) ,β (1) q β T ω ( β ) . It implies that m (cid:48) ,β (1) = f ( m ,β (1)) for all β such that η ( β ) = 1. (cid:3) Positivity.
Let L ⊂ X be a Lagrangian submanifold, and consider a clas-sically minimal model of its Fukaya A ∞ algebra over C ∗ = H ∗ ( L ; Λ rel ), for somechoice of a compatible almost complex structure. Recall that C ∗ = H ∗ ( L ; R ); itsupports the topological part of the A ∞ algebra, { m k } k ≥ . The topological dif-ferential m vanishes by the minimality condition. Recall that S k denotes thesymmetric group. The lemma below says that the Ψ-invariant of Lagrangian sub-manifolds is non-trivial; if the lemma were false, the Ψ-invariant would vanish. Thefollowing is shown in [29, Theorem A3.19]. Lemma 5.3.
For all k ≥ and all c , . . . , c k ∈ C odd it holds that (cid:88) σ ∈ S k m k ( c σ (1) , . . . , c σ ( k ) ) = 0 . Remark . The case k = 2 follows from the fact that the product on H ∗ ( L ; R )is anti-commutative on odd degree elements. If L is topologically formal, e.g. L isthe n -torus, the full statement also quickly follows from Theorem 3.7. Remark . We provide an argument for completeness. By the invariance of Ψshown in Section 4, if the lemma holds true for some minimal A ∞ algebra C ∗ , itholds for any other minimal A ∞ algebra quasi-isomorphic to it. Let C ∗ dR ( L ; R ) bethe de Rham complex of the space of differential forms on L , considered as an A ∞ al-gebra with the de Rham differential, exterior product µ and trivial higher structureoperations. By Theorem 3.1, there is an A ∞ quasi-isomorphism C ∗ → C ∗ dR ( L ; R ).Moreover, starting from C ∗ dR ( L ; R ), one can construct a minimal A ∞ algebraquasi-isomorphic to it (hence quasi-isomorphic to C ∗ ) explicitly by the Kontsevich perturbation formula. Looking at the formula, see e.g. [33, Proposition 6], onesees that starting with µ that is skew-symmetric on odd-degree elements, the sym-metrisations of the A ∞ operations on the minimal model also vanish on odd-degreeelements. Corollary 5.4.
For any Lagrangian submanifold L ⊂ X , Ψ( L ) is strictly positive.Proof. This follows from Theorems 4.8,3.7 and Lemma 5.3. (cid:3)
General shape bound.
From now on, η L shall denote the indicator functionof the Fukaya algebra of L ⊂ X . The theorem below is the main bound on star-shape in terms of the Ψ-function. It says that every class β such that η L ( β ) = 1constrains Sh (cid:63)L ( X ) to an affine half-space bounded by the hyperplane in H ( L ; R )which is a translate of the annihilator of ∂β ∈ H ( L ; Z ). If the indicator function η L equals 1 on several classes β with different boundaries, the shape consequentlybecomes constrained to the intersection of the corresponding half-spaces. Theorem 5.5.
Let X be a symplectic manifold and L ⊂ X an oriented spin La-grangian submanifold. Then, for any β ∈ H ( X, L ; Z ) such that η L ( β ) = 1 , thefollowing holds. Sh (cid:63)L ( X ) ⊂ { f ∈ H ( L ; R ) : Ψ( L ) + f · ∂β > } . Proof.
Suppose η L ( β ) = 1, and consider a star-isotopy { L t } t ∈ [0 , , L = L withtotal flux f ∈ H ( L ; R ). We begin as in the proof of Theorem 4.9. By definition ofstar-isotopy: Flux( { L t } t ∈ [0 ,(cid:15) ] ) = (cid:15) · f , (cid:15) ∈ [0 , . Let { m tk } k ≥ be the structure maps of the Fukaya algebra of L t . Recall Fukaya’strick used in the proof of proof of Theorem 4.9: there exists a sufficiently small t > (cid:15) < t , m (cid:15)k,β (cid:15) ( c (cid:15) , . . . , c (cid:15)k ) does not depend on (cid:15). So by Theorem 4.8, Ψ( L (cid:15) ) ≤ ω ( β (cid:15) ) . But ω ( β (cid:15) ) = ω ( β ) + (cid:15) ( f · ∂β ) = Ψ( L ) + (cid:15) f · ∂β. It follows that Ψ( L (cid:15) ) ≤ Ψ( L ) + (cid:15) ( f · ∂β ) , (cid:15) ∈ [0 , t ] . Because Ψ( L t ) is concave and continuous in (cid:15) , while the right hand side dependslinearly on (cid:15) , the same bound holds globally in time:Ψ( L t ) ≤ Ψ( L ) + t ( f · ∂β ) , t ∈ [0 , . The theorem follows by taking t = 1 and using the fact that Ψ( L ) is positive. (cid:3) Here is a convenient reformulation of Theorem 5.5. Define the low-area Newtonpolytope to be the following convex hull: P lowL = Conv { ∂β ∈ H ( L ; Z ) : η L ( β ) = 1 } ⊂ H ( L ; R ) . Now consider its (open) dual polytope:( P lowL ) ∨ = (cid:110) α ∈ H ( L ; R ) : α · a > − ∀ a ∈ P lowL (cid:111) ⊂ H ( L ; R ) . EOMETRY OF FLUX 25
Theorem 5.5 may be rewritten as follows:(5.1) Sh (cid:63)L ( X ) ⊂ Ψ( L ) · ( P lowL ) ∨ , where Ψ( L ) is used as a scaling factor.5.5. Landau-Ginzburg potential.
Let L ⊂ X be a monotone Lagrangian sub-manifold and J a tame almost complex structure. The Landau-Ginzburg potential[2, 3, 29] is the following Laurent polynomial in d = dim H ( L ; R ) variables:(5.2) W L ( J ) = (cid:88) β ∈ H ( X,L ) : µ ( β )=2 M β ( J ) · x ∂β . Here M β ( J ) counts J -holomorphic disks in class β passing through a specifiedpoint on L . We use the notation x l = x l . . . x l d d where l i are the coordinates of anintegral vector l ∈ H ( L ; Z ) /T ors ∼ = Z d in a chosen basis. Since L is monotone, itsLG potential does not depend on J and on Hamiltonian isotopies of L .Consider the Newton polytope of W L , denoted by P L . Explicitly: P L = Conv { ∂β ∈ H ( L ; Z ) : µ ( β ) = 2 , M β ( J ) (cid:54) = 0 } ⊂ H ( L ; R ) . The convex hull is taken inside H ( L ; R ), where we consider each [ ∂β ] ∈ H ( L ; Z )as a point in H ( L ; R ) via the obvious map H ( L ; Z ) → H ( L ; R ). We define P L = H ( L ; Z ) if the considered set is empty. Corollary 5.6.
Let L ⊂ X be a monotone Lagrangian submanifold with mono-tonicity constant c , i.e. ω = cµ ∈ H ( X, L ) . Then Sh (cid:63)L ( X ) ⊂ c · P ∨ L . Proof.
Since L is monotone (and orientable), Maslov index 2 classes have lowestpositive symplectic area in H ( X, L ; Z ). Note that when µ ( β ) = 2, M β ( J )contributes to the m -operation of the Fukaya algebra as follows: m ,β (1) = T ω ( β ) q β M β ( J ) · [1 L ]where [1 L ] ∈ H ( L ; R ) is the unit.In view of this and Lemma 5.3, we conclude the following about the characteristicfunction η L . If µ ( β ) = 2 and M β ( J ) (cid:54) = 0, one has η L ( β ) = 1. By definition, theclasses ∂β for such β span P L . Furthermore, Ψ( L ) = ω ( β ) = 2 c . The statementfollows from Theorem 5.5. (cid:3) A modification in dimension four.
Suppose dim X = 4 and L ⊂ X is aLagrangian submanifold. Fixing a compatible almost complex structure J , defineΨ ( L ) as the lowest area among Maslov index 2 classes β such that M β ( J ) (cid:54) = 0.Let P low, L be the convex hull of the boundaries ∂β of such classes. We claim thatwith these adjustments, it still holds that Sh (cid:63)L ( X ) ⊂ Ψ( L )( P low, L ) ∨ . Indeed, for a generic path J t of almost complex structures, all J t -holomorphic diskshave Maslov index ≥
0, hence disks of Maslov index > Computations of shape
In this section we study the Ψ-function and shapes for
Gelfand-Cetlin fibrations ,which are generalisations of toric fibrations. Next, we compute shapes and star-shapes of Clifford and Chekanov tori in C n ; and study the wild behaviour of (non-star) shape in C P .6.1. Gelfand-Cetlin fibrations.
Only a small fraction of Fano varieties are toric,but one may broaden this class by allowing more singular Lagrangian torus fibra-tions which are reminiscent of the toric ones. We shall work with a class of fibrationswhich we call Gelfand-Cetlin fibrations. The name is derived from Gelfand-Cetlinfibrations on partial flag varieties [34, 13]; fibrations with similar properties on C P and C P × C P appeared in [47]. More generally, any toric Fano degeneration givesrise to a Gelfand-Cetlin fibration by the result of [30].We axiomatise the general properties of those constructions in the followingnotion. Definition 6.1.
A Gelfand-Cetlin fibration (GCF) on a symplectic 2 n -manifold X is given by a continuous map µ : X → P = Im( µ ) ⊂ R n , called the moment map,whose image is a compact convex lattice polytope P with the following property.Denote by P codim ≥ the union of all faces of P of dimension at most n − µ is a smooth map over P \ P codim ≥ , and is an actualtoric fibration over it. This means that over the interior of P , µ is a Lagrangiantorus fibration without singularities, and over the open parts of the facets of P itand has standard elliptic corank one singularities.Note that in most examples, µ is not smooth (only continuous) over P codim ≥ ,and the preimage of any point in P codim ≥ is a smooth isotropic submanifold of X .Observe that it is a reasonable conjecture that every Fano variety admits a toricFano degeneration, and if this holds true, it follows every Fano variety admits aGelfand-Cetlin fibration.Recall the statement of Theorem B: Theorem 6.2.
Let X be a Fano variety, X → P ⊂ R n a Gelfand-Cetlin fibration,and L ⊂ X its monotone Lagrangian fibre.Let P ∨ L ⊂ H ( L ; R ) be the interior of the dual of the Newton polytope associatedwith the Landau-Ginzburg potential of L (Section 5.5). Let c be the monotonicityconstant of X , and assume P is translated so that the origin corresponds to thefibre L . Then the following three subsets of H ( L ; R ) ∼ = R n coincide: c · P ∨ L = P = Sh (cid:63)L ( X ) . Remark . The statement is also true if dim X = 4, L is the monotone fibre ofan almost toric fibration over a disk, and P is the polytope of the limiting orbifold[45, Definition 2.14]. We leave the obvious modifications to the reader.Using the fact that µ is the standard toric fibration away from P codim ≥ , thenormal segment I from the point x L = µ ( L ) onto that facet determines a Maslovindex 2 class in H ( X, L ; Z ); see Figure 4. Using the identification H ( L ; R ) ∼ = R n coming from the embedding P ⊂ R n , we see that the boundary of β is given by theexterior normal to the chosen facet. Given a class β arising from this constructionusing some facet, we will denote that facet by P β . EOMETRY OF FLUX 27
Lemma 6.3.
For each facet of P , the corresponding class β ∈ H ( X, L ; Z ) satisfies η L ( β ) = 1 where η L is the characteristic function from Section 5. We need some preliminary lemmas first. For each x ∈ I , denote by L x ⊂ X the fibre over it, and by β x ∈ H ( X, L x ; Z ) the class obtained from a class β ∈ H ( X, L ; Z ) by continuity.Pick a facet P β of the moment polytope, and consider a star-isotopy { L t } t ∈ [0 , corresponding to the segment I ∼ = [0 , ⊂ P starting from the monotone torus L = L , and going towards P β in the direction normal to it. Here β stands for theclass in H ( X, L ; Z ) corresponding to the chosen facet. Lemma 6.4.
For any t ∈ I \ { } sufficiently close to the facet in question, thereexists a compatible almost complex structure J (cid:48) for which the algebraic count of J (cid:48) -holomorphic disks in class β t passing through a fixed point on L t equals one.Moreover, β t has minimal area among all J (cid:48) -holomorphic disks on ( X, L t ) . Thus, Ψ( L t ) = ω ( β t ) and η L t ( β t ) = 1 .Proof. Denote by o the endpoint I ∩ P β . There is a symplectomorphism betweenthe µ -preimage of a neighbourhood U of o in P and a neighbourhood of T n − × { } inside T ∗ T n − × C with the standard symplectic form; see Figure 4. Moreover, one can arrange thissymplectomorphism to map L t to a product torus of the form T n − × { a circle } .The class β t is represented in H ( U, L t ) and is identified in this model with thedisk class in the second C -factor. Moreover, for the standard Liouville structureon T ∗ T n − × C , the algebraic count of disks in ( U, L t ) equals one, by an explicitcomputation. We want to show that for some J (cid:48) on X , there are no holomorphicdisks in the same class that escape U . Figure 4.
Domains U and V used for stretching in the proof ofLemma 6.4 resp. Lemma 6.3. The dotted point corresponds to themonotone fibre L .The idea is to fix U and consider almost complex structures J (cid:48) which are suffi-ciently stretched around ∂U , with respect to some Liouville form on U . Note thatas t → o , ω ( β t ) →
0, in particular this area becomes eventually smaller than the ac-tion of any 1-periodic Reeb orbit in ∂U . Now suppose that for each almost complexstructure in the stretching sequence, there is a holomorphic disk in ( X, L t ) in class β t that escapes U . In the SFT limit, such disks converge to a holomorphic buildingwith a non-trivial holomorphic piece u (cid:48) in X \ U having punctures asymptotic toReeb chords in ∂U . In particular, u (cid:48) defines a 2-chain in ( X \ U, ∂U ).Let ω be the initial symplectic form on X . First, we have that ω ( u (cid:48) ) < ω ( β t ) . But the Lemma 6.5 below gives a bound in the other direction; this contradictionproves Lemma 6.4.As in the proof of Corollary 5.6, our computation of Maslov index 2 disks showsthat m ,β t (1) (cid:54) = 0 for the stretched almost complex structure. To argue thatΨ( L t ) = ω ( β t ) and η L t ( β t ) = 1, we must show that m ,β t (1) (cid:54) = 0 in some classicallyminimal model, while strictly speaking we have computed m ,β t (1) = [ L t ] as thefundamental chain in some (not necessarily minimal) chain model, depending onthe setting of the Fukaya algebra. In the case of the stabilising divisor approach, onecan take a perfect Morse function on L t which automatically gives the computationin a minimal model. In general, the application of homological perturbation lemmawill take m ,β t (1) to its cohomology class (the fundamental class). (cid:3) Remark . If one uses the stabilising divisor approach to Fukaya algebras, oneneeds to make sure that the above SFT stretchings are compatible with keepingthe stabilising divisor complex. The simplest way to ensure this is by imposingan extra condition in the definition of the Gelfand-Cetlin fibration, which is againsatisfied in examples and should be generally satisfied for the fibrations arising bythe general mechanism of [30]. The condition is that X has a smooth anticanonicaldivisor projecting, after a suitable Hamiltonian isotopy, to an arbitrarily smallneighbourhood of ∂ P , and which coincides with the preimages of the facets of P away from an arbitrarily small neighbourhood of the set of codimension ≥ P . This divisor should be stabilising for the monotone torus (and hence it willbe stabilising for any torus which is the preimage of an interior point of P , aftera Hamiltonian isotopy if necessary). This way, the neighbourhood in Figure 4(left) intersects the divisor in the standard way which makes it possible to stretchthe almost complex structure keeping it complex. The neighbourhood in Figure 4(right) does not intersect the divisor at all, which again makes consistent stretchingpossible. Lemma 6.5.
In the setting of the previous proof, it holds that ω ( u (cid:48) ) greater thanthe sum of the actions of its asymptotic Reeb orbits.Proof. Consider the space( X \ U, ω ∞ + ) ∼ = ( X \ U, ω ) ∪ ([ −∞ , × ∂U, d ( e t θ ))obtained by attaching the infinite negative Liouville collar to ( X \ U, ω ). Here θ is the Liouville contact form on ∂U , dθ = ω | ∂U . By the construction of neck-stretching, u (cid:48) is a curve which is holomorphic with respect to a cylindrical almostcomplex structure taming ω ∞ + . It implies that ω ∞ + ( u (cid:48) ) >
0. Finally, one has that ω ( u (cid:48) ) = ω ∞ + ( u (cid:48) ) + (cid:80) i (cid:82) γ i θ > (cid:80) i (cid:82) γ i θ where λ i are the asymptotic Reeb orbits of u (cid:48) and (cid:82) γ i θ are their actions. (cid:3) Proof of Lemma 6.3.
In the proof of Lemma 6.4 we have shown that Maslov index 2disks β t satisfy η L t ( β t ) = 1.Consider the domain V ⊂ X which is the µ -preimage of a convex open neigh-bourhood of the segment [ x L = 0 , t ] ⊂ I ⊂ P connecting x L to the point t thatis sufficiently close to the facet of P , so that Lemma 6.4 applies. See Figure 4.Then V is a Weinstein neighbourhood of L ⊂ X , which is moreover a Liouvilleneighbourhood. By Lemma 2.3, one finds an almost complex structure J (cid:48)(cid:48) on X ,sufficiently stretched around V , for which the fibres over the segment [ x L , t ] bound EOMETRY OF FLUX 29 no holomorphic disks of non-positive Maslov index. Note that this stretching hap-pens along a different domain than considered in the proof of Lemma 6.4.So Maslov index 2 disks undergo no bubbling as we move the Lagrangian torusfrom L to L t along the segment. Since η L t ( β t ) = 1, it follows that η L ( β ) = 1. (cid:3) Proof of Proposition C.
Let { L t } t ∈ [0 , be the star-isotopy corresponding to a seg-ment starting from the monotone torus L = L going towards a codimension onefacet of P in the normal direction. Let β ∈ H ( X, L ; Z ) be class of the corre-sponding the Maslov index 2 disk, and β t ∈ H ( X, L t ; Z ) the continuation of thisclass.Observe that showing that the values of Ψ on fibres are not below the conespecified in Proposition C is equivalent to showing thatΨ( t ) = Ψ( { L t } ) ≥ ω ( β t ) . Assume for a contradiction that Ψ( t ) < ω ( β t ). There are two posibilities.The first possibility is that for some t ∈ [0 , t ) atthat point is smaller than the derivative of ω ( β t ). In this case concavity of Ψ forcesΨ( s ) < t < s <
1, contradicting the positivity of Ψ.The second possibility is that the left derivative of Ψ( t ) at some point is greateror equal to the derivative of ω ( β t ). In this case we would get 0 < Ψ(0) < ω ( β ),contradicting the fact that L is orientable monotone Lagrangian and β has Maslovindex 2.To prove the desired equality, it remains to check it for some t , by concavity ofΨ( t ). For t close to 1, we have that Ψ( t ) = ω ( β t ) by Lemma 6.4. (cid:3) Proof of Theorem 6.2.
Clearly, P ⊂ Sh (cid:63)L ( X ). To show the converse, assume thatthere exists a star-isotopy with flux leaving P . This would mean Ψ( t ) ≤ t during this isotopy, by concavity of Ψ and the fact that it tends to zeroat the boundary of P . It follows that P = Sh (cid:63)L ( X ). Now by Lemma 6.3 andCorollary 5.6, 2 c · P ∨ L = P . (cid:3) Using a similar argument as above we can show a result stronger than Lemma 6.3,saying that for any Gelfand-Cetlin toric fibre L x , x ∈ P , Ψ is realized by diskswith boundary in L x corresponding to the facets that have the least area. Thisallows to get bounds on star-shapes relative to L x , see Corollary 6.7 below. Lemma 6.6.
Let { β i } be the set of classes in H ( X, L ; Z ) corresponding to thefacets of P , as described above. For each x ∈ P , consider the correspondingclasses { β xi } in H ( X, L x ; Z ) as above. Then for all β xi ∈ B = { β xi | ω ( β xi ) ≤ ω ( β xj ) , ∀ j } , we have that η L x ( β xi ) = 1 .Proof. For each x ∈ P , we can consider a Liouville neighbourhood V of L con-taining L x and an almost complex structure J as in Lemma 2.3, so that there isa correspondence between Maslov index 2 J -holomorphic disks with boundary on L and on L x . Since, by Proposition C, Ψ( L x ) = ω ( β xi ) for all β xi ∈ B , it followsthat η L x ( β xi ) = 1. (cid:3) Let B = β , . . . , β k be the corresponding classes in H ( X, L x ; Z ), with the samearea Ψ( L x ) (this means that the ray from x L in the direction of x intersects acodim = k facet). Corollary 6.7.
It holds that Sh (cid:63)L x ( X ) ⊂ (cid:92) i B β i , where B β i = { f ∈ H ( X ; R ) : Ψ + f · ∂β i > } . (cid:3) Proof.
It follows from Lemma 6.6, Theorem 5.5, and arguments similar to theabove. (cid:3)
Example . Figure 5 shows bounds on star-shapes of various toric fibres in C P coming from Corollary 6.7. For non-monotone fibres, we do not know whether theyare sharp. Figure 5.
Toric fibres in C P equidistantly close to 3, 2 and 1side(s) of the triangle (top row), and the corresponding bounds onthe star-shape (bottom row).6.2. Shapes in complex space.
Consider the product Lagrangian torus T r = S ( r ) × · · · × S ( r n ) ⊂ C n . Here r = ( r , . . . , r n ), r i > S ( r i ) = {| z | = r i } ⊂ C is the circle of radius r i . Theorem 6.8.
Identify H ( T r ; R ) ∼ = R n using the standard basis. For any n ≥ ,it holds that:(i) Sh T r ( C n ) = R n \ {− r + t ( − , . . . , −
1) : t ≥ } ;(ii) Sh (cid:63)T r ( C n ) = (cid:26) ( x , . . . , x n ) − r : x i ∈ R and x i > if r i = min j =1 ,...,n r j (cid:27) . Partial estimates on these shapes have been obtained earlier in [24, Theorem 1.15,Corollary 1.17, Corollary 1.18].
Example . When r = ( r, . . . , r ), one has Sh (cid:63)T r ( C n ) = ( R > − r ) n . This is the interior of the moment polytope ( R ≥ ) n of the standard toric fibrationon C n , translated so that T r becomes the preimage of the origin. In particular, allpossible star-fluxes can be achieved by the obvious isotopies among toric fibres. Proof of Theorem 6.8 (i).
Consider the standard toric fibration C n → ( R ≥ ) n ; thepoint r is the image of T r . Since, Sh T r ( C n ) are all the same, up to translation, itis enough to prove the result for a specific r . For convenience, we take a monotonefibre corresponding to r = ( r, . . . , r ). EOMETRY OF FLUX 31
Figure 6.
The “segment” isotopy { ˜ L t } and the “loop” isotopy { L t } in an almost toric fibration on C .First, let us show that Sh T r ( C n ) ⊂ R n \ {− r + t ( − , . . . , −
1) : t ≥ } . Suppose that there is a Lagrangian isotopy { L s } starting from L = T r and withflux − r + t ( − , . . . , − ∈ R n . Then the Lagrangian torus L satisfies[ ω ] = − tµ/ ∈ H ( C n , L ; R ) . This contradicts the established Audin conjecture (whose proof for C n read-ily adopts from [15]) asserting that L bounds a Maslov index 2 disk of positivesymplectic area.Now let us restrict to the case n = 2. Consider the star-isotopy˜ L t = T r + t v , t ∈ [0 , , v ∈ { ( x , x ) − r : x i > } ⊂ R . We have that v + r ∈ ( R > ) , see Figure 6, and tautologically v = Flux( { ˜ L t } ).Consider an almost toric fibration (ATF) obtained by applying a sufficientlysmall nodal trade [42, Section 6] to the standard toric fibration on C . For addi-tional background on nodal trade, see e.g. [43, Section 2], [2, 3]. One can ensurethat nodal trade does not modify the fibres ˜ L t = T r + t v , for t ∈ [0 , { L t } be a Lagrangian isotopy from T r to itself given by a loop in the baseof our ATF starting at r and going once around the nodal singularity, say in thecounter-clockwise direction, as in Figure 6. More generally, for each k ∈ Z considerthe Lagrangian isotopy { L kt } given by a loop in the base of our ATF with wrappingnumber k around the node.The isotopy { L t } induces a monodromy M T : H ( T r ; Z ) → H ( T r ; Z )whose matrix is the transpose of the monodromy M of the affine structure on thebase around the considered loop. See [42, Section 4] and [43, Section 2.3]. Usingthe standard identifications H ( T r ; R ) ∼ = T r ( R > ) ∼ = R ∼ = ( R ) ∗ ∼ = T ∗ r ( R > ) ∼ = H ( T r ; R ) , the monodromy matrix M is explicitly given by M = (cid:18) − (cid:19) . Let α be a class in H ( C , T r ; Z ) corresponding to the vanishing cycle associatedwith the nodal fibre, i.e., α can be represented by a disk projecting onto a segmentconnecting r to the node. Note that ∂α corresponds to the invariant cycle (1 , M , up to sign. So when we follow the isotopy { L t } , ∂α t closes up to a nunnho-mologous cycle in C , hence has zero area. Now, consider the cycle γ ∈ H ( T r ; Z )corresponding to (1 , { L t } , let this cycle sweep a cylinder with ends on cycles γ and γ − ∂α , the latter one in class corresponding to (0 , − C , we can just add a representative ofthe class α . Then the area of the cylinder, which is Flux( { L t } ) · γ , equals − ω ( α ).Because we chose r so that L r is monotone, ω ( α ) = 0. Indeed, α is Maslov 0 andcan be represented by a Lagrangian disk. Hence, Flux( { L t } ) = 0.Consider the concatenation { ˜ L t } ∗ { L kt } . This is a Lagrangian isotopy which firstfollows the “loop” { L kt } and then “segment” { ˜ L t } . Since { L kt } has zero flux, onehas that Flux( { ˜ L t } ∗ { L kt } ) = M k v . Recall that the vector v may be freely chosen from the domain Q = { ( x , x ) − r : x i > } . Again, we take r = ( r, . . . , r ), so it is invariant under M . We have that (cid:91) k ∈ Z M k Q = R \ {− r + t ( − , −
1) : t ≥ } ⊂ Sh T r ( C ) . Indeed, this follows by noting that M k = (cid:18) − k k − k k + 1 (cid:19) , and that the columns [1 ∓ k, ∓ k ] → | k | ( − , −
1) as k → ±∞ . This completes theproof of Theorem 6.8 (i) for n = 2.For a general n ≥
2, consider the splitting C n = C × C n − , and the SYZ fibrationwhich is the product of the previously considered ATF on the C -factor with thestandard toric fibration on the C n − -factor.There is a loop in the base of this SYZ fibration starting at r whose monodromyis the block matrix (cid:18) M
00 Id n − (cid:19) where M appears above. Arguing as before, we conclude that(6.1) R n \ {− r + ( − t, − t, x , . . . , x n ) : t ≥ , x k ≤ for k ≥ } ⊂ Sh T r ( C n ) . The argument can be applied to any pair of coordinates instead of the first twoones. The union of sets as in (6.1) arising this way covers the whole claimed shape: (cid:83) i,j ( R n \ {− r + ( x , . . . , x n ) : x i = x j ≤ , x k ≤ for k (cid:54) = i, j } )= R n \ (cid:84) i,j {− r + ( x , . . . , x n ) : x i = x j ≤ , x k ≤ for k (cid:54) = i, j } = R n \ {− r + t ( − , . . . , − ∈ R n : t ≥ } . The result follows. (cid:3)
Proof of Theorem 6.8 (ii).
The inclusion of the star-shape into the desired set(6.2) Sh (cid:63)T r ( C n ) ⊂ (cid:26) ( x , . . . , x n ) − r : x i ∈ R and x i > r i = min j =1 ,...,n r j (cid:27) follows from Theorem 5.5.To prove the reverse inclusion, we again start with n = 2. The case r = r isclear, since the shape in question { ( x , x ) − r : x i > } is realised by isotopies inthe standard toric fibration. Assuming r < r , the set which we must prove tocoincide with star-shape is { ( x , x ) − r : x > } . EOMETRY OF FLUX 33
Figure 7.
Left: the standard toric fibration on C . Middle andright: two diagrams representing the same almost toric fibrationwith one nodal fibre, with two different ways of making a cut.To compare, isotopies within the standard toric fibration achieve flux of the form( x , x ) − r where both x , x >
0. Figure 7 shows star-isotopies that achieve theremaining flux, i.e. of the form ( x , x ) − r where x > x ≤
0. This completesthe proof for n = 2.Unlike the proof of Theorem 6.8 (i), in higher dimensions it will not be enoughto consider SYZ fibrations which almost look like the product of the 4-dimensionalATF with the standard toric fibration; we must consider a larger class of SYZfibrations that exist on C n .Let us discuss the case n = 3; the details in the general case are analogous. Themonotone case r = r = r is again clear. Now, the case r = r < r is preciselythe one when considering the SYZ fibration from Theorem 6.8 (i) is sufficient.Looking at the product of the 4-dimensional ATF on C with the standard toricfibration on C , we obtain any flux of the form ( x , x , x ) − r , where x , x > x ≤
0, see Figure 8. The remaining flux is realised by isotopies in the standardtoric fibration, and we conclude that Sh (cid:63)T r ( C ) = { ( x , x , x ) − r : x , x > } , as desired. Figure 8.
Left: the standard toric fibration on C . Middle andright: SYZ fibration which is the product of the 4-dimensional ATFwith the standard toric fibration on C . Middle and right show thesame fibration with two different ways of making a cut. We move to the most complicated case r < r ≤ r . We must show that Sh (cid:63)T r ( C ) = { ( x , x , x ) − r : x > } , and the constructions we discussed so far miss out the subset where x , x ≤ c >
0, whose fibres are parametrised by ( r , µ , µ ) and are given by:(6.3) T c r ,µ ,µ = { ( z , z , z ) : z i ∈ C , | z z z − c | = r ,π ( | z | − | z | ) = µ , π ( | z | − | z | ) = µ } ⊂ C , see Figure 9. We point out that ( r , µ , µ ) are not locally affine coordinates onthe base, although ( µ , µ ) are a part of locally affine coordinates.A non-singular torus T c r ,µ ,µ can be understood as follows. Consider the map f : C → C , ( z , z , z ) (cid:55)→ z z z whose fibres are invariant under the T -action( e iθ , e iθ ) · ( z , z , z ) = ( e i ( θ + θ ) z , e − iθ z , e − iθ z ) . Its moment map is ( µ , µ ) = ( π ( | z | − | z | ) , π ( | z | − | z | )). Then T c r ,µ ,µ isthe parallel transport of an orbit of this T -action with respect to the symplecticfibration f , over the radius- r circle centred at c .Because f has a singular fibre over 0, some Lagrangians T cc,µ ,µ will be singular.This happens precisely when µ = 0 , µ < , or µ < , µ = 0 , or µ = µ > , or µ = µ = 0 (the most degenerate case) . (6.4)All other fibres are smooth Lagrangian tori, for r >
0. Observe that our SYZfibration extends over r = 0, where it becomes a singular Lagrangian T -fibrationon z z z = c . Also recall that this construction depends on the parameter c , andthe limiting case c = 0 is actually the standard toric fibration on C . Figure 9.
Left: the standard toric fibration on C . Middle andright: the SYZ fibration described by T c r ,µ ,µ , for some fixed c . Thered curves show the discriminant locus. The most singular fibre T cc, , is marked by the node. The node moves with c in the directionof the dashed line. Middle and right show two ways of making cuts,which are the shaded surfaces.The complement of ( R ≥ ) to the discriminant locus of the constructed fibra-tion carries a natural affine structure. Since this affine structure has monodromy EOMETRY OF FLUX 35 around the discriminant locus, it is not globally isomorphic to one induced fromthe standard affine structure on ( R ≥ ) . However, it is isomorphic to the standardone in the complement of a codimension-one set called a cut . There are variousways of making a cut, and two of them are shown in Figure 9.Finally, let us discuss the effect of changing the parameter c . It correspondsto sliding the dashed segment in Figure 9; this is a higher-dimensional version ofnodal slide. Intuitively, by taking c to be sufficiently large (i.e. sliding the nodesufficiently far towards infinity), one can see the existence of a star-isotopy from T r to with any flux of the form ( x , x , x ) − r ∈ R such that x > T r , r = ( r , r , r ), r < r ≤ r , and observe that T r = T r ,µ ,µ , r = (cid:16) r r r π (cid:17) / , µ = r − r µ = r − r . Note that µ ≤ µ <
0. Denote by β ∈ H ( C , T r ) the class of holomorphic diskswith area r . We will build a star-isotopy of the form T c ( t ) r ( t ) ,µ ( t ) ,µ ( t ) with flux t (( x , x , x ) − r ), x >
0. Note that in order for T c ( t ) r ( t ) ,µ ( t ) ,µ ( t ) to be a star-isotopy,the relative class β ( t ) ∈ H (cid:16) C , T c ( t ) r ( t ) ,µ ( t ) ,µ ( t ) (cid:17) , which is the continuous extension of β , must satisfy ω ( β ( t )) = r + t ( x − r ). Wearrange:(I) µ ( t ) = r − r + t [ x − x − ( r − r )];(II) µ ( t ) = r − r + t [ x − x − ( r − r )];(III) c ( t ) = ψ ( t ) c , where c is a large real number and ψ ( t ) is a non-decreasingsmooth cutoff function: it satisfies ψ (0) = 0 and identically equals 1 for t ≥ (cid:15) , where (cid:15) is sufficiently small;(IV) r ( t ) is chosen so that ω ( β ( t )) = r + t ( x − r ).We first set (I), (II) and choose (cid:15) small enough to ensure µ j ( t ) < t ∈ [0 , (cid:15) ] , j = 1 ,
2. We need now to set the endpoint of our isotopy by choosing c andthe corresponding r (1) < c . We can make the area of the corresponding β class ina torus of the form T c ˜ r ,µ (1) ,µ (1) as big as we want, in particular bigger than x , bytaking c sufficiently large and then ˜ r < c sufficiently close to c . Taking such c ,we may take r (1) < c , so that for T c r (1) ,µ (1) ,µ (1) , we have ω ( β (1)) = x .Now we choose our cutoff function ψ ( t ), setting item (III) of our desired list.Since x >
0, the expression r + t ( x − r ) is non-negative, so we can find r ( t ) toensure we have (IV).Our setup guarantees that T c ( t ) r ( t ) ,µ ( t ) ,µ ( t ) is a smooth torus for all t ∈ [0 , T c ( t ) r ( t ) ,µ ( t ) ,µ ( t ) could be non-singular only at the moment t when c ( t ) = r ( t ), but our choice of ψ ( t ) ensures that t < (cid:15) . This implies that µ ( t ) < µ ( t ) < T c ( t ) r ( t ) ,µ ( t ) ,µ ( t ) is smooth, recall (6.4). This finishes theproof of Theorem 6.8 ( ii ) for n = 3. Conditions (I),(II) and (IV) ensures we havea star-isotopy.The situation in higher dimensions is very similar to the n = 3 case. Assumethat r = · · · = r k +1 < r k +2 ≤ · · · ≤ r n . The monotone case n = k + 1 is trivial, sowe assume n > k + 1. Let us split C n as C n = C k × C n − k , take the standard toric fibration in the C k -factor and the following SYZ fibration in the C n − k factor. Its fibres T c r ,µ ,...,µ n − k − are defined analogously to the previous construction, using the auxiliary symplecticfibration f ( z , . . . , z n − k ) = z . . . z n − k , the corresponding T n − k − -action, and thesimilar parallel transport. One again uses c as a parameter of the construction.Given ( x , . . . , x n ) ∈ R n with x , . . . , x k +1 >
0, we can construct a star-isotopyfrom T r = T ( r ,...,r k ) × T ( r k +1 ,...,r n ) to a torus of the form T ( x ,...,x k ) × T c r ,µ ,...,µ n − k − , such that the flux of this isotopy equals ( x , . . . , x n ) − r . Indeed, using the conditionthat x , . . . , x k >
0, first consider a star-isotopy from T ( r ,...,r k ) to T ( x ,...,x k ) in the C k -factor through toric fibres. Using x k +1 >
0, there is now a star-isotopy from T ( r k +1 ,...,r n ) to T c r ,µ ,...,µ n − k − in the C n − k -factor, analogously to what we did in thecase of C . (cid:3) Theorem 6.9.
For n ≥ , let Θ n − ( r ) ⊂ C n be the Chekanov torus introduced in [10] , bounding a Maslov index 2 disk of symplectic area r . The following holds.(i) Sh Θ n − ( r ) ( C n ) = R n \ {− r + t ( − , . . . , −
1) : t ≥ } where r = ( r, . . . , r ) ;(ii) Sh (cid:63) Θ n − ( r ) ( C n ) = { ( x , . . . , x n ) − ( r, . . . , r ) : x > , x i ∈ R } , the half-spacebounded by the hyperplane x = − r .Proof. The tori Θ n − ( r ) are Hamiltonian isotopic to the tori of the form T c r , ,..., described above, provided that r < c . It is shown in [3, Section 3.3] that thereis a unique family of holomorphic disks with boundary on T c r , ,..., , and each diskprojects via f ( z , . . . , z n ) = z · · · z n isomorphically onto the disk of radius r centredat c . (The values of c, r are such that these disks have area r .) Note that T c r , ,..., is Lagrangian isotopic with zero flux to the product torus T r / /π ) n , ,..., = S ( r ) × · · · × S ( r ) , Now (i) follows from Theorem 6.8 (i). For (ii), observe that the proof of Theorem6.8 (ii) achieves star-flux of any desired form from the statement. The reverseinclusion follows from Theorem 5.5. (cid:3)
We note that Theorem 6.9 improves the computation in [24, Theorem 1.19].6.3.
Wild shapes of toric manifolds.
In contrast to star-shapes, the (non-star)shapes of compact toric manifolds behave wildly. The idea is that in toric manifolds,there exist loops of embedded Lagrangian tori with various monodromies, and thesemonodromies together generate big subgroups of SL ( n, Z ). We shall illustrate thephenomenon by looking at C P .What is perhaps more surprising, tori T in compact toric varieties usually possess unbounded product neighbourhoods T × Q . Figure 1 from the introduction showsan example for C P , where Q ⊂ R is the unbounded open set shown on the left.Such products cannot be convex or Liouville with respect to the zero-section, byTheorem 2.2. Corollary 6.10.
The symplectic neighbourhood T × Q does not admit a Liouvillestructure making T × { pt } exact, where pt is marked in Figure 1 (it is sent to themonotone fibre under the above embedding). (cid:3) EOMETRY OF FLUX 37
We continue to focus on C P . The monotone Clifford torus T is the fibre cor-responding to the barycentre of the standard moment triangle P of C P . Let usapply nodal trades to each of the three vertices of P . Let P be the interior of P .For the cuts shown in Figure 1, the monodromies around the nodes are respectivelygiven by: M = (cid:18) −
11 0 (cid:19) , M = (cid:18) − − (cid:19) , M = (cid:18) − − (cid:19) . Consider the subgroup G C P < SL (2 , Z ), G C P = (cid:104) M , M , M (cid:105) , generated bythe M i . Revisiting the proof of Theorem 6.8 ( i ) for n = 2, one concludes that Sh T ( C P ) contains the orbit of P under the total monodromy group action: G C P · P = { g ( x ) : g ∈ G C P , x ∈ P } ⊂ R . First, let us check that this orbit is unbounded. If Q ⊂ R is the domain shownin Figure 1, by consecutively applying the monodromies one sees that Q ⊂ P ∪ ∞ (cid:91) k =1 M k · · · M · P ⊂ G C P , x ∈ P , where the subscripts are taken modulo 3: M i = M j if and only if i ≡ j mod 3.Next comes a question we were not able to answer. Question 6.11.
Is the orbit G C P · P dense in R ? Although we do not have an answer, it will be useful to pursue this question. Tothis end, one computes M M M = (cid:18) −
90 1 (cid:19) . Conjugating by P = (cid:18) − (cid:19) we get: P M P − = (cid:18) (cid:19) , P M P − = (cid:18) − − (cid:19) , P M M M P − = (cid:18) (cid:19) . So P G C P P − is generated by the three matrices above. In particular, G C P con-tains a subgroup isomorphic to G where:(6.5) G k = (cid:28) t = (cid:18) (cid:19) , h k = (cid:18) k (cid:19)(cid:29) . Let G be a locally compact Lie group with the right-invariant Haar measure µ .A discrete subgroup Γ of G is called a lattice [25, Section 1.5 b] if the inducedmeasure on G/ Γ has finite volume. The Haar measure on
P SL (2 , R ) is inducedfrom the hyperbolic metric on H ∼ = P SL (2 , R ) /P SO (2 , R ), so Γ < P SL (2 , R ) is alattice if and only if the induced action of Γ on H produces the quotient H / Γ offinite area.Let us view S as the projectivisation of the plane: S = proj( R ) ∼ = P SL (2 , R ) /U ,where U < P SL (2 , R ) is the subgroup of upper-triangular matrices. Howe-Moore ergodicity theorem implies that the action of any lattice Γ < P SL (2 , R ) on proj( R )is ergodic; see [25, Theorem 3.3.1, Corollary 3.3.2, Proposition 4.1.1].Now suppose D ⊂ R is any open subset containing the origin, and Γ
The subgroup G k < P SL (2 , R ) is a lattice if and only if < | k | ≤ .Proof. For k = 0, one has H /G = { ( x, y ) ∈ H : − / ≤ x ≤ / } / ( − / , y ) ∼ (1 / , y )which is of infinite volume. We claim that for k (cid:54) = 0, the fundamental domain ofthe action of G k on H is: D G k = { ( x, y ) ∈ H : − / ≤ x ≤ / , (cid:107) ( x ± /k, y ) (cid:107) ≥ / | k |} . We are using the upper half-plane model for the hyperbolic plane. Indeed, since t (6.5) acts by integer translation in the coordinate x ∈ H , we may assume − / ≤ x ≤ /
2. Next, the y -coordinate of h nk · ( x, y ) equals y ( nkx + 1) + ( nky ) . So ( x, y ) is the representative of its (cid:104) h k (cid:105) -orbit with the largest value of y if and onlyif ( nkx +1) +( nky ) ≥ n , equivalently, if and only if (cid:107) ( x ± /k, y ) (cid:107) ≥ / | k | . It explains that D G k is a fundamental domain. Finally, D G k ⊂ H has finite volumeif and only if 0 < | k | ≤ (cid:3) As we have seen above, G C P is generated by G and (cid:0) − − (cid:1) . We do not knowwhether G C P is a lattice, so we could not answer Question 6.11. However, we cannow answer the analogous question for some other symplectic 4-manifolds. Corollary 6.13.
Let X = Bl k C P be the blowup of C P at k ≥ points, withany (not necessarily monotone) symplectic form. Let L a fibre of an almost toricfibration on X whose base is diffeomorphic to a disk (e.g. a fibre of a toric fibration).Then Sh L ( X ) is dense in R .Remark . A symplectic manifold admitting an almost toric fibration with basehomeomorphic to a disk is diffeomorphic to Bl k C P or C P × C P by [32]. Proof.
Consider an almost toric fibration from the statement, and let P be its base.Performing small nodal trades when necessary, we may assume that the preimageof the boundary of P is a smooth elliptic curve representing the anticanonical class − K X [42, Proposition 8.2]. Consider the loop in P which goes once around theboundary ∂ P sufficiently closely to it, and encloses all singularities of the almosttoric fibration. The affine monodromy around this loop is conjugate to (cid:18) k (cid:19) , because it has an eigenvector given by the fibre cycle of the boundary elliptic curve.Furthermore, it can be shown that k = K X ; see e.g [39]. Following the proof ofTheorem 6.8 (i), one argues that G K X · P ⊂ Sh L ( X ) . If 0 (cid:54) = | K X | ≤
4, the result follows from Proposition 6.12 and the Howe-Mooretheorem, in particular it hods for Bl C P . EOMETRY OF FLUX 39
Figure 10.
Almost toric blowup of the A k − ATF on Bl k − C P (left) to the A k ATF on Bl k C P (middle and right). The comple-ment of a neighbourhood of a cut in the A k ATF (the triangularcut in the middle diagram) embeds into the A k − ATF on Bl k C P (left diagram).In general, denote by G X < SL (2 , Z ) the group generated by all monodromies ofan almost toric fibration on X as above. We claim that G Bl m C P is a subgroup of G Bl k C P for m ≤ k . This implies that the result holds for Bl k C P , k ≥
5. Indeed,starting with an ATF A k on ( Bl k C P , ω ), one deduces from [32, Theorem 6.1] thatthere is a different ATF A k on ( Bl k C P , ω (cid:48) ) satisfying: A k is obtained from anATF A k − on Bl k − C P via almost toric blowup ([32, Section 4.2], see also [48,Example 4.16], [42, Section 5.4], and Figure 10); and A k is obtained from the ATF A k by deforming ω to ω (cid:48) and applying nodal slides. In particular, they have thesame groups of monodromies. By disallowing to travel around the distinguishednodal fibre coming from the almost toric blowup, one gets an embedding of themonodromy group of A k − into one of A k , which is the same as for the initial ATF A k . (cid:3) Space of Lagrangian tori in C P Given a symplectic manifold X , the space of all (not necessarily monotone)Lagrangian embeddings of a torus into X is usually non-Hausdorff. Despite theindications that this space should be in some way related to the rigid-analyticmirror of X (if it exists), we do not seem to have a rigorous understanding ofthis connection so far. More basically, there is a lack of examples in the literaturecomputing these spaces. We shall study this question for C P . Recall that allsymplectomorphisms of C P are Hamiltonian.In [43, 44], it is shown that monotone tori in C P are associated with Markovtriples. We recall that a Markov triple ( a, b, c ) is a triple of positive integers satis-fying the Markov equation:(7.1) a + b + c = 3 abc. All Markov triples are assumed to be unordered. They form the vertices of theinfinite Markov tree with root (1 , , a, b, c ) → ( a, b, ab − c ) . Besides the univalent vertex (1 , ,
1) and the bivalent vertex (1 , , (1 , , , , , , , ,
29) (1 , , , , , , , , , , The Markov tree.There is an almost toric fibration (ATF) on C P corresponding to each Markovtriple ( a, b, c ), constructed in [43, 44]. Its base can be represented by a triangle(with cuts) whose sides have affine lengths ( a , b , c ). Imposing restrictions on thecuts, one get that the base diagram representing an ATF containing the monotonefibre T ( a , b , c ), uniquely determine the above mentioned triangle, up to SL (2 , Z ),c.f. [43, 44]. Slightly abusing terminology, we call it the moment triangle associatedto T ( a , b , c ). We shall call an ( a, b, c )-ATF any ATF containing T ( a , b , c ) asa monotone fibre.From now, we maintain the following agreement: the nodes of these fibrationsare assumed to be slided arbitrarily close to the vertices of the moment triangle .So when we speak of a fibre of the ( a, b, c )-ATF, we always mean the preimage ofa point in the base triangle with respect to an ATF whose nodes are closer to thevertices than the point in question. More formally, the fibres of the ( a, b, c )-ATFare the regular fibres of the corresponding fibration on the weighted projectivespace, pulled back to C P via smoothing (which defines them up to Hamiltonianisotopy).By [44], two monotone tori corresponding to different Markov triples are notHamiltonian isotopic to each other. Our aim is to study all (not necessarily mono-tone) fibres of all the ATFs together, modulo symplectomorphisms of C P . Theyform a non-Hausdorff topological space: H = { T ⊂ C P a Lag. torus fibre of the ( a, b, c )-ATF for some Markov triple } / ∼ where T ∼ T is there exists a symplectomorphism of C P taking T to T . It is aplausible but hard conjecture that any Lagrangian torus in C P is actually isotopicto some ( a, b, c )-ATF fibre. If this is true, then H is the space of all Lagrangiantori in C P up to symplectomorphism.We shall study H with the help of the numerical invariant arising from theremark made in Section 5.6:(7.3) Ξ ( L ) = (cid:88) β : µ ( β )=2 ,ω ( β )=Ψ ( L ) M β ( J ) ∈ Z , where M β ( J ) is the number of J -holomorphic disks in class β of Maslov index 2,passing through a fixed point on L . Observe that we are only counting disks oflowest area Ψ ( L ). EOMETRY OF FLUX 41
We start by analysing the space of Lagrangian fibres of the standard toric fi-bration on C P up to symplectomorphism. In the above terminology, this is an(1 , , S of symplectomorphisms permuting the homogeneous coordinates on C P . Thisleaves us with a “one-sixth” slice of the initial moment triangle. That slice is aclosed triangle with one edge removed, see Figure 11. We will now show that toricfibres corresponding to different points in this slice are not related by symplecto-morphisms of Symp( C P ). Figure 11.
The moment polytope of C P . Fibres over the shadedregion represent fibres of the toric fibration modulo action ofSymp( C P ). Proposition 7.1.
Let T ( x ,y ) and T ( x ,y ) be toric fibres over distinct points ( x , y ) (cid:54) =( x , y ) belonging to the shaded region of Figure 11. Then there is no symplecto-morphism of C P taking T ( x ,y ) to T ( x ,y ) .Proof. We normalise the symplectic form so that the area of the complex lineequals 1. Assume that there is a symplectomorphism φ ∈ Symp( C P ) such that φ ( T ( x ,y ) ) = T ( x ,y ) . We aim to show that ( x , y ) = ( x , y ). The condition that( x i , y i ) belongs to the shaded region means:(7.4) 0 < x i ≤ / , x i ≤ y i ≤ − x i . Since there is only one monotone fibre T (1 / , / , we may assume x i < / i = 1 ,
2. The torus T ( x i ,y i ) bounds Maslov index 2 holomorphic disks in threerelative classes α i , β i , H − α i − β i and areas x i , y i and 1 − x i − y i , respectively. Bythe 4-dimensional modifications of our results mentioned in Section 5.6, the countsof minimal area holomorphic disks are invariant under symplectomorphisms.If x = y , there are two minimal area holomorphic disks in classes α , β , so φ must send them to classes α , β . In particular, x = y = x = y .Now consider the case x i < y i . Now α i is the unique class supporting theminimal area holomorphic disk. So we must have φ ∗ ( α ) = α and x = x .Assume, without loss of generality, that y ≥ y . We have:0 < x = x < / x = x < y ≤ y ≤ − x − x . (7.5)Since φ ∗ ( β ) has Maslov index 2, and { α , β , H } generate H ( C P , T ( x ,y ) ; Z ),one may write φ ∗ ( β ) = β + k ( H − α ) + l ( α − β ) for some k, l ∈ Z . Since ∂α · ∂β = 1 and φ preserves the intersection form on thetori up to sign, ∂φ ∗ ( β ) · ∂φ ∗ ( α ) = ± ∂ ( β + k ( H − α ) + l ( α − β )) · ∂α = 1 − l. Let us first analyse the case ∂φ ∗ ( β ) · ∂φ ∗ ( α ) = −
1, which means l = 2. Then φ ∗ ( β ) = β + k ( H − α ) + 2( α − β ), and by computing symplectic area:(7.6) y = 2 x − y + k (1 − x ) . We get from (7.5), (7.6) that2 x < y + y = 2 x + k (1 − x ) ≤ − x . Since 1 − x >
0, we obtain k ≥ k ≤ k = 1 and y = y = (1 − x ) / ∂φ ∗ ( β ) · ∂φ ∗ ( α ) = 1, which means l = 0. In thiscase, φ ∗ ( β ) = β + k ( H − α ), and by computing symplectic area:(7.7) y = y + k (1 − x ) . Since 1 − x > y ≥ y , we have k ≥
0. From (7.5), (7.7), we get x + k (1 − x ) < y + k (1 − x ) = y ≤ − x . We conclude that k < /
2. Hence k = 0 and y = y . (cid:3) Below is a useful corollary of the previous proof.
Corollary 7.2.
Let T ⊂ C P be a Lagrangian torus for which there is a uniqueclass α ∈ H ( C P , T ) realising Ψ ( T ) , i.e. satisfying µ ( α ) = 2 , ω ( α ) = Ψ ( T ) ,such that that the count of holomorphic disks in class α through a fixed point of L is non-zero.Assume that there is another class β ∈ H ( C P , T ) with ∂β · ∂α = ± and suchthat the areas x = ω ( α ) , y = ω ( β ) satisfy inequalities (7.4) . Then the only possibletoric fibre in the shaded region of Figure 11 that could be symplectomorphic to T is T ( x,y ) . (cid:3) The following notation will help us describe the space H . Consider the ( a, b, c )-ATF, and mark the vertices of the moment triangle by the corresponding Markovnumbers. Consider three line segments connecting a vertex to the opposite edgevia the barycentre. The barycentre divides each segment (say, corresponding tothe vertex a ) into two pieces; for a (cid:54) = 1, we call them β a and γ a where β a is thesegment containing a , see Figure 12. We name β and γ only the segments showingin Figure 11. We call the fibres over these segments fibres of type β a , γ a , etc. . Proposition 7.3. If L is a β a -type fibre of the ( a, b, c ) -ATF, then Ξ ( L ) = 2 a . If L is a γ a -type fibre, then Ξ ( L ) = 1 .Proof. By the proof of Theorem 2.2, there is an almost complex structure forwhich L has the same enumerative geometry as the monotone torus T ( a , b , c ).The potential of the latter is computed by the wall-crossing formula [35] and itsNewton polytope is dual to the moment triangle of the ATF [44].There is a unique term in the potential that becomes a lowest area holomorphicdisk on a γ a -type fibre; it corresponds to the vertex of the Newton polytope, andthe count of disks in that class equals 1. Next, those terms of the potential thatbecome lowest area disks on a β a -type fibre correspond to an edge of the Newton EOMETRY OF FLUX 43
Figure 12.
Special fibres of type β and γ type in an ( a, b, c )-ATF. Figure 13.
Left and middle: (1 , , , , a + 1 lattice points and the coefficients in front of thecorresponding monomials are the binomial coefficients; their sum is 2 a . (cid:3) Proposition 7.4.
Any γ a -type fibre is equivalent to a fibre of the (1 , , -ATF ora γ -type fibre of the (1 , , -ATF.Proof. If L is a γ a -type fibre of the ( a, b, c )-ATF, then it is also a fibre of one of thetwo mutated ATFs corresponding to ( a, ac − b, c ) or ( a, b, ab − c ), see (7.2) and [44,Proposition 2.4]. Now note that L can be also seen as a fibre of an (3 bc − a, b, c )-typeATF if (breaking the standing convention) we allow to slide the node associated with γ a pass the monotone fibre, but not pass L . This way, we are able to mutatethe ( a, b, c )-type ATF all the way to an (1 , , L as a fibre. The result follows, noting that fibres ofan (1 , , T (1 , ,
1) fibre) are precisely the fibres of the (1 , , γ -fibres. (cid:3) Denote by H ( a, b, c ) the space of all fibres of the ( a, b, c )-ATF modulo symplec-tomorphisms of C P . Let ∆ be a closed triangle minus an edge, which is affinelyisomorphic to one of the six triangles in Figure 11, which describes H (1 , , a, b, c )-ATF in order to describe H ( a, b, c ). In order to keep track of special fibres in an embedding of ∆, morespecifically γ a ’s and β a ’s type fibres, we develop the following notation. Let x , y , w , w , w some be half-open segments in ∆ starting at the vertex and ending atthe missing edge. Among them, x and y must be the edges of ∆, but w , w , w can be arbitrary and go through the interior. We then denote by ∆ w ,w ,w x,y , thetriangle ∆ labeled by x , y , w , w , w . The set of w i is also allowed to be empty,in which case they do not appear in the notation. We can write H (1 , , ∼ = ∆ β ,γ , since we can embed ∆ in the toric (1 , , β and γ type fibres.Let us see how H ( a, b, c ) changes as we mutate the Markov triple, beginning withan analysis of how H (1 , ,
2) differs from H (1 , , , , γ or β , according toour notation.By Proposition 7.3 the β -type fibres have invariant Ξ = 2 = 4. So these fibresare not equivalent to any toric fibre. Next, by Corollary 7.2, a γ -type fibre canonly be equivalent to a toric fibre if it is of β -type, but their Ξ -invariants equal 1and 2 respectively. Therefore, the γ -type fibres are also not equivalent to a toricfibre.The three shaded triangles in the middle diagram of Figure 13 are embeddingsof ∆ accordingly labeled as ∆ β ,γ , ∆ β ,β , ∆ γ ,γ .Let’s introduce further notation. Given a finite set of labeled triangles { ∆ w i ,w i ,w i x i ,y i ; i =1 , . . . , k } , we define∆ w ,w ,w x ,y ∨ · · · ∨ ∆ w k ,w k ,w k x k ,y k = (cid:113) ki =1 ∆ w i ,w i ,w i x i ,y i / ∼ where we have for p ∈ ∆ w i ,w i ,w i x i ,y i and q ∈ ∆ w j ,w j ,w j x j ,y j , p ∼ q if:(i) p and q are the vertices of ∆ w i ,w i ,w i x i ,y i , respectively, ∆ w j ,w j ,w j x j ,y j ; or(ii) p / ∈ x i ∪ y i ∪ w i ∪ w i ∪ w i , q / ∈ x j ∪ y j ∪ w j ∪ w j ∪ w j , and p, q correspond tothe same underlying point of ∆; or(iii) p ∈ x i ∪ y i ∪ w i ∪ w i ∪ w i , q ∈ x j ∪ y j ∪ w j ∪ w j ∪ w j , p, q correspond to thesame underlying point of ∆, and their segments labels match for i and j . EOMETRY OF FLUX 45
Using this notation, we get that H (1 , , ∼ = ∆ β ,γ ∨ ∆ β ,β ∨ ∆ γ ,γ . The vertex corresponds to the monotone Chekanov torus T (1 , , ). Now, if weforget about the embeddings of ∆ in the (1 , , H (1 , , H (1 , , ∼ = ∆ β ,γ ∨ ∆ γ ,β . We represent H (1 , ,
2) in the second diagram in Figure 14, by superposing thesetwo triangles and keeping track of the labels.
Figure 14.
Spaces H ( a, b, c ) of Lagrangian almost toric fibres.Now, consider the diagram in Figure 13 representing the (1 , , β having invariant Ξ = 2 by Proposition 7.3. So they are notequivalent to any of the previously considered fibres. On the other hand, followingthe proof of Proposition 7.4, we see that the γ -type fibres are equivalent to γ -typefibres. So the new triangles that arise in the (1 , , β β ,β and ∆ γ ,γ , where the fibres of the latter can de identified with the fibres of ∆ γ ,γ .We arrive at the description: H (1 , , ∼ = ∆ β β ,β ∨ ∆ γ ,γ , see the third diagram in Figure 14.In general, the spaces H ( a, b, c ) follow the same pattern. Theorem 7.5.
Besides the spaces H (1 , , , H (1 , , , H (1 , , described above,the spaces H ( a, b, c ) are given by:(i) H (1 , b, c ) ∼ = ∆ β b ,β c β ,γ ∨ ∆ γ ,γ ;(ii) H (2 , b, c ) ∼ = ∆ β b ,β c γ ,β ∨ ∆ γ ,γ ; (iii) H ( a, b, c ) ∼ = ∆ β a ,β b ,β c γ ,γ ∨ ∆ γ ,γ for a, b, c > .Moreover, assuming a ≤ b < c (cid:54) = 2 and after applying congruences, the segmentcorresponding to β c lies between the segments corresponding to β a and β b . Figure 15.
The triangle ∆ β a ,β b ,β c γ ,γ embedded into an ( a, b, c )-ATF. Proof.
Using Propositions 7.3, 7.4, we are able to give a complete description ofwhat happens to the space of tori as one mutates from the ( a, b, c )-ATF to the( a, b, ab − c )-ATF. Namely, this mutation introduces a new family of β c (cid:48) -typefibres, while the γ c (cid:48) -type fibres are not new: they are either toric or of γ -type.Furthermore, the β c -type fibres disappear, unless c = a or c = b as in the mutations H (1 , , → H (1 , , → H (1 , , a, b, c >
2, this means we are justreplacing ∆ β a ,β b ,β c γ ,γ , by ∆ β a ,β b ,β (cid:48) c γ ,γ as we change from H ( a, b, c ) to H ( a, b, c (cid:48) ), seeFigure 15. The remaining cases are equally easy to consider: just note that, if ourmutation is increasing the Markov sum a + b + c [31, Section 3.7][45, Proposition 4.9]then, after (1 , , β type fibres will lie inside the consecutive modificationsof ∆ β β ,β , see the third diagram of Figure 13.The last part of the claim can be proved inductively, we still work in the settingof increasing the Markov sum. In Figure 15, this means that c > a, b and to increasethe Markov sum we mutate either a or b , creating a (cid:48) > c, b , resp. b (cid:48) > c, a , with β a (cid:48) appearing between β b , β c , resp. β b (cid:48) appearing between β a , β c . (cid:3) Joining all the spaces H ( a, b, c ) together, one obtains the following descriptionof the full space H . We develop another notation ˙ ∨ for describing this space.Before describing ˙ ∨ , we point out that we interpret a wedge of labeled trian-gles, as a “labelled non-Hausdorff triangle”, that projects to ∆. For instance, H (1 , ,
2) = ∆ β ,γ ∨ ∆ γ ,β , has over the bottom edge of ∆, two non-Hausdorffopen segments, one receiving the label β and the other γ , see Figure 14. Also wepoint out that, most of the points are thought to have empty label. For instance,in H (1 , ,
5) = ∆ β β ,β ∨ ∆ γ ,γ , we have two non-Hausdorff open segments project-ing over the segment in ∆ corresponding to β . One of these segments receive EOMETRY OF FLUX 47 the β label (corresponding to β -type tori) and the other receive the empty label(corresponding to tori in the toric diagram), see again Figure 14.With the above in mind, we set ˙ ∨ between wedges of labelled triangles in thesame way as ∨ with the exception that the vertices are not identified – also replace“same underlying point of ∆”, by “projecting to the same point in ∆” in items (ii)and (iii).For instance, if we consider H (1 , ,
1) ˙ ∨H (1 , ,
2) = ∆ β ,γ ˙ ∨ (∆ β ,γ ∨ ∆ γ ,β ) , we have two non-Hausdorff vertices (corresponding to the Clifford and Chekanovmonotone tori), and two pair of non-Hausdorff edges ( β , γ ) and ( β , γ ). Allremaining points are separable. Note also that, if we approach the vertex, say overthe β edge, we have two possible limits. So the space is not the same as two copiesof ∆ identified over the interior.Before stating the Theorem, we recall the Markov conjecture, that says that aMarkov triple is uniquely determined by its biggest Markov number. Theorem 7.6.
Assume that the Markov conjecture holds (otherwise see Remark7.1). The space H of all almost toric fibres of C P modulo symplectomorphisms isthe following non-Hausdorff space: H = ˙ (cid:95) ( a,b,c ) Markov triple H ( a, b, c ) Note that the vertex of each H ( a, b, c ) -space is not identified with the analogousvertex for any other Markov triple, as these vertices correspond to the differentmonotone tori T ( a , b , c ) .Remark . Each time a Markov number c appears in the Markov tree, it generatesan infinite ray obtained by mutating the other two Markov numbers indefinitely.The Markov conjecture implies that there is a unique such ray involving the number c . If the Markov conjecture does not hold, in Theorem 7.6, the β c correspondingto different rays associated to c , should not be identified, in other words, the labelshould be indexed by the corresponding ray, instead of the Markov number.8. Calabi-Yau SYZ fibres are unobstructed
Mirror symmetry and SYZ fibrations.
The Strominger-Yau-Zaslow con-jecture [41] aims to provide a geometric construction of mirrors of symplectic man-ifolds. In one of its forms, it says that every closed symplectic Calabi-Yau manifold Y is supposed to carry a Lagrangian torus fibration with singularites, called an SYZ fibration , and that the mirror variety can be recovered from the SYZ fibrationby properly dualising and regluing it.Assuming an SYZ fibration on Y has been constructed, a technique called FamilyFloer homology is expected to prove homological mirror symmetry. It was envi-sioned by Fukaya [26] and recently advanced by Abouzaid [1]. In its present formit applies when the SYZ fibration on Y is non-singular ; the main examples areAbelian varieties. To further extend this technique, one needs to learn how toinclude singular fibres into the story. Much more basically, one needs to know thatsmooth SYZ fibres are unobstructed, i.e. have non-trivial Floer theory. Althoughthis fact is fully expected by the mirror symmetry intuition, it does not seem tohave been shown rigorously. We will do so now, as an application of the Ψ-invariantand its concavity property. Definition 8.1.
Let Y be a compact symplectic Calabi-Yau 2 n -manifold. We sayit satisfies the SYZ hypothesis if the following holds.(1) There is a topological n -manifold B without boundary called the base and aclosed codimension two subspace ∆ ⊂ B called the discriminant locus suchthat B ◦ := B \ ∆ is a smooth manifold with an integral affine structure;(2) For every sufficiently small (cid:15) > ∆ (cid:15) ⊂ B , called a thickening or a localised amoeba of ∆, which belongs to an (cid:15) -neighbourhood of∆ and such that there is a symplectic embedding ι (cid:15) : X ( B \ ∆ (cid:15) ) → Y where X ( B \ ∆ (cid:15) ) is the symplectic manifold canonically associated with thebase B \ ∆ (cid:15) with its non-singular affine structure which is the restriction of thegiven one on B ◦ . Namely, X ( B \ ∆ (cid:15) ) = T ∗ ( B \ ∆ (cid:15) ) Z T ∗ ( B \ ∆ (cid:15) ) , i.e. X is defined as the symplectic manifold which has a non-singular Lagrangiantorus fibration with base B \ ∆ (cid:15) .(3) The following compatibility up to Hamiltonian isotopy holds. For all sufficientlysmall (cid:15) , (cid:15) > p ∈ B ◦ away from a max { (cid:15) , (cid:15) } -neighbourhood of ∆,the images ι (cid:15) ( T p ) and ι (cid:15) ( T p ) are Hamiltonian isotopic Lagrangian tori in Y .Here T p ⊂ X ( B \ ∆ (cid:15) i ) is the Lagrangian torus fibre over the point p . Thesefibres have vanishing Maslov class.In general, one expects all closed Calabi-Yau manifolds to admit such fibrations,although this has not been rigorously verified in general. One expects that toricdenenerations provide such fibrations. The dimension 3 case has been studied in[6, 7, 5]. Remark . Having to thicken the codimension two discriminant locus is an essen-tial feature of the existing examples, in dimension at least 3. Without thickeningthe discriminant, the best one can hope to construct is a non-smooth continuous fibration π : Y → B , having toric fibres away from ∆.8.2. SYZ fibres are unobstructed.
Although Y may not possess a smooth La-grangian torus fibration over the whole B ◦ = B \ ∆, the SYZ hypothesis ensures thatup to Hamiltonian itotopy, there is a well-defined Lagrangian torus correspondingto any point in B ◦ . Definition 8.2.
Let Y be a Calabi-Yau manifold satisfying the SYZ hypothesisand p ∈ B ◦ . The Ham-fibre F p ∈ Y is the Hamiltonian isotopy class of a Lagrangiantorus in Y defined by: F p = ι (cid:15) ( T p ) for any sufficiently small (cid:15). Here where T p ∈ X ( B \ ∆ (cid:15) ) is the standard fibre. Compatibility up to Hamiltonianisotopy implies that the Hamiltonian isotopy class F p is well-defined.For every p ∈ B ◦ , consider the numberΨ( F p ) ∈ (0 , + ∞ ] , the Z -graded version of the invariant from Section 4, that is, defined as Ψ( F p ) =inf { val m ( e b ) : b ∈ H ( F p ; Λ rel + ) } . Because L has vanishing Maslov class, the prop-erties of Ψ from Section 4 remain the same. EOMETRY OF FLUX 49
Since Ψ is invariant under Hamiltonian isotopies, we get a well-defined functionΨ : B ◦ → (0 , + ∞ ] . Theorem 8.3.
Let Y be a Calabi-Yau manifold satisfying the SYZ hypothesis.Then the Ψ -function is identically constant on the base: Ψ( p ) ≡ λ for all p ∈ B ◦ , where λ ∈ (0 , + ∞ ] . Moreover:— If λ = + ∞ , every Ham-fibre F p is unobstructed. Namely, for any compati-ble almost complex structure, its Fukaya A ∞ algebra has vanishing curvature: m (1) = 0 . Furthermore, it holds that H ( F p , Λ + ) = MC ( F p ) where MC ( F p ) is the space of ( Z -graded) bounding cochains, or Maurer-Cartanelements, modulo gauge equivalence.— If λ (cid:54) = + ∞ , the lowest-area non-trivial symmetrised m k,β -operation from The-orem 4.8 arises from disks in a class β ∈ H ( X, F p ; Z ) such that [ ∂β ] = 0 ∈ H ( F p ; Z ) . This case is impossible if the relative divisor axiom from Defini-tion 8.4 holds. The relative divisor axiom is a fully expected property of the Fukaya algebrawhose proof, however, requires an extra layer of detail compared to the generalsetup of the Fukaya category. Since it does not seem to have appeared in theliterature in full generality, we decided to state the result accordingly.
Proof.
The main step is to prove that Ψ : B ◦ → (0 , + ∞ ] is identically constant. IfΨ achieves value + ∞ at come point, then it is identically equal to + ∞ by concavity.So we may assume that Ψ < + ∞ everywhere on B ◦ .Fix a point p ∈ B ◦ and S n − = P orient ( T p B )be the set of lines through p . Let N be a positive number and define the shade ofthe thickened discriminant Shade( ∆ (cid:15) , N ) ⊂ S n − be the set of oriented affine lines l : R → B ◦ such l (0) = p and l ([ − N, N ]) ∩ ∆ (cid:15) (cid:54) = ∅ .We also add to the shade the directions of all affine lines starting from p that hitthe discriminant ∆ itself, hence do not exist in B ◦ .Let µ be standard Lebesgue measure on S n − . For a fixed N there is a “finitenumber of copies” of the thickening that can contribute to the shade, µ (Shade( ∆ (cid:15) , N )) → (cid:15) → . Therefore (cid:92) (cid:15)> Shade( ∆ (cid:15) , N ) ⊂ S n − has measure zero for any fixed N .We wish to prove that Ψ is constant. Assume it is not, let p ∈ B ◦ be a pointsuch that Ψ is not constant on a convex neighborhood U ⊂ B ◦ of it, and assumethat U is contained in an affine chart. Let p (cid:48) ∈ U be such that Ψ( p ) > Ψ( p (cid:48) ), andlet v = p (cid:48) − p . Fix N = Ψ( p )Ψ( p ) − Ψ( p (cid:48) ) . By the discussion of shades, one can find a point p (cid:48) as above and some (cid:15) > t (cid:55)→ p + tv, t ∈ [ − N, N ]exists in B ◦ and avoids ∆ (cid:15) . Using the SYZ hypothesis, fix an SYZ fibration on Y whose singularity locus is contained in ∆ (cid:15) , then the above segment describes aLagrangian star-isotopy in X . Now consider f : [ − N, N ] → (0 , + ∞ ] , f ( t ) = Ψ( p + tv ) , t ∈ [ − N, N ] . This function is positive and concave by Theorem 4.9. It equals Ψ( p ) at t = 0 andΨ( p (cid:48) ) at t = 1. Hence satisfies the linear bound0 < f ( t ) ≤ Ψ( p ) + t (Ψ( p (cid:48) ) − Ψ( p )) for all t ∈ [ − N, (cid:116) [1 , N ] . Now put t = N . The right hand side vanishes, giving a contradiction. This provesthat Ψ is constant.To conclude the proof, first suppose that Ψ ≡ + ∞ . The claim is that F p is fullyalgebraically unobstructed. Let { m k } k ≥ be the Fukaya A ∞ algebra operations on L p . If m (1) (cid:54) = 0 , then val( m (1)) < + ∞ , and hence Ψ( p ) ≤ val( m (1)) < + ∞ , which is not the case. Second, considering b ∈ H ( F b , Λ rel + ) we must have m ( e b ) = 0 , since otherwise Ψ( p ) ≤ val( m ( e b )) < + ∞ . This means that b is a Maurer-Cartanelement.Finally, suppose that Ψ ≡ λ < + ∞ is a finite constant on B ◦ . Let p ∈ B ◦ be anypoint. By Theorem 4.8, there exists a class β ∈ H ( X, F p ; Z ) such that ω ( β ) = λ ;furthermore there exist k ≥ c i ∈ H ( F p ; R ) such that (cid:88) σ ∈ S k m k,β ( c σ (1) , . . . , c σ ( k ) ) (cid:54) = 0 . Now consider small fluxes F p + (cid:15)v within a neighbourhood of p , where v ∈ T p B ◦ .Extending the classes β and c i continuously to F p + (cid:15)v , the operations above remainnon-trivial for each fibre F p + (cid:15)v by Fukaya’s trick, as in the proof of Theorem 4.9.On the other hand, the area of the class β p + (cid:15)v continuously obtained from β equals ω ( β p + (cid:15)v ) = ω ( β ) + (cid:15) ( v · ∂β ) = λ + (cid:15) ( v · ∂β ) . Again by Theorem 4.8, Ψ( F p + (cid:15)v ) is bounded above by the right hand side. SinceΨ identically equal to λ , this is only possible when ∂β = 0. The clause involvingthe divisor axiom follows directly from its definition, see below. (cid:3) Relative divisor axiom.
We finish with a discussion of the relative divisoraxiom which was mentioned in Theorem 8.3.
Definition 8.4.
Let L ⊂ X be a Lagrangian submanifold. We say that it satisfiesthe relative divisor axiom if its Fukaya A ∞ algebra admits a classically minimalmodel over H ∗ ( L ; Λ ) such that the following property holds for all c , . . . , c k ∈ H ( L ; R ).(8.1) (cid:88) σ ∈ S k m k,β ( c σ (1) , . . . , c σ ( k ) ) = (cid:104) c · [ ∂β ] (cid:105) . . . (cid:104) c k · [ ∂β ] (cid:105) · m ,β (1) ∈ H ( L ; Λ + ) . The numbers (cid:104) c i · [ ∂β ] (cid:105) ∈ R are the pairings. Additionally, we add the conditionthat(8.2) m ,β (1) = 0 whenever [ ∂β ] = 0 ∈ H ( L ; R ) . EOMETRY OF FLUX 51
The existence of a minimal model satisfying (8.1) for general Lagrangian sub-manifolds follows from [27, 11]. The existence of a model satisfying both (8.1) and(8.2) has been written down for Lagrangians in a Calabi-Yau 3-fold [27, 28], butthe same methods are expected to work in general. We can conclude the following.
Corollary 8.5.
Ham-fibres of Calabi-Yau threefolds satisfying the SYZ hypothesisare unobstructed. (cid:3)
For the sake of completeness, let us briefly mention what goes into the proof of thedivisor axiom. For (8.1), one needs to arrange the evaluation maps at the boundarymarked points of moduli spaces of (perturbed) holomorphic disks to commute withforgetting a boundary marked point. The existence of a perturbation scheme withthis property is explained in [27]. Once the compatibility is achieved, the proof of(8.1) is a formal argument, see [11, Section 6].In the case when L is 3-dimensional, one can use the additional Poincar´e duality H ( L ; R ) ∼ = H ( L ; R ) for a slightly different look on the divisor axiom. Namely,one considers a perturbation scheme which guarantees that the A ∞ operationsare cyclic , see [29, Section 3.6.4]; such perturbations were shown to exist again in[27]. Given that, (8.1) follows by a slightly different formal argument [29, Propo-sition 3.6.50]. Now, in the 3-dimensional case one can introduce the numbers m − ,β ∈ R , β ∈ H ( X, L ; Z ) , µ ( β ) = 0 , see [28], with the property that (cid:104) m ,β (1) , ρ (cid:105) = ( ρ · [ ∂β ]) m − ,β . This in particular implies (8.2). Roughly speaking, m − ,β counts the number ofdisks in class β without any marked points and taken modulo automorphisms;when dim L = 3, such Maslov index 0 disks are rigid. One way of proving (8.2) inall dimensions would be to generalise m − ,β in a suitable way. References [1] M. Abouzaid. Family Floer cohomology and mirror symmetry.
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