aa r X i v : . [ m a t h . S G ] J a n HOMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES
ABIGAIL WARD
Abstract.
We show homological mirror symmetry results relating coherent analytic sheaves on some com-plex elliptic surfaces and objects of certain Fukaya categories. We first define the notion of a non-algebraicLandau-Ginzburg model on R × ( S ) and its associated Fukaya category, and show that non-K¨ahler sur-faces obtained by performing two logarithmic transformations to the product of the projective plane andan elliptic curve have non-algebraic Landau-Ginzburg models as their mirror spaces; this class of surfaceincludes the classical Hopf surface S × S and other elliptic primary and secondary Hopf surfaces. We alsodefine localization maps from the Fukaya categories associated to the Landau-Ginzburg models to partiallywrapped and fully wrapped categories. We show mirror symmetry results that relate the partially wrappedand fully wrapped categories to spaces of coherent analytic sheaves on open submanifolds of the compactcomplex surfaces in question, and we use these results to sketch a proof of a full HMS result. Contents
1. Introduction 12. Classification of candidate S τ ( A )
63. Landau-Ginzburg models 94. Admissible Lagrangians and perturbing Hamiltonians 115. A ∞ structures 196. Fukaya categories 417. Categories of line bundles 458. Mirror symmetry for D τ ( m, k ) and C ∗ × E τ Introduction
For q ∈ C ∗ with ∣ q ∣ <
1, there is a free action of Z on C n ∖ { } given by scaling by q . When n =
1, thequotient space C ∗ / ( z ∼ qz ) is the elliptic curve E τ with modular parameter τ satisfying q = e πiτ . When n =
2, the quotient is the
Hopf surface : S τ = C ∖ { }/ ( z ∼ q z ) . The map z ↦ ( log ∣ z ∣ , z ∣ z ∣ ) ∈ ( R /∣ πiτ ∣ Z , C ) is a diffeomorphism S τ → S × S . Since H ( S τ , Z ) =
0, the Hopf surface is non-K¨ahler and non-algebraic.The purpose of this work is to demonstrate that despite its lack of algebraic or symplectic structure, theHopf surface still exhibits homological mirror symmetry behavior; that is, there exists a mirror space Y to S τ and a Fukaya category F τ associated to Y such that there is a correspondence between coherent analyticsheaves on S τ and objects of F τ .The starting point of our mirror symmetry correspondence is the observation that there exists an anti-canonical divisor D on S τ such that S τ ∖ D is a more familiar complex surface to which we can associate apredicted mirror space. Indeed, let D ⊂ S τ denote the image of the two coordinate axes { z = } ⊔ { z = } in C ∖ { } under the quotient map C ∖ { } → S τ ; D consists of the disjoint union of two copies of the ellipticcurve E τ . There is a meromorphic volume form on S τ given by dz ∧ dz z z that has simple poles on the components of D . Moreover, S τ ∖ D ≅ C ∗ × E τ is an open Calabi-Yau manifoldwith a symplectic form which is compatible with its complex structure, and a fibration by special Lagrangiantori. Following the Strominger-Yau-Zaslow philosophy ([27]), one predicts that the mirror to C ∗ × E τ is themoduli space of these tori equipped with a unitary flat connection, which is parametrized by C ∗ × E τ , where E τ is the elliptic curve mirror to E τ .We then recall the following heuristic for performing mirror symmetry relative to an anti-canonical divisor(see [6]). Let ( X, D X ) be a pair consisting of a K¨ahler manifold X and an anti-canonical divisor D X with theproperty that X ∖ D X is open Calabi-Yau and admits a fibration by Lagrangian tori. For such a pair ( X, D X ) ,the mirror operation to compactifying X ∖ D X by D X is introducing a Landau-Ginzburg superpotential W on the open SYZ mirror ( X ∖ D ) ∨ which encodes the data of Maslov index 2 discs that pass through D .To this Landau-Ginzburg model one associates a Fukaya-Seidel category denoted F S (( X ∖ D ) ∨ , W ) ; theobjects of this A ∞ category are Lagrangians in ( X ∖ D ) ∨ that obey boundary conditions specified by W .This method gives a mirror space to various non-Calabi-Yau X , e.g., for toric varieties (in [1] and laterwork), and one can then verify the mirror correspondence by checking the predictions of homological mirrorsymmetry.Inspired by this philosophy, we construct a non-algebraic Landau-Ginzburg model associated to the pair ( S, D ) and other pairs consisting of a complex surface S τ ( A ) and an anti-canonical divisor D A such that S τ ( A ) ∖ D A = C ∗ × E τ . Each such compactification of C ∗ × E τ that we consider is obtained by applying twologarithmic transformations to P × E over two points in P , and the index A records which compactificationwe obtain in this way. Recall that the logarithmic transformation is an operation on elliptic complex surfaceswhich takes an elliptic surface X with a fibration f ∶ X → B to a new elliptic surface f ′ ∶ X ′ → B , with theproperty that there exists a point b ∈ B such that there is a biholomorphism X ∖ f − ( b ) → X ′ ∖ ( f ′ ) − ( b ) . Topologically, this transformation amounts to performing a surgery in which one removes a neighborhood U = D × f − ( b ) of the fiber f − ( b ) and glues the neighborhood back in via a diffeomorphism ϕ ∶ ∂U = T → T . One imposes conditions on ϕ so that the resulting space has a complex structure. The logarithmictransformation has two parameters ( m, k ) which are a pair of integers such that ( m, k ) is a primitive elementin Z and m >
0; the parameters ( , ) give the trivial case where the diffeomorphism ϕ is the identity. We let D ( m, k ) denote the result of performing a logarithmic transformation with parameters ( m, k ) at the originat C × E . By performing two logarithmic transformations to P × E , one over each pole, with parameters ( m , k ) and ( m ∞ , k ∞ ) , we obtain an elliptic surface S τ (( m , k ) , ( m ∞ , k ∞ )) ; we let A denote the pair (( m , k ) , ( m ∞ , k ∞ )) . In this notation, we may write the Hopf surface S τ as S τ = S τ (( , ) , ( , )) . Fix the anti-canonical divisor D P × E = ({ } × E τ ) ∪ ({ ∞ } × E τ ) ⊂ P × E τ ; let D A denote the union of thefibers over 0 and ∞ in S ( A ) . By design we have S τ ( A ) ∖ D A = ( P × E τ ) ∖ D P × E τ = C ∗ × E τ . When ( m , k ) = ( m ∞ , − k ∞ ) , the resulting surface is a quotient of P × E by the action of a finite group;otherwise, the resulting surface is non-algebraic.We then associate to each surface S τ ( A ) a non-algebraic Landau-Ginzburg model denoted by ( Y, A ) andan associated Fukaya category F ( A ) . Here Y = ( R × ( S ) , dt ∧ dϕ t + ds ∧ dϕ s ) should be thought of asthe mirror of C ∗ × E τ . These categories are defined over the Novikov ring with formal variable T ; we showthat there are “evaluation maps” (determined by T ↦ e πiτ ) from F ( A ) to C -linear categories denoted by H F τ . The construction bypasses the process of defining of a Landau-Ginzburg superpotential by countingholomorphic discs in S τ ( A ) as we would be unable to define the energy (symplectic area) of any such disc.Instead, we specify boundary conditions that Lagrangians will need to satisfy at each end of Y . OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 3
Definition 1.0.1.
A type- A Lagrangian in Y satisfies the following condition: there exists a T > m ϕ t + k ϕ s vanishes identically when restricted to L ∩ ( −∞ , − T ] × ( S ) , and(2) − m ∞ ϕ t + k ∞ ϕ s vanishes identically when restricted to L ∩ [ T, ∞ ) × ( S ) .The objects of F ( A ) are (roughly speaking) type- A Lagrangians which are linear at infinity and havewell-defined Floer theory. The construction of the category is similar to that in [10] which associates to aLandau-Ginzburg model a stopped Fukaya category.We conjecture the following:
Conjecture 1.0.1.
There is a full and faithful functor D b Coh an S τ ( A ) → D π F τ ( A ) . Our first theorem provides evidence of this conjecture:
Theorem 1.0.2.
For each A such that S τ ( A ) is not an algebraic surface, there is a full and faithful functor (3) Φ A ∶ Pic an S τ ( A ) → H F τ ( A ) The image of each line bundle is a Lagrangian section. This mirror correspondence confirms the expecta-tion from previous instances of SYZ mirror symmetry (e.g., in the family Floer program of Abouzaid, see [2])that holomorphic vector bundles of rank n are mirror to Lagrangian multi-sections that have intersectionnumber n with a fiber.However, the derived category of S τ ( A ) is not (split-)generated by line bundles and torsion sheaves,so one cannot prove Conjecture 1.0.1 using this collection of objects. This motivates us to work towardsa more structural understanding of the objects of the categories of the conjecture. To this end, we firstdescribe sheaves on S τ ( A ) using a cover of S τ ( A ) that reflects its construction in terms of the logarithmictransformations. Let D to be the reduced fiber lying over 0 and D ∞ the reduced fiber lying over 0. Thereare restriction functors Coh an S τ ( A ) → Coh an ( S τ ∖ D ) = Coh an D τ ( m ∞ , k ∞ ) and Coh an S τ ( A ) → Coh an ( S τ ∖ D ∞ ) = Coh an D τ ( m , k ) . We define subcategories C τ ( m, k ) ⊂ Pic an D τ ( m, k ) and C τ ⊂ Pic an C ∗ × E ; the first consists of line bundleswhich can extend from D τ ( m, k ) to some S τ ( A ) , and the second consists of line bundles that can extendfrom C ∗ τ × E to some compactification S τ ( A ) . Thus a line bundle on S τ ( A ) can be described as a pair ofobjects of C τ ( m , k ) and C τ ( m ∞ , k ∞ ) that restrict to the same object of C τ , along with gluing data.In parallel, we decompose the boundary conditions that a Lagrangian L ∈ F ( A ) satisfies. This leads us todefine two Fukaya categories with relaxed boundary conditions: the “partially wrapped” category F ( m, k ) isdefined by relaxing the boundary conditions at one end, and the “fully wrapped” category F W r by relaxingthem at both ends: Ob F ( m, k ) = ⋃ ( m ∞ ,k ∞ ) Ob F (( m, k ) , ( m ∞ , k ∞ )) and Ob F W r = ⋃ (( m ,k ) , ( m ∞ ,k ∞ )) Ob F (( m , k ) , ( m ∞ , k ∞ )) . There are localization maps(4) F (( m , k ) , ( m ∞ , k ∞ )) → F ( m , k ) → F W r ;(5) F (( m , k ) , ( m ∞ , k ∞ )) → F ( m ∞ , k ∞ ) → F W r and(6) F ( m, k ) → F W r . ABIGAIL WARD
On objects, the first and third maps are inclusions; the second differs by an interchange of the ends of Y and can be expressed as the composition of maps F (( m , k ) , ( m ∞ , k ∞ )) → F (( m ∞ , k ∞ ) , ( m , k )) → F ( m ∞ , k ∞ ) . One expects objects of F ( m, k ) and F W r to correspond to coherent analytic sheaves on D τ ( m, k ) and C ∗ × E τ , respectively. One also expects the localization maps of Fukaya categories to be mirror to localizationmaps of sheaves. These expectations are confirmed with the following theorems: Theorem 1.0.3. (a) For each ( m, k ) there is a full and faithful functor Φ ( m,k ) ∶ C τ ( m, k ) → H F τ ( m, k ) . (b) There is a full and faithful functor Φ W r ∶ C τ → H F τ, W r Theorem 1.0.4.
The following diagram of categories commutes: (7) Pic S τ ( A ) C τ ( m , k ) H F τ ( A ) H F τ ( m , k ) C τ ( m ∞ , k ∞ ) C τ H F τ ( m ∞ , k ∞ ) H F τ, W r where each arrow is given by the functor between the corresponding categories explained above. These theorems provide an explicit and geometric example of a homological mirror symmetry correspon-dence in the case when the complex side is not K¨ahler and admits no symplectic structure. Mirror symmetryfor non-K¨ahler space has been investigated in some other settings. Most recently, in [5], ´Alvarez-C´onsul, deArriba de la Hera, and Garcia-Fernandez show that certain homogeneous surfaces, including the Hopf surface,admit ( , ) mirrors with isomorphic half-twisted models. In [17] Lau, Tseng, and Yau use a Fourier-Mukaitransform to obtain a correspondence between different Ramond-Ramond flux terms for pairs of Type IIAand Type IIB supersymmetric systems, and in [23] Popovici shows that there is a classical mirror symmetrycorrespondence for the Iwasawa surface in the sense that there is an isomorphism between the Gauduchoncone (parametrizing the space of symplectic structures) and a space of complex deformations of the samesurface. In [2], Abouzaid sketches the construction of a candidate SYZ mirror to the Kodaira-Thurstonsurface using the existence of a fibration of the surface by smooth Lagrangian tori, and states that theLagrangian sections of the SYZ fibration should correspond to coherent sheaves twisted by a gerbe on anabelian surface. Our construction bears most resemblance to this last work.In the remainder of the introduction, we give more details of the construction of the Fukaya categories,as well as a proposed proof of Conjecture 1.0.1.1.1. Mirror symmetry for compact surfaces.
The Fukaya-Seidel category associated to a Landau-Ginzburg model was constructed by Seidel ([25], [26]) in the case where the the superpotential is a Lefschetzfibration. There have been recent formulations of this category in terms of stops on a Liouville domainin [29] and in terms of Liouville sectors in [11]. Our construction is closest in spirit to the formulationusing a Liouville sector with stops. To the symplectic manifold Y = R × ( S ) equipped with the standardsymplectic form ω = dt ∧ dϕ t + ds ∧ dϕ s , we associate an A ∞ category denoted F ( A ) with objects linear type- A Lagrangians with unitary flat connection. This category is morally similar to the monomially admissibleFukaya category introduced in work of Hanlon ([13]), in which one demands that Lagrangians have specifiedradial behavior in certain regions. The actual construction of the category proceeds through a localizationprocedure similar to that in [13] or [11] and inspired by work of Abouzaid and Seidel. There is a quasi-isomorphism Hom ∗F( A ) ( L, L ′ ) → CF ∗ ( ψ H ( L ) , ψ H ′ ( L ′ )) where H and H ′ are two Hamiltonians chosen so that ψ H ( L ) and ψ H ′ ( L ′ ) intersect transversely, such thataway from a compact set each Hamiltonian is equal to ε ∣ t ∣ for some ε , and such that ψ H ( L ) and ψ H ′ ( L ′ ) OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 5 are in correct position . Note that at infinity we always perturb in the t direction, so the direction we perturbin is independent of the choice of boundary condition.The conditions on H and H ′ ensure that ψ H ( L ) and ψ H ′ ( L ′ ) are disjoint away from a compact set,so Hom ∗F( A ) ( L, L ′ ) is necessarily finite-dimensional. This agrees with the expectation from the mirrorsymmetry correspondence: for any two coherent analytic sheaves F and G on a compact complex space X , Hom D b Coh X ( F , G [ i ]) is finite-dimensional by Cartan-Serre finiteness ([9]). This is in contrast to thesituation when X is non-compact.1.2. Mirror symmetry for D τ ( m, k ) and C ∗ × E τ . The definition of F ( m, k ) and F W r originates fromthe requirement that the diagram of Theorem 1.0.4 accommodate any pair A = (( m , k ) , ( m ∞ , k ∞ )) . Thecategory F W r is an enlargement of the wrapped category which is the expected mirror to C ∗ × E τ . Thisenlargement does not respect any notion of a“cylindrical Lagrangian;” the objects that make up F ( W r ) cannot be simultaneously invariant under the flow of any “Liouville vector field”. Rather, the objects of F W r are those which are invariant under some linear outward pointing vector field.The categories F ( m, k ) and F W r are computed by a categorical localization process. The morphismscan be computed by “wrapping at an end;” this amounts to perturbing by Hamiltonians which are close toquadratic functions of t at the end where the boundary conditions were relaxed.1.2.1. Convergence.
There are several technical challenges that arise in this process. Since R × ( S ) is notan exact symplectic manifold, we cannot directly apply results from [11]. We rectify some of the problemsthat arise by keeping track of the areas of holomorphic discs and by imposing strong geometric conditionson our Lagrangians (for example, we demand that the Lagrangians are exact when lifted to the universalcover R ). However, because of the lack of exactness (and the resulting lack of a priori energy bounds onJ-holomorphic curves), we can no longer bound the image of discs which are counted in the A ∞ productsfor a fixed set of generators of the Floer complexes; thus the output of a µ n product can (and will) havenonzero coefficients on infinitely many generators. We accommodate this possibility by completing the Floercomplex to allow infinite series. We first define all Floer complexes over the Novikov ring. In this case, thecompletion is equal to the following: ̂ CF ∗ ( L, K ) = ⎧⎪⎪⎨⎪⎪⎩ ∑ y i ∈ L ∩ K c i y i ∣ c i ∈ Λ , lim t ( y i )→±∞ v ( c i ) → ∞ ⎫⎪⎪⎬⎪⎪⎭ . In Section 4 we prove that this completion is closed under taking A ∞ products.The mirror symmetry statements in Theorem 1.0.2 are equivalences of C -linear categories. These cate-gories require a slightly different completion of the Floer complex, which is described in Section 5.3. Fromthe point of view of mirror symmetry, the completion process is a natural operation which is mirror tocompleting the space of algebraic functions on D τ ( m, k ) and C ∗ × E τ to the space of holomorphic functions.We use the basic fact from complex analysis that one can express the domain of convergence of a Laurentseries on C ∗ using the limit of the n th roots of the absolute value of its coefficients. We also show that wehave necessary convergence results for the µ products.1.3. Proposed proof of mirror symmetry.
One hopes that the diagram of Theorem 1.0.4 can be up-graded to a proof of Conjecture 1.0.1. We expect that F ( A ) is the pull back of the diagram of gluing F ( m , k ) and F ( m ∞ , k ∞ ) by F W r . To obtain a proof, we would then express D b Coh an S τ ( A ) as the pullback of a diagram that glues the subcategories of D b Coh an D τ ( m , k ) and D b Coh an D τ ( m ∞ , k ∞ ) gener-ated by objects in C m ,k (denoted D τ ( m , k ) ) and C ( m ∞ , k ∞ ) (denoted D τ ( m ∞ , k ∞ ) ) via the category D τ which is the subcategory of D b Coh an C ∗ × E τ generated by C . The maps C τ ( m , k ) → H F sq ( m , k ) and C τ → H F s W r,q could be upgraded to maps D τ ( m , k ) → D π F τ ( m , k ) and D → D π ( F τ, W r ) . Gluingderived categories is in general a process requiring some care, but in this case the fact that there are onlytwo categories being glued simplifies the computations significantly ([16]).In this work, we only compute on the level of cohomology categories; this allows us to compute only µ and µ . The proposed proof would require calculating the higher products µ n in F W r , which would requiresignificantly more computation. It would also require proving that the categories D m,k are large enough –that is, that D τ ( m, k ) contains the image of the map D b Coh an S τ (( m, k ) , ( m ∞ , k ∞ )) → D b Coh an D τ ( m, k ) ABIGAIL WARD induced by restriction for all pairs (( m, k ) , ( m ∞ , k ∞ )) .The idea of proving mirror symmetry diagrammatically is not new; it is found in, e.g., work of Lee [18](although in this work is in the “other direction” – the Landau Ginzburg model is found on the complexside of the mirror pair).1.4. Organization.
The paper is organized as follows. In Section 2 we classify the complex surfaces that weconsider. In Section 3, we give a more complete description of the SYZ heuristics that guide our construction.Section 4 is dedicated to the description of the Lagrangians that make up the objects of the Fukaya categoriesthat we describe; Section 5 describes Floer complexes that will be used to define the Hom spaces in thesecategories. In Section 6 we define the Fukaya categories. Section 7 describes the categories which will makeup the complex side of the mirror symmetry correspondence. Sections 8 and 9 are dedicated to the proofof Theorems 1.0.2 and 1.0.3. In the final chapter of the main text, Section 10, we relate all the categoriesvia Theorem 1.0.4. Appendix A sketches how the mirror correspondence could be expanded to include sometorsion sheaves.The reader who wishes to understand the flavor of the mirror symmetry equivalences is advised to readSection 3 for intuition and Sections 8 and 9 to see the mirror symmetry correspondence. Example 9.0.1treats mirror symmetry for the Hopf surface, and Example 8.3.1 is a simple example of how calculationsin the wrapped category reflect the behavior of analytic line bundles on the surfaces; these two examplescapture most aspects of the mirror phenomena.1.5.
Acknowledgements.
This paper was completed as part of the author’s Stanford University doctoralthesis supervised by Denis Auroux, and she would like first and foremost to thank him for his guidance,support, and generosity, as well as the many ideas he contributed to this work. She was co-advised byRavi Vakil, who provided invaluable insight. She would also like to thank Sheel Ganatra, Fran¸cois Greer,and Andrew Hanlon for helpful conversations and suggestions. The author was supported by NSF grants1147470 and 2002183, by the Stanford DARE and EDGE fellowships, and by the Simons Foundation grant“Homological Mirror Symmetry and Applications.”2.
Classification of candidate S τ ( A ) Logarithmic transformations.
Let ( m, k ) ∈ Z be primitive with m >
0. The logarithmic transfor-mation with parameters ( m, k ) replaces a smooth fiber E b on an elliptically fibered surface S → B by afiber with multiplicity m . Around a fiber E b ⊂ S , there exists a neighborhood U which we may write as U = ∆ v × C ∗ / (( v ′ , w ) ∼ ( v ′ , q ( v ′ ) w )) where ∆ v is a disc in C centered around zero and 0 < ∣ q ( v ′ ) ∣ <
1. Webase change by writing v ′ = v m and write U ′ = U v / (( v, w ) ∼ ( e πi / m v, e − πik / m w )) . The fibration U ′ → ∆ v has a multiple fiber E ′ b at 0. There is a biholomorphic map U ′ ∖ E ′ b → U ∖ E b given by(8) ( v, w ) ↦ ( v m , v k w ) and gluing in U ′ via this map produces a new elliptic surface S ′ with the property that there is a biholo-morphism S ∖ E b → S ′ ∖ E b . Note that we allow the case where m =
1. For an elliptic surface S → B with a multiple fiber E b , there exists a unique k mod m for which we can perform an inverse logarithmictransformation to produce a new surface S ′ with no multiple fiber over b . Remark 2.1.1. If k ≡ k ′ mod m , then the surfaces obtained by performing logarithmic transformations withparameters ( m, k ) and ( m, k ′ ) are locally biholomorphic near the log-transformed fiber but will in generalnot be globally biholomorphic.Let e πiτ = q ∈ C ∗ with ∣ q ∣ <
1, and let E τ denote the elliptic curve with modular parameter τ . Let D τ ( m, k ) = C × E τ / (( v, w ) ∼ ( e πi / m v, e − πik / m w )) ; OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 7 when we write D τ ( m, k ) we will always mean this complex space along with this choice of coordinates. Let i m,k ∶ C ∗ × E τ → D τ ( m, k ) denote the inclusion. In coordinates i m,k is given by ( z, x ) ↦ ( z / m , z − k / m x ) . In writing this map, and similar maps throughout this section, we make a consistent choice of the m -th rootin both factors and then note that the map is invariant under such a consistent choice.2.2. Constructing S τ ( A ) . Consider the surface S τ ( A ) obtained by performing log transforms with pa-rameters ( m , k ) and ( m ∞ , k ∞ ) to P × E τ over 0 and ∞ respectively. (In this subsection we will fre-quently omit the τ .) It will be useful to consider S τ ( A ) as the union of D τ ( m , k ) and D τ ( m ∞ , k ∞ ) glued over their common open set C ∗ × E , via the transition map specified by (8). The transition map D τ ( m , k ) ∖ { v = } → D τ ( m ∞ , k ∞ ) is given by ( v, w ) ↦ ( v − m / m ∞ , v k +( m k ∞ / m ∞ ) w ) . By construction, each logarithmically transformed surface is elliptic and fibers over P ; denote the map S τ ( A ) → P by π . We now investigate the properties of the surfaces S ( A ) . Remark 2.2.1.
Each S ( A ) has Kodaira dimension −∞ . The canonical bundle is given by K S ( A ) = O S ( A ) ( − ( m + m ∞ ) [ F ]) where [ F ] is the class of the smooth fiber (see [8]). In other words, the reduced preimage D A = π − ({ } ∪ { ∞ }) satisfies [ D A ] ∈ ∣ − K S ( A ) ∣ . Remark 2.2.2.
The surfaces of the form S (( m, k ) , ( m, − k )) are quotients of P × E by Z / m Z and as suchare algebraic. In this work we do not prove mirror symmetry results for these algebraic surfaces, but thepredicted mirror spaces do agree with the mirror spaces we obtain. Remark 2.2.3.
The map S ( A ) → P is more properly thought of as a map to the weighted projective space P ( m , m ∞ ) . Proposition 2.2.1.
For each A , π ( S ( A )) = coker ⎛⎜⎝ m − m ∞ k k ∞
00 0 0 ⎞⎟⎠ ∶ Z → Z . Proof.
Denote the matrix in the statement of the proposition by M . We apply the Seifert-Van Kampentheorem to the cover of S ( A ) by D ( m , k ) and D ( m ∞ , k ∞ ) glued over C ∗ × E . Choose the following basisfor π ( C ∗ × E ) = Z : ( , , ) = {( e πit , ) ∣ t ∈ [ , ]} ;(9) ( , , ) = {( , e πit ) ∣ t ∈ [ , ]} ;(10) ( , , ) = {( , + t ( q − )) ∣ t ∈ [ , ]} . (11)The inclusion C ∗ × E ↪ D ( m , k ) amounts topologically to attaching a two-cell to the class ( m , k , ) ; theinclusion C ∗ × E ↪ D ( m ∞ , k ∞ ) attaches a two-cell to the class ( − m ∞ , k ∞ , ) (the negative sign originatesfrom the change in coordinates z ↦ z − ). Thus we may write π ( S ( A )) = Z / ( m , k , ) ⋆ Z Z / ( − m ∞ , k ∞ , ) = coker M. (cid:3) Proposition 2.2.2.
For A with det ( m − m ∞ k k ∞ ) = n ≠ , the universal cover of S τ ( A ) is C ∖ { } . The fundamental group π ( S τ ( A )) , written as a quotient of Z ,acts freely on the universal cover by ( a, b, c ) ⋅ ( z , z ) = ( exp ( πi ( k ∞ a + m ∞ b + m ∞ cτ ) n ) z , exp ( πi ( − k a + m b + cm τ ) n ) z ) , ABIGAIL WARD and S τ ( A ) is the quotient of C ∖ { } by this action.Proof. Note that the kernel of the map ρ ∶ Z → ( C ∗ ) ⊂ GL ( C ) given by ρ ( a, b, c ) = ( exp ( πi ( k ∞ a + m ∞ b + m ∞ cτ ) n ) , exp ( πi ( − k a + m b + cm τ ) n )) is equal to the image of the map M ∶ Z → Z . Moreover,Im ( ρ ) ∩ ({ } × C ∗ ∪ C ∗ × { }) = ( , ) . Thus ρ descends to the quotient to give a free action of coker M on C ∖ { } .Let S = { z ≠ } and S ∞ = { z ≠ } . Define a map f ∶ S → D τ ( m , k ) by f ( z , z ) = ( z z − m ∞ / m , z n / m ) . We first verify that this map is independent of the choice of branch cut of the m th root. Indeed, note that k m ∞ ≡ n mod m , so ( exp ( − πim ∞ m ) z z − m ∞ / m , exp ( πinm ) z n / m ) = ( exp ( − πim ∞ m ) z z − m ∞ / m , exp ( πik m ∞ m ) z n / m ) ∼ ( z z − m ∞ / m , z n / m ) . We also note that f (( a, b, c ) ⋅ ( z , z )) = ( exp ( πiam ) z z − m ∞ / m , exp πiτcn exp ( − πik am ) z n / m ) ; ∼ f ( z , z ) ;so the map is well-defined. f has an inverse given by(12) f − ( v, w ) = ( vw m ∞ / n , w m / n ) which one can also check is well-defined. We similarly define a map g ∶ S → D τ ( m ∞ , k ∞ ) by g ( z , z ) = ( z − m / m ∞ z , z n / m ∞ ) with inverse(13) g − ( v, w ) = ( w m ∞ / n , vw m / n ) . It remains to show that the two maps agree over S ∩ S ∞ , i.e., that f − ○ i m ,k ( z, x ) = g − ○ i m ∞ ,k ∞ ( z − , x ) .We check that f − ○ ( i m ,k ) ( z, x ) = f − ( z / m , z − k / m x ) = ( z k ∞ / n x m ∞ / n , z − k / n x m / n ) (14) g − ○ ( i m ∞ ,k ∞ ) ( z − , x ) = g − ( z − / m ∞ , z k ∞ / m ∞ x ) = ( z k ∞ / n x m ∞ / n , z − k / n x m / n ) . (15) (cid:3) Denote f − ○ ( i m ,k ) ∶ C ∗ × E τ → S τ ( A ) by i A . Remark 2.2.4 (Terminology) . Any surface that is the quotient of C ∖ { } by a free action of a discretegroup G is called a Hopf surface. A primary Hopf surface is one where G = Z ; topologically these surfacesare all S × S . A secondary Hopf surface is one where G is not Z ; equivalently, one which is not topologically S × S ; equivalently, one with torsion in its fundamental group. For more background, see [15]. Thus allthe surfaces we consider are Hopf surfaces which are also elliptic. Remark 2.2.5.
In this notation, the classical Hopf surface S is S ( A ) where A = (( , ) , ( , )) . OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 9 Landau-Ginzburg models
In this section, we explain in more detail the idea that lead us to define the Fukaya category based on thephilosophy of performing mirror symmetry “relative to an anti-canonical divisor,” given in work of Auroux([6]). Although these heuristics provide the motivation for the definition of the category, they do not enterinto the proofs of any of the theorems in this paper.Let π ∶ S ( A ) → P denote the map which furnishes the fibration. Let D denote π − ({ }) with its reducedscheme structure and D ∞ = π − ({ ∞ }) with its reduced scheme structure, so D A = D ⊔ D ∞ . By design, S A ∖ D A = C ∗ × E , and D A is anti-canonical. The standard symplectic form on C ∗ × E is compatible withthe complex structure on S A ∖ D A = C ∗ × E ; moreover, C ∗ × E admits an SYZ fibration C ∗ × E → R × S given by ( z, x ) ↦ ( log ∣ z ∣ , log ∣ x ∣) ∈ R × ( R /∣ πiτ ∣ Z ) . Each fiber L ∣ z ∣ = e t , ∣ x ∣ = e s = L t,s bounds no holomorphic discs in C ∗ × E . Guided by the hope that we canconstruct a mirror space to S ( A ) by correcting the mirror to S ( A ) ∖ D A , we define a SYZ mirror Y consisting of the dual fibration over C ∗ × E . This consists of the points Y = {( L t,s , ∇ )} where ∇ is a unitary flat connection up to gauge equivalence on the rank-one trivial bundle on L t,s . Fixingthe symplectic structure and complex structure on Y described in [6], one obtains that Y = R × S × S × S . The coordinates on Y are ( t, s, ϕ t , ϕ s ) ; the ϕ t and ϕ s encode the holonomy of ∇ around the loops γ = { e iθ t e t , e s ∣ θ t ∈ [ , π )} and γ = { e t , e iθ s e s ∣ θ s ∈ [ , π )} , respectively. The symplectic form on Y is givenby ω = dt ∧ dϕ t + ds ∧ dϕ s , and the complex structure J is given by J ( ∂ / ∂t ) = ∂ / ∂ϕ t , J ( ∂ / ∂s ) = ∂ / ∂ϕ s .Note that each L t,s is a totally real submanifold of S ( A ) which bounds the following holomorphic discspassing through D A : u t,s ∶ D → D ( m , k ) ⊂ S ( A ) ; u t,s ∞ ∶ D → D ( m ∞ , k ∞ ) ⊂ S ( A ) .u t,s is given by z ↦ ( e t / m z, e s − k t / m ) ; u t,s ∞ is given by the analogous map to D ( m ∞ , k ∞ ) . The boundaries of the discs give elements [ ∂u ] , [ ∂u ∞ ] ∈ π ( L ) ⊂ π ( C ∗ × E ) . In the basis for π ( C ∗ × E ) of Proposition 2.2.1, [ ∂u ] = ( m , k , ) and [ ∂u ∞ ] = ( − m ∞ , k ∞ , ) (one notes that these discs represent the cells we attached in calculating the fundamentalgroup).From a dimension counting argument similar to that of Lemma 3.1 of [6], we expect that these are the onlyrigid holomorphic discs passing through the point ( e t , e s ) on L t,s . If S ( A ) were K¨ahler and the symplecticform ω on S ( A ) ∖ D A were inherited from a symplectic form ω on S ( A ) , we would define a superpotentialrecording each disc: W ( L, ∇ ) = e − ω ( u t,s [ D ]) hol ∇ [( m , k )] + e − ω ( u t,s ∞ [ D ]) hol ∇ [( − m ∞ , k ∞ )] . We would then define a Fukaya-Seidel category associated to the Landau Ginzburg model ( Y, W ) withobjects the Lagrangians which fiber over paths asymptotic to the positive real axis under W . This wouldgive conditions on the Lagrangians at infinity; in the region where each monomial corresponding to a termrecording a disc u in the superpotential dominated, the Lagrangians would be restrained to the region wherehol ∇ ([ ∂u ]) is close to the identity.Of course we cannot define such a W because there is no symplectic form ω ! Thus the failure of S ( A ) to bea symplectic manifold is reflected in the failure of the superpotential W to be well-defined, and in particularin the failure of each monomial term to have a well-defined norm. However, the standard symplectic form onthe universal cover endows S ( A ) with a natural conformal symplectic structure – that is, S ( A ) is locally asymplectic manifold where the symplectic form is defined only up to multiplication by a constant – and thetori L t,s are Lagrangian for the conformal symplectic form. We can use this structure to define a “relativeenergy” of the two discs via the universal cover; this will reflect the intuition that the e − ω ( u [ D ]) hol ∇ [( m , k )] term should dominate as t → −∞ , and the e − ω ( u [ D ]) hol ∇ [( − m ∞ , k ∞ )] term should dominate as t → ∞ . Let ˜ u t,s and ˜ u t,s ∞ be lifts of the discs u t,s and u t,s ∞ to C ∖ { } so that ˜ u t,s ( ) = ˜ u t,s ∞ ( ) . After applying thechange of coordinates of Equations (12), (13), (14), (15), we obtain:˜ u t,s ( z ) = ( exp ( πiτ ℓ m ∞ n ) exp ( πiℓ m ∞ n + tk ∞ + sm ∞ n ) z, exp ( πiτ ℓ m n ) exp ( πiℓ m n + − tk + sm n )) ;˜ u t,s ∞ ( z ) = ( exp ( πiτ ℓ m ∞ n ) exp ( πiℓ m ∞ n + tk ∞ + sm ∞ n ) , exp ( πiτ ℓ m n ) exp ( πiℓ m n + − tk + sm n ) z ) , where ℓ , ℓ are integers depending on the choice of lift. We can then calculate the energy of these lifts ofthe discs using the standard symplectic form ω C = i ( dz ∧ dz + dz ∧ dz ) on C ∖ { } , which is the form which provides the conformal symplectic structure on S ( A ) : we obtain A ( t, s ) = ∫ D ( ˜ u t,s ) ∗ ω C = π exp ( πiτ ℓ m ∞ n ) exp ( ( tk ∞ + sm ∞ ) n ) ; A ∞ ( t, s ) = ∫ D ( ˜ u t,s ∞ ) ∗ ω C = π exp ( πiτ ℓ m n ) exp ( ( − tk + sm ) n ) . We observe that the ratio A ( t, s ) m A ∞ ( t, s ) m ∞ = exp ( t ) ;is independent of the choice of lift and of s . Thus as t → −∞ , one can say that A ( t, s ) is becoming smallrelative to A ∞ ; as t → ∞ , A ∞ ( t, s ) is becoming small relative to A .We thus define a Fukaya category, denoted F ( A ) , with objects Lagrangians L in Y so that for t << ( e πi ( m ϕ t + k ϕ s ) ) = ∈ R / π Z when restricted to L and for t >> ( e πi ( − m ∞ ϕ t + k ∞ ϕ s ) ) = ∈ R / π Z when restricted to L , with the expectation that objects of such a category will correspond to coherent sheaveson S ( A ) . A formal definition of this A ∞ category will be given in Section 4.3.1. Stop removal and (partially-)wrapped categories.
There are restriction mapsCoh an S ( A ) → Coh an ( S ( A ) ∖ D ∞ ) = Coh an ( D ( m , k )) Coh an S ( A ) → Coh an ( S ( A ) ∖ D ) = Coh an D ( m ∞ , k ∞ ) ;Following the “stop removal” philosophy of [29] or [11], we expect that the first map corresponds to alocalization map F ( A ) → F ( m , k ) , where F ( m , k ) is a partially wrapped category with objects whichare Lagrangians which satisfy m ϕ t + k ϕ s → t <<
0; and similarly for the second map after aninterchange of the ends Y . Roughly speaking, F ( m , k ) should be the partially wrapped category associatedto a superpotential which counts discs that pass through D A ∖ D ∞ = D . Similarly, we expect that theinclusion C ∗ × E ⊂ D ( m , k ) corresponds to a localization map F ( m , k ) → F W r from a partially wrappedcategory to a fully wrapped category. We will define such localizations and show that there is a diagram ofthe form OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 11 (18) F ( m , k ) F ( A ) F W r F ( m ∞ , k ∞ ) These maps differ from the stop removal maps found in [10] in that they are not essentially surjective;we will show that the boundary conditions impose conditions on the homology classes of the Lagrangians ineach category. 4.
Admissible Lagrangians and perturbing Hamiltonians
Let Y = ( R × ( S ) , dt ∧ dϕ t + ds ∧ dϕ s ) , with coordinates ( t, s, ϕ t , ϕ s ) . Let π ∶ R → Y denote the coveringmap. The map Y → R × S given by the projection to the first two coordinates provides the SYZ fibrationseen in Section 3 on C ∗ × E ; we refer to the ϕ t and ϕ s coordinates as the angular coordinates. The term Lagrangian section refers to a section of this SYZ fibration. We will define three Fukaya categories on Y with increasingly relaxed boundary conditions on their objects. These boundary conditions will control thebehavior of the angular coordinates as ∣ t ∣ grows large.Fix the following notation for the two ends of Y ∶ for T > Y − T = {( t, s, ϕ t , ϕ s ) ∈ Y ∣ t < − T } ; Y + T = {( t, s, ϕ t , ϕ s ) ∈ Y ∣ t > T } . Let Y T = Y + T ∪ Y − T . For a subset N ⊂ Y , let N ± T = N ∩ Y ± T and let N T = N ∩ Y T . Let t, s, ϕ t , ϕ s ∶ Y → R denotethe corresponding coordinate functions.Define basis elements of R : ˆ e t = ( , , , ) , ˆ e s = ( , , , ) , ˆ e ϕ t = ( , , , ) , ˆ e ϕ s = ( , , , ) . Within thissection, let d ∶ R × R → R denote the Euclidean distance function.4.1. Admissible planar Lagrangians.Definition 4.1.1 (Admissible Lagrangians) . Let L be a Lagrangian submanifold of Y with following prop-erties: ● A lift of L to R is a section of the fibration R → R given by ( t, ˜ s, ̃ ϕ t , ̃ ϕ s ) → ( t, ˜ s ) . ● µ ( L ) = ∈ H ( L, Z ) . ● ( planarity ) L is planar at the ends of Y : that is, there exists T > L + T to R lieson the points of an affine plane P + , and similarly for L − T .Then L is an admissible Lagrangian . Definition 4.1.2.
Let ( m , k ) and ( m ∞ , k ∞ ) be two primitive elements in Z such that m , m ∞ >
0, andlet A denote the pair (( m , k ) , ( m ∞ , k ∞ )) . L is a type- A Lagrangian if L is an admissible Lagrangian suchthat there exists T > m ϕ t + k ϕ s ≡ ∈ R / Z on L T − ;(19) − m ∞ ϕ t + k ∞ ϕ s ≡ ∈ R / Z on L T + . (20)We will refer to this property – i.e., the property that the projection of the ends of L to the ( ϕ t , ϕ s ) planelie in a line – as linearity . We call the bounded region in which L does not satisfy the linearity conditionsthe non-linear region of L , and its complement the linear region of L .Let G ( A ) denote the set of type- A planar Lagrangians equipped with a grading (i.e., a lift of the phasemap L → S from the complex volume form on Y ) and a choice of Pin structures as in [25], with each Lagrangian remembering the choice of A . We will frequently abuse notation by suppressing the choice ofgrading and Pin structures, and denote an element of G by L . Let G ( m, k ) = ⋃ ( m ′ ,k ′ ) G (( m, k ) , ( m ′ , k ′ )) G = ⋃ ( A ) G ( A ) . Thus G is the set of linear admissible Lagrangians.4.2. Topology of Lagrangians.
The conditions above give control over the behavior of the Lagrangiansat the ends of Y . Let L be an admissible planar Lagrangian of type A . By the linearity and planarity of L ,as well as the fact that L is Lagrangian, there exists d − r − ∈ Q , E − ∈ R , T > L − T to R lieson the points of the affine plane P − = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ t ⎛⎜⎜⎜⎜⎝ − d − r − k m d − r − k ⎞⎟⎟⎟⎟⎠ + s ⎛⎜⎜⎜⎜⎝ d − r − k − d − r − m ⎞⎟⎟⎟⎟⎠ + E − ∣ t, s ∈ R ⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭ and L + T similarly lies on an affine plane P + . By considering the set L ∩ { t = t } for any fixed t , we obtainan element [ γ L ] ∈ H ( T ) . Using the natural basis for H ( T ) from the coordinates ( s, ϕ t , ϕ s ) , for aLagrangian with P + and P − of the above form and d ± r ± reduced with r ± >
0, we obtain that [ γ L ] = ( r − , d − k , − d − m ) = ( r + , − d + k ∞ , − d + m ∞ ) . We conclude that when m ∞ k + m k ∞ ≠ d + = d − = r + = r − =
1. In all cases we can write [ γ L ] as ( r, dk , − dm ) . This homology class is an invariant of the Lagrangian under any perturbation . Definition 4.2.1.
With L as above, let the quantity dr denote the slope of the Lagrangian, and let mdr denote the total slope of the Lagrangian L . Definition 4.2.2.
Let
L, L ′ ∈ G be in correct position. Fix lifts ˜ L and ˜ L ′ of L and ˜ L ′ ; define a subgroupΓ ( L, L ′ ) ⊂ Z by Γ ( L, L ′ ) = { ˆ n ∈ Z ∣ π ( ˜ L ∩ ˜ L ′ ) = π ( ˜ L ∩ ( ˜ L ′ − ˆ n ))} . Note that Γ ( L, L ′ ) is a topological invariant of L and L ′ . Definition 4.2.3 (Interpolating Functions) . We define the following functions, which will be necessary inthe subsequent lemma and throughout.(a) For a and b in R , choose monotonic functions ρ ba ∶ t → R such that ρ ba = ⎧⎪⎪⎨⎪⎪⎩ a t ≤ − b t ≥ , ; ∣ dρ ba ( t ) dt ∣ ≤ ∣ b − a ∣ ; ρ ba + ρ b ′ a ′ = ρ b + b ′ a + a ′ ρ ba ( ) = a + b ρ ba is linear with slope b − a on ( − / , / ) ;(b) Let α + , α − , ε + , ε − > α − + ε − = α + + ε + . Let σ α + ,ε + α − ,ε − ∶ R → R be a smooth function with σ α + ,ε + α − ,ε − ( t ) = ⎧⎪⎪⎨⎪⎪⎩( α + + ε + ) t t ∈ [ − / , / ] α ± t + ε ± ∣ t ∣ ± t ∈ [ , ∞ ) ;such that ∣ dσ α + ,ε + α − ,ε − ( t ) dt ∣ > α + t + ε + OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 13 for t ≥ /
2, and similarly for t < − /
2; and σ α + ,ε + α − ,ε − ( t ) + σ α ′+ ,ε ′+ α ′− ,ε ′− ( t ) = σ α + + α ′+ ,ε + + ε ′+ α − + α ′− ,ε − + ε ′− ( t ) . (c) Let χ ∶ R → [ , ] be a symmetric function with χ ( t ) = ⎧⎪⎪⎨⎪⎪⎩ ∣ t ∣ < ∣ t ∣ > . so that for t >
0, 0 ≤ χ ( t ) ≤ ∣ dχ ( t )/ dt ∣ < . For R ∈ R ∗ , let χ R ( t ) = χ ( t / R ) . Definition 4.2.4 (Cylic covers) . For r ∈ Z , let Y r → Y denote the r -fold cover where the deck transformationgroup Z / r Z acts as ( t, s, ϕ t , ϕ s ) → ( t, s + , ϕ t , ϕ s ) . Let C r = R × R / r Z ; S r = R / r Z , so S r includes into C r and Y r fibers over C r . Lemma 4.2.1.
Let L ∈ G ( A ) with γ L = [ r, dk, − dm ] , let ˜ L be a lift of L to R = T ∗ R , and let h ∶ R → R be a function so that ˜ L is the graph of the differential dh in R = T ∗ R . If d = , then we may write h as afunction of the following form: (21) h = as + tρ b + b − ( t ) + ρ c ( t ) + G ( t, ˜ s ) where a, b − , b + , c ∈ R , and G ∶ R → R is a function which vanishes for ∣ t ∣ >> with the property that G ( t, ˜ s + ) = G ( t, s ) .If d ≠ , we can write h in the following form: (22) h = − dm ( s − ( k / m ) t ) r + a ( s + ( k / m ) t ) + tρ b + b − ( t ) + ρ c ( t ) + G ( t, ˜ s ) where b − , b + ∈ m Z , c ∈ R , and G ∶ R → R is a function which vanishes for ∣ t ∣ >> with the property that G ( t, ˜ s + r ) = G ( t, s ) .Proof. Assume first that d =
0. Then necessarily r =
1, and L is a Lagrangian section. Write˜ L = { t, s, g ( t, s ) , g ( t, s ) ∣ ( t, s ) ∈ R × R } . We have seen that for ∣ t ∣ >> g and g take constant values; let g = b − for t << g = b + for t >>
0, andlet g = a − for t << g = a + for t >>
0. We can assume that h = b − t + a − s for t << I ∶ R → R defined by I ( t ) = ∫ g ( t, s ) ds. Note that I ( t ) = a − for t <<
0, and I ( t ) = a + for t >>
0. Also note that dIdt = ∫ ∂g ∂t ( t, s ) ds (23) = ∫ ∂g ∂s ( t, s ) ds (24) = . (25)Thus I ( t ) is constant with value a = a − = a + . Let c = ∫ ∞−∞ g ( t, ) − ddt ( tρ b + b − ( t )) dt ;this quantity is well-defined since the integrand is bounded and vanishes outside of a compact set. Nowconsider the real-valued function G defined by G ( t, s ) = h ( t, s ) − ( as + tρ b + b − ( t ) + ρ c ( t )) . We wish to show that G is a compactly supported function on R × R / Z . By construction, G vanishesidentically for t <<
0. For t >>
0, note that G ( t, s ) = ∫ t −∞ ( g ( τ, ) − ddτ ( τ ρ b + b − ( τ ) + ρ c ( τ ))) dτ + ∫ s ( g ( t, σ ) − a ) dσ = . Finally note that G ( t, s + ) − G ( t, s ) = − a + ∫ s + s g ( t, σ ) dσ = − a + I ( t ) = . Thus G is of the desired form.Now assume that ˜ L is a Lagrangian section and d ≠
0. Then necessarily ( m , − k ) = ( m ∞ , k ∞ ) = ( m, − k ) .We can write ˜ L = { t, s, g ( t, s ) , g ( t, s ) ∣ ( t, s ) ∈ R × R } where for ∣ t ∣ >> g = dkr ( s − ( k / m ) t ) + b ± g = − dmr ( s − ( k / m ) t ) + a ± . We can assume that for t << h = − dm r ( s − ( k / m ) t ) + b − t + a − s. Let h ′ = h + dm r ( s − ( k / m ) t ) g ′ = g − dkr ( s − ( k / m ) t ) g ′ = g + dmr ( s − ( k / m ) t ) . Then we may apply the above arguments to h ′ , g ′ , g ′ to conclude again that h ′ = as + tρ ( b + ) ′ ( b − ) ′ ( t ) + ρ c ( t ) + G ( t, ˜ s ) . Let ( b ± ) ′ = b ± + a ( k / m ) . The boundary conditions imply that m (( b ± ) ′ + a ( k / m )) + k a ∈ Z so m ( b ± ) ′ ∈ Z . Thus we may write h in the desired form: h = h ′ − dm r ( s − ( k / m ) t ) = − dm r ( s − ( k / m ) t ) + a ( s − ( k / m ) t ) + tρ ( b + ) ′ ( b − ) ′ ( t ) + ρ c + G ( t, ˜ s ) . Finally, assume that d ≠
0, but r ≠
1. Let L r denote a lift of L to Y r ; note that L r is a section of thefibration Y r → C r . Then the above methods will apply, and we can conclude that h is of the desired form. (cid:3) Perturbing Hamiltonians.
For a Hamiltonian function H ∶ Y → R , we denote the time-1 flow of theHamiltonian vector field X H by ψ H . We first define the class of Hamiltonians that we will use to perturbour Lagrangians. Definition 4.3.1. A perturbing Hamiltonian is a function H ∶ Y → R depending only on t and s such thatthere exists T > H ( t, s ) = α + t + ε + ∣ t ∣ + f + ( t, s ) on Y + T H ( t, s ) = α − t + ε − ∣ t ∣ + f − ( t, s ) on Y − T . where α ± ∈ Q , α ± , ε ± ≥ α − + ε + = α − + ε − , and f ± ∶ R × S → R are smooth functions that satisfy the following constraints:(a) there exists N ± ∈ N such that f ± ( t + mN ± , s ) = f ± ( t, s ) for all t, s ∈ Y T , for all m ∈ Z .(b) there exists t > T and κ ± > f ± ( t, s ) = ∣ t − t ∣ < κ ± . (Note that then f ( t, s ) = t > T such that there exists m ∈ N such that ∣ t − t + mN ∣ < κ .) For each such f , fix a κ ( f ± ) < OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 15 (c) ∥ f ± ∥ C < min { κ ( f ± ) , α ± } . We will use f ± ( H ) , α ± ( H ) , and ε ± ( H ) to mean the corresponding parameter of H at infinity. We definea partial order on perturbing Hamiltonians by setting H ′ > H if H ′ > H outside of a compact set. Definition 4.3.2.
A perturbing Hamiltonian is(a) a type- A Hamiltonian if H satisfies α ± ( H ) = f ± ( H ) = ≤ ε − < m ;0 ≤ ε + < m ∞ . (b) a type- ( m, k ) Hamiltonian if H satisfies α − ( H ) = f − ( H ) = . ≤ ε − ( H ) < m . Remark 4.3.1 (Universal notation) . In the next section we will define Fukaya categories F ( A ) , F ( m , k ) ,and F W r , the objects of which will (roughly) correspond to Lagrangians in G ( A ) , G ( m , k ) , and G , respec-tively. We will frequently have statements that will have versions for each category. We will shorten thesestatements by referring to the labelling by I ; the absence of a label implies that we are working in the W r setting.In each case, we refer to the ends of Y where the boundary conditions are relaxed as the wrapped ends , sothere are no wrapped ends in the A setting, one wrapped end in the ( m, k ) setting, and two wrapped endsin the W r setting.Let G P ( I ) denote the set of Lagrangians which may be written as ψ H ( L ) , where H is a type- I Hamil-tonian. Note that G P ( A ) ⊂ G P ( m , k ) ⊂ G P . We now begin investigating the properties of such perturbedLagrangians. Proposition 4.3.1 (Periodicity of Lagrangians at ends) . Let L ∈ G P . Then there exists T > and N ∈ N such that L ± T ± N ˆ e t = L ± T ± N . Proof.
This follows from the planarity of the Lagrangians, the rationality of α ± , and the periodicity of the f ± . (cid:3) Proposition 4.3.2. (a) Let H and H be type- A Hamiltonians such that H > H and let L and L betype- A Lagrangians. Then ψ H ( L ) and ψ H ( L ) are disjoint outside of a compact set.(b) Let H and H be Hamiltonians which are ( m, k ) -suitable for type- ( m, k ) Lagrangians L and L suchthat H > H . Then ψ H ( L ) and ψ H ( L ) are disjoint on Y − T .Proof. Note that points ( t, s, ϕ t , ϕ s ) on ψ H i ( L i ) with t << m ϕ t + k ϕ s = m ε − ( H i ) . Since H > H , 0 < ε − ( H ) − ε − ( H ) < m , and the two Lagrangians are disjoint for t <<
0. The same reasoning shows disjointness for t >> A setting. (cid:3) We cannot ensure disjointness of perturbed Lagrangians at an end with relaxed boundary conditions;instead we ensure transversality. We first note that any Lagrangian in G P is close to a plane determined bythe parameters α ± and ε ± . Let L ∈ G , and let H be a perturbing Hamiltonian. We can write H = H ′ + f ,where H ′ is quadratic at the ends, and the decomposition is unique at the ends. Then ψ H ′ ( L ) is planar atinfinity: there exists T > ψ H ′ (̃ L ) to R , there exist affine planes P + , P − ⊂ R suchthat ψ H ′ ( ˜ L ± T ) ⊂ P ± . Concretely, we may write(26) P ± = ⎧⎪⎪⎨⎪⎪⎩ x, ⎛⎝ − dr k m + α ± ( H ) dr k dr k − dr m ⎞⎠ x + E i ∣ x ∈ R ⎫⎪⎪⎬⎪⎪⎭ . Definition 4.3.3.
For a Lagrangian ˜ L in R which is the lift of a Lagrangian L in G P , we call the planes P + and P − the corresponding planes to ˜ L and to L ; we will also refer to the corresponding planes of theLagrangian L itself. In the latter case, the ambiguity in the lift will be immaterial.Since H ′ and f Poisson-commute, we may write ψ H ( L ± T ) = ψ f ( ψ H ′ ( L ± T )) . The C bounds on f bound the Hausdorff distance d H between a perturbed Lagrangian ψ H ( L ) and itscorresponding planes at infinity:(27) d H ( P ± T , ψ H ( L ± T )) ≤ ( ψ H ′ ( L ± T ) − ψ H ( L ± T )) ≤ ∥ ψ f ± − Id Y ± ∥ C ≤ ∥ Df ± ∥ C ≤ κ ± . Correct positioning of Lagrangians.
We now investigate to what extent we can ensure transversalityby imposing conditions on the perturbing Hamiltonians. The following lemma is key to many of the followingarguments:
Lemma 4.4.1.
Let L and L ′ be two admissible Hamiltonians of type A = (( m , k ) , ( m ∞ , k ∞ )) and A ′ = (( m ′ , k ′ ) , ( m ′∞ , k ′∞ )) and slope n and n ′ respectively.(a) If m n ≠ m ′ n ′ or k n ≠ k ′ n ′ , then there exists a constant K ( L, L ′ ) such that for any two perturbingHamiltonians H, H ′ such that α ± ( H ) − α ± ( H ′ ) > K ( L, L ′ ) , the corresponding planes P ± and ( P ± ) ′ of ψ H ( L ) and ψ H ′ ( L ′ ) intersect transversely.(b) If m n = m ′ n ′ and k n = k ′ n ′ then if H > H ′ , the corresponding planes P ± and ( P ± ) ′ of ψ H ( L ) and ψ H ′ ( L ′ ) intersect in a line with a constant value of t if α ± ( H ) − α ′± ( H ′ ) > . In this case set K ( L, L ′ ) = .Proof. Let P and P ′ denote the corresponding planes to the perturbed Lagrangians at the ± t << α = α ± ( H ) and α ′ = α ± ( H ′ ) , and let M ( α ) and M ′ ( α ′ ) be the symmetric 2 × p ∈ π ( P ∩ P ′ ) , P and P ′ intersect in a dimension 2 − rk ( M ( α ) − M ′ ( α )) set. Wenow decompose M ( α ) − M ′ ( α ′ ) as the sum of two matrices, only one of which depends on the quantities α and α ′ . Let µ = nm − n ′ m ′ , and κ = nk − n ′ k ′ . When µ ≠
0, we can write M ( α ) − M ′ ( α ′ ) = ( − ( κ ) / µ κκ − µ ) + ( n ( k ) / ( m ) − n ′ ( k ′ ) / ( m ′ ) + α − α ′
00 0 ) (28) ∶ = M + M ( α − α ′ ) ;(29)when µ = M ( α ) − M ′ ( α ′ ) = ( n ( k ) /( m ) − n ′ ( k ′ ) /( m ′ ) κκ ) + ( α − α ′
00 0 ) (30) ∶ = M + M ( α − α ′ ) . (31)Using this decomposition, one obtains thatdet ( M ( α ) − M ′ ( α ′ )) = − ( m ′ k − m k ′ ) nn ′ m m ′ + ( α − α ′ ) ( nm − n ′ m ′ ) = f ( α − α ′ ) from which we may deduce ( a ) . In the case where µ ≠
0, take K ( L, L ′ ) to be the solution to the equation f ( K ( L, L ′ )) =0; otherwise take K ( L, L ′ ) = µ = κ =
0, then M vanishes, so M ( α ) − M ′ ( α ′ ) has rank 1 if α − α ′ ≠
0. In this case one observes that
T P ∩ T P ′ ∣ p is spanned by ( , , nk , − nm ) ∈ R . (cid:3) Definition 4.4.1.
Let L and L ′ be two admissible Hamiltonians of type A = (( m , k ) , ( m ∞ , k ∞ )) and A ′ = (( m ′ , k ′ ) , ( m ′∞ , k ′∞ )) . Let H and H ′ be two Hamiltonians perturbing L and L ′ . The ordered pair ofLagrangians ( ψ H ( L ) , ψ H ′ ( L ′ )) is in correct position if ψ H ( L ) and ψ H ′ ( L ′ ) intersect transversely, if for allbut finitely many points y ∈ L ∩ L ′ , y lifts to a point ˜ y ∈ ˜ L ∩ ˜ L ′ with ∣ π ( ˜ L ∩ ˜ L ′ ) ∣ ≤
2, and:
OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 17 (a) In the case where A = A ′ : if H − H ′ > H − H ′ is an admissible Hamiltonian, the corresponding planesof ψ H ( L ) and ψ H ′ ( L ′ ) do not coincide at either end, and any two lifts ψ H ( ˜ L ) and ψ H ′ ( ˜ L ′ ) to theuniversal cover are disjoint away from a compact set.(b) In the case where A ≠ A ′ : if H − H ′ > H − H ′ is an admissible Hamiltonian, α − ( H ) − α − ( H ′ ) > K ( L, L ′ ) if ( m , k ) ≠ ( m ′ , k ′ ) , α + ( H ) − α + ( H ′ ) > K ( L, L ′ ) ,and any two lifts ψ H ( ˜ L ) and ψ H ′ ( ˜ L ′ ) to the universalcover are disjoint away from a compact set.An ordered collection { L , L , L , . . . , L n } is in correct position if each pair ( L i , L i + ) is in correct positionfor 0 ≤ i ≤ n − Definition 4.4.2.
Let L and L ′ be in correct position. We say that L and L ′ are essentially transverse atan end of Y if their corresponding planes intersect in a point on that end, essentially non-transverse at anend if their corresponding planes intersect in a line, and essentially disjoint at an end if their correspondingplanes do not intersect. Note that in the fully wrapped case, L and L ′ are either essentially transverse atboth ends of Y , or essentially non-transverse at both ends of Y . In this case being essentially transverse isequivalent to satisfying [ γ L ] ≠ [ γ L ′ ] .We also have a notion of two lifts ˜ L and ˜ L ′ being essentially transverse in any setting. Given two suchlifts ˜ L and ˜ L ′ , ˜ L and ˜ L ′ are I -essentially transverse , I -essentially non-transverse , and I -essentially disjoint if there exists a type- I Hamiltonian H such that ψ H ( ˜ L ) and ˜ L ′ intersect in a point, intersect in a line, ordo not intersect, respectively.4.5. Associated quadratic forms and action.
Let
L, L ′ ∈ G P be two Lagrangians in correct position.Fix lifts ˜ L and ˜ L ′ of L and L ′ in R , and corresponding planes ( P ± ) , ( P ± ) ′ of these lifts. Definition 4.5.1. (a) Let µ and κ be defined as above for the Lagrangians L and L ′ . Define the coordinatefunction w L,L ′ ∶ R → R by w L,L ′ = s − ( κ / µ ) t when µ ≠
0; otherwise let w L,L ′ = s .(b) Consider the quadratic form defined by x T ( M − M ′ ) x ; denote this form by Q L,L ′ , + ( x ) , and define Q L,L ′ , − ( x ) similarly. We can write this sum of two quadratic forms using the decomposition of Equations(28) and (30): let Q βL,L ′ , + ( x ) = x T M x , and Q αL,L ′ , + = x T M x . Then except in the case where µ = κ ≠
0, the coordinates ( w L,L ′ , t ) diagonalize the quadratic forms. In this case we can write Q L,L ′ , ± ( x ) = Q βL,L ′ , ± ( x ) + Q αL,L ′ , ± ( x ) Q βL,L ′ , ± ( x ) = βw L,L ′ ( x ) Q αL,L ′ , ± ( x ) = α ± t ( x ) . Remark 4.5.1.
The quadratic form Q βL,L ( x ) is an invariant of the homology classes [ γ L ] , [ γ L ′ ] . Thus to anytwo Lagrangians, we can associate such a quadratic form without the data of the perturbing Hamiltonians. Remark 4.5.2.
On any end of Y , Q L,L ′ , ± is degenerate as a quadratic form on R if and only L and L ′ are essentially non-transverse at that end. On the wrapped end(s) of Y , Q L,L ′ , ± has signature ( , ) when mn − m ′ n ′ <
0, signature ( , ) when mn − m ′ n < mn − m ′ n ′ = kd − k ′ n ′ ≠
0, and is degenerate andpositive semi-definite when mn − mn ′ = kn − k ′ n ′ =
0. In the fully wrapped setting, the signatures of Q L,L ′ , + and Q L,L ′ , − agree. Lemma 4.5.1. (a) Let h and h ′ be two primitives for ˜ L and ˜ L ′ . Then for ± t >> , we can write ( h − h ′ ) ( t, s ) = Q L,L ′ , ± ( t, s ) + at + bs + ρ c ( t ) + f ( t, s ) + G ( t, s ) ; = αt + Q βL,L ′ ( t, s ) + at + bs + ρ c ( t ) + f ( t, s ) + G ( t, s ) , where a, b, c ∈ R , α ≥ , ∥ f ∥ C < , and G ( t, s ) is a compactly supported function on C R for some R ∈ N .On any end with wrapped boundary conditions of the form mϕ t + kϕ s ≡ , this takes the form βw + aw + ρ c ( t ) + f ( t, s ) + G ( t, s ) . (b) For ± t >> , h − h ′ can be rewritten as Q L,L ′ , ± ( y ) + Q L,L ′ , ± ( y − y ) + ρ c ( t ) + G ( t, s ) + f ( t, s ) where y is an intersection point of the corresponding planes P + of ˜ L and ( P + ) ′ of ˜ L ′ .Proof. (a) follows from Lemma 4.2.1 and the decomposition of Definition 4.5.1. (b) follows from the factthat any function F ∶ R → R of the form F ( x ) = x T M x + ( e , e ) ⋅ x can be expanded around a criticalpoint x as x T M x + ( x − x ) T M ( x − x ) . (cid:3) Definition 4.5.2.
Let y ∈ L ∩ L lift to ˜ y ∈ ˜ L ∩ ˜ L ′ . Let h and h ′ be primitives for ˜ L and ˜ L ′ of the formprescribed by 4.2.1. Define S R ( ˜ y ) by S R ( ˜ y ) = ( h − h ′ ) ( ˜ y ) . Definition 4.5.3.
Recall the quadratic form Q αL,L ′ +± given in the decomposition of Equation (28). Definethe action of a point y ∈ ( L ∩ L ′ ) with ± t ( y ) > S ( y ) = Q αL,L ′ , ± ( ˜ y ) = α ± t . Note that this action is defined regardless of the chosen lift of ˜ y , since all lifts occur at the same t -value. Proposition 4.5.2. (a) There exists C ∈ R such that for all ˆ n ∈ Z , for all ˜ y ∈ ˜ L ∩ ( ˜ L ′ − ˆ n ) with ± t ( y ) ≥ , d ( ˜ y, ( P ± ) ∩ (( P ± ) ′ − ˆ n )) < C . (b) There exists C > such that for all ˜ y ∈ ˜ L ∩ ( ˜ L ′ − ˆ n ) with ± t ( ˜ y ) > , ∣ S R ( ˜ y ) − Q L,L ′ , ± ( ˜ y )∣ < C . (c) If L and L ′ are essentially non-transverse at an end, then there exists C > such that for all ˜ y ∈ ˜ L ∩ ˜ L ′ , ∣ S R ( ˜ y ) − S ( y )∣ < C . Proof. (a) This follows from Equation (27): there exists D > y , there exist x ∈ P ± , x ′ ∈ ( P ± ) ′ − ˆ n such that d ( ˜ y, x ) , d ( ˜ y, x ′ ) < D . There is a constant D > n such that forany pairs of points x ∈ P ± , x ′ ∈ ( P ± ) ′ − ˆ n with d ( x, x ′ ) < C , d ( x, ( P ± ∩ (( P ± ) ′ − ˆ n ))) < CD . Taking C = D D + D gives the result.(b) The quantity ∣ S R ( ˜ y ) − Q L,L ′ , ± ( ˜ y )∣ depends only on the equivalence class ˆ n ∈ Z / Γ ( L, L ′ ) . If L and L ′ are essentially disjoint on the ± t → ∞ end, then there are only finitely many classes ˆ n ∈ Z / Γ ( L, L ′ ) sothat ˜ L and ˜ L ′ − ˆ n intersect and satisfy ˜ L ∩ ( ˜ ′ L − ˆ n ) ⊂ Y ± . Otherwise, by Lemma 4.5.1, we can write D + = max ∣ x ∣ < C ( Q L,L ′ , + ( x )) + ∥ ρ c + G ( t, s ) + f ( t, s ) ∥ C . We similarly obtain a bound D − on the t > Y . Take C = max { D + , D − } .(c) This follows from (a) and (b), using the fact that in this case Q αL,L ′ , ± = Q L,L ′ , ± . (cid:3) Lemma 4.5.3.
Fix a lift ˜ L of L , and let H be a Hamiltonian such that the set ( ψ H ( L ) , L, L ′ ) is in correctposition. Assume also that Q β is a degenerate quadratic form equal to βw L,L ′ for some β ∈ R , where w L,L ′ is the coordinate defined in Lemma 4.5.1. Then there exists D > such that for all ˆ n ∈ Z , if y ∈ ˜ L ∩ ( ˜ L ′ − ˆ n ) ,and y ∈ ψ H ( ˜ L ) ∩ ( ˜ L ′ + ˆ n ) , then ∣( S R ( y ) − S ( y )) − ( S R ( y ) − S ( y ))∣ < D. If we further assume that L and L ′ lift to planes in R , then ∣( S R ( y ) − S ( y )) − ( S R ( y ) − S ( y ))∣ = . Proof.
Let P ± , P ± H , and ( P ′ ) ± denote the ( ± ) − corresponding planes of ψ H ( ˜ L ) , ˜ L , and ˜ L ′ , respectively. FromProposition 4.5.2, it suffices to show that for all ˆ n ∈ Z , for x ∈ P ± ∩ (( P ′ ) ± − ˆ n ) , x ∈ ( P ± H ( P ′ ) ± − ˆ n ) , ∣( Q L,L ′ , + ( x ) − Q αL,L ′ , + ( x )) − ( Q ψ H ( L ) ,L ′ , + ( x ) − Q αψ H ( L ) ,L ′ , + ( x ))∣ = ∣ Q βL,L ′ ( x ) − Q βL,L ′ ( x )∣ = β ( w L,L ′ ( x ) − w L,L ′ ( x ) ) = OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 19
First assume that t ( y ) >
0. Lift to the universal cover, and work in the t, w
L,L ′ coordinates. In thesecoordinates, the intersection of the corresponding planes P and P ′ + ˆ n is given by x = ( t , w ) solving ( α β ) ( t w ) + ( e e ) = P H and P ′ + ˆ n is given by x = ( t , w ) solving ( α + α + ( H ) β ) ( t w ) + ( e + ε + ( H ) e ) = . Thus βw L,L ′ ( x ) = βw L,L ′ ( x ) ; when β ≠ w L,L ′ ( x ) = w L,L ′ ( x ) . When L and L ′ are planar, one has x = y , x = y , and S R ( x i ) = Q L,L ′ , + ( x i ) = Q L,L ′ , − ( x i ) . (cid:3) A ∞ structures J-holomorphic curves.
The Fukaya categories we consider are A ∞ categories where the higher prod-ucts µ n are defined by counts of pseudo-holomorphic curves. In our case, the definition of the µ n productsin our A ∞ categories requires the use of specific almost complex structures, the definition of which reflectsthe controls we impose on the ends of Y . Definition 5.1.1. An eventually constant almost complex structure is an ω -compatible almost complexstructure for which there exists T and J + , J − ∈ J ( R , ω std ) for which for all points p ∈ Y + T , J ∣ p = J + , andfor all points p ∈ Y − T , J ∣ p = J − , using the natural trivialization T Y = Y × R . Let J EC denote the set ofeventually constant J .Recall that the choice of an almost complex structure J on Y will induce a metric g J defined by g J ( v, w ) = ω ( v, Jw ) ; an almost complex structure is eventually constant if and only if the metric tensor is eventuallyconstant. Proposition 5.1.1.
The space of eventually constant almost complex structures is contractible.Proof.
The linear homotopy between any J and a fixed J , constructed using the identification of J with g J , is through the space of eventually constant almost complex structures. (cid:3) We will use the notation g E to denote the Euclidean metric on Y pushed forward from the Euclideanmetric on R . We use d J ( ⋅ , ⋅ ) to denote the distance function induced from J , and d E ( ⋅ , ⋅ ) to denote thedistance function induced by g E . We use the notation B J ( y, R ) ⊂ Y to denote the g J ball of radius R around y ′ , i.e. the set of points y ′ ∈ Y such that d J ( y, y ′ ) < R , and we use the notation B E ( y, R ) to denote theEuclidean ball centered at y. Note that at each point y ∈ Y there exists a constant D y < v, w ∈ T Y ∣ y ,(32) 1 D y g E ( v, w ) > g J ( v, w ) > D y g E ( v, w ) . Let J be a family of eventually constant complex structures over a compact base; then there exists a universalconstant D J such that (32) holds with D J at all points, for all J ∈ J . Then B J ( y, RD ) ⊆ B E ( y, R ) for all y ∈ Y .For a compact family J , and collection of Lagrangians L , . . . , L n ∈ G P , L i = ψ H i ( L ′ i ) , fix T lin ( L , L . . . , L n , J ) > L ′ i , H i ,and J ∈ J are satisfied on Y T lin ( L , L . . . , L n , J ) .5.1.1. Bounds on J-holomorphic curves.
Our boundary conditions can be used to derive bounds which willcontrol the image of any J -holomorphic curve in a fixed homology class. These bounds will be usefulthroughout, and in particular will be used to show that the moduli spaces we use to define our structuremaps are well-behaved.Let ( L , . . . , L n ) be a collection of Lagrangians G P ( I ) in correct position, and let y i ∈ L i ∩ L i + , ≤ i ≤ n − y n ∈ L ∩ L n . Let S be a disc with boundary marked points { p , . . . , p n } , and let ∂ i S denote the boundarycomponent of S so that p i − , p i ∈ ∂ i S . Let J = { J s } s ∈ S be a family of eventually constant almost complex structures parametrized by S such that there exists J such that J s = J outside of a compact subset of Y forall J s . Let u ∶ S → Y be a map satisfying ∂ J u = u ( ∂ i S ) ∈ L i (34) u ( p i ) = y i . (35)We will use counts of such J -holomorphic u to define the A ∞ structures on our Fukaya categories. Thefeatures of our Lagrangians give us controls over the behavior of such u . In the A setting, we can ensurethat J -holomorphic curves lie within a compact region in the same way that one does when defining aFukaya-Seidel category: Lemma 5.1.2.
Let L , L . . . , L n ∈ G P ( A ) be a collection of transverse Lagrangians in G P ( A ) in cor-rect position. Then the image of any u satisfying Equations (33) , (34) and (35) is contained within Y ∖ Y T lin ( L ,L ...,L n , J ) .Proof. Let J + ∶ R → R denote the complex structure on R so that J s ∣ y = J + for any s ∈ S and point y such that t ( y ) > T lin ( L , L . . . , L n , J ) , and g J + denote the induced metric on R . The non-degeneracy of ω implies that there exists v ∈ R such that for any w = ( r , r , r , r ) ∈ R , ω ( v, w ) = m ∞ r − k ∞ r for all w ∈ R . Consider the linear map Π v ∶ R → C defined by Π v ( w ) = ( g J + ( v, w ) , ω ( v, w )) . Note thatΠ v ( J + w ) = ( ω ( v, J + J + w ) , ω ( v, J + w )) = ( − g J + ( v, w ) , ω ( v, w )) = i Π ( w ) . so Π v is J + -holomorphic. Now, observe that the image of any lift (̃ L j ) + T lin ( L ,L ...,L n , J ) under Π v is a ray (bythe fact that L j is planar at infinity and Lagrangian), which is contained within the line { z ∈ C ∣ Im z = m ∞ ε + ( H j ) + n } for some n depending on the lift; when i ≠ j these lines are disjoint by the conditions on the perturbingHamiltonians H j . Since the compositionΠ z ○ ˜ u ∶ ˜ u − ({ t ≥ T lin ( L , L . . . , L n , J )}) → C is holomorphic on the interior of S and has compact image with boundary contained within a finite collectionof parallel lines, we conclude that ˜ u − ({ t ≥ T lin ( L , L . . . , L n , J )}) = ∅ . Thus the image of u is containedwithin the region { y ∈ Y ∣ t ( y ) < T lin ( L , L . . . , L n , J )} . Applying the same argument at the negative endgives the result. (cid:3) After we relax the boundary conditions, we can no longer constrain all holomorphic curves to lie within acompact set via a holomorphic projection. We instead rely on the following bound, which we obtain by usingthe planar regions of each Lagrangian which are ensured by the vanishing conditions on the non-quadraticpart of the perturbing Hamiltonians.
Lemma 5.1.3.
Let L , L . . . , L n ∈ G P be a collection of transverse Lagrangians in G P in correct position;write L i = ψ H i ( L ′ i ) . Let y i ∈ L i ∩ L i + , ≤ i ≤ n , and let u ∶ S → Y be a J -holomorphic map satisfyingEquations (33) , (34) , and (35) for J = { J s } s ∈ S . For T > , let ℓ + T ( u ) denote the length of the interval ( t ○ u ) − ( Y + T ) ⊂ R , and ℓ − T ( u ) denote the length of the interval ( t ○ u ) − ( Y − T ) ⊂ R . Let ℓ T ( u ) = ℓ + T ( u ) + ℓ − T ( u ) . Then there exists R > , M > , and T B > T lin ( L , L . . . , L n , J ) , depending only on the collection ( L , L . . . , L n , J ) , such that for all such u with ℓ T B ( u ) > Mℓ T B ( u ) ≤ R ∫ S u ∗ ω. Proof.
We first fix several pieces of geometric data about the collection ( L , L . . . , L n , J ) . First, let ε > ≤ i < j ≤ n , two lifts ̃ L i and ̃ L j are a distance at least ε away from each otheroutside of a compact set. For two corresponding planes which have translates that intersect, observe thatthere exists C ij > P ± i ( a i , b i ) , P ± j ( a j , b j ) which intersect at a fixed t -value t i,j,a i ,b j ,a j ,b j , for all y ∈ P ± i ( a i , b i ) , C ij ∣ t ( y ) − t i,j,a i ,b j ,a j ,b j ∣ < d E ( y, P ± j ( a j , b j )) . OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 21
Let C = min { C ij } .Consider now the collection of small perturbing functions f i + = f + ( H + i ) . Let N i denote the period of f i + ,and let κ i denote κ ( f i + ) . Note that for every point y ∈ L i with t ( y ) > T lin ( L , L . . . , L n , J ) , d E ( y, P + i ) < κ i .Let N = N ⋅ N ⋯ N n > . Finally let γ = min { κ i / , ε, δ } . Let J + ∈ End ( R ) denote the limiting complex structure of the family J at the t >> Y . For eachaffine corresponding plane P + i ⊂ R , choose v i ∈ R with g J + ( v i , v i ) = v i is g J + -orthogonal to thetranslate of P + i passing though the origin. Let Π v i ∶ R → C denote the map defined above:Π v i ( v ) = ( g J + ( v i , v ) , ω ( v i , v )) . Then there exists ι ∈ R such that Π v i ( P + i ) = { z ∈ C ∣ Re ( z ) = ι } , i.e., P i maps to a vertical line under Π v i . Let w i and ι ′ be such that ω ( v i , w i ) = g J + ( v i , Jw i ) = g J + ( w i , w i ) = g J + ( P + i , w i ) = ι ′ . Then the holomorphic lines spanned by v i and w i are orthogonal. One can then verify that the mapΠ v i ⊕ Π w i ∶ ( R , ω, J + ) → ( C ⊕ C , ω C ⊕ ω C , i ⊕ i ) is an isomorphism of K¨ahler vector spaces, and the image of P + i under such a map is the product of twovertical lines.Now, let u ∶ S → Y be such a J -holomorphic map so that u − ( Y + T lin ( L ,L ...,L n ,J ) ) ≠ ∅ . Lift u to a map˜ u ∶ S → R with boundary on lifts ̃ L i , and let P + i be the corresponding planes to these lifts. When P + i and P + j intersect, let t ij = t ( P + i ∩ P + j ) .Note that u satisfies a maximum principle for the t -coordinate by the argument above, so any t -valueachieved by u ( S ) is achieved by a point z ∈ ∂S . For each i , 0 ≤ i ≤ n , consider the interval I i = t ( ∂ i S ) ⊂ R ;let ℓ i denote the length of the interval I i . Note that n ∑ i = ℓ i ≥ ℓ + T lin ( L ,L ...,L n , J ) ( u ) , so there exists some i such that ℓ ( I i ) ≥ ℓ + T lin ( L ,L ...,L n , J ) ( u ) n . Then there exists some subinterval I ⊂ I i of the form I = [ t ij , t ij ′ ] with j, j ′ ≠ i so that t ij ′′ ∉ ( t ij , t ij ′ ) forall j ′′ ≠ i , and ℓ ( I ) ≥ ℓ ( I i ) n ≥ ℓ + T lin ( L ,L ...,L n , J ) ( u ) n . We now choose a sequence of points y , . . . , y K ∈ ˜ u ( ∂ i S ) such that ● t ( y k ) ∈ I for all k ; ● t ( y k ) = ( t ) i + m k N , m k ∈ N , and m k ≠ m k ′ for k ≠ k ′ ; ● ∣ t ( y k ) − t ij ∣ > γ + δC for i ≠ j ● K is large: K ≥ ℓ ( I ) N − N − γ + δC ≥ ℓ + T lin ( L ,L ...,L n , J ) ( u ) N n − N − γ + δC . Since ∣ f i ∣ < κ i , we observe that if y ∈ ̃ L i satisfies d E ( y, y k ) < κ , then y ∈ P + i . Note also that for all j ≠ i , d E ( y k , P j ) ≥ C ∣ t ( y k ) − t ij ∣ > γ + δ so d E ( y k , ̃ L i ) > d ( y k , P i ) − δ > γ. Thus each y k is a distance at least γ away from ̃ L j for all j ≠ i , and is a distance at least κ > γ away fromthe non-planar region of ̃ L i .Now, for each y k , consider the set Q k = B E ( y k , D J + γ ) . By construction, Q k ∩ Q k ′ = ∅ if k ≠ k ′ , and Q k ∩ ̃ L i ⊂ P + i .We also conclude that Q k ∩ ̃ L j = ∅ for j ≠ i . Finally, note that ̃ L i is planar within Q k , i.e. ̃ L i ∩ Q k = P + i ∩ Q k . Let U k = ˜ u − ( Q k ) ⊂ S . The above shows that ∂S ∩ U k ⊂ ∂ i S , and ˜ u ( ∂S ∩ U k ) ⊂ P + i . By Gromov’smonotonicity lemma ([12]) and the reflection principle, we can bound ∫ U k ˜ u ∗ ω ≥ π ( D + J γ ) . Note that the quantity D J + γ depends only on J + and the Lagrangians L , . . . , L n , and the choice of L i .Letting R denote the minimum over this quantity Dγ over such a choice of I and over both ends, and taking T B = T gives the result. (cid:3) Corollary 5.1.4.
Let A > , let S be a compact family of discs with n boundary marked points, let J ∶ S →J EC be a family of eventually constant almost complex structures, and let y ∈ Y . Then there exists a compactregion V ( A, y, S , J ) ⊂ ( Y ) such that all u ∶ S → Y which satisfy Equations (33) , (34) , (35) with y ∈ u ( S ) , S ∈ S , and which have ∫ S u ∗ ω < A , satisfy u ( S ) ⊂ V ( A, y, S , J ) . Proof.
We can find a universal R > T B so that Lemma 5.1.3 is satisfied for this family. Then if y ′ ∈ Y has ∣ t ( y ′ ) ∣ > ( T B + ∣ t ( y ) ∣ + + A ) ( R + ) and y ′ ∈ u ( S ) , then ∫ S u ∗ ω ≥ ℓ T B ( u ) R > A. (cid:3) Corollary 5.1.5.
The spaces M A ( y , . . . , y n , [ β ] , J ) defined by ⋃ ω ([ β ]) < A M A ( y , . . . , y n , [ β ] , J ) . are compact.Proof. We have excluded sphere and disc bubbling by our conditions on the Lagrangians. It suffices to showthat all ( r, u ) ∈ M A ( y , . . . , y n , [ β ] , J ) are contained within some compact region V A ⊂ Y . But this followsfrom the fact that the image of each such u passes through the point y and must have ℓ T B ( u ) ≤ Rω ([ β ]) < RA where T B and R are above. The result then follows by Gromov compactness. (cid:3) Moduli spaces of J holomorphic curves. The definition of the moduli spaces that we use to definethe µ n will be exactly the same as in the literature (c.f. [25]), but our boundary conditions force us toslightly adapt some standard arguments that are used to show that counts are well-defined. Let { L σ } σ ∈ Σ be a set Lagrangians in G P indexed by a partially ordered set Σ with the property that any collection ( L σ , L σ , . . . L σ n ) with σ > σ > . . . > σ n is in correct position. We inductively define compactifications ofmoduli spaces of J -holomorphic curves using this set of Lagrangians.We first define the moduli spaces of bigons. Let L , L ∈ { L σ } σ ∈ Σ be two Lagrangians with L > L . Let y , y ∈ L ∩ L . Let [ β ] ∈ π ( M, L , L ) . Let J = { J x } x ∈ [ , ] ∈ J EC be a family of eventually constantalmost complex structures with the property that there exists J ∈ J EC such that for all x ∈ [ , ] , J x = J outside of a compact set. OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 23
Let ̂ M ( y , y , [ β ] , J ) denote the moduli space of solutions u ∶ R × [ , ] → Y in the class [ β ] ∈ π ( M, L , L ) to the equation ∂ J u = u ( R × { }) ⊂ L , u ( R × { }) ⊂ L ;(37) lim t →∞ u ( t, x ) = y , lim t →−∞ u ( t, x ) = y . (38)Let M ( y , y , J ) denote the quotient of ̂ M ( y , y , J ) by the action of R by re-parametrization. For generic J ( L , L ) , the moduli space M ( y , y , [ β ] , J ( L , L )) is smooth, and admits a Gromov compactificationby broken strips (since we have excluded sphere or disc bubbling by the condition that ⟨ ω, π ( Y, L i )⟩ = u lie within a fixed compact region by Corollary 5.1.4).We then define moduli spaces which will we use to define higher products in the Fukaya categories. Let R n + be the Deligne-Mumford-Stasheff compactification of the space of discs with n + S n + denote the universal family over the disc. Denote the boundarymarked points on such a disc by p , . . . , p n . We choose strip-like ends ε n + j ∶ R + × [ , ] × R n + → S n + and ε n + j ∶ R − × [ , ] × R n + → S n + which parametrize the disc near the marked points. We demand that these ends are compatible near theboundary strata: near each boundary face we have gluing maps ( R, ∞ ) × R n + × R k + × → R n + k , and we require that on these boundary collars ε n + kj agrees with the induced strip-like end for sufficientlylarge R .Assume that we have chosen domain dependent almost complex structures J L ,L for each pair of La-grangians in the set of Lagrangians we are considering such that the moduli spaces of maps M ( y , y , [ β ] , J ( L , L )) above are compact manifolds of the expected dimension. For each set of Lagrangians L , . . . , L n ∈ { L σ } σ ∈ Σ with L > . . . > L n , we now choose domain-dependent eventually constant almost complex structures J ( L , . . . , L n ) ∶ S → J EC inductively (see [25]), demanding compatibility: we require that J ( L , . . . , L n ) agree near each boundary puncture with the complex structure given by the strip-like coordinates, and thatabove R n + × R k + , J ( L , L , ⋯ , L n + k ) be compatible with the decomposition of S n + k . We also demand thatin each fiber of S n + over R n + the almost complex structures be equal to a fixed almost complex structureaway from a compact set. (This is an implicit assumption on all the families of eventually constant almostcomplex structures we will consider.) Note that for each such set ( L , . . . , L n ) , there exists a uniform T suchthat for each s ∈ S , J ( L , . . . L n ) s is linear on Y T .Now, let L , L , . . . , L n be such a collection of Lagrangians. Let y n ∈ L i − ∩ L i for 0 ≤ i ≤ n −
1, and let y n ∈ L ∩ L n . Let [ β ] ∈ π ( Y, L , . . . , L n ) . Define M ( y , . . . , y n − , y n , [ β ] , J ( L , . . . , L d )) to be the set ofpairs ( r, u ) where r ∈ R n + and u is a map from the fiber in S n + over r satisfies which Equations (33), (34),(35) for J ( L , . . . , L n ) . For generic J ( L , . . . , L n ) , this space is smooth and compact (where again we relyon Lemma 5.1.3 to prove compactness).5.2. A ∞ products. Throughout this subsection, assume that any collection ( L , . . . , L n , J ) is a collection ofLagrangians in I with L > L > . . . > L n , and J = J ( L , . . . , L n ) is a domain-dependent family of eventuallyconstant complex structures so that the moduli spaces M ( L , . . . , L n , [ β ] , J ) are well-defined.Recall the definition of the Novikov field Λ:Λ = { ∑ n ∈ N ξ n t λ n ∣ ξ n ∈ C , λ n ∈ R , λ n < λ n + , λ n → ∞ . } The Novikov field has a valuation v ∶ Λ → R v ( ∑ n ∈ N ξ n t λ n ) = min n { λ n ∣ ξ n ≠ } which has the property that for all a, b ∈ Λ, v ( a + b ) ≥ min ( v ( a ) , v ( b )) and v ( ab ) = v ( a ) + v ( b ) . For L , L ∈ G P ( I ) such that the pair ( L , L ) is in correct position, define the following Λ module: CF ∗ ( L , L ) = ⊕ y n ∈ L ∩ L Λ ⋅ y n This module has a Z -grading which originates from the grading on the Lagrangians L and L (see [7]). Remark 5.2.1.
Since our Lagrangians lift to sections of the fibration T ∗ R → R , they have a canonicalgrading from the Morse indices of the critical points of their primitives; see [24]. Definition 5.2.1 (Topological grading) . CF ∗ ( L , L ) has an additional grading. Fix lifts ˜ L and ˜ L of L and L ; recall that we define Γ ( L , L ) ⊂ Z byΓ ( L , L ) = { ˆ n ∈ Z ∣ π ( ˜ L ∩ ˜ L ) = π ( ˜ L ∩ ( ˜ L − ˆ n ))} . Define a Z / Γ ( L , L ) grading on CF ∗ ( L , L ) by setting CF ∗ ( L , L ) ˆ n = ⊕ y ∈ π ( ˜ L ∩ ( ˜ L − ˆ n )) Λ y. For y ∈ CF ∗ ( L , L ) , we will use y ˆ n to denote the projection of y to the ˆ n graded piece. We will also write Y ⊃ ( L ∩ L ) ˆ n = π ( ˜ L ∩ ( ˜ L − ˆ n )) . Completions of the Floer complexes.
Throughout this subsection, let L , L ∈ G P be two La-grangians in correct position. We define two completions of the Floer complex CF ∗ ( L , L ) ; along the waywe will use the following definitions: Definition 5.3.1.
For y ∈ L ∩ L , we used the boldfaced y to denote the action-corrected generator T − S ( y ) y . Definition 5.3.2. r ∈ R , r >
0, and define the following C -subalgebra of the Novikov field:Λ r = { ∑ c n T λ n ∣ λ n ∈ R , ∑ ∣ c n ∣ e − πrλ n < ∞ } . Define an evaluation map ev τ ∶ Λ r → C for every τ ∈ C with Im τ ≥ r byev τ ( ∑ c n T λ n ) = ∑ c n e πiτλ n . For r >
0, define ev r = ev ir .We first define a completion over Λ: Definition 5.3.3. (39) ̂ CF ∗ ( L , L ) = ⎧⎪⎪⎨⎪⎪⎩ ∑ y i ∈ L ∩ L c i y i ∣ c i ∈ Λ , lim t ( y i ) →±∞ v ( c i ) → ∞ ⎫⎪⎪⎬⎪⎪⎭ . Definition 5.3.4.
Define the following Λ r -submodule of ̂ CF ∗ ( L , L ) : ̂ CF ∗ r ( L , L ) = ⎧⎪⎪⎨⎪⎪⎩ ∑ y i ∈ L ∩ L c i y i ∣ c i ∈ Λ r , lim ∣ t ( y i )∣ →∞ ∣ ev r ( c i ) ∣ /∣ t ( y i )∣ = ⎫⎪⎪⎬⎪⎪⎭ . The periodicity of the Lagrangians L and L allows us to define an indexing by a subset of Z on the setof generators y ∈ L ∩ L so that there exists constants M, C > M i − C < t ( y i ) < M i + C . Thenthe following follows from the fact that a Laurent series ∑ i ∈ Z a i z i converges everywhere on C ∗ if and only iflim ∣ i ∣ →∞ ∣ a i ∣ − / i = Proposition 5.3.1. ∑ y i ∈ L ∩ L c i y i defines an element of ̂ CF ∗ r ( L , L ) if and only ∑ ev r ( c j ) e − πcrt ( y j ) converges for all c ∈ R . Definition 5.3.5.
For τ ∈ C with Im τ >
0, define a graded C -complex ̂ CF ∗ τ ( L , L ) via the map ev τ on ̂ CF ∗ Im τ ( L , L ) . OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 25
Convergence of Floer products in the Novikov ring completion.Proposition 5.4.1. (a) Let y i ∈ L i ∩ L i + , ≤ i ≤ n − , y n ∈ L ∩ L n . Then the sum ∑ [ β ] ∈ π ( Y,L ,L ,...,L n ) T ω ([ β ]) M ( y , . . . y n − , y n , [ β ] , J ) , converges in Λ .(b) For a collection of intersection points y i ∈ L i ∩ L i + , ≤ i < n , the sum (40) ∑ y ∈ L ∩ L n , ([ β ]) ∈ π ( Y,L ,...,L n ) T ω ([ β ]) M ( y , . . . , y, [ β ] , J ) y converges in ̂ CF ∗ ( L , L n ) . Proof. (a) The statement that the sum converges can be rephrased as the following: for each A >
0, thereare only finitely many homotopy classes with ω ([ β ]) < A . This follows from Corollary 5.1.4 and Gromovcompactness.(b) Again let A >
0. Let T B = T B ( L , . . . , L n , J ) > R = R ( L , . . . , L n , J ) > y ∈ L ∩ L n with ∣ t ( y ) ∣ > ( T B + ∣ t ( y ) ∣ + A + ) ( R + ) , for all u contributing to the sum ∑ [ β ] ∈ π ( Y,L ,L ,...,L n ) T ω ([ β ]) M ( y , . . . y n − , y, [ β ] , J ) ,ω ([ u ]) ≥ ℓ T B ( u ) R ≥ A. Then v ⎛⎝ ∑ [ β ] T ω ([ β ]) M ( y , . . . y n − , p, [ β ] , J )⎞⎠ ≥ min [ β ] ω ([ β ]) ≥ A. Thus as ∣ t ( y ) ∣ → ∞ , the valuation of the coefficient on y becomes large, so the sum converges. (cid:3) For such a collection of intersection points y , . . . , y n − , define µ n ( y n − , . . . , y ) = ∑ y n ∈ L ∩ L n , [ β ] ∈ π ( Y,L ,L ,...,L n ) T ω ([ β ]) M ( y , . . . y n − , y n , [ β ] , J ) y n . Proposition 5.4.2.
The µ n can be extended linearly to maps µ n ∶ ̂ CF ∗ ( L n − , L n ) ⊗ ⋯ ⊗ ̂ CF ∗ ( L , L ) → ̂ CF ∗ ( L , L n ) for any collection ( L , . . . , L n , J ) ; that is, given a collection w i ∈ ̂ CF ∗ ( L i , L i + ) , ≤ i ≤ n − , with w i = ∑ y ( m,i ) ∈ L i ∩ L i + c ( m,i ) y ( m,i ) , the sum defined by ∑ m ,...,m n − ( c ( m , ) ⋯ c ( m n − ,n − ) ) µ n ( y ( m n − ,n − ) , . . . , y ( m , ) ) converges in ̂ CF ∗ ( L , L n ) .Proof. Let C ∈ R be such that C < C < min ( m,i ) v ( c ( m,i ) ) ;then nC < v ( c ( m , ) ⋯ c ( m n − i ,n − ) ) for all collections ( m , ) , . . . , ( m n − i , n − ) . Let y ∈ L ∩ L n . The coefficient of y in the above sum is givenby ∑ m ,...,m n − ( c ( m , ) ⋯ c ( m n − i ,n − ) ) T ω ([ β ]) M ( y ( m , ) , . . . , y ( m n − ,n − ) , y, β, J ) . We first show that this converges in Λ for all y ; the proof is similar to the proof of part (b) in Proposition5.4.1. Let A >
0. Then for any collection of ( y ( m , ) , . . . , y ( m n − ,n ) ) for which one of the points satisfies ∣ t ( y ( m i ,i ) ) ∣ > ( T B + ∣ t ( y ) ∣ + + A ) ( R + ) , any u ∶ S → Y contributing to the count ∑ [ β ] T ω ([ β ]) M ( y , . . . y n − , y, [ β ] , J ) , satisfies ω ([ u ]) ≥ A. Then for such a collection y ( m , ) , . . . , y ( m n − ,n − ) of intersection points, v ⎛⎝ ∑ [ β ] ( c ( m , ) ⋯ c ( m n − i ,n − ) ) T ω ([ β ]) M ( y ( m , ) , . . . , y ( m n − ,n − ) , y, β, J )⎞⎠ ≥ A + nC. This condition is satisfied by all but finitely many such collections of endpoints, and we conclude that thecoefficient on each y ∈ L ∩ L n converges.Now again let A >
0. Let t A be such that if ∣ t ( y ( m,i ) ) ∣ > t A , c ( m,i ) > A . There are finitely many y ( m,i ) with ∣ t ( y ( m,i ) ) ∣ < t A , so there exists t ′ A such that if a J -holomorphic curve u contributing to the sum has u − ( Y t ′ A ) ≠ ∅ and u ( p i ) = y ( m,i ) for some such y ( m,i ) , then ω [ u ] > A . Now let y ∈ L ∩ L n have ∣ t ( y ) ∣ > t ′ A .Then we can separate the sum calculating the coefficient of y into two parts, where one part is the sum overcurves where all boundary marked points go to points close to y , and the other is the sum over the othercurves, which are forced to have large image when projected to R under t ∶ Y → R and thus large area. ∑ m ,...,m n − ( c ( m , ) ⋯ c ( m n − i ,n − ) ) T ω ([ β ]) M ( y ( m , ) , . . . , y ( m n − ,n ) , p, β, J ) = ∑ ∣ t ( y mi )∣ < t A for all i ( c ( m , ) ⋯ c ( m n − i ,n − ) ) T ω ([ β ]) M ( y ( m , ) , . . . , y ( m n − ,n ) , p, β, J ) +∑ ∣ t ( y mi )∣ ≥ t A for some i ( c ( m , ) ⋯ c ( m n − i ,n − ) ) T ω ([ β ]) M ( y ( m , ) , . . . , y ( m n − ,n ) , p, β, J ) . Label the first sum Σ , and the second sum Σ . Each term contributing to Σ has valuation at least A + nC by the condition that t ( y ) > t ′ A , and every term contributing to Σ has valuation at least A + nC by thecondition on t A . We conclude that v ( c ( y )) ≥ min { v ( Σ ) , v ( Σ )} ≥ A + nC. Since there are only finitely many y ∈ L ∩ L n with ∣ t ( y ) ∣ < t A , we conclude that there are only finitely manyterms with valuation less than A + nC . The convergence of the sum follows. (cid:3) A ∞ relations. Since we have excluded disc and sphere bubbling by the assumptions on our La-grangians, the µ n products will satisfy the A ∞ relations (see [25]) for any collection of perturbed Hamiltoniansin correct position with compatible choices of data in the definition of the moduli spaces. Remark 5.4.1.
In fact, the A ∞ relations hold for complexes defined over Λ Z , the Novikov ring with coeffi-cients in Z , since the counts of holomorphic discs of any given class are valued in Z . Remark 5.4.2.
Let ̂ HF ∗ ( L , L ) denote the cohomology of the complex ̂ CF ∗ ( L , L ) . Note that µ induces an associative product ̂ HF ∗ ( L , L ) ⊗ ̂ HF ∗ ( L , L ) → ̂ HF ∗ ( L , L ) ; we write this product as [ a ] ⋅ [ b ] = [ µ ( a, b )] .5.5. Continuation elements.
We relate our Floer complexes via continuation elements, with the aim offormally inverting the morphisms given by multiplication with these elements when we define our Fukayacategories. Given L an admissible planar Lagrangian, and H , H ∶ Y → R Hamiltonians suitable for L sothat the pair ( ψ H ( L ) , ψ H ( L )) is in correct position, one would like to define continuation elements [ c H → H ] ∈ ̂ HF ∗ ( ψ H ( L ) , ψ H ( L )) OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 27 by counting maps with moving boundary conditions that interpolate between ψ H ( L ) and ψ H ( L ) . Toimplement this, we use moduli spaces of such curves in two situations where those moduli spaces are wellbehaved. Compactly supported Hamiltonians.
Let
L, L ′ ∈ G P , and let H be a compactly supported Hamiltonian on Y suitable for L such that the pairs ( L, L ′ ) and CF ∗ ( ψ H ( L ) , L ′ ) are in correct position. We define acontinuation map Φ H ∶ ̂ CF ∗ ( L, L ′ ) → ̂ CF ∗ ( ψ H ( L ) , L ′ ) . Let S denote the strip R × [ , ] . Let ρ ∶ R → [ , ] be a surjective increasing function such that ρ ( s ) = s <<
0, and ρ ( s ) = s >>
0. Let β ∈ Ω ( S ) be a one-form satisfying β ∣ R × { } = dρβ ∣ R × { } = . Fix a domain-dependent eventually constant complex structure J on Y .For y ∈ ψ H ( L ) , y ∈ ψ H ( L ) , consider the set of maps u ∶ S → Y such that ( du − X H ⊗ β ) , J = u ( s, ) ∈ ψ ρ ( s ) H ( L ) (42) u ( s, ) ∈ L (43)with lim s →−∞ u ( s, ⋅ ) = y (44) lim s →∞ u ( s, ⋅ ) = y . (45)For such u , define the geometric and topological energy of such maps: E geom ( u ) = ∫ S ∥ du − X H ⊗ β ∥ = ∫ S u ∗ ω − d ( u ∗ H ) ∧ β ; E top ( u ) = ∫ S u ∗ ω − d ( u ∗ Hβ ) = E geom ( u ) − ∫ S u ∗ Hdβ.
We now use the assumption that our Hamiltonians are exact when lifted to the universal cover to boundthe topological energy of u : lifting u to the universal cover and expressing ω = dθ , θ ∣ L ′ = dh , θ ∣ L ′ = dh ′ forsome functions h, h ′ ∶ R → R , we use Stokes’ theorem to write ∫ S u ∗ ω − d ( u ∗ Hβ ) = − ∫ ∂S u ∗ θ − u ∗ Hβ = − ∫ R × { } u ∗ dh ′ − ∫ R × { } ( u ∗ ( dh ) + ρu ∗ ( dH ) + u ∗ Hdρ ) = − ∫ R × { } u ∗ dh ′ + ∫ R × { } ( u ∗ ( dh ) + ρu ∗ ( dH )) = h ′ ( y ) − h ′ ( y ) − h ( y ) + h ( y ) + H ( y ) = S R ( ˜ y ) − S R ( ˜ y ) . Note that the quantity S R ( ˜ y ) − S R ( ˜ y ) is independent of the choice of lift of ˜ y . Denote this quantity S ( y , y ) . Then we obtain a bound on the geometric energy of u :(46) ∣ E geom ( u ) ∣ − ∣ S ( y , y ) ∣ ≤ ∣ ∫ S u ∗ Hdβ ∣ ≤ max y ∈ Y ∣ H ∣ ∫ ∂S β = max ∣ H ∣ . Away from the support of H , Equation (41) reduces to the unperturbed Cauchy-Riemann equation, so in thisregion, a solution u will satisfy the maximum principle. Moreover, in this region, the Hamiltonian isotopy isconstant. Thus the methods of Lemma 5.1.3 can be used in combination with the geometric energy boundabove to produce a compact region containing the image of any such u . Then for generic J , the modulispace M ( y , y , β, J ) of such maps is a smooth manifold with dimension depending on the degrees of y and y as generators of the Floer complex, and this manifold admits a compactification M ( y , y , β, J ) . Define Φ H on the generators of CF ∗ ( ψ H ( L ) , L ) byΦ H ( y ) = ∑ y ∈ ψ H ( L ) ∩ L,u ∈ M ( y,y ,β, J ) T E geom ( u ) y where as usual the sum is over all y such that M ( y, y , β, J ) is zero-dimensional, that is, the generators y ∈ CF ∗ ( ψ H ( L ) , L ) with the same degree as y . Lemma 5.5.1.
For any compactly supported H , L, L ′ ∈ G P with ( L, L ′ ) and ( ψ H ( L ) , L ′ ) in correct position,and corresponding map Φ H defined on the generators of CF ∗ ( L, L ′ ) as above:(a) There exists T > with Supp H ⊂ Y T such that Φ H ( y ) = y for all y ∈ ( L ∩ L ′ ) T .(b) Φ H extends to a map ̂ CF ∗ ( L, L ′ ) → ̂ CF ∗ ( ψ H ( L ) , L ′ ) .(c) Φ H extends to a map ̂ CF ∗ r ( L, L ′ ) → ̂ CF ∗ r ( ψ H ( L ) , L ′ ) .(d) Φ H is a chain map.(e) Φ H is a quasi-isomorphism, with quasi-inverse Φ − H .(f ) For two compactly supported Hamiltonians H and H , Φ H ○ Φ H agrees with Φ H + H on cohomology.Proof. By choosing lifts appropriately, we can define the topological Z / Γ ( L, L ′ ) grading of Definition 5.2.1on the two Floer complexes so that the continuation map must respect the grading. Then for each ˆ n ∈ Z / Γ,we get an induced map Φ ˆ nH ∶ CF ∗ ( L, L ′ ) ˆ n → CF ∗ ( ψ H ( L ) , L ′ ) ˆ n For all ˆ n , the complexes CF ∗ ( L, L ′ ) ˆ n and CF ∗ ( ψ H ( L ) , L ′ ) ˆ n have finitely many generators. Thus on eachgraded piece, (d), (e) and (f) hold by standard arguments. Thus it suffices to prove (a),(b), and (c).From Equation (46), we know that the energy of any strip connecting generators y ∈ CF ∗ ( L, L ′ ) ˆ n and y ∈ CF ∗ ( ψ H ( L ) , L ′ ) ˆ n is bounded by ∣ S ( y , y ) ∣ + max ∣ H ∣ . By Lemma 4.5.3, we know that there exists D > α ∈ R such that ∣ S ( y , y ) ∣ < ∣ S ( y ) − S ( y ) ∣ + D = ∣ Q αL,L ′ , ± ( y ) − Q αL,L ′ , ± ( y )∣ + D < ∣ α ( t ( y ) − t ( y ) )∣ + D But α ( t ( y ) − t ( y ) ) is uniformly bounded for all such pairs ( y , y ) . So we obtain a uniform bound onthe geometric energy E geom ( u ) for all u counted by Φ H .Away from the support of H , Equation (41) reduces to the non-perturbed J -holomorphic equation. Inthis region methods of Lemma 5.1.3 will apply, so we obtain a similar bound of the form ℓ T B ( u ) ≤ RE geom ( u ) . for some T B so that Supp H ∩ Y T B = ∅ . Then using the uniform bound on E geom ( u ) , we can find T > T B suchthat for all y ∈ ( L ∩ L ′ ) T and maps u ∈ M ( y , y , β ) , the image of u must be disjoint from the support of H .But in this region, any u ∶ R × [ , ] counted by the continuation map must be constant; any non-constant u can re-parametrized by translation by s , and thus will come in a higher-dimensional family. Thus thecontinuation map is the identity on generators outside of a compact set, so (b) follows.Now, note that for any y ∈ L ∩ L ′ , Φ n ( y ) counts only finitely many u since the moduli spaces M ( y, y , β ) are empty for all but finitely many n , and each zero dimensional moduli space is compact. Thus for any T ,for a collection c i ∈ Λ r , Φ n ⎛⎝ ∑ y i ∈ Y ∖ Y T c i y i ⎞⎠ ∈ CF ∗ R ( ψ H ( L ) , L ′ ) ⊂ ̂ CF ∗ r ( ψ H ( L ) , L ′ ) . Combining this with (a) shows that that Φ H extends to a map ̂ CF ∗ ( L, L ′ ) → ̂ CF ∗ ( ψ H ( L ) , L ′ ) . Finally,note that for any generator which is stationary under the flow of H , the action of that generator remainsunchanged. Thus the extension restricts to a map ̂ CF ∗ r ( L, L ′ ) → ̂ CF ∗ r ( ψ H ( L ) , L ′ ) . (cid:3) OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 29
Hamiltonians with a unique minimum.
Let L ∈ G ( W r ) , and let H and H be two Hamiltonians suitablefor L such that the function H − H = H has a unique minimum at t = s = R × S with H ( ) =
0, and such that the pair ( ψ H ( L ) , ψ H ( L )) is in correct position.Define the continuation element c H → H ∈ CF ∗ ( ψ H ( L ) , ψ H ( L )) by c H → H = ∑ y n ∈ L ∩ { t = s = } y n , i.e., by the sum of the points fixed by the Hamiltonian isotopy ψ tH ( L ) at the minimum of H . We use thenotation c H to denote the continuation element c → H here and throughout. Perturbing Hamiltonians.
Let L be an admissible planar Lagrangian, and let H , H ∶ Y → R be perturbingHamiltonians suitable for L so that the pair ( ψ H ( L ) , ψ H ( L )) is in correct position. Let H = H − H , so ψ H ( ψ H ( L )) = ψ H ( L ) . We now fix a way to decompose H into a compactly supported Hamiltonian anda Hamiltonian which has a unique minimum as a function H ∶ R × S → R at ( t, s ) = ( , ) .Recall that we can write H at the two ends of Y as H ± ( t, s ) = α ± t + ε ± ∣ t ∣ + f ± ( t, s ) . Let a = α + + ε + = α − + ε − . Let σ α + ,ε + α − ,ε − , χ, ρ ∶ R → R be extrapolating functions as described in Definition 4.2.3.Define(47) H Q ( t, s ) = σ α + ,ε + α − ,ε − ( t ) + aχ ( t ) ( cos ( πs ) + ) π + ρ ( − t + ) f − ( t, s ) + ρ ( t − ) f + ( t, s ) . Note that the map H ↦ H Q is linear in H . We have ∂H Q ( t, s ) ∂t = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩ ( α + + ε + ) t t ∈ [ , / ) ; dσ α + ,ε + α − ,ε − ( t ) dt + a ( sin ( πs ) + ) π dχ ( t ) dt + ∂∂t ( ρ ( t − ) f + ( t, s )) t ∈ [ / , ) ;2 α + t + ε + + ∂f ( t, s ) ∂t t ∈ [ , ∞ ) . Recall that ∥ f + ∥ C < α /
8, and α + , ε + >
0; from this we see that the partial derivative is non-vanishing for t ∈ [ , / ) and t ∈ [ , ∞ ) . We can also bound the partial derivative away from 0 for t ∈ [ / , ) : ∣ ∂H ( t, s ) ∂t ∣ = ∣ dσ α + ,ε + α − ,ε − ( t ) dt + a ( sin ( πs ) + ) π dχ ( t ) dt + dρ ( t − ) dt f + ( t, s ) + ( ∂f + ( t, s ) ∂t ) ρ ( t − )∣ ≥ α + t + ε + − α + + ε + π − ∥ f + ∥ C > . From this we conclude that the partial derivative is non-vanishing for 1 ≤ t ≤
2. The same analysis for the t < t <
0. We conclude that the uniqueminimum of H occurs at ( t, s ) = ( , ) .Let H K be the compactly supported Hamiltonian such that H = H Q + H K . We define a continuation element c H ∈ CF ∗ ( ψ H ( L ) , L ) by c H = Φ H K ( c H Q ) . Properties of the continuation elements.
Lemma 5.5.2. (a) For all Hamiltonians H perturbing L ∈ G P , µ ( c H ) = .(b) Let H and H ′ be two Hamiltonians perturbing L in G P . Then ( H + H ) Q = H Q + H ′ Q and µ ( c H Q , c H ′ Q ) = c ( H + H ′ ) Q . (c) Let H , H , and H be Hamiltonians such that the set ( ψ H ( L ) , ψ H ( L ) , ψ H ( L )) is in correct position.Then in HF ∗ ( ψ H ( L ) , ψ H ( L )) , [ c H → H ] ⋅ [ c H → H ] = [ c H → H ] . Proof. (a) Since Φ H K is a chain map, it suffices to show that µ ( c H Q ) =
0. Note that we can compute thedifferential µ ( c H Q ) in any cover of Y . Lift ψ H ( L ) and ψ H K ( L ) to T ∗ C R , where r is defined using theclass [ γ L ] = [( r, dk, dm )] ; the lifts of ψ H and ψ H I ( L ) are then sections of the fibration T ∗ C R → C R . Let G ∶ C R → R be the compactly supported function that appears in the primitive of L ∈ T ∗ R in Lemma4.2.1. The symplectomorphism ψ − G − H K ∶ T ∗ C R → T ∗ C R . induces an isomorphism of chain complexes ̂ CF ∗ ( ψ H ( L ) , L ) → ̂ CF ∗ ( ψ H ( ψ − G − H K ( L )) , ψ − G − H K ( L )) . Note that ψ H ( ψ − G − H K ( L )) and ψ − G − H K ( L ) are both product Lagrangians when restricted to theproduct submanifold ([ − / , / ] × R , dt ∧ dϕ t ) ⊕ ( S R × R , ds ∧ dϕ s ) ⊂ T ∗ ( R × S r ) . Choose an almost complex structure J which respects this splitting; recall that the differentials on theFloer complexes can be computed up to homotopy using any almost complex structure that ensuresregularity. Any holomorphic disc that contributes to µ ( c H ) can be constrained to lie in ([ − / , / ] × S , dt ∧ dϕ t ) ⊕ ( S R × S , ds ∧ dϕ s ) ⊂ Y R by using the similar methods of the proof of Lemma 5.1.3 to obtain a priori constraints on the imageof u based on ω ( u ) = H Q ( y ′ ) , and then writing H Q = ( H Q n + . . . + H Q n ) , choosing n large and applying(a). But in this region all intersection points lie at t = ϕ t =
0, so applying the maximum principle to theholomorphic map π ( t,ϕ t ) ○ u allows us to deduce that π ( t,ϕ t ) ○ u is constant. Thus any u must split as u = { } × ( π s,ϕ s ○ u ) . Such maps are automatically regular since all holomorphic maps to C are regular. Then the problemreduces to showing that the sum of the index 0 intersection points is closed in T ∗ S R . Labelling theintersection points by their s value, we have ψ ( y n ) = ( T A ) y n + / − ( T A ) y n − / , for n ∈ Z / R Z , where A = H ( / , ) − H ( , ) . Summing over all the intersection points gives the result.(b) The map H ↦ H Q is linear in H , so the first equality follows.Note that H Q and H Q + H ′ Q each have minima only at t = s =
0. We can use the exactness of L whenlifted to R to express the energy of any u ∶ D → Y contributing to µ ( c H Q , c H Q → H Q + H ′ Q ) with u ( p ) = y , u ( p ) = y , u ( p ) = y as ∫ D u ∗ ω = − ( H Q + H ′ Q ) ( y ) . Since H Q + Q ′ ( y ) ≥
0, we must have that ∫ D u ∗ ω =
0, so such u must be constant; thus y = y = y mustbe a fixed point of the isotopy ϕ tH ( L ) . We conclude that µ ( c H Q , c H Q → H Q + H ′ Q ) = ∑ y ∈ L ∩ { t = s = } y = c H → H . (c) By gluing the domain of the maps counted byΦ ( H − H ) K ( c H → H + ( H − H ) Q ) ;Φ ( H − H ) K ( c H → H + ( H − H ) Q ) , the space of discs u ∶ S → Y counted by µ ( c H → H , c H → H ) can be deformed to the space of mapscounted by Ψ ( H ) K ( c ( H ) Q ) . (cid:3) OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 31
Lemma 5.5.3.
Let H and H be type- I Hamiltonians perturbing a type- I Lagrangian L , and let H ′ be atype-I Hamiltonian perturbing a type- I Lagrangian L ′ . Assume that the set ( ψ H ( L ) , ψ H ( L ) , ψ H ′ ( L ′ )) is in correct position. Also assume that α ± ( H ) − α ± ( H ′ ) > K ± ( L, L ′ ) on the wrapped end(s) of Y . Thenthe multiplication map defines a map µ ( ⋅ , c H → H ) ∶ CF ∗ ( ψ H ( L ) , ψ H ′ ( L ′ )) → CF ∗ ( ψ H ( L ) , ψ H ′ ( L ′ )) which is a quasi-isomorphism. Similarly, if the set ( ψ H ′ ( L ′ ) , ψ H ( L ) , ψ H ( L )) is in correct position and α ± ( H ′ ) − α ± ( H ) > K ± ( L ′ , L ) on the wrapped end(s) of Y then the map µ ( c H → H , ⋅ ) ∶ CF ∗ ( ψ H ′ ( L ′ ) , ψ H ( L )) → CF ∗ ( ψ H ′ ( L ′ ) , ψ H ( L )) is a quasi-isomorphism.Moreover, the quasi-isomorphisms defined above extend to maps µ ( ⋅ , c H → H ) ∶ ̂ CF ∗ ( ψ H ( L ) , ψ H ′ ( L ′ )) → ̂ CF ∗ ( ψ H ( L ) , ψ H ′ ( L ′ )) ,µ ( c H → H , ⋅ ) ∶ ̂ CF ∗ ( ψ H ′ ( L ′ ) , ψ H ( L )) → ̂ CF ∗ ( ψ H ′ ( L ′ ) , ψ H ( L )) which are injective on cohomology, and restrict to quasi-isomorphisms µ ( ⋅ , c H → H ) ∶ ̂ CF ∗ r ( ψ H ( L ) , ψ H ′ ( L ′ )) → ̂ CF ∗ r ( ψ H ( L ) , ψ H ′ ( L ′ )) ,µ ( c H → H , ⋅ ) ∶ ̂ CF ∗ r ( ψ H ′ ( L ′ ) , ψ H ( L )) → ̂ CF ∗ r ( ψ H ′ ( L ′ ) , ψ H ( L )) . Proof.
We prove the first case; the second can be proved with the same reasoning. Change notation slightly:let H − H = H , and let L now denote the perturbed Lagrangian ψ H ( L ) and L ′ now denote the perturbedLagrangian ψ H ′ ( L ′ ) .We have seen that if G is a compactly supported Hamiltonian, the mapΦ G ∶ ̂ CF ∗ ( L, L ′ ) → ̂ CF ∗ ( ψ G ( L ) , L ′ ) is a quasi-isomorphism. Using the decomposition [ c H ] = Ψ H K ( H Q ) , we can reduce to the case when H = H Q ,so H is a Hamiltonian of the form given by Equation (47).Note that the multiplication with the continuation element respects the Z / Γ grading on the Floer com-plexes. We first show that ⋅ [ c H ] ˆ n ∶ HF ∗ ( L, L ′ ) ˆ n → HF ∗ ( ψ H ( L ) , L ′ ) ˆ n is an isomorphism for each ˆ n ∈ Z .Recall that we associate to L and L ′ elements [ γ L ] , [ γ L ′ ] ∈ H ( T ) of the form ( r, dk, dm ) and ( r ′ , d ′ k ′ , d ′ m ′ ) .Let R = rr ′ , and consider the cyclic cover Y R → Y of Remark 4.2.4. Let L R and L ′ R be lifts of L and L ′ to Y R , respectively, which lift further to ˜ L and ˜ L ′ ; note that both L R and L ′ R are sections of the fibration Y R → C R . Let c H R ∈ ̂ CF ∗ ( L R , L ′ R ) = π − R ( c H ) . The natural identifications CF ∗ ( L, L ′ ) ˆ n ≅ CF ∗ ( L R , L ′ R ) ˆ n CF ∗ ( ψ H ( L ) , L ′ ) ˆ n ≅ CF ∗ ( ψ H ( L R ) , L ′ R ) ˆ n commute with the induced map ⋅ [ c H R ] ˆ n ∶ HF ∗ ( L R , L ′ R ) ˆ n → HF ∗ ( ψ H ( L R ) , L ′ R ) ˆ n . Thus it suffices to showthat this second map is an isomorphism.Let ˆ n ∈ Z . Let h be such that ˜ L is the graph of dh , and let h ′ be such that ˜ L ′ − ˆ n is the graph of dh ′ ˆ n .Note that ̂ CF ∗ ( L, L ′ ) ˆ n is generated by points of L R ∩ L ′ R that lie above critical points of h − h ′ . Assumenow that that h − h ′ and h − h ′ + H have a critical point at t = s =
0, and at that point both functionsvanish. Let y ∈ ̂ CF ∗ ( L R , L ′ R ) ˆ n and y ∈ ̂ CF ∗ ( L, L ′ R ) ˆ n , denote the corresponding generators above thiscritical point and assume that y and y have the same degree. Then the constant solution is the only diskwhich contributes to the product µ ( y , c H R ) , since all of the disks contributing to this product must havesymplectic area zero. We conclude that µ ( y , c H R ) = y .Note that for i ≠ , ,
2, the Floer complexes vanish in degree i . We will show that for i = , , G ∶ Y R → R so that the functions h − h ′ − G and h − h ′ + H − G have this property, and such that the corresponding generators y ∈ CF i ( L R , ψ G ( L ′ R )) ˆ n and y ∈ CF i ( ψ H ( L R ) , ψ G ( L ′ R )) ˆ n are closed and span rank-1 modules HF i ( L R , ψ G ( L ′ R )) ˆ n and HF i ( ψ H ( L R ) , ψ G ( L ′ R )) ˆ n respectively. Then the map ⋅ [ c H R ] ˆ n ∶ HF ∗ ( L R , ψ G ( L ′ R )) ˆ n → HF ∗ ( L R , ψ G ( L ′ R )) ˆ n is an isomorphism. We then use the commutativity of the below diagram to conclude that ⋅ [ c H ] ˆ n is anisomorphism: HF i ( ˜ L, ˜ L ′ ) ˆ n HF i ( ψ H ( ˜ L ) , ˜ L ′ ) ˆ n HF i ( L R , ψ G ( L ′ R )) ˆ n HF i ( ψ H ( L R ) , ψ G ( L R )) ˆ n . ⋅ [ c H ] ˆ n Φ G ≅ Φ G ≅ ⋅ [ c HR ] ˆ n We distinguish the cases [ γ L ] = [ γ L ′ ] and [ γ L ] ≠ [ γ L ′ ] , and work in the A , ( m, k ) , and W r settings separately. [ γ L ] ≠ [ γ L ′ ] , A setting . Let λ = k ( dr − d ′ r ′ ) ; by assumption λ ≠
0. Let ε = ε ( H ) , ε = ε ( H ) − ε ( H ′ ) , ε = ε ( H ) − εH ′ . By writing H = n ( n H ) for some large n , we can assume that ε = ε − ε < ε ∣ k / m ∣ + . and ∣ λ ∣ > πε (∣ k / m ∣ + ) . Note that ε ( n H ) = ε ( H ) n , so this step involves only one choice of n . Using Lemmas 4.5.1 and 4.2.1, we canwrite h − h ′ = λw + aw + ϕ ( t ) + G ( t, s ) where w = s + ( k / m ) t , G is a compactly supported function on C R , and ϕ ∶ R → R is a function with thefollowing properties: ● There exists c ∈ R and T > t < − T , ϕ ( t ) = ( b − − ε ) t and for t > T , ϕ ( t ) = ( b + + ε ) t + c. where b + , b − ∈ m Z . ● If ( b − − ε ) ( b + + ε ) >
0, then ∣ dϕ ( t ) dt ∣ > ε (∣ k / m ∣ + ) for t ∈ [ − , ] , and ∣ dϕdt ∣ > ε everywhere. ● If ( b − − ε ) ( b + + ε ) <
0, then ϕ ( t ) = ( M / ) t for t ∈ [ − , ] , where M has the same sign as ( b + + ε ) ,and satisfies(48) ∣ + k λm (∣ M ∣ + ε ) ∣ < ∣ dϕ ( t ) dt ∣ > ε for t ∉ [ − , ] . OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 33
Let g = h − h ′ − G . By construction ∂g / ∂t is non-vanishing if ( b − − ε ) ( b + + ε ) >
0, and vanishes onlyat zero if ( b − − ε ) ( b + + ε ) <
0. In the former case let G = G . In the latter case, consider the compactlysupported diffeomorphism Θ ∶ C R → C R defined byΘ ( t, w ) = ( t, w − a λ χ T / ε ( t )) . Note that the function g ○ Θ ∶ C R → R has a unique critical point at w = t =
0. Moreover, G = g ○ Θ − g is acompactly supported function on Y R . Let G = G + G . We conclude that ˜ L and ψ G ( ˜ L ′ ) intersect in R onlyat t = w =
0. In both cases µ vanishes on ̂ CF ∗ ( L R , ψ G ( L ′ R )) ˆ n ; the complex is either zero or concentratedin degree 0 , , or 2.We wish to show that the same holds for ̂ CF ∗ ( ψ H ( L R ) , ψ G ( L ′ R )) ˆ n . First assume that ( b − − ε ) ( b − + ε ) > . In this case we can write ∂∂t ( g + H ) = ddt ( ϕ + σ εε ) + εk sin ( π ( w − ( k / m ) t )) χ ( t ) m + ε ( cos ( π ( w − ( k / m ) t )) + ) π dχ ( t ) dt > ∣ dϕdt ( t )∣ − ε (∣ k / m ∣ + ) > . Thus ̂ CF ∗ ( L R , ψ − G ( L ′ R )) ˆ n = . In this case the continuation map is trivially an isomorphism.Now assume that ( b − − ε ) ( b − + ε ) <
0. Then for t ∈ [ − T / ε, T / ε ] , w − a λ χ T / ε ( t ) = w − a λ , so in thisregion we can write: ∂∂t ( g + H ) = ddt ( ϕ + σ εε ) + εk sin ( π ( w − ( k / m ) t )) χ ( t ) m − ε ( cos ( π ( w − ( k / m ) t )) + ) π dχ ( t ) dt For t ∈ [ − / , / ] , we have the expressions ∂∂t ( g + H ) = M t + εt + εk sin ( π ( w − ( k / m ) t )) m∂∂w ( g + H ) = λw − ε ( sin ( π ( w − ( k / m ) t ))) . We can conclude that any critical points of g + H in this region satisfy2 ( k / m ) λw = ( M + ε ) t. But we then can bound ∂∂w ( g + H ) away from zero in this for such t, w : any solutions to ∂∂w ( g + H ) mustsatisfy(49) 2 λw − ε ( sin ( π ( + k λm ( M + ε ) ) w )) = . Using the inequality ∣ sin ( x ) ∣ < ∣ x ∣ , and the bound from Equation (48), we write ε ( sin ( π ( + k λm ( M + ε ) ) w )) < πε ∣ w ∣ < λw, so there are no solutions to Equation (49) in this region other than at the origin.For t with 1 / < ∣ t ∣ <
2, we have ∣ ∂∂t ( g + H ) ( t, w )∣ > M t − ∣ dσ εε ( t ) dt + εk sin ( π ( w − ( k / m ) t )) χ ( t ) m − ε ( cos ( π ( w − ( k / m ) t )) + ) π dχ ( t ) dt ∣ > M − ε ( + ∣ k / m ∣) > . For t with ∣ t ∣ >
2, we have ∂∂t ( g + H ) ( t ) = dϕ ( t, w ) dt + ε − a λ Tε ∂ ( g + H ) ∂w so it suffices to show that dϕ ( t,w ) dt + ε is non-vanishing; this is true since ∣ dϕ ( t ) dt ∣ > ε for t ∉ [ − , ] The samemethods show that ∂ ( g + H )( t,w ) ∂t ≠ t < − /
2. We conclude that g + H has a unique critical point at ( t = w = ) . Hence ϕ H ( L R ) and ψ − G ( L ′ R ) intersect only at t = w =
0, and CF i ( ψ H ( L R ) , ψ − G ( L R )) ˆ n isconcentrated in degree i for i = , , or 2.We now consider the map µ ( ⋅ , c ˜ H ) ∶ CF i ( L R , ψ − G ( L R )) ˆ n → CF i ( ψ H ( L R ) , ψ − G ( L R )) ˆ n . By the exact-ness of the lifts of the Hamiltonians and the vanishing of H at its minima, the area of any J -holomorphiccurve contributing to this product is zero; thus the product counts only the constant solution over t = w = [ γ L ] ≠ [ γ L ′ ] , W r setting . The proof in this setting is similar. The fact that L and L ′ are in correct positionimplies that when L and L ′ do not have the same total slope, h − h ′ is of the form(50) h − h ′ = λw + at + bw + ϕ ( t ) + G ( t, s ) + f ( t, s ) , where: λ > w = w L,L ′ = s − ct for some c ∈ R ; a > G is compactly supported on C R ; f satisfies ∣ f ( t ) ∣ < α = min { , α , − , α , + } and f ( t ) is zero for t ∈ [ − , ] , and ∣ ∂f∂w ∣ ≤ α , + ∣ ∂f∂t ∣ ≤ ( + ∣ c ∣) α , (52) ∣ ∂f∂t ∣ ≤ ∣ at ∣ ;(53)and ϕ ∶ R → R is a function with the following properties: ϕ ( t ) is identically zero for t ∈ [ − , ] and satisfies − at − > dϕdt ( t ) for all t > at + < dϕdt ( t ) for all t < . We also assume(54) 2 π ∣( α + + ε + ) ( a + α + + ε + ) ( + ∣ c ∣)∣ < λ and(55) a − ( α + + ε + ) > . Let g = h − h ′ − G . Then by construction ∂g / ∂t > ( t, w ) = ( , w ) . Again let Θ ∶ C R → C R be defined by ( t, w ) ↦ ( t, w − w χ ε / T ( t )) where T >
2, and let − G = g − g ○ Θ. Then g − G has a unique critical point at t = s =
0. Set G = G + G ; then CF i ( ψ H ( L R ) , ψ G ( L R )) ˆ n .We wish to show that the same holds for CF i ( ψ H ( L R ) , ψ G ( L R )) ˆ n . Write g ○ Θ + H = λ ( w ′ ) + at + b ( w ′ ) + ϕ ( t ) + f ( t, w ′ ) + σ α + ,ε + α − ,ε − ( t ) − χ ( t ) ( α + + ε + ) cos ( π ( w ′ − ct )) + π + ρ ( − t − ) f − ( t, w ′ ) + ρ ( t + ) f + ( t, w ′ ) , where w ′ = w − ( χ T ( t ) b / λ ) . Note that for t ∈ [ − T, T ] , ( t, w ′ ) = ( t, w + C ) for some constant C .For t ∈ [ − / , / ] , ϕ ( t ) = ρ ( t + ) = ρ ( t ± ) = , OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 35 and χ ( t ) =
1. Thus in this region ∂ ( g ○ Θ + H ) ( t, w ′ ) ∂t = ( a + ε + + α + ) t + ( α + + ε + ) c sin ( π ( w − ct )) ∂∂w ( g ○ Θ + H ) = λw − ( α + + ε + ) sin ( π ( w − ct )) . The same methods as in the above, now using the bound from Equation (54), show that g − G − H has nocritical points away from t = t with 1 / < t <
2, we can bound ∂∂t ( g − G + H ) below: ∂ ( g ○ Θ + H ) ( t, w ) ∂t = at + ∂f ( t, w ) ∂t + dσ α + ,ε + α − ,ε − ( t ) dt + dχ ( t ) dt ( α + + ε + ) cos ( π ( w − ct )) + π + χ ( t ) ( α + + ε + ) sin ( π ( w − ct )) + dρ ( t + ) dt f + ( t, w ) + ρ ( t + ) ∂f + ( t, w ) ∂t > at + α + t + ε + − ( α + + ε + ) − α + ≥ a + ( α + + ε + ) − α + > . For t with t >
2, we have ∂ ( g ○ Θ + H ) ( t, w ) ∂t = ∂ ( g + H ) ( t, w ) ∂t − b λT dχ t ( t ) dt ∂ ( g + H ) ( t, w ′ ) ∂w∂ ( g ○ Θ + H ) ( t, w ) ∂w = ∂ ( g + H ) ( t, w ′ ) ∂w . Thus it suffices to show that ∂ ( g + H )( t,w ) ∂t is non-vanishing in this region. We have that ∂ ( g + H ) ( t, w ′ ) ∂t = at + ∂f ( t, w ′ ) ∂t + α ′ t + ∂f + ( t, w ′ ) ∂t ≥ at − α + > . Similar methods will show that there are no critical points in the t < g ○ Θ + H occurs at t = w =
0. Again this allows us to conclude that the map µ ( ⋅ , c H R ) ∶ ̂ CF i ( L R , ψ G ( L ′ R )) ˆ n → ̂ CF i ( ψ H ( L R ) , ψ G ( L ′ R )) ˆ n is an isomorphism, and thus ⋅ [ c H ] ˆ n is a quasi-isomorphism.When L and L ′ have the same total slope, we can write(56) h − h ′ = at + b ( ts ) + cs + ϕ ( t ) + G ( t, s ) + f ( t, s ) , for some a, b ≠
0. In this case we again choose a compactly supported diffeomorphism Θ ∶ C R → C R that takesthe unique critical point of the function at + b ( ts ) + cs to ( , ) , and show that for carefully chosen Θ, theminima of ( h − h ′ − G ) ○ Θ and ( h − h ′ − G ) ○ Θ + H both lie above t = s =
0. Let G = G + ( h − h ′ − G ) ○ Θ − h − h ′ shows that µ ( ⋅ , c ˜ H ) ∶ CF i ( L R , ψ − G ( L R )) ˆ n → CF i ( ψ H ( L R ) , ψ − G ( L R )) ˆ n . By the exactness of the lifts ofthe Hamiltonians and the vanishing of H at its minima, the area of any J -holomorphic curve contributing tothis product is zero; thus the product counts only the constant solution over t = w =
0. This map is evidentlyan isomorphism. [ γ L ] = [ γ L ′ ] , A setting. Again we lift to Y R , and perturb L R and L ′ R by a compactly supported Hamiltonian.For ∣ t ∣ >>
0, we can write h − h ′ = as + b ± t + ε ∣ t ∣ . Assume first that a ≠
0. Then we assume that ∣ ε ∣ < a ; we further assume that 2 Rε < ε . We can write ( h − h ′ ) ( t, s ) = as + ϕ ( t ) + σ ε ε ( t ) χ ( tt ) + ε ( cos ( πs ) + ) π + G ( t, s ) ; ( h − h ′ + H ) ( t, s ) = as + ϕ ( t ) + σ ε ε ( t ) + G ( t, s ) . where G ( t, s ) is compactly supported on C R . Let g = h − h ′ − G . Then ∂g∂s is nonvanishing everywhere, as is ∂g + H∂s . We conclude that ̂ CF ∗ ( L R , ψ G ( L ′ R )) ˆ n = ̂ CF ∗ ( ψ H ( L R ) , ψ G ( L ′ R )) ˆ n = , so the multiplication map ̂ CF ∗ ( L R , ψ G ( L ′ R )) ˆ n → ̂ CF ∗ ( ψ H ( L R ) , ψ G ( L ′ R )) ˆ n is trivially an isomorphism.Now assume a =
0. Then, similarly as in the first case, we can write ( h − h ′ + H ) ( t, s ) = ϕ ( t ) + σ ε ε ( t ) + cχ T ( t ) ε ( cos ( πs / R ) + ) π + χ ( t ) ε ( cos ( πs ) + ) π + G ( t, s ) ;and g = h − h ′ − G where ϕ , T and c are such that ∂g∂t is either non-vanishing (in which case the Floercomplexes are both zero and the map is trivially an isomorphism) or vanishes only at 0, and ∂g∂s vanishesonly at s = , R /
2, and ∂g + H∂t has the same behavior, and G is a compactly supported function on C R . Thenthe Floer complexes ̂ CF ∗ ( L R , ψ G ( L ′ R )) ˆ n and ̂ CF ∗ ( ψ H ( L R ) , ψ G ′ ( L ′ R )) ˆ n are concentrated in degrees i and i + i = i =
1, and are each rank one in each non-zero degree. We can show that the differentialson these complexes vanish using the same methods as in the proof of Lemma 4.2.1 (b). Thus as gradedΛ-modules, ̂ HF ∗ ( L R , ψ G ( L ′ R )) ˆ n = ̂ CF ∗ ( L R , ψ G ( L ′ R )) ˆ n ̂ HF ∗ ( ψ H ( L R ) , ψ G ( L ′ R )) ˆ n = ̂ CF ∗ ( ψ H ( L R ) , ψ G ( L ′ R )) ˆ n . Thus the multiplication by the continuation map furnishes an isomorphism ̂ HF i ( L R , ψ G ( L ′ R )) ˆ n ≅ ̂ HF i ( ψ H ( L R ) , ψ G ( L ′ R )) ˆ n . We conclude that ⋅ [ c H ] i ˆ n is an isomorphism for each ˆ n ∈ Z . We can similarly deform L R and L ′ R via acompactly supported Hamiltonian to ensure that the generators in degree i + t = s = ⋅ [ c H ] i + n is an isomorphism. Other settings.
The proof in the [ γ L ] = [ γ L ′ ] case in the W r setting is similar; again at the ends of C R the function h − h ′ will be (a small perturbation of) a quadratic function in t and a linear function in s , andwe can deform h and h ′ by compactly supported functions so that the generators of the Floer complexes lieabove t = s = t .Finally, in all cases, the proof in the partially wrapped ( m, k ) setting can be obtained by combining themethods of the proofs above, treating each end of Y R separately. Extension to completions.
We now show that these isomorphisms extend to the completions ̂ CF ∗ ( L, L ′ ) and ̂ CF ∗ r ( L, L ′ ) . Note that for each y ∈ CF ∗ ( L, L ′ ) , the output µ ( y, c H Q ) counts only finitely many u ∶ D → Y ; thus the statement is evidently true in the finite A setting. We now show controls on the outputsfor y ∈ ( L ∩ L ′ ) T for some T > µ ( y ) = y ∈ CF ∗ ( L, L ′ ) . This follows from the factthat L and L ′ are in correct position, so for all but finitely many ˆ n ∈ Z / Γ, ∣ ˜ L ∩ ˜ L ′ ˆ n ∣ = , χ ( CF ∗ ( L, L ′ ) ˆ n ) is equal to 1 or 0 for all ˆ n . Thus away from a finite set A of ˆ n ∈ Z / Γ, the map µ ( ⋅ , c H Q ) ˆ n ∶ CF ∗ ( L, L ′ ) ˆ n → CF ∗ ( ψ H ( L ) , L ′ ) ˆ n computes the induced map on homology.Let ˆ n ∉ A , let y ∈ CF ∗ ( L, L ′ ) ˆ n and y H Q be the generator of the same degree in CF ∗ ( ψ H Q ( L ) , L ′ ) ˆ n ; thenthe map y ↦ y H Q defines a bijection { ⋃ ˆ n ∉ A ( ψ H ( L ) ∩ L ′ ) ˆ n } ↔ { ⋃ ˆ n ∉ A ( L ∩ L ′ ) ˆ n } . Since µ ( ⋅ , c H Q ) ˆ n is an isomorphism for ˆ n ∉ A and respects the degrees of the generators, so µ ( y, c H Q ) ˆ n = cy H Q for some c ∈ Λ ∗ . For any u ∶ D → Y contributing to µ ( y, c H Q ) , ∫ D u ∗ ω = S R ( y ) − S R ( y H Q ) by Stokes’ theorem. Thus µ ( y, c H Q ) ˆ n = nT S R ( y ) − S R ( y HQ ) y H Q OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 37 for some n ∈ N . But since the quasi-isomorphisms described above exist over Λ Z (see Remark 5.4.1), we musthave that n ∈ GL ( Z ) = { , − } . Since S R ( y ) − S R ( y H Q ) > y ∈ L ∩ L ′ ∖ X , the map µ extends toa map ̂ CF ∗ ( L, L ′ ) → ̂ CF ∗ ( ψ H ( L ) , L ′ ) .Now let y = ∑ i c i y i ∈ ̂ CF ∗ ( L, L ′ ; Λ r ) and let µ ( y, c H Q ) be the formal sum ∑ i µ ( c i y i , c H Q ) . We nowshow that this sum defines an element of ̂ CF ∗ r ( ψ H ( L ) , L ′ ) if and only if y ∈ ̂ CF ∗ r ( L, L ′ ) . Note that µ ( y, c H Q ) = ∑ ˆ n ∈ A c i µ ( y i , c H Q ) ˆ n + ∑ ˆ n ∉ A c i T S R ( y i ) − S R (( y i ) HQ ) − S ( y i ) − S (( y i ) HQ ) ( y i ) H Q . The first sum is finite; write the second sum as ∑ i d i ( y i ) H Q .By Lemma 4.5.3, there exists D > ∣ S R ( y i ) − S R (( y i ) H Q ) − S ( y i ) + S (( y i ) H Q )∣ < D for all i ;by the linearity of the equations governing the intersection of the corresponding planes, there exists C , C > y i with ∣ t ( y i ) ∣ sufficiently large, C (∣ t (( y i ) H Q )∣ − ) < ∣ t ( y i ) ∣ < C (∣ t (( y i ) H Q )∣ + ) . Let R = e − πr . Thenlim ∣ t ( y i )∣ → ∞ ∣ R D − C ev r ( c i ) ( C ∣ t ( y i )∣) − ∣ ≤ lim ∣ t ( y i )∣ → ∞ ∣ ev r ( d i ) ∣ t (( y i ) HQ )∣ − ∣ ≤ lim ∣ t ( y i )∣ → ∞ ∣ R − D − C ev r ( c i ) ( C ∣ t ( y i )∣) − ∣ so the middle term vanishes if and only if lim ∣ t ( y i )∣ → ∞ ev r ( c i ) ∣ t ( y i )∣ − vanishes; thus µ ( y, c H Q ) defines anelement of ̂ CF ∗ r ( ψ H ( L ) , L ′ ) if and only if y converges in ̂ CF ∗ r ( L, L ′ ) . We conclude that the map µ ( ⋅ , c H Q ) is a quasi-isomorphism. (cid:3) Convergence of products over C . We now show that the µ products on cohomology converge over C . We first prove a collection of propositions which have proofs that use similar methods to those in theproof in Lemma 5.5.3. From the proof of the lemma, we can conclude the following: Proposition 5.6.1. (a) For x = ∑ i a i x i ∈ ̂ CF ( L , L ; Λ r ) , x defines an element of the completion ̂ CF ∗ r ( L , L ) if [ x − x ] ′ = for some x ′ ∈ ̂ CF ∗ r ( L , L ) .(b) Let x = ∑ i a i x i ∈ ̂ CF ∗ ( L , L ; Λ r ) , and assume that there exists a perturbing type- W r Hamiltonian H such that µ ( x, c H ) ∈ ̂ CF ∗ r ( ψ H ( L ) , L ) . Then x ∈ ̂ CF ∗ r ( L , L ; Λ r ) . Lemma 5.6.2.
Let ( L , L , L ) ∈ G P ( I ) be Lagrangians in correct position in F ( I ) such that the differential µ vanishes on CF ∗ ( L , L ) , CF ∗ ( L , L ) , and CF ∗ ( L , L ) . Let ˆ n , ˆ n ∈ Z . Let x ∈ ( L ∩ L ) ˆ n and x ∈ ( L ∩ L ) ˆ n , and let x lift to a point ˜ x . Then if ( L ∩ L ) ˆ n + ˆ n is non-empty, and if the Floercomplex CF ∗ ( L , L ) ˆ n + ˆ n is generated in degree deg x + deg x by a point x , the signed number of mapscounted by µ ( x , x ) that lift to maps u ∶ S → R with u ( p ) = ˜ x and u ( ∂ S ) ⊂ ˜ L , u ( ∂ S ) ⊂ ˜ L − ˆ n , u ( ∂ S ) ⊂ ( ˜ L − ˆ n − ˆ n ) is ± .Proof. We have seen that there exists R ∈ N and Hamiltonians G , G ∶ Y R → R , which are compactlysupported on Y R such that ˜ L , ̃ ψ G ( L ) , ̃ ψ G ( L ) each intersect at over t = s =
0, andΦ − G ( x ) = T A y ∈ CF ∗ ( L , ψ G ( L )) ;Φ G − G ( x ) = T A y ∈ CF ∗ ( ψ G ( L ) , ψ G ( L )) ;Φ − G ( x ) = T A y ∈ CF ∗ ( L , ψ G ( L )) ;and Φ − G ( ˜ x ) = T A ˜ y ;where y i denotes the corresponding generator of the Floer complex over t = s =
0. Then any disc counted by µ ( y , y ) with the prescribed boundary conditions must have energy 0 and hence be constant, so µ ( y , y ) ˆ n + ˆ n = y . Thus we must have that µ ( x , x ) ˆ n + ˆ n = nT A x = T − A − A ψ G ( y ) for some n ∈ Z , where A = − S R ( x ) + S R ( x ) + S R ( x ) , and n is the signed number of discs counted by µ . But then since Ψ G is an isomorphism over Λ Z , we conclude that n = ± (cid:3) Corollary 5.6.3. If L , L , L ∈ G P are such that the set ( L , L , L ) is in correct position, L and L areessentially non-transverse or L and L are essentially non-transverse, and the differentials of the complexes CF ∗ ( L , L ) , CF ∗ ( L , L ) , and CF ∗ ( L , L ) vanish, then for each x ∈ L ∩ L , z ∈ L ∩ L , µ ( z, x ) countsat most one J -holomorphic map. Moreover, there exists a uniform C > such that for all such x, z whichbound a J -holomorphic triangle with end point y ∈ L ∩ L , µ ( z , x ) = T A y where ∣ A ∣ ≤ C . There also exists D > such that if µ ( z , x ) = T A y , ∣ t ( y ) ∣ < D (∣ t ( z ) + t ( x )∣ + ) .Proof. We apply the previous lemma to prove the first statement when L and L are essentially non-transverse. Let x ∈ L ∩ L , z ∈ L ∩ L be such that µ ( z, x ) counts at least one map. Let ˆ n , ˆ n ′ ∈ Z be suchthat z lifts to points ˜ z, ˜ z ′ in ( ˜ L + ˆ n ) ∩ ( ˜ L − ˆ n ) and ( ˜ L + ˆ n ) ∩ ( ˜ L − ˆ n ′ ) respectively. Then ˆ n − ˆ n ′ definesan element of the group Γ ( L , L ) ⊂ Z of Definition 5.2.1. Since L and L are essentially non-transverse, [ γ L ] = [ γ L ] , so Γ ( L , L ) = Γ ( L , L ) since the stabilizer subgroups are topological invariants. But thenˆ n + ˆ n − ˆ n + ˆ n ′ = ˆ n − ˆ n ′ ∈ Γ ( L , L ) . Thus µ ( z, x ) is concentrated in CF ∗ ( L , L ) ˆ n + ˆ n ; by the previouslemma the signed number of maps counted by µ ( z, x ) is 1.Now, fix x and z , and let u ∶ S → R be the unique triangle which contributes to µ ( z , x ) = T A y ; lift u toa map S → R which maps boundary marked points to ˜ x , ˜ z ,and ˜ y . Apply Stokes’ theorem to write A = ( S R ( ˜ x ) − S ( x )) + ( S R ( ˜ z ) − S ( z )) − ( S R ( ˜ y ) − S ( y )) . Since L and L are essentially non-transverse, there exists a Hamiltonian H ∶ R → R which is C -closeto a quadratic function of t such that ψ H ( L ) = ψ H ( L ) . We can thus apply Lemma 4.5.3; there exists auniform C > ∣ ( S R ( z ) − S ( z )) − ( S R ( y ) − S ( y )) ∣ < C for all z and y . We apply Lemma 4.5.2(c) to bound ∣ ( S R ( x ) − S ( x )) ∣ < C . Then taking C = C + C gives the second statement.The third statement follows from linear algebra. All points are close to the intersection points of the cor-responding planes of the Lagrangians, and the t -value at which two planes P and P intersect is determinedby a linear expression in the intersection points of the planes P and P and the planes P and P . (cid:3) Proposition 5.6.4.
Let L , L , L ∈ G P be such that the set ( L , L , L ) is in correct position. Let x ∈ ̂ CF ∗ r ( L , L ) , z ∈ ̂ CF ∗ r ( L , L ) satisfy µ ( x ) = , µ ( z ) = . Then µ ( z, x ) converges in ̂ CF ∗ r ( L , L ) .Proof. First assume that L and L are essentially non-transverse, and that the differential vanishes on thecomplexes CF ∗ ( L , L ) , CF ∗ ( L , L ) , and CF ∗ ( L , L ) . Let C and D be as in the above proposition. Let R = e − πr . Write x = ∑ i a i x i , z = ∑ j b j z j . If µ ( x i , z j ) is non-vanishing, then we can write µ ( x i , z j ) = C ij T A i j y ij for some y ij ∈ L ∩ L where C ij = C ij = y ij is unique. Write then µ ( x, z ) = ∑ i,j a i b j C ij T A ij y ij , We now show this sum satisfies the criterion of Proposition 5.3.1, i.e., for all c ∈ R , ∑ i,j ∣ R A ij ev r ( a i b j ) R ct ( y ij ) ∣ < ∞ . Note that ∑ i,j ∣ R A ij ev r ( a i b j ) R − ∣ ct ( y ij )∣ ∣ ≤ R − C − ∑ i,j R − ∣ cDt ( x i )∣ R − ∣ cDt ( z j )∣ ∣ ev r ( a i b j ) ∣ = R − C − ( ∑ i R − ∣ cDt ( x i )∣ ∣ ev r ( a i ) ∣) ⎛⎝ ∑ j R − ∣ cDt ( z j )∣ ∣ ev r ( b j )∣⎞⎠ < ∞ . The same method will show convergence for all c < OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 39
Now assume that L , L , and L are planar Lagrangians which each intersect transversely with [ γ L i ] = [ r i , d i m i , d i k i ] . Now fix several pieces of data. Let M , M , M ∈ Sym × ( R ) , E , E , E ∈ R be such thatfor i = , , L i = {( X, M i X + E i ) ∣ X ∈ R . } Recall from Definition 4.5.1 that to each pair of Lagrangians we associate quadratic forms and coordinates w L i ,L j . Denote these quadratic forms Q L ,L , Q L ,L , Q L ,L by Q , Q , and Q respectively. Let Q i = Q αi + Q βi , and let w i be the coordinate associated to the pair of Lagrangians in question.Let d ′ i = d i / r i , i = , ,
2. We now show that µ ( x, z ) ≠ ( d ′ m − d ′ m )( d ′ m − d ′ m )( d ′ m − d ′ m ) > . Note that for all pairs of transverse planar Lagrangians L and L ′ , CF ∗ ( L, L ′ ) = i , where i = Q L,L ′ . Assume that ( d ′ m − d ′ m ) =
0. Then forall x i ∈ L ∩ L , det ( M − M ) <
0, so deg ( x i ) =
1. Then the non-vanishing of the product implies that z is non-zero in degree 0, hence concentrated in degree zero. But this implies that d ′ m − d ′ m >
0, hence d ′ m − d ′ m >
0, so the output is concentrated in degree zero; this contradicts the fact that µ ( x, z ) isconcentrated in deg ( z ) + deg ( x ) = B ∶ R × R → R as follows: for a pair ( z, x ) , let B ( z, x ) denote the unique point in R such that there is a planar triangle ∆ ( x, z ) in R with sides lying in translates of the planar Lagrangians L i and vertices lying over z, x, and B ( z, x ) . Explicitly, B ( z, x ) is given by ( M − M ) − (( M − M ) z + ( M − M ) x ) . Define A ( z, x ) to be the symplectic area of the triangle ∆ ( z, x ) . Extend B and A to maps on R × R bycomposing with the projection to the first two coordinates.Let x i ∈ L ∩ L and z j ∈ L ∩ L ; fix ˆ n , ˆ n ∈ Z such that ˜ x i ∈ ˜ L − ˆ n , ˜ z j ∈ ( L − ˆ n ) ∩ ( L − ˆ n − ˆ n ) satisfy 0 < w ( ˜ x i ) , w ( ˜ z j ) <
1. Using Proposition 5.6.2, we can write the product µ ( z j , x i ) as the sum ofproducts corresponding to different choices of the lift of z j : µ ( z j , x i ) = ∑ n ∈ N µ ( z j , x i ) ˆ n + ˆ n + n ˆ e s , where µ ( z j , x i ) ˆ n + ˆ n + n ˆ e s = ± T A ijn y ijn . Here y ijn = π ( B ( ˜x i , ˜z j − nˆe s )) denotes the action-corrected generator of CF ∗ ( L , L ) ˆ n + ˆ n + n ˆ e s . For c ∈ R ,define the following function:(57) f ij ( c ) = ∑ n R A ijn + ct ( y ijn ) . Then we can write f ij ( c ) as f ij ( c ) = R − ( S ( x i ) + S ( z j )) ∑ n ∈ N R S ( B ( ˜ z j + n ˆ e s , ˜ x i )) R A ( ˜ z j + n ˆ e s ,x ) R ct ( B ( ˜ z j + n ˆ e s , ˜ z )) . Expanding the exponent on R , applying Stokes theorem to find A ( ˜ z j + n ˆ e s , x ) , and expressing the differences S − S R in terms of the degenerate quadratic forms Q βi allows us to write the coefficient on R on the n thterm as: − S ( x i ) − S ( z j ) + S ( B ( ˜ z j + n ˆ e s , ˜ x i )) + A ( ˜ z j + n ˆ e s , ˜ x i ) + ct ( B ( ˜ z j + n ˆ e s , ˜ x i )) = − S ( x i ) − S ( z j ) + S ( B ( ˜ z j + n ˆ e s , ˜ x i )) + S R ( ˜ x i ) + S R ( ˜ z i + n ˆ e s ) − S R ( B ( ˜ z j + n ˆ e s , ˜ x i )) + c t ( B ( ˜ z j + n ˆ e s , ˜ x i )) = − Q β ( ˜ x i ) − Q β ( ˜ z j + n ˆ e s ) + Q w ( B ( ˜ z j + n ˆ e s , ˜ x i )) + c tB ( ˜ z j + n ˆ e s , ˜ x i ) . A calculation then shows that this expression is equal to ( ( d ′ m − d ′ m ) ( d ′ m − d ′ m ) ( d ′ m − d ′ m ) ) ( n − w ( ˜ x i ) + w ( ˜ z j )) + ct ( B ( ˜ z j + n ˆ e s , ˜ x i )) . Since m = ( d ′ m − d ′ m ) ( d ′ m − d ′ m ) ( d ′ m − d m ) − is positive by assumption, f ij ( c ) < ∞ for all c .This proves that µ ( z j , x i ) defines an element of the completion. Now, for x ∈ ̂ CF ∗ r ( L , L ) , z ∈ ̂ CF ∗ r ( L , L ) , write x = ∑ i a i x i z = ∑ j b j z j . Define f x,z ( c ) = ∑ i,j,n ∈ Z ev r ( a i ) ev r ( b j ) R A ijn + cy ijn ;we wish to show that f x,z ( c ) < ∞ for all c ∈ R . By the linearity of B , there exists D > ∣ t ( B ( ˜ z j + n ˆ e s , ˜ x )) ∣ < D (∣ t ( z )∣ + D ∣ t ( x )∣ + D ∣ n ∣ + ) for all i, j, n ∈ Z . Then we can write f x,z ( c ) = ∑ i,j,n ∈ Z ∣ R m ( n − w ( ˜ x i ) − w ( ˜ z j )) / ev r ( a i b i ) R ct ( B ( ˜ z j + n ˆ e s , ˜ x i )) ∣ < R − m ∑ n ⎛⎝ R mn / R − − ∣ Dcn ∣ ⎛⎝ ∑ i,j ∣ ev r ( a i b j ) R Dc ∣ t ( x i )∣ R Dc ∣ t ( z j )∣ )RRRRRRRRRRR⎞⎠ ≤ R − m ( ∑ n R mn / q − − ∣ Dcn ∣ ) ( ∑ i ∣ ev r ( a i ) R − ∣ Dct ( x i )∣ ∣) ⎛⎝ ∑ j ∣ ev r ( b j ) R − ∣ Dct ( b j )∣ ∣⎞⎠ < ∞ . To prove the general case, note that there exist type- W r Hamiltonians H , H , and H such that the set ( ψ H ( L ) , ψ H ( L ) , ψ H ( L )) is in correct position in F ( W r ) and such that ψ H ( L ) , ψ H ( L ) and ψ H ( L ) satisfy the conditions prescribed above (so each Lagrangian is planar if the Lagrangians are essentiallytranverse, and are small perturbations of planar Lagrangians otherwise). LetΦ ∶ ̂ CF ∗ r ( L , L ) → ̂ CF ∗ r ( ψ H ( L ) , ψ H ( L )) ;Φ ∶ ̂ CF ∗ r ( L , L ) → ̂ CF ∗ r ( ψ H ( L ) , ψ H ( L )) ;Φ ∶ ̂ CF ∗ r ( L , L ) → ̂ CF ∗ r ( ψ H ( L ) , ψ H ( L )) , denote the quasi-isomorphisms from multiplication furnished by the continuation elements. Then [ Φ H ( µ ( z, x ))] = [ µ ( Φ ( z ) , Φ ( x ))] as classes in HF ∗ ( ψ H ( L ) , ψ H ( L )) . We have shown that µ ( Φ ( z ) , Φ ( x )) ∈ ̂ CF ∗ r ( ψ H ( L ) , ψ H ( L )) . Appealing to Proposition 5.6.1, we deduce that Φ H ( µ ( z, x )) ∈ ̂ CF ∗ r ( ψ H ( L ) , ψ H ( L )) ; then the sameproposition allows us to conclude that µ ( z, x ) is an element of the completion ̂ CF ∗ r ( L , L ) . (cid:3) Unitary flat connections.
We will enlarge our Fukaya categories by equipping our Lagrangians withlocal systems (see [7]). Let ∇ be a unitary flat connection on the trivial rank-one vector bundle V → L , anddenote this data by ( L, ∇ ) . Define ̂ CF ∗ (( L, ∇ ) , ( L ′ , ∇ ′ )) = ⊕ y ∈ L ∩ L ′ Hom ( V ∣ y , V ′ ∣ y ) and weight each disc u contributing to a product µ n by the monodromy e πiν ∶ ( V ) ∣ y out → ( V ) n ∣ y out obtained by parallel transport around its boundary: µ n ( y n − , . . . , y ) = ∑ [ u ] M ( y , . . . , y n − , y out ) T ω [ u ] e πiν y out Since ∇ is unitary, adding such coefficients does not affect the analysis of the convergence of the µ n overΛ or over C ; we can thus define a completion ̂ CF ∗ (( L, ∇ ) , ( L, ∇ ′ )) as in Equation (39). (In general, in allproofs of convergence results, we will take Lagrangians without local systems.) OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 41
We say that a collection of Lagrangians with local systems {( L , ∇ ) , . . . , ( L n , ∇ n )} is in correct positionif the set {( L , . . . , L n )} is in correct position. We can relate Floer complexes with continuation elementsdefined in the same way above, but now weighting by the parallel transport of the connections. Proposition 5.7.1.
Let L and L ′ be in G P ( I ) . Let ( ψ H ( L ) , ∇ ) and ( ψ H ′ ( L ′ ) , ∇ ′ ) be in correct position,and assume that α ± ( H ) − α ± ( H ′ ) > K ± ( L, L ′ ) on the wrapped end(s) of Y . Let ˆ n ∈ Z . Then if ˜ L and ˜ L ′ are I -essentially transverse, HF ∗ (( ψ H ( L ) , ∇ ) , ( ψ H ′ ( L ′ ) , ∇ ′ )) ˆ n = Λ . If ˜ L and ˜ L ′ − ˆ n are I -essentially non-transverse and the monodromies of ∇ and ∇ around the loop [( , , )] ∈ π ( Y ) = π ( T ) agree, then HF ∗ (( ψ H ( L ) , ∇ ) , ( ψ H ′ ( L ′ ) , ∇ ′ )) ˆ n = Λ ⊕ Λ where there is a generator in degrees i and i + for i = or . Otherwise HF ∗ (( ψ H ( L ) , ∇ ) , ( ψ H ′ ( L ′ ) , ∇ ′ )) ˆ n = . Proof.
We have seen that if ˜ L and ˜ L ′ − ˆ n are I -essentially transverse, there exists some type- I Hamiltonian H ′′ such that ∣ ψ H ′′ ○ ψ H ( ˜ L ) ∩ ψ H ′ ( ˜ L ′ ) − ˆ n ∣ =
1; then HF ∗ (( ψ H ( L ) , ∇ ) , ( ψ H ′ ( L ′ ) , ∇ ′ )) ˆ n ≅ HF ∗ (( ψ H ′′ ○ ψ H ( L ) , ∇ ) , ( ψ H ′ ( L ′ ) , ∇ ′ )) ˆ n = Λ . If ˜ L and ˜ L ′ are I -essentially non-transverse, we have seen there exists some type- I Hamiltonian H ′′ suchthat CF ∗ ( ψ H ′′ ○ ψ H ( ˜ L ) , ψ H ′ ( ˜ L ′ )) ˆ n = Λ y i ⊕ Λ y i + , with generators of degree i and i + i = ∇ and ∇ around the loop [( , , )] by e πiν and e πiν ′ respectively. Then µ ( y i ) = T A ( e πiν − e πiν ′ ) ;note that this differential vanishes if and only e πiν = e πiν ′ . (cid:3) Definition 5.7.1 (Complexified action corrections) . Let ( L , ∇ ) , ( L , ∇ ) ∈ G P . Let w L ,L denote thecoordinate function associated to the pair of Lagrangians. Then we can always decompose ∇ − ∇ as(58) ∇ − ∇ = πi ( B d w L ,L + A d t ) . For y a generator of CF ∗ ( L , L ) , define the complexified action-corrected generator y = e − πiAt ( y ) T − S ( y ) y. Fukaya categories
Definition of Fukaya categories.
We define three variants of the Fukaya categories via the localiza-tion approach found in unpublished work of Abouzaid and Seidel ([4]), which has recently been used in otherworks, e.g. [11]. This approach is well-suited to our purposes as it allows us to relate our categories easilythrough localization maps following the work of Ganatra, Pardon, and Shende in [10].For I = A , ( m, k ) , or W r , consider a set of Lagrangians { L σ } σ ∈ Σ I with L σ ∈ G ( I ) which contains at leastone representative of each class of type-I Lagrangians up to compactly supported Hamiltonian isotopies.Then choose for each L σ , σ ∈ Σ I , an I -cofinal sequence H σ, , H σ, , . . . of perturbing type- I Hamiltonianswith the property that any set { ψ H σ ,i ( L σ ) , ψ H σ ,i ( L σ ) , . . . , ψ H σk,ik ( L σ k )} with i > i > . . . > i k is in correct position. By I -cofinal we mean a sequence of perturbing type-I Hamiltonianswith the property that for any generic L ′ ∈ G P ( I ) , there exists n ∈ N such that the set ( ψ H σ,n ( L σ ) , L ′ ) is incorrect position. Definition of O I . Let O I be the A ∞ category with objects ( L σ , ∇ , i ) where σ ∈ Σ I , ∇ is a flat unitaryconnection on V → L the rank-one trivial C -bundle, and i ∈ N . In the following, we often simplify notationby writing ( L , ∇ , i ) as an object L ; when unambiguous, we refer to the entire set as well as its gradingdata and Spin structures as the “Lagrangian.” The reader should keep in mind that when referring to theLagrangian, we mean this entire set of data.Let ( L σ , ∇ , i ) > ( L σ , ∇ , i ) when i > i . The morphisms in the directed category O I are defined by: O I (( L σ , ∇ , i ) , ( L σ , ∇ , i )) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩̂ CF ∗ (( ψ H σ ,i ( L σ ) , ∇ ) , ( ψ H σ ,j ( L σ ) , ∇ )) i > i Λ { e } ( L σ , ∇ , i ) = ( L σ , ∇ , i ) . Here { e } ∈ ̂ CF ∗ (( ψ H ( L ) , ∇ ) , ( ψ H ( L ) , ∇ )) is defined to be a formal element of degree zero. The higherproducts µ n ∶ O I ( L n − , L n ) ⊗ ⋯ ⊗ O I ( L , L ) → O I ( L , L n ) are defined when L ≥ L ≥ . . . ≥ L n , using compatible choices of strip-like ends and almost-complex structuresas described in 5.1.2. As previously noted, the µ n will satisfy the A ∞ relations for any finite collection ofLagrangians where each pair ( L i , L j ) with i > j is in correct position.Let σ ∈ Σ and let ∇ be a flat unitary connection on L σ . Let L iσ be shorthand for the data ( ψ H ( σ,i ) ( L σ ) , ∇ ) ,when used in Floer complexes.6.1.2. Localization.
For j > i , let c (( L σ , ∇ ) ,j,i ) ∈ ̂ CF ∗ ( L jσ , L iσ ) denote the continuation element. Denote the set of all such quasi-units by Q I : Q I = { c (( L, ∇ ) ,j,i ) } ( L, ∇ ) ∈ O ( I ) ,j > i . We define F ( I ) = O I [ Q − ] by taking the A ∞ quotient category in the sense of Lyubashenko-Ovsienko([19]). This defines a category with the same objects as O I . The morphisms between two objects arecalculated by considering the composition of functors. O I → Tw O I → Tw O I / Cones ( Q I ) . For any pair ( L σ , L σ ′ ) ∈ Ob F ( I ) , the Λ-modules ̂ HF ∗ ( L i + jσ , L jσ ′ ) , i ≥
0, form a directed system wherethe maps between objects are furnished by the continuation elements. We use the following lemma from thework of Ganatra-Pardon-Shende (Lemma 3.57 in [11]);
Proposition 6.1.1. H ∗ Hom F ( I ) ( L σ , L σ ′ ) ≅ lim i → ∞ ̂ HF ∗ ( L j + iσ , L jσ ′ ) . for any j . Remark 6.1.1.
While we use completed Floer complexes in every variant of our Fukaya categories, themorphism complexes that appear for the unwrapped category are finitely generated free Λ-modules, so theuse of the completion is immaterial in that instance. In the partially wrapped category we have seen thatthe completion is not necessary in the sense: the subspace of ̂ CF ∗ consisting of finite sums of generators isclosed under the µ n . In contrast, it is essential to work with the completion for the fully wrapped category.6.2. Invariance of choices.Proposition 6.2.1.
The choices involved in the construction of F ( I ) are immaterial: different choices ofstrip-like ends, complex structures, and 1-forms β used to define continuation elements will produce quasi-isomorphic A ∞ categories.Proof. This is a standard fact that follows from the contractibility of the space of choices of such data;see [11]. (cid:3)
The following proposition and its proof is identical to Proposition 3.39 in [11]; we record the proof heresince it will also be used below.
OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 43
Proposition 6.2.2.
The A ∞ quasi-equivalence class of F ( I ) is independent of choices of cofinal sequencesor isotopy classes.Proof. The idea is that given two choices Σ I , Σ ′ I and choices of cofinal sequences, we can embed each Fukayacategory into a larger category containing both categories, and each embedding will be a quasi-equivalence.The observation in [11] is that one can define a wrapped category with only a partial ordering on itselements. Let N be a partially ordered set, and let O I be a partially ordered set consisting of Lagrangians ( L σ , ∇ , a ) with a ∈ N , with associated perturbing Hamiltonians H σ,a . Assume that O I has the property thatevery totally ordered collection in O I is in correct position. Assume also that for each Lagrangian L , thereis a sequence L < L ( ) < L ( ) < . . . which is I -cofinal. Let Q I be a set of quasi-units so each continuationelement corresponding to the positive perturbations L ( i ) → L ( j ) is an element of Q I . Then we may localize O I at Q I to construct a category F ( I ) . For each L, L ′ ∈ O I , the maps below are isomorphisms (Lemma3.37, [11]):(59) lim i ̂ HF ∗ ( L ( i ) , L ) → lim i H ∗ Hom F ( I ) ( L ( i ) , L ) ← H ∗ ( Hom F ( I ) ( L, L ′ )) . Now, let O I , and O ′ I denote two such categories constructed with two choices of Σ I . Let the objects of O ′′ I consist of the set Z + ∪ O I ⊔ O I with the lexicographical partial order. Find a sequence of Hamiltoniansfor each object in O I ∪ O ′ I which is cofinal with respect to this partial ordering; let Q ′′ I be the correspondingcontinuation elements. Let F ′′ ( I ) be the localization of O ′′ at Q ′′ I ∪ Q I ∪ Q ′′ I . Then there are natural functors F ( I ) → F ′′ and F ′ ( I ) → F ′′ ( I ) , each of which is a quasi-equivalence by Equation (59). (cid:3) Localization functors.
Let A = (( m , k ) , ( m ∞ , m k ∞ )) . We now define localization functors F ( A ) → F ( m , k ) → F W r . The functor F ( A ) → F ( m , k ) is constructed as follows: given O A and a set of quasi-units Q A , weconstruct a category O ( m ,k ) with a partial ordering and a set of continuation morphism Q ( m ,k ) such thateach element ( L, ∇ , i ) ∈ O A is an element of O ( m , k ) , and Q A ⊂ Q ( m ,k ) , and the choice of strip-like endsand complex structures are compatible. We can then use this set of data to define F ( m , k ) . Similarly, wechoose O W r to include O ( m ,k ) and Q ( m ,k ) ⊂ Q W r . Then there exist localization morphisms ι A ∶ F ( A ) → F ( m , k ) ;(60) ι ( m,k ) ∶ F ( m, k ) → F W r (61)which are the inclusion on objects. (Compare [10], Section 5.) Remark 6.3.1.
Let Ψ be the symplectomorphism Y → Y given by ( t, s, ϕ t , ϕ s ) ↦ ( − t, s, − ϕ t , ϕ s ) . Then Ψinduces equivalences of categories F W r → F W r and F (( m , k ) , ( m ∞ , k ∞ )) → F (( m ∞ , k ∞ ) , ( m , k )) . When all categories are defined via compatible choices, the following diagram commutes: F ( m , k ) F ( A ) F W r F ( m ∞ , k ∞ ) i ( m ,k ) i A i A ○ Ψ ∗ Ψ ∗ ○ i ( m ∞ ,k ∞) In particular, there is a diagram of categories H F ( m , k ) H F ( A ) H F W r H F ( m ∞ , k ∞ ) independent of the choices involved in the creation. Remark 6.3.2. (a) For all n ∈ Z , there is an equivalence of A ∞ categories I n ∶ F ( m , k ) → F ( m , k + nm ) for n ∈ Z .(b) For all n ∈ Z , there is an equivalence of A ∞ categories I n ∶ F (( m , k ) , ( m ∞ , k ∞ n )) → F (( m , k + nm ) , ( m ∞ , k ∞ − nm ∞ )) . Moreover, for each n the following diagram commutes:(62) F (( m , k ) , ( m ∞ , k ∞ )) F (( m , k + nm ) , ( m ∞ , k ∞ − nm ∞ )) F ( m , k ) F ( m , k + nm ) I n I n where the vertical arrows are the localization functors.The equivalences I n are induced by the symplectomorphism Y → Y given by ( t, s, ϕ t , ϕ s ) ↦ ( t, s − nt, ϕ t + nϕ s , ϕ s ) ;some slight care must be taken to ensure that the perturbing Hamiltonians used to define the category aresent to perturbing Hamiltonians. Definition 6.3.1.
For τ ∈ C with Im τ >
0, define H F τ ( I ) to be the C -linear cohomology category wherethe morphisms are defined via the map ev τ ∶ Λ e − πi Im τ → C .The following lemma will be essential as we perform calculations in our categories: Proposition 6.3.1.
There is a quasi-isomorphism ̂ CF ∗ τ ( ψ H ( L σ ) , ψ H ′ ( L σ ′ )) ≅ Hom ∗ F τ ( I ) ( L σ , L ′ σ ) for any I -type H, H ′ such that the pair ( ψ H ( L σ ) , ψ H ′ ( L σ ′ )) is in correct position and H and H ′ satisfy theconditions of Lemma 5.5.3.Proof. By the I -cofinality of the sequences { L ( n ) σ } n ∈ N we can always find n > ( ψ H σ,n ( L σ ) , ψ H ( L σ ) , ψ H ′ ( L σ ′ )) is in correct position. Then by Lemma 5.5.3, multiplication by the continuation elements give quasi-isomorphisms ̂ CF ∗ τ ( ψ H ( L σ ) , ψ H ′ ( L σ ′ )) → ̂ CF ∗ τ ( L nσ , ψ H ′ ( L σ ′ )) → ̂ CF ∗ τ ( L nσ , L σ ′ ) . Moreover, by the same lemma we know that for all n ′ > n , the maps ̂ CF ∗ τ ( L nσ , L σ ′ ) → ̂ CF ∗ τ ( L n ′ σ , L σ ′ ) are quasi-isomorphisms. Thus we can use Equation (59) to deduce that the map ̂ CF ∗ τ ( L nσ , L σ ′ ) → Hom ∗ F τ ( I ) ( L σ , L σ ′ ) is a quasi-isomorphism. (cid:3) OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 45
Remark 6.3.3 (Stops) . The category F ( A ) could equivalently be defined in terms of stops `a la Sylvan([29]) or Ganatra, Pardon and Shende ([10]), after reworking some definitions to allow for non-Liouvillemanifolds.The“boundary at ∞ ” of Y is given by ({ − } ⊔ { }) × ( S ) = ( S ) ⊔ ( S ) with coordinates ( s, ϕ t , ϕ s ) at each end. We insert two ( S ) stops given by the set of points {( s, ϕ t , ϕ s ) ∣ m ϕ t + k ϕ s = } . at the negative end and {( s, ϕ t , ϕ s ) ∣ − m ∞ ϕ t + k ∞ ϕ s = } . at the positive end. This reflects the Landau-Ginzburg heuristics found in Section 3. In this setting, thelocalization maps can be thought of as “stop removal” maps.7. Categories of line bundles
The main theorems of this work exhibit a correspondence between categories of Lagrangian sections in F and categories of line bundles on the complex surfaces S τ ( A ) , D τ ( m, k ) , and C ∗ × E τ . For a complexmanifold X , Pic an ( X ) can be given the structure of a category by taking Hom ( L , L ′ ) = Ext ( L , L ′ ) , wherethe composition of morphisms is given by the Yoneda product. The categories of line bundles we considerare subcategories of Pic an ( X ) ; in the case of the open surfaces we consider the category of all line bundleswhich can extend to a compactification S τ ( A ) , and in the case of the compact surfaces we take our categoryto be the entire analytic Picard group.7.1. Open surfaces.
We first investigate line bundles on the open elliptic surfaces D τ ( m, k ) and C ∗ × E τ ;we will exhibit admissible Lagrangians mirror to these line bundles. Our methods are inspired by Polischukand Zaslow’s work on mirror symmetry for the elliptic curve ([22]), and we largely follow the notation in thiswork throughout. Let L τ, be the degree one line bundle on E τ called L in [22] with total spaceTot ( L τ, ) = ( C ∗ × C / ( x, t ) ∼ ( e πiτ x, e − πiτ x − t )) . Definition of the line bundles L τ ( m, k, d, ξ ) . For a pair of coprime integers ( m, k ) with m >
0, let g m ∶ D τ ( m, k ) → C × E mτ denote the degree- m map defined by ( v, w ) ↦ ( v m , w m ) . As in Section 2, let i m,k ∶ C ∗ × E τ → D τ ( m, k ) denote the inclusion map. Now consider the map ( π E mτ ○ g m ○ i m,k ) ∗ ∶ Pic an E mτ → Pic an C ∗ × E τ . Recall that any degree d line bundle L ∈ Pic an E mτ can be written as T ∗ ξ − ( L mτ, ) L d − mτ, where T ξ − denotesthe translation map z ↦ ξ − z (written multiplicatively), and ξ ∈ C ∗ . Let ( η, ν ) be in R . Define L τ ( m, k, d, e πi ( τη + ν ) ) = ( π E mτ ○ g m ) ∗ ( T ∗ e − πim ( τη + ν ) ( L mτ, ) ⋅ L d − mτ, ) ∈ Pic an D τ ( m, k ) L τ ( m, k, d, e πi ( τη + ν ) ) = ( π E mτ ○ g m ○ i m,k ) ∗ ( T ∗ e − πim ( τη + ν ) ( L mτ, ) ⋅ L d − mτ, ) ∈ Pic an C ∗ × E τ . In coordinates, ( g m ○ i m,k ) ( z, x ) = ( z, z − k x m ) . Let ξ = e πi ( τη + ν ) . Letting i z ∶ E τ → C ∗ × E τ denote the inclusion map of the fiber over z ∈ C ∗ , we have thefollowing:(63) i ∗ z ( L τ ( m, k, d, ξ )) = T ∗ ξ − z kd ( L τ, ) ⋅ L md − τ, . We use this fiber-wise description to extend the notation L τ ( m, k, d, ξ ) to the case where ( m, k ) is notprimitive in Z , so L τ ( m, k, d, ξ ) describes a line bundle that restricts in this way to each fiber; this isjustified with the following lemma: Lemma 7.1.1.
If a line bundle L on C ∗ × E τ satisfies i ∗ z L = O E for all z , then L = O . The most notable difference is that we reverse the sign conventions in the mirror symmetry correspondence: where weconsider Lagrangian L on E τ in this work, the authors of [22] would consider the Lagrangian − L . We also change notation forthe translation factor: our τη + ν is their τα + β . Proof.
This follows from the fact that Pic an C ∗ =
0. Let L be such a line bundle. Consider the restriction L ∣ C ∗ × { } ; let s ∶ C ∗ × { } → L be a non-vanishing section. On each fiber there exists a unique section s z ∈ H ( E, i ∗ z L ) such that s z ( ) = s ( z ) . Then we can define a global non-vanishing section σ ∈ H ( C ∗× E, L ) by σ ( z, t ) = s z ( t ) . (cid:3) Remark 7.1.1.
Allowing ( m, k ) to be non-coprime is useful, but it introduces a source of non-uniquenessin the labelling beyond that of the choice of ξ , namely that L ( mj, kj, d, ξ ) = L ( m, k, dj, ξ ) for any integer j . In particular, L ( , , d, ξ ) = L ( m, k, , ξ ) for any ( m, k ) . Note also that this implies that we may alwaysrewrite the label on L ( mj, kj, d, ξ ) = L ( m, k, dj, ξ ) so that ( m, k ) are co-prime and m > Lemma 7.1.2.
Assume that either m d + m d ≠ , or m d + m d = k d + k d = . Then L ( m , k , d , ξ ) ⊗ L ( m , k , d , ξ ) = L ( m d + m d , k d + k d , , ξ ξ ) . In the case where d = d = d , L ( m , k , d, ξ ) ⊗ L ( m , k , d, ξ ) = L ( m + m , k + k , d, ξ ξ ) . In the case where k = k = k and m = m = m , L ( m, k, d , ξ ) ⊗ L ( m, k, d , ξ ) = L ( m, k, d + d , ξ ξ ) . Proof.
From Equation 63, we know that L ( m , k , d , ξ ) ⊗ L ( m , k , d , ξ ) ⊗ L ( k d + k d , m d + m d , , ξ ξ ) − ∣ E τ = O ∣ E τ ;the result then follows from Lemma 7.1.1. (cid:3) Proposition 7.1.3.
Let L be a line bundle on C ∗ × E τ which is degree zero when restricted to the fiber.Then L gives a section of the relative Jacobian J ( C ∗ × E τ / C ∗ ) = J ( E τ ) × C ∗ , i.e. (via the Abel-Jacobi map)a holomorphic map f ∶ C ∗ → E τ . L extends to a line bundle over C × E τ if and only if the map f extendsacross the origin.Proof. The forward direction is clear. Let C ⊂ J ( C ∗ × E τ / C ∗ ) be the graph of f . Let C ⊂ C × E τ be thecompactification of C determined by the extension of f . Let π denote the projection π ∶ C × C ( C × E τ ) → C × E τ . Let P be the Poincar´e line bundle on J ( C × E τ / C ) × C ( C × E τ ) .Then the map π is an isomorphism, so π ∗ ( P ∣ C ) is a line bundle on C × E τ . Moreover, i ∗ z ( π ∗ ( P ∣ C )) = i ∗ z L by construction, so by Lemma 7.1.1, π ∗ ( P ∣ C ) = L . (cid:3) Corollary 7.1.4.
Parametrize
P ic d ( E τ ) via the isomorphism E → P ic d ( E τ ) given by z ↦ T ∗ z L τ, ⋅ L dτ, .Then a line bundle L ∈ Pic an ( C × E ∗ τ ) which is degree d on each fiber defines a map f ∶ C → E τ ≅ P ic d ( E τ ) .Such a line bundle L will extend to a line bundle on C × E τ if and only if the function f extends across theorigin. Corollary 7.1.5.
Let L be a line bundle on C ∗ × E τ of degree d on each fiber. Then there exists a linebundle L on D τ ( m, k ) with i ∗ ( m,k ) L = L if and only m divides d and the map f ∶ C ∗ → E τ ≅ Pic d E τ has theproperty that z kd f ( z m ) ∶ C ∗ → Pic d ( E τ ) ≅ E τ defines a holomorphic map which extends across the origin.Proof. Recall that the divisor class of the central fiber is m [ F ] , where [ F ] is the class of the generic fiber;thus any line bundle which extends across the central fiber must have degree divisible by m on the genericfiber.Assume that L is degree d = d ′ m on each fiber. First assume that ( i − m,k ) ∗ L extends to L ∈ Pic an D τ ( m, k ) .Recall that D τ ( m, k ) = C × E τ / (( v, w ) ∼ ( e πi / m v, e − πik / m w )) ; pull back L to L ′ ∈ Pic an C × E τ via thequotient map. Then we obtain a holomorphic map h ∶ C → E ≅ Pic d ( E τ ) . On C ∗ ⊂ C , h ( v ) = v kd f ( v m ) ;we conclude that z kd f ( z m ) extends across the origin. OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 47
Conversely, assume that m divides d and h ( z ) = z kd f ( z m ) extends across the origin. Consider the linebundle L ′ of degree d / m on C × E mτ determined by the map h ∶ C → E mτ . Again let g denote the m -to-onemap g m ∶ D τ ( m, k ) → C × E mτ given by ( z, t ) ↦ ( z m , t m ) . Note that i ∗ ( m,k ) ( g ∗ m L ′ ) ∣ C ∗ × E τ = L ; thus g ∗ m L ′ gives the desired extension. (cid:3) Corollary 7.1.6.
For d ≠ , L ( m , k , d, ξ ) extends to D ( m , k ) if and only if k m = k m .Proof. The corresponding map f ∶ C ∗ → Pic dm ( E ) is given by z ↦ ξz − dk , so L ( m , k , d, ξ ) extends to D ( m , k ) if and only if the map ξz k dm z − d k m extends across the origin. The monomial map z n from C ∗ → E lifts to a map C ∗ → C ∗ ; this map extends to a map C → C ∗ if and only if n =
0. We conclude that L ( m , k , d, ξ ) extends if and only if k m = k m . (cid:3) Corollary 7.1.7.
A line bundle L on C ∗ × E τ extends to a compactification S τ ( A ) of C ∗ × E τ if and onlyif L = L τ ( m, k, d, ξ ) for some ( m, k, d, ξ ) such that md = only if k = .Proof. When md ≠ L = L τ ( m, k, d, ξ ) extends to P × E τ / Z / m Z , where Z / m Z acts by ([ z ∶ z ] , t ) ↦ ( e πi / m [ z ∶ z ] , e − πik / m t ) ; we recognize this quotient as S (( m, k ) , ( − m, k )) . When md = L τ extends toany S τ ( A ) as π ∗ E τ ( T ∗ ξ − L τ, ⋅ L τ, ) .Now let L ∈ Pic d ′ an C ∗ × E be a line bundle which extends to a compactification S ( A ) . First assume that d ′ ≠
0. Then if L extends to a compactification S ( A ) , [ E ] ∈ H ( S ( A )) ≠ c ( L ) ⋅ [ E ] = d ′ ≠
0. Then S ( A ) is algebraic, so A = (( m, k ) , ( m, − k )) . Then since the map f ∶ C ∗ → E has the property that z kd f ( z m ) extends across the origin, f extends to a map f ∶ C ∗ → C ∗ , and can be written as a Laurent series f = ∑ n ∈ Z f n z n . Now apply Corollary 7.1.5. Since z kd f ( z m ) extends over the origin, f n ≠ n = − dk / m and f n = n with kd + mn <
0; since z kd f ( z m ) extends over infinity, f n = n with kd + mn >
0. We conclude that f = ξz − d / mk for some ξ ∈ C ; then L = L ( m, k, d, ξ ) .Now assume that d =
0, and again consider the corresponding function f ∶ C ∗ → E and its Laurentseries f = ∑ n ∈ Z f n z n . Using Corollary 7.1.5, we conclude that f n = n ≠
0. Thus f is constant, so L = L ( m , k , , f ) for any m , k . (cid:3) Definition 7.1.1.
Let C τ denote the full subcategory of C ⊂ Coh an C ∗ × E τ generated by the set of linebundles L τ ( m, k, d, ξ ) . Let C τ ( m, k ) denote the full subcategory of Coh an C ∗ × E τ with objects L τ ( m, k, d, ξ ) .7.1.2. Morphisms in C τ and C τ ( m, k ) . Let L , L ′ ∈ C τ . ThenHom C ( L , L ′ ) = H ( C ∗ × E τ , L ′ ⊗ L − ) . So it suffices to compute the global sections H ( C ∗ × E, L ( m, k, d, ξ )) . Fix a line bundle L = L ( m, k, d, ξ ) ;note that sections exist of L only if c ( L ) . [ E ] = md ≥
0. Thus when d < H ( C ∗ × E τ , L ) = d ≥
0. We express the total space of L as a quotient:Tot ( L ( m, k, d, ξ )) = ( C ∗ × C ∗ × C ) / (( z, x, t ) ∼ ( z, e πiτ x, ξz − kd x md e πiτmd t )) . The global sections σ ∈ H ( C ∗ × E τ , L ) can be expressed in terms of convergent power series σ ( z, x ) = ∑ j,a ∈ Z C j,a z j x a which are equivariant under this Z action on C ∗ × C ∗ × C .When d =
0, we must have C j,a = aτ / ≡ ( τ η + ν ) mod Z ; we conclude that sections exist only when η = N ∈ Z , ν ∈ Z . Let σ L , ( j,N ) denote the monomial z j x N . In this case we can write any global section σ as the product of x N with a globally convergent Laurent series: H ( C ∗ × E, L ) = ⎧⎪⎪⎨⎪⎪⎩⎛⎝ ∑ j c j z j ⎞⎠ x N ∣ lim ∣ j ∣ → ∞ ∣ c j ∣ /∣ j ∣ = ⎫⎪⎪⎬⎪⎪⎭ (64) = ⎧⎪⎪⎨⎪⎪⎩ ∑ j c j σ L , ( j,N ) ∣ lim ∣ j ∣ → ∞ ∣ c j ∣ /∣ j ∣ = ⎫⎪⎪⎬⎪⎪⎭ . (65) Remark 7.1.2.
Any degree zero line bundle with sections is isomorphic to O E × C ∗ ; however, it will beconvenient to keep the extra data provided by ξ as we write the restriction maps Pic S τ ( A ) → D τ ( m, k ) → C × E τ .When d ≥
0, recall that sections of line bundles over the elliptic curve can be expressed in terms of theclassical theta functions: ϑ [ c ′ , c ′′ ] ( q, x ) = ∑ ℓ ∈ Z q ( ℓ + c ′) ( e πic ′′ x ) ( ℓ + c ′ ) The theta function ϑ [ , ] ( q, x ) is a section of L . This observation allows us to find a Fr´echet space basisfor H ( C ∗ × E, L ) . Let e πi ( τη + ν ) = ξ , and let σ L , ( j,a ) = e πiτη md e πiνηmd z ( ka / m + j ) ϑ [ amd , ] ( q md , ξz − kd x md ) (66) = e πiτη md e πiνηmd ∑ ℓ ∈ Z q md ( ℓ + amd ) / ξ ℓ + amd z − kdℓ + j x mdℓ + a ;(67)note that this power series defines an equivariant function which restricts to a theta function on each fiber.Further note that σ L , ( j,a ) = σ L , ( j + kdn,a − mdn ) for any n ∈ Z ; thus these sections are indexed by the quotient Z /( kd, − md ) ≅ Z ⊕ Z / d , with the isomorphism realized by ( j, a ) ↦ ( mj + ka, a ) . The factor ( mj + ka ) controls the power of z multiplying the theta function, and any globally convergent equivariant power series σ decomposes as σ = md − ∑ a = g a ( z ) z ka / m ϑ [ amd , ] ( q md , ξz − kd x md ) . Thus we can write again write the space of global sections in terms of everywhere convergent Laurent series:(68) H ( C ∗ × E, L ( m, k, d, ξ )) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ∑ ( j,a ) ∈ Z /( kd, − md ) c ( j,a ) σ L , ( j,a ) ∣ lim ∣ mj + ka ∣ → ∞ ∣ c ( j,a ) ∣ / mj + ka = ⎫⎪⎪⎪⎬⎪⎪⎪⎭ . Multiplication of sections.
Let L i = L ( m i , k i , d i , ξ i ) , i = , ,
2, and let e πi ( τη i + ν i ) = ξ i . We nowinvestigate the Yoneda productExt ( L , L ) × Ext ( L , L ) → Ext ( L , L ) ;using the identification Ext ( L , L ′ ) = H ( C ∗ × E τ , L − L ′ ) , we can identify this with the product H = H ( C ∗ × E τ , L − L ) ⊗ H ( C ∗ × E τ , L − L ) → H ( C ∗ × E τ , L − L ) . We investigate the product σ L − L , ( j ,b ) ⋅ σ L − L , ( j ,a ) . When m i + d i + − m i d i > i = L − i + L i has negative degree, so the space of sections is empty. When one of the bundles is of degree zero, i.e.when m i d i = m i + d i + for some i , the multiplication is simple:(69) σ L − L , ( j ,b ) ⋅ σ L − L , ( j ,a ) = σ L − L , ( j + j ,a + b ) . Now assume both line bundles are of positive degree. Let m d − m d = m and k d − k d = k . When ( m d − m d ) < ( m d − m d ) <
0, we can use Equation (66) to write(70) σ L − L , ( j ,a ) ⋅ σ L − L , ( j ,b ) = ∑ n ∈ Z e πi ( τK ( j + j − kn,a + b + mn ) + θ ( j + j − kn,a + b + mn ) ) σ L − L , ( j + j − kn,a + b + mn ) OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 49 where K ( j + j ′ ,a + b ′ ) = ( ( a + η − η ) m d − m d + ( b ′ + η − η ) m d − m d − ( a + b ′ + η − η ) m d − m d ) ;(71) θ ( j + j ′ ,a + b ′ ) = ( ν − ν ) ( a + η − η m d − m d ) + ( ν − ν ) ( b ′ + η − η m d − m d ) − ( ν − ν ) ( a + b ′ + η − η m d − m d ) . (72) Remark 7.1.3.
Note that when ( m , k ) = ( m , k ) = ( m , k ) , the image of the homomorphism Z → Z n ↦ n ( − k, m ) = ( − k ( d − d ) n, m ( d − d ) n ) is contained within the torsion part of Z /( − k ( d − d ) , m ( d − d )) . Thus the sum above has non-zerocoefficients on only finitely many generators; the coefficients on those finitely many generators are thetafunctions in z . This reflects the fact that the sections σ L − L ( j ,a ) and σ L − L , ( j ,b ) are both algebraic sections of algebraic line bundles on the quasi-projective variety D τ ( m, k ) .7.2. Calculations over D τ ( m, k ) . Consider the image of the restriction map H ( D τ ( m, k ) , L ( m, k, d, ξ )) → H ( C ∗ × E τ , L ( m, k, d, ξ )) ;we say a global section extends over D τ ( m, k ) if it lies in the image of this map. Proposition 7.2.1. σ L , ( j,a ) extends over D τ ( m, k ) if and only if mj + ka ≥ .Proof. Write such a section as a sum of monomials: σ L , ( j,a ) = ∑ ℓ ∈ Z C ( ℓ ) z − kdℓ + j x mdℓ + a ∶ (when d = C ( ℓ ) = ℓ ≠
0; when d > C ( ℓ ) are the coefficients of the corresponding ϑ function). After changing to the ( v, w ) coordinates on C × E τ , we write σ L , ( j,a ) = v mj + ka ∑ ℓ ∈ Z w mdℓ + a ∶ Such section will extend over v = mj + ka >
0. Let g denote the generator of Z / m Z acting on C × E ;note that g ⋅ v mi + ka w mdj + a = e πika / m e − πika / m v mi + ka w mdj + a = v mi + ka w mdj + a so any power series of this form is invariant under the action of Z / m Z , and thus defines a section of L ( m, k, d, ξ ) . Thus σ L , ( j,a ) is in the image of the restriction map ι ( m, k ) ∗ ∶ H ( D τ ( m, k ) , L ) → H ( C ∗ × E τ , L ) .Let σ L , ( j,a ) satisfy ι ∗ ( m,k ) ( σ L , ( j,a ) ) = σ L , ( j,a ) ; we can then write H ( D τ ( m, k ) , L ( m, k, d, ξ )) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ∑ ( j,a ) ∈ Z /( − kd,md ) c ( j,a ) σ ( j,a ) ∣ mj + ka > , lim mj + ka → ∞ ∣ c ( j,a ) ∣ ∣ mj + ka ∣ − = ⎫⎪⎪⎪⎬⎪⎪⎪⎭ . (cid:3) The theta function identities of the previous subsection govern the multiplications of sections over the D τ ( m, k ) . We conclude Proposition 7.2.2. i ∗ ( m,k ) ∶ C τ ( m, k ) → C τ is an embedding of categories, i.e., is both faithful and injective on objects. Compact surfaces S τ ( A ) . One obtains a description of the Picard groups of the surfaces S τ ( A ) fromthe calculation of the fundamental groups of the surfaces in question. Let A = (( m , k ) , ( m ∞ , k ∞ )) , andlet n = m k ∞ + m ∞ k ≠
0; recall that the condition that n ≠ S τ ( A ) is not algebraic.Recall that π ( S τ ( A )) = coker ⎛⎜⎝ m − m ∞ k k ∞
00 0 0 ⎞⎟⎠ ∶ Z → Z , and π ( S τ ( A )) acts freely on the universal cover C ∖ { } by ( a, b, c ) ⋅ ( z , z ) = ( exp ( πi ( k ∞ a + m ∞ b + m ∞ cτ ) n ) z , exp ( πi ( − k a + m b + m cτ ) n ) z ) . Let Λ ⊂ Z denote the lattice spanned by the columns of the matrix ( m − m ∞ k k ∞ ) , and let Λ ∗ ⊂ Q denote the dual lattice. Elements of Λ ∗ are those elements ( f, g ) ∈ Q that satisfy m f + k g ∈ Z ;(73) − m ∞ f + k ∞ g ∈ Z . (74)It is known (see [20]) that any line bundle on a Hopf surface has trivial pullback to the universalcover; thus there is a map Hom ( π ( S τ ( A )) , C ∗ ) → Pic S τ ( A ) described as follows: given an element γ of Hom ( π ( S τ ( A )) , C ∗ ) , we define a π ( S τ ( A )) action on ( C ∖ { }) × C by ℓ ⋅ ( z , z , t ) = ( ℓ ( z , z ) , γ − ( ℓ ) t ) . This defines a holomorphic line bundle L γ on S τ ( A ) with total spaceTot L γ = (( C ∖ { }) × C ) / π ( S τ ( A )) . It is also known that this map Hom ( π ( S τ ( A )) , C ∗ ) → Pic S τ ( A ) is an isomorphism. NoteHom ( π ( S τ ( A )) , C ∗ ) = Hom ( Z ⊕ Z / Λ , C ∗ ) ≅ C ∗ ⊕ ( Λ ∗ / Z ) . The canonical isomorphism is given by ( ξ, f, g ) ↦ (( a, b, c ) ↦ ξ c e πifa e πigb ) . Let L ( ξ, f, g ) denote the line bundle corresponding to ( ξ, f, g ) ∈ C ∗ × Λ ∗ . Write γ = ( ξ, f, g ) . Thecohomology of the line bundles can be calculated via techniques similar to that in [20], which we summarizebelow. The sections of L ( γ ) are given by holomorphic functions on C ∖ { } (equivalent, by Hartog’s lemma,to holomorphic functions on C ) which are equivariant under the action of the fundamental group. The setof such functions is given by the monomials z n z n , where n and n are positive integers satisfyingexp ( πiτ ( m ∞ n + m n ) n ) = ξ ;(75) exp ( πi ( k ∞ n − k n ) n ) = exp ( πif ) ;(76) exp ( πi ( m ∞ n + m n ) n ) = exp ( πig ) . (77)Denote the number of such pairs ( n , n ) by N γ . From Remark 2.2.1, we know that the dualizing sheaf is O S ( A ) ( − ( m + m ∞ ) [ F ]) = L ( − k ∞ + k n , − m ∞ − m n , e πiτ ( − m ∞− m n ) ) . Write γ = L ( − k ∞ + k n , − m ∞ + m n , e πiτ ( − m ∞+ m n ) ) ; then by Serre duality h ( L γ ) = h ( L γ − γ ) . One obtainsfrom the Hirzeburch-Riemann-Roch formula that χ ( L ) = L ∈ Pic ( S τ ( A )) . Thus we can deduce(78) h i ( L ( γ )) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ N γ i = N γ + N g − γ i = N γ − γ i = . OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 51
Note that for all L , h and h cannot both be non-vanishing.7.3.1. Products on global sections.
The product H ( S τ ( A ) , L ( γ )) ⊗ H ( S τ ( A ) , L ( g ′ )) → H ( S τ ( A ) , L ( gg ′ )) is induced by the product on the ring of holomorphic functions O C : z n z n ⋅ z n ′ z n ′ = z n + n ′ z n + n ′ . Restriction maps.
The restriction maps Pic S τ ( A ) → Pic an D τ ( m , k ) → Pic an ( C ∗ × E τ ) induced bythe inclusion map ι m,k can be calculated using the change of coordinates of Equation (14). L ( f, g, e πi ( τλ + θ ) ) ↦ L ( m, k, , e πi ( τλ + θ − τg ) ) ↦ L ( m, k, , e πi ( τλ + θ − τg ) ) . The induced maps on global sections are given by(79) z n z n ↦ v n w ( m ∞ n + m n )/ n ↦ z ( k ∞ n − k n )/ n x ( m ∞ n + m n )/ n . Mirror symmetry for D τ ( m, k ) and C ∗ × E τ We now prove mirror symmetry results for the open surfaces D τ ( m, k ) and C ∗ × E τ . Let M ( m, k, d, α ) = ( − α dkdk − dm ) , and when m ≠
0, let E ( m, k, η ) = (( k / m ) ηη ) ;Let η, ν ∈ R . Define the following type- ( m, k ) Lagrangian section and choice of lift to R by L ( m,k,d,e πi ( τη + ν ) ) = ({( Y, M ( m, k, d, dk / m ) Y − E ( m, k, η )) ∣ Y ∈ R } , d − πiν (( k / m ) d t + d s )) . This section is dedicated to the proofs of the following theorems:
Theorem.
There is a full and faithful functorΦ W r ∶ C τ → H F τ, W r realized on objects by Φ ( W r ) ( L τ ( m, k, d, e πi ( τη + ν ) )) = [ L ( m,k,d,e πi ( τη + ν ) ) ] . Theorem.
For each ( m, k ) there is a full and faithful functorΦ ( m,k ) ∶ C τ ( m, k ) → H F τ ( m, k ) realized on objects by Φ ( m,k ) ( L τ ( m, k, d, e πi ( τη + ν ) )) = [ L ( m,k,d,e πi ( τη + ν ) ) ] . Theorem.
The following diagram commutes: C τ ( m, k ) C τ H F τ ( m, k ) H F τ, W r . i ∗( m,k ) Φ ( m,k ) Φ (W r ) ι ( m,k ) Calculations in H F τ, W r . We first calculate in the wrapped category H F τ, W r . Perturbing Hamiltonians.
For each triple of Lagrangians ( L , L , L ) , choose type- W r Hamiltonians H , H , H satisfying the following conditions:(a) The set ( ψ H ( L ) , ψ H ( L ) , ψ H ( L )) is in correct position.(b) H i = H α i ( t ) + f i ( t, s ) for some f ( t, s ) with ∥ f i ∥ C < α i ( t ) − α j ( t ) > K ( L i , L j ) for each i < j .(d) For each degree zero generator y ∈ CF ( ψ H i ( L i ) , ψ H j ( L j )) with i < j , f i ( y ) − f j ( y ) = ψ H i ( ˜ L i ) and ψ H j ( ˜ L j ) − ˆ n are not essentially disjoint, so they are not disjoint afterperturbing by any compactly supported Hamiltonian, ∣ ψ H i ( ˜ L i ) ∩ ψ H j ( ˜ L j ) − ˆ n ∣ = ∣ ψ H i ( ˜ L i ) ∩ ψ H j ( ˜ L j ) − ˆ n ∣ ⊆ P i ∩ ( P j + ˆ n ) , where P i and P j are the corresponding planes of the respectiveLagrangians, and equality holds when the two planes intersect transversely (i.e., when [ γ L ] ≠ [ γ L ] ).(g) When P i and P j do not intersect transversely, the degree zero generators y ∈ ψ H i ( ˜ L i ) ∩ ( ψ H j ( ˜ L j ) − ˆ n ) have s ( y ) ∈ N .Note that these conditions ensure that S R ( y ) = Q L i ,L j ( y ) for all y since equality holds for all points on theintersection points of the corresponding planes.By Proposition 6.3.1, for such choices of Hamiltonian, the natural map CF ∗ ( ψ H ( L ) , ψ H ( L )) → Hom ∗ F W r ( L , L ) is a quasi-isomorphism; thus we can compute the cohomology category Hom H F W r ( L , L ) using such choicesof Hamiltonians. Remark 8.1.1 (Notation) . Let L = L ( m ,k ,d ,e πi ( τη + ν ) ) , L = L ( m ,k ,d ,e πi ( τη + ν ) ) .(a) Let y ∈ CF ∗ τ ( ψ H ( L ) , ψ H ( L )) denote the action-corrected generator y = e − πi ( At ( y ) + τS ( y )) y , thatis, the image of the action-corrected generator e − πiAt ( y ) T − S ( y ) y ∈ CF ∗ ( ψ H ( L ) , ψ H ( L )) under theevaluation map ev τ ∶ CF ∗ ( ψ H ( L ) , ψ H ( L )) → CF ∗ τ ( ψ H ( L ) , ψ H ( L )) .(b) The stabilizer subgroup Γ ( L , L ) ⊂ Z is equal to Z ( ˆ e s + k d ˆ e ϕ t − m d ˆ e ϕ s ) ⊕ Z (( k d − k d ) ˆ e ϕ t − ( m d − m d ) ˆ e ϕ s ) . Thus the the map Z → Z defined by ( j, a ) ↦ je ϕ t + ae ϕ s descends to an isomorphism Z /(( k d − k d ) , − ( m d − m d )) ≅ Z / Γ ( L , L ) , the indexing group for the topological grading on the Floer complexes. Let y H ,H ( j,a ) ∈ R denote agenerator of CF ∗ ( ψ H ( ˜ L ) , ( ψ H ( ˜ L ) − je ϕ t − ae ϕ s )) ( j,a ) ; let y H ,H ( j,a ) = π Y ( y H ,H ( j,a ) ) denote the corresponding generator of the Floer complex. Remark 8.1.2.
The conditions on the Hamiltonians ensure that for any y = ( Y, Φ ) ∈ ψ H ( ˜ L ) ∩ ( ψ H ( ˜ L ) − je ϕ t − ae ϕ s ) ,Y satisfies the equation(80) ( − d ( k ) m + d ( k ) m + α − α k d − k d k d − k d − m d + m d ) Y − ( j − k m ( η ) + k m ( η ) a − ( η ) + η ) = , as this is the equation which describes the intersection of the corresponding planes.8.1.2. Morphisms.
Let L = L ( m ,k ,d ,e πi ( τη + ν ) ) , L = L ( m ,k ,d ,e πi ( τη + ν ) ) . We now calculateHom H F τ, W r ( L , L ) ≅ ̂ HF τ ( ψ H ( L ) , ψ H ( L )) . OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 53
Case 1: L and L represent the same homology class. The conditions on H and H ensure that CF iτ ( ψ H ( L ) , ψ H ( L )) ( j,a ) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ C ⋅ y H ,H ( j,a ) i = C ⋅ z H ,H ( j,a ) i = ψ H ( ˜ L ) and ψ H ( ˜ L ) + j ˆ e ϕ t + a ˆ e ϕ s are not essentially disjoint, and CF ∗ ( ψ H ( L ) , ψ H ( L )) ( j,a ) = HF ( ψ H ( L ) , ψ H ( L )) ( j,a ) = ⎧⎪⎪⎨⎪⎪⎩ Λ y H ,H ( j,a ) ν − ν ∈ Z ν − ν ∉ Z . It now remains to calculate the ( j, a ) ∈ Z such that ψ H ( ˜ L ) and ψ H ˜ L ) − j ˆ e ϕ t − a ˆ e ϕ s intersect. Notethat if η − η ∉ Z , ψ H ( L ) ∩ ( ψ H ( L ) − i ˆ e ϕ t − a ˆ e ϕ s ) = ∅ for all ( j, a ) ∈ Z . When η − η = N ∈ Z , ∣ ψ H ( L ) ∩ ( ψ H ( L ) − i ˆ e ϕ t − a ˆ e ϕ s ) ∣ = a = N .Let t j = t ( y H ,H ( j,N ) ) ; from Equation (80), we read off that t j = j ( α − α ) − . We conclude thatHom H F τ, W r ( L , L ) ≅ ̂ HF τ ( ψ H ( L ) , ψ H ( L )) ;(81) = ⎧⎪⎪⎨⎪⎪⎩ ∑ j ∈ Z c j y ( j,N ) ∣ lim ∣ t j ∣ → ∞ ∣ c j ∣ ( t j ) − = ⎫⎪⎪⎬⎪⎪⎭ (82) = ⎧⎪⎪⎨⎪⎪⎩ ∑ j ∈ Z c j y ( j,N ) ∣ lim ∣ j ∣ → ∞ ∣ c j ∣ j − = ⎫⎪⎪⎬⎪⎪⎭ . (83) Case 2: L and L represent different homology classes. Now assume that [ γ L ] ≠ [ γ L ] . Let α ( H i ) = α i + d i ( k i ) / m i . Let m = m d − m d , k = k d − k d , d = Q ψ H ( L ) ,ψ H ( L ) of Definition 4.5.1 is given by Q ψ H ( L ) ,ψ H ( L ) = x T M ( m, k, d, α − α ) x ;its decomposition is given by Q α ( L ,L ) + Q β ( L ,L ) = αt + Q β ( L ,L ) . In this case the Floer complex CF ( ψ H ( L ) , ψ H ( L )) is concentrated in a single degree i . i is zero if andonly the signature of Q ψ H ( L ) ,ψ H ( L ) is ( , ) , if and only if m d < m d . In this case CF ( ψ H ( L ) , ψ H ( L )) ( j,a ) is nonzero for all ( j, a ) since two transverse planes will always intersect, and generated by a single intersectionpoint. We write CF τ ( ψ H ( L ) , ψ H ( L )) ( j,a ) = C y H ,H ( j,a ) . Write t j,a = t ( y H ,H ( j,a ) ) ; from Equation (80), we read off that(84) t ( y H ,H ( j,a ) ) = c ( mj + mk ) + c . for two constants c and c , with c >
0. We conclude thatHom H F τ, W r ( L , L ) = ̂ HF τ ( ψ H ( L ) , ψ H ( L )) ( j,a ) ;(85) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ∑ ( j,a ) ∈ Z /( − k,m ) c ( j,a ) y H ,H ( j,a ) ∣ lim ∣ t j,a ∣ → ∞ ∣ c ( j,a ) ∣ ( t j,a ) − ⎫⎪⎪⎪⎬⎪⎪⎪⎭ = = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ∑ ( j,a ) ∈ Z /( − k,m ) c ( j,a ) y H ,H ( j,a ) ∣ lim ∣ mj + ka ∣ → ∞ ∣ c ( j,a ) ∣ ∣ mj + ka ∣ − = ⎫⎪⎪⎪⎬⎪⎪⎪⎭ . (87)8.1.3. Independence of choices of the Hamiltonian.
Proposition 8.1.1.
Let L and L be as above. For any choices of H , H , H ′ , H ′ where H and H satisfythe conditions above, [ y H ,H ( j,a ) ] = [ y H ′ ,H ′ ( j,a ) ] as classes in Hom H F τ ( L , L ) .Proof. For any set of Hamiltonians H , H , H ′ , H ′ , there exists a set of Hamiltonians H ′′ , H ′′ such that thesets ( ψ H ′′ ( L ) , ψ H ′′ ( L ) , ψ H , ψ H ) and ( ψ H ′′ ( L ) , ψ H ′′ ( L ) , ψ H ′ ( L ) , ψ H ( L )) are both in correct position;thus it suffices to show that in the case that ( ψ H ′ ( L ) , ψ H ′ ( L ) , ψ H ( L ) , ψ H ′ ( L )) is in correct position(88) µ ( c H → H ′ , y H ′ ,H ′ ( j,a ) ) = µ ( y H ,H ( j,a ) , c H → H ′ ) ∈ CF ∗ ( ψ H ′ ( L ) , ψ H ( L )) . Write(89) ∇ − ∇ = πi (( ν − ν ) d w + A d t ) as in Equation (58). We have seen that µ ( y H ,H ( j,a ) , c H → H ′ ) counts one triangle with area S R ( y H ,H ( j,a ) ) − S R ( y H ′ ,H ( j,a ) ) ;the monodromy around this triangle is given byexp ( πi ( B ( w ( y H ,H ( j,a ) ) − w ( y H ′ ,H ( j,a ) )) + A ( t ( y H ,H ( j,a ) ) − t ( y H ′ ,H ( j,a ) )))) . Recall that w ( y H ,H ( j,a ) ) = w ( y H ′ ,H ( j,a ) ) ; thus this quantity is equal toexp ( πiA ( t ( y H ,H ( j,a ) ) − t ( y H ′ ,H ( j,a ) ))) . Since y H ,H ( j,a ) = e − πi ( τS ( y H ,H ( j,a ) ) + At ( y H ,H ( j,a ) )) y H ,H ( j,a ) , and similarly for y H ′ ,H ( j,a ) , we write µ ( y H ,H ( j,a ) , c H → H ′ ) = e πiτ ( S R ( y H ,H ( j,a ) ) − S ( y H ,H ( j,a ) )) − ( S R ( y H ′ ,H ( j,a ) ) − S ( y H ′ ,H ( j,a ) )) y H ′ ,H ( j,a ) . When [ γ L ] = [ γ L ] , S R ( y H ,H ( j,a ) ) − S ( y H ,H ( j,a ) ) = S R ( y H ′ ,H ( j,a ) ) − S ( y H ′ ,H ( j,a ) ) = [ γ L ] ≠ [ γ L ] we can apply Lemma 4.5.3 to conclude ( S R ( y H ,H ( j,a ) ) − S ( y H ,H ( j,a ) )) − ( S R ( y H ′ ,H ( j,a ) ) − S ( y H ′ ,H ( j,a ) )) = . Thus µ ( y H ,H ( j,a ) , c H → H ′ ) = y H ′ ,H ( j,a ) . We can repeat this process on the left to show that µ ( c H → H ′ , y H ′ ,H ′ ( j,a ) ) = y H ′ ,H ( j,a ) ; we conclude that theequality of Equation (88) holds. (cid:3) OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 55
Definition 8.1.1.
Let [ y H ,H ( j,a ) ] = [ y ( j,a ) ] ∈ Hom H F τ, W r ( L , L ) . We will also denote this generator by [ y L ,L , ( j,a ) ] when it is useful to record the extra data.8.1.4. µ calculations. Let L i = L ( m i ,k i ,d i ,e πi ( τηi + νi ) ) , i = , , [ x ( j ,a ) ] ∈ Hom H F τ, W r ( L , L ) ; [ z ( j ,b ) ] ∈ Hom H F τ, W r ( L , L ) ; [ y ( j ,c ) ] ∈ Hom H F τ, W r ( L , L ) . Let w , w , and w be the coordinates associated to the pairs ( L , L ) , ( L , L ) , and ( L , L ) (so w = s − ( k d − k d )/( m d − m d ) t when m d − m d ≠
0, and so on). Write [ x ( j ,a ) ] ⋅ [ z ( j ,b ) ] = ∑ ( j ,c ) A ( j ,c ) [ y ( j ,c ) ] . We now calculate the coefficients in the above equation. Choose representative Hamiltonians H , H , H and generators x ( j ,a ) , z ( j ,b ) . First assume that m i d i − m i + d i + = i = ,
1; then Hom H F τ ( L i , L i + ) ≠ k i d i − k i + d i + = µ ( z ( j ,b ) , x ( j ,a ) ) counts only a single J -holomorphic map which has vanishing action-corrected area; thus(90) µ ( z ( j ,b ) , x ( j ,a ) ) = y ( j + j ,a + b ) . Otherwise, fix a lift x ( j ,a ) = ˜ L ∩ ( ˜ L − ( j ˆ e ϕ t , a ˆ e ϕ s )) . Let k = k d − k d , m = m d − m d . UsingProposition 5.6.2, write the product as the sum of products corresponding to different choices of the lift of z ( j,a ) : let ( − k, m ) = ( k d − k d , m d − m d ) , and write(91) µ ( z ( j ,b ) , x ( j ,a ) ) = ∑ ℓ ∈ Z e πi ( τC ( j + j − kℓ,a + b + mℓ ) + θ ( j + j − kℓ,a + b + mℓ ) ) y ( j + j − kℓ,a + b + mℓ ) The coefficients C ( j + j − kℓ,a + b + mℓ ) and θ ( j + j − kℓ,a + b + mℓ ) record the action-corrected area and monodromy ofthe unique J-holomorphic map with the two boundary marked points mapping to the intersection points x ( j ,a ) = ˜ L ∩ ( ˜ L − ( j ˆ e ϕ t , a ˆ e ϕ s )) ; z ( j ′ ,b ′ ) = ( ˜ L − ( j ˆ e ϕ t , a ˆ e ϕ s )) ∩ ( ˜ L − ( j + j ′ ) ˆ e ϕ t − ( a + b ′ ) ˆ e ϕ s ) ; y ( j + j ′ ,a + b ′ ) = ˜ L ∩ ( ˜ L − ( j + j ′ ) ˆ e ϕ t − ( a + b ′ ) ˆ e ϕ s ) . We conclude that in H F τ, W r , [ z ( j ,b ) ] ⋅ [ x ( j ,a ) ] = ∑ ℓ ∈ Z e πi ( τC ( j + j − kℓ,a + b + mℓ ) + θ ( j + j − kℓ,a + b + mℓ ) ) [ y ( j + j − kℓ,a + b + mℓ ) ] . We now find the coefficients on the terms above. Write C ( j ,a ) , ( j ′ ,b ′ ) (92) = S R ( x ( j ,a ) ) − S ( x ( j ,a ) ) + S R ( z ( j ′ ,b ′ ) ) − S ( z ( j ′ ,b ′ ) ) − S R ( y ( j + j ′ ,a + b ′ ) ) + S ( y ( j + j ′ ,a + b ′ ) ) (93) = Q βL ,L ( x ( j ,a ) ) + Q βL ,L ( z ( j , ′ b ′ ) ) − Q βL ,L ( y ( j + j ′ ,a + b ′ ) ) (94) = (( m d − m d ) ( w ( x ( j ,a ) )) + ( m d − m d ) ( w ( z ( j ′ ,b ′ ) )) − ( m d − m d ) ( w ( y ( j + j ′ ,a + b ′ ) )) ) . (95) Write x ( j ,a ) = ( X, Φ ) . Let α = α − α + ( k d − k d ) m d − m d . Note that X = ( x t , x s ) is the solution to the equation0 = (( − ( k d − k d ) m d − m d k d − k d k d − k d − m d + m d ) + ( α
00 0 )) X − ( j − k / m ( η ) + k / m ( η ) a − ( η ) + η ) = (( k d − k d ) w ( x ( j ,a ) ) + αt ( x ( j ,a ) ) − ( m d + m d ) w ( x ( j ,a ) ) ) + ( j − k / m ( η ) + k / m ( η ) a − η + η ) . From this, we can read off w ( x ( j ,a ) ) = − a − η + η m d − m d . We can similarly show that w ( z ( j ′ ,b ′ ) ) = − b ′ − η + η m d − m d ; w ( y ( j ,a ) + ( j ′ ,b ′ ) ) = − a + b ′ − η + η m d − m d ;Thus we can write the term in Equation (92) as(96) C ( j ,a ) , ( j ′ ,b ′ ) = ( ( a + η − η ) m d − m d + ( b ′ + η − η ) m d − m d − ( a + b ′ + η − η ) m d − m d ) Using the decomposition of ∇ of Equation (89), write θ = ( ν − ν ) w ( x ( j ,a ) ) + ( ν − ν ) w ( z ( j ′ ,b ′ ) ) + ( ν − ν ) w ( y ( j + j ′ ,a + b ′ ) ) (97) = ( ν − ν ) ( a + η − η m d − m d ) + ( ν − ν ) ( b ′ + η − η m d − m d ) + ( ν − ν ) ( a + b ′ + η − η m d − m d ) . (98)8.2. Calculations in H F τ ( m, k ) . We now calculate in the partially wrapped category H F τ ( m, k ) . Con-sider the collection L ( m,k,d,η ) for fixed k , m . For each pair L and L of such Lagrangians, choose type- ( m, k ) Hamiltonians H and H with the following properties:(a) The set ( ψ H ( L ) , ψ H ( L )) is in correct position.(b) There are type- W r Hamiltonians H ′ , H ′ satisfying constraints (a)-(f) of Section 8.1.1 so that H = H ′ on Y + T , where T is such that π Y ({ ˜ L ∩ ( ˜ L − ( j ˆ e ϕ s , a ˆ e ϕ t )) ∣ mj + ka ≥ }) ⊆ Y + T π Y ({ ˜ L ∩ ( ˜ L − ( j ˆ e ϕ s , a ˆ e ϕ t )) ∣ mj + ka < }) ⊆ Y − T ;we see that such a T is guaranteed to exist by Equation (80). We also demand that H ( T − t ) = H ( T + t ) .(c) ψ H ( L ) and ψ H ( L ) are disjoint on Y − T for all i < j .There is a natural inclusion of complexes ι i,j ∶ CF ∗ τ ( ψ H ( L ) , ψ H ( L )) → CF ∗ τ ( ψ H ′ ( L ) , ψ H ′ ( L )) for i < j induced by the inclusion of generators which respects the actions S R and S , thus a natural inclusion ι i,j ∶ ̂ HF τ ( ψ H ( L ) , ψ H ( L )) → ̂ HF τ ( ψ H ′ ( L ) , ψ H ′ ( L )) . Let y ( j,a ) denote the generator of CF ∗ τ ( ψ H ( L ) , ψ H ( L )) which maps to y ( j,a ) ; then using Proposition6.3.1 we can writeHom H F τ ( m,k ) ( L , L ) ≅ ̂ HF τ ( ψ H ( L ) , ψ H ( L )) ≅ ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ∑ ( j,a ) ,mj + ka ≥ c ( j,a ) y ( j,a ) ∣ lim mj + ka → ∞ c ( mj + ka ) − ( j,a ) = ⎫⎪⎪⎪⎬⎪⎪⎪⎭ . Proposition 8.2.1.
The inclusion of complexes ι ( , ) ∶ CF ∗ τ ( ψ H ( L ) , ψ H ( L )) → CF ∗ τ ( ψ H ′ ( L ) , ψ H ′ ( L ′ )) agrees with the map Hom H F τ ( m,k ) ( L , L ) → Hom H F τ, W r ( L , L ) . OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 57
Proof.
It suffices to show that in the case H = H ′ =
0, there exists type- W r Hamiltonians G and G ′ suchthat G + H = G ′ + H ′ , such that the sets ( ψ G + H ( L ) , ψ H ( L ) , L ) and ( ψ G ′ + H ′ ( L ) , ψ H ′ ( L ) , L ) are in correct position, and such that the following diagram commutes: CF ∗ τ ( ψ H ( L ) , L ) CF ∗ τ ( ψ H ′ ( L ) , L ) CF ∗ τ ( ψ G + H ( L ) , L ) CF ∗ τ ( ψ G ′ + H ′ ( L ) , L ) . ι , µ ( c G , ⋅ ) µ ( c G ′ , ⋅ ) = Let T > H = H ′ on Y + T . Then defining G and G ′ by G ( t ) = H ( T + ( T − t )) + f and G ′ ( t ) = H ( T − ∣ T − t ∣) + f , where f is a small function supported on a neighborhood of T such that H ( T − ∣ T − t ∣) + f function, provides such G and G ′ . (Here we use the fact that H ′ satisfies H ′ ( T − t ) = H ′ ( T + t ) .) Note thatfor t > T , H ( t ) = H ′ ( t ) and G ( t ) = G ′ ( t ) . Let y ( j,a ) denote the unique generator of CF ( ψ H ( L ) , L ) ( j,a ) ;let y G ( j,a ) denote the unique generator of CF ( ψ G + H ( L ) , L ) ( j,a ) . Then µ ( y ( j,a ) , c G ) = µ ( ι , ( y ( j,a ) ) , c G ′ ) = y G ( j,a ) ;in both cases the areas of the discs contributing to the count defining µ are cancelled by the change inaction of the generators. (cid:3) We conclude the following:
Proposition 8.2.2. ι ( m,k ) ∶ H F τ ( m, k ) → H F τ, W r is an embedding of categories, i.e., is both faithful and injective on objects. Mirror symmetry statements.
We now prove the theorems in the beginning of the section, whichare restated below with proofs; the proofs amount to matching up the structure coefficients.
Theorem 8.3.1.
There is a full and faithful functor Φ W r ∶ C τ → H F τ, W r realized on objects by Φ W r ( L τ ( m, k, d, e πi ( τη + ν ) )) = [ L ( m,k,d,e πi ( τη + ν ) ) ] . Proof.
Let L i = L ( m i ,k i ,d i ,e πi ( τηi + νi ) ) , i = , L i = L ( m,k,d,e πi ( τηi + νi ) ) . Having defined Φ W r ( L i ) = [ L i ] ,define for σ L − L , ( j,a ) ∈ Ext ( L , L ) , Φ W r ( σ L − L , ( j,a ) ) = [ y L ,L , ( j,a ) ] . Identifying Equations (64) and Equations (83), and (68) and (87), shows that this induces an isomorphismHom H F τ ( Φ W r ( L ) , Φ W r ( L )) = Hom H F τ ( Φ W r ( L ) , Φ W r ( L )) . for all pairs of Lagrangians. By identifying Equations (69) and (90), (71) and (96), and (72) and (98), wesee that this isomorphism respects the composition of morphisms. (cid:3) By restricting the domain of the functor Φ W on objects, and using the fact that the localization functors i ( m,k ) ∶ H F τ ( m, k ) → H F τ, W r ; ι ∗ ( m,k ) ∶ C τ ( m, k ) → C τ are both embeddings of categories, we obtain the following theorems: Theorem 8.3.2.
There is a full and faithful functor Φ ( m,k ) ∶ C τ ( m, k ) → H F τ ( m, k ) realized on objects by Φ ( m,k ) ( L τ ( m, k, d, e πi ( τη + ν ) )) = [ L ( m,k,d,e πi ( τη + ν ) ) ] . The functor above takes σ L − L , ( j,a ) to [ y L ,L , ( j,a ) ] . Theorem 8.3.3.
The following diagram commutes: C τ ( m, k ) C τ H F τ ( m, k ) H F τ, W r . i ∗( m,k ) Φ ( m,k ) Φ (W r ) ι ( m,k ) Example 8.3.1.
Consider the line bundles L = L ( , , , ) and L ′ = L ( , − , , ) on C ∗ × E τ ; in less arcanenotation, these are, respectively, the line bundles π ∗ E τ ( L τ, ) and f ∗ ( π ∗ E τ ( L τ, )) , where f ∶ C ∗ × E τ → C ∗ × E τ is given by f ( z, x ) = ( z, zx ) . (Note that this is the change of coordinate map by which C ∗ × E τ is gluedto itself to produce the standard Hopf surface.) There does not exist an algebraic chart on C ∗ × E τ forwhich L and L ′ are both algebraic line bundles.) Let q = e πiτ , and recall that the standard theta function ϑ [ , ]( q, x ) is a section of L τ, . Thus H ( C × E τ , L ) = { p ( z ) ⋅ ϑ [ , ]( q, x ) ∣ p ∶ C ∗ → C holomorphic } . The Fr´echet basis for H ( C × E τ , L ) of Equation (66) is given by { σ ( j,a ) } ( j,a ) ∈ Z /( , ) where σ ( j,a ) = z j ϑ [ a, ] ( q, x ) = ∑ ℓ ∈ Z q ( ℓ + a ) / z j x ℓ + a . Pulling back through f gives a basis { σ ′ ( j,a ) } ( j,a ) ∈ Z /( , ) for the global sections of L ′ : σ ′ ( j,a ) = z − a f ∗ ( σ ( j,a ) ) = z j ϑ [ a, ] ϑ ( q, zx ) = ∑ ℓ ∈ Z q ( ℓ + a ) / z ℓ + j x ℓ + a . Let L ′′ = L ⊗ L ′ = L ( , − , , ) . The basis { σ ′′ ( j,a ) } ( j,a ) ∈ Z /( , ) for the sections of L ′′ given by Equation(66) is σ ′′ ( j,a ) = z ( a + j ) ϑ [ a / , ] ( q , zx ) = ∑ ℓ ∈ Z q ( ℓ + a ) z ℓ + j x ℓ + a . Standard theta function identities (or the calculations of Subsection 7.1) give σ ( , ) ⋅ σ ′ ( , ) = ϑ [ , ]( q, x ) ⋅ ϑ [ , ]( q, zx ) ;(99) = ϑ [ , ]( q , z ) ϑ [ , ]( q , zx ) + ϑ [ / , ] ( q , z ) ϑ [ / , ] ( q , zx ) ;(100) = ∑ n ∈ Z q n ( ∑ ℓ ∈ Z q ( ℓ + n / ) z kℓ + n x ℓ + n ) ;(101) = ∑ n ∈ Z q n σ ′′ ( n,n ) ;(102)the third equality follows from expanding the power series in z .We now view the multiplication of sections as the product Ext ( O , L ′ ) ⊗ Ext ( L − , O ) → Ext ( L − , L ′ ) and calculate the product of the mirror elements on the symplectic side. The Lagrangians mirror to the linebundles L − , O , and L ′ are given by L ( , , − , ) , L ( , , , ) , and L ( , − , , ) ; denote these L − , L O , L ′ re-spectively. Let ( L − ) − α and ( L ′ ) α ′ denote the image of the Lagrangians under perturbation by the quadratic OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 59 y out,n ˜ L O ( ˜ L − ) − α ˜ y ( , ) ( ˜ L ′ ) α − n ˆ e ϕ t − n ˆ e ϕ s ˜ y ′ ( , ) + n ˆ e s Figure 1.
A triangle contributing to µ ( y ( , ) , y ′ ( , ) ) Hamiltonians − αt / α ′ t /
2, where α, α ′ >
1; so ( L − ) − α = {( Y, ( − α − ) Y ) ∣ Y ∈ R } ; ( L ′ ) α ′ = {( Y, ( α ′ + ) Y ) ∣ Y ∈ R } . By Proposition 6.3.1, we can identify the relevant morphism spaces with the Floer complexes HF ∗ (( L − ) − α , L O ) , HF ∗ ( L O , ( L ′ ) α ′ ) , and HF ∗ (( L − ) − α , ( L ′ ) α ′ ) . Under the functor of Theorem 8.3.1,Φ W r ( σ ( , ) ) = [ y ( , ) ] ;Φ W r ( σ ′ ( , ) ) = [ y ′ ( , ) ] , where y ( , ) = e − πiτS ( y ( , ) ) y ( , ) and y ′ ( , ) = e − πiτS ( y ′( , ) ) y ′ ( , ) are the generators at ( , , , ) ∈ Y of theFloer complexes, weighted by action. The Floer product µ ( y ( , ) , y ′ ( , ) ) counts triangles homotopic in˜ Y = R to the planar triangles shown in Figure 1, where we have fixed lifts ˜ y ( , ) and ˜ y ′ ( , ) at ( , , , ) ,as well as lifts of the Lagrangians. There is one such triangle ∆ n for each n ∈ Z . We can solve for thecoordinates of ˜ y out,n to obtain˜ y out,n = ( n ( + ( α + α ′ )) − , n ( α + α ′ )( + ( α + α ′ )) − , n, n ) ;recalling that the generators are indexed by their radial coordinates, we see that y out,n = y ′′ ( n,n ) . The action-corrected symplectic area of ∆ n is equal to B ( ∆ n ) = ω ( ∆ n ) − S ( y ( , ) ) − S ( y ( , ) ) + S ( y ′′ ( n,n ) ) ; = αn ( + α ) + n ( + α ) = n . We conclude that [ y ( , ) ] ⋅ [ y ′ ( , ) ] = ∑ n ∈ Z q n [ y ′′ ( n,n ) ] , and comparing this expression to Equation (99), we verify that the two products agree. Mirror symmetry for compact surfaces
We now prove a mirror symmetry statement for the non-algebraic compact surfaces S τ ( A ) . Let A = (( m , k ) , ( m ∞ , k ∞ )) ; A = ( m − m ∞ k k ∞ ) ; n = det A = m k ∞ + m ∞ k . Let ( f, g, e πiτλ e πiθ ) ∈ Q × C ∗ define a representation of π ( S τ ( A )) and thus a line bundle L ( f, g, e πi ( λτ + θ ) ) on S τ ( A ) via the correspondence of Section 7.3. Define the following type- A Lagrangian section: L ( f,g,e πi ( λτ + θ ) ) = ({( ts ) , ( − λρ k ∞ / m ∞ − k / m ( t ) − λ ) + ( fg ) ∣ ( ts ) ∈ R × S } , d + πiθ d s ) . The remainder of this section is dedicated to the proof of the following theorem:
Theorem 9.0.1.
For all S τ ( A ) which are not algebraic surfaces, i.e., for such surfaces with n ≠ , there isa full and faithful functor Φ A ∶ Pic ( S τ ( A )) → H F τ ( A ) . realized on objects by Φ A ( L ( f, g, e πi ( τλ + θ ) )) = [ L ( f,g,e πi ( τλ + θ )) ] . As in the previous section, we define the map using specific choices of perturbing Hamiltonians. For eachtriple of Lagrangians L , L , L , with L i = L γ i ; γ i = ( f i , g i , e πiτλ i e πiθ i ) , ≤ i ≤ , define L W ri = L ( m ∞ , − k ∞ , ,e πiτ ( λi − gi ) e πiθ ) = L ( m ,k ,e πiτ ( λi − gi ) e πiθ ) , and choose type- A Hamiltonians H i , 0 ≤ i ≤
2, such that the following conditions are satisfied:(a) The set ( ψ H ( L ) , ψ H ( L ) , ψ H ( L )) is in correct position.(b) The function ∂H ( t, s ) ∂t − λρ m ∞ / k ∞ − m / k ( t ) controlling the ϕ t coordinate is a monotonic function for all s ∈ S .(c) There exist type- W r Hamiltonians H ′ , H ′ , H ′ satisfying conditions (a)-(f) of Subsection 8.1.1 such that ( ψ H i ( L i )) ∩ ([ − , ]) = ( ψ H ′ i ( L W ri )) ∩ ([ − , ]) .(d) When two lifts ψ H i ( ˜ L i ) and ψ H j ( ˜ L j ) − j ˆ e ϕ t − a ˆ e ϕ s are essentially disjoint, so they are disjoint afterperturbing by some compactly supported Hamiltonian, they are disjoint.These conditions ensure that for y ∈ L i ∩ L j , S R ( y ) = Q ψ Hi ( L W ri ) ,ψ Hj ( L W rj ) ( y ) . We use the isomorphism HF ( ψ H i ( L i ) , ψ H j ( L j )) ≅ Ð→ Hom H F τ ( A ) ( L , L ) to compute the cohomology category Hom H F τ ( A ) ( L , L ) . Lemma 9.0.2.
Let γ − γ = ( f, g, e πi ( τλ + θ ) ) . There is a bijection between solutions ( n , n ) ∈ N to Equa-tions (75) , (76) and (77) applied to ( f, g, e πi ( τλ + θ ) ) , and closed generators of degree zero of ψ H ( L ) and ψ H ( L ) , where H is a Hamiltonian of the above form.Proof. Note that when λ, θ ∉ Z , both sets are empty: if λ ∉ Z , ψ H ( L ) ∩ ψ H ( L ) = ∅ , and if λ ∉ Z , θ ∈ Z ,then there are no closed generators of degree zero by Proposition 5.7.1. Otherwise, observe that we havechosen H and H such that when 0 > − λ ( k ∞ m ∞ + k m ) = − λnm m ∞ , OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 61 i.e. when λ and n have the same sign, there is a bijection { intersection points y j,N of degree zero } ↔ { j ∈ Z ∩ I } where I is the interval I = [ − λk ∞ m ∞ − f − ε ( H ) + ε ( H ) , λk m − f + ε ( H ) − ε ( H )] . We claim that the map(103) j ↦ A (( jN ) + ( fg )) realizes the desired bijection.Note that solutions in Z to Equations (75), (76) and (77) are given by elements ( ℓ , ℓ ) ∈ Z such thatfor some j ∈ Z , 1 n ( k ∞ − k m ∞ m ) ( ℓ ℓ ) − ( fg ) = − ( jN ) . Denote this set of solutions Σ. We rewrite the above equation as A − ( ℓ ℓ ) = ( fg ) − ( jN ) . If we omit the requirement that ℓ , ℓ , and j are integers, then the set of solutions to this equation in R isgiven by a line in R ; the set of solutions ( ℓ , ℓ ) ∈ ( R + ) is given by a (potentially empty) segment S of thisline. Let I ′ = { x ∈ R ∣ A − ( ℓ ℓ ) = ( fg ) − ( xN ) , ( ℓ ℓ ) ∈ S. } .I ′ is non empty if and only if λnm ∞ > λnm >
0, if and only if λ and n have the same sign. In this casethe endpoints of I correspond to the solutions x to the equation above when ( ℓ , ℓ ) = ( λnm ∞ , ) and ( , λnm ) ;thus I ′ = [ − λk ∞ m ∞ − f, λk m − f ] . We now show that if j is an integer, A (( jN ) + ( fg )) ∈ Z . Note that A (( jN ) + ( fg )) = ( m ( j + f ) + k ( g + N ) − m ∞ ( j + f ) + k ∞ ( g + N )) . Then the fact that the pair ( f, g ) satisfies Equations (73) and (74) implies that ( ℓ , ℓ ) ∈ Z . Thus the mapof Equation (103) is a bijection from I ′ ∩ Z to Σ.It remains to show only that I ′ ∩ Z = I ∩ Z . The ( m , k ) boundary conditions ensure that m ( λk m − f ) + kN ∈ N . Since N is an integer, ( λk m − f ) ∈ m Z . Since ε ( H ) < m , there are no integers in the interval ( λk m − f, λk m − f + ε ( H )] ;similarly, we deduce that there are no integers in the interval [ λk ∞ m ∞ − f − ε ( H ) , λk ∞ m ∞ − f ) . We conclude that I ′ ∩ Z = I ∩ Z . (cid:3) Corollary 9.0.3.
Let L ( γ ) , L ( γ ) ∈ Pic S τ ( A ) . There is an isomorphism of C -vector spaces ϕ ∶ Ext ( L ( γ ) , L ( γ )) ≅ HF τ ( ψ H ( L γ ) , ψ H ( L γ )) for H , H as above.Proof of Theorem 9.0.1. Define the map on objects byΦ ( L ( γ )) = [ L γ ] . Consider bundles L ( γ i ) ∈ Pic ( S τ ( A )) , Lagrangians L γ i , and Hamiltonians H i for 0 ≤ i ≤ HF ( ψ H i ( L γ i ) , ψ H j ( L γ j )) ≅ Ð → Hom H F τ ( A ) ( L γ i , L γ j ) , i < j. Let s H ,H n ,n = [ e − πiτS R ( ϕ ( z n z n )) ϕ ( z n z n )] where ϕ is the isomorphism defined above. Then define thefunctor Φ A on the generators by Φ A ( z n z n ) = s H ,H n ,n . Note that by the assumptions on H and H , S R ( z n z n ) is independent of the choice of lifts of s , since itdepends only on t ; using the methods of Proposition 8.1.1, one can also show that this class is independentof the choice of the Hamiltonians H and H satisfying our assumptions.We now show that the map Φ respects the composition of morphisms. We have seen that the Yonedaproduct H ( S τ ( A ) , L ( γ − γ )) ⊗ H ( S τ ( A ) , L ( γ − γ )) → H ( S τ ( A ) , L ( γ − γ )) , is given by z n z n ⋅ z ℓ z ℓ = z n + ℓ z n + ℓ . By Lemma 5.6.2, the multiplication map µ ( ϕ ( z n z n ) , ϕ ( z ℓ z ℓ )) is equal to e πi ∆ y , where y is thegenerator in degree zero in CF ∗ ( ψ H ( L γ ) , ψ H ( L γ )) ( A − (( n + ℓ ,n + ℓ ) − ( f ,g ) + ( f ,g ))) , and ∆ encodes the area and monodromy around a planar triangle. Note that this generator is equal to ϕ ( z n + ℓ z n + ℓ ) . The action-corrected area and monodromy of the triangle ∆ vanish, so e πi ∆ y = s H ,H n + ℓ ,n + ℓ .Thus [ Φ ( z n z n )] ⋅ [ Φ ( z ℓ z ℓ )] = [ s n ,n ] = Φ A ( z n z n z ℓ z ℓ ) . (cid:3) Remark 9.0.1.
Note that ϕ ( z n z n ) is the generator of CF τ ( ψ H ( L γ ) , ψ H ( L γ )) ( A − (( n ,n ) − ( f,g ))) = CF τ ( ψ H ( L γ ) , ψ H ( L γ )) ( j,N ) for any choice of H , H . Remark 9.0.2.
With slightly more work, the proof above can be extended to give an isomorphism on thecohomology categories H ∗ F τ ( A ) → Pic S τ ( A ) , where the Hom spaces of the category Pic S τ ( A ) are givenby the full Ext groups. Example 9.0.1 (Hopf surface) . In the case that A = (( , ) , ( , )) , S τ ( A ) is the classical Hopf surface C ∖ { }/ ( z ∼ e πiτ z ) ; denote this surface by S . We have seen above that Pic an S ≅ Hom ( π ( S ) , C ) ≅ C ∗ ,with the isomorphism realized by ξ ↦ L ( ξ ) whereTot L ( ξ ) = C ∖ { } × C / (( z , t ) ∼ ( e πiτ z , ξt )) . Each L ( ξ ) has vanishing cohomology unless ξ = e πiτk for k ∈ N , in which case L ( e πiτk ) ≅ π ∗ A ( O ( k )) and H ∗ ( S, L ( ξ )) = H ∗ ( S ; C ) ⊗ H ∗ ( P , O ( k )) . In this case the global sections of L ( ξ ) are given by the monomials z n z n with n + n = k ; note that theseare the pullbacks under π A of the monomials which give a basis for the global sections of O ( k ) .The mirror to the Hopf surface is the non-algebraic Landau-Ginzburg model ( Y, A ) . The Lagrangianmirror to L ( e πiτ ( k + iθ ) ) is L ( e πi ( τk + θ ) ) = ({( t, s, − kρ ( t ) , − k ) ∣ ( t, s ) ∈ R × S } , d + e πiθ d s ) . OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 63
Figure 2.
We calculate the complex CF ∗ ( ψ H ( L O ) , L ( ξ ) ) by choosing a perturbing Hamil-tonian H such that the two Lagrangians split as product Lagrangians in the region con-taining all intersection points and holomorphic curves contributing to the differential. Thecase ξ = q is pictured; in this case there are six intersection points after perturbation. Thedifferential counts two discs contained in the fiber of the projection map for each intersectionpoint y i with ind y i = ind z i −
1. Here the two discs contributing to µ ( y ) , which cancel, areshown.where ρ ∶ R → [ , ] is a smooth increasing function interpolating between 0 and 1. Denote the mirror to thestructure sheaf by L O .We now calculate the morphism spaces Hom H ∗ F τ ( A ) ( L O , L ( q k e πiθ ) ) , which we expect to be isomorphicto the cohomology spaces H ∗ ( S, L ( q k e πiθ )) . Recall that there is an isomorphismHom H ∗ F τ ( A ) ( L O , L ( q k e πiθ ) ) ≅ HF ∗ ( ψ H ( L O ) , L ( q k e πiθ ) ) for a perturbing type- A Hamiltonian H ∶ Y → R . When k ∉ Z , H can be chosen such that the twoLagrangians, which are disjoint before perturbation, remain disjoint. Otherwise, we can choose H suchthat the calculations of the Floer complexes and Floer product essentially reduce to calculations involvingproduct Lagrangians in T ∗ S × T ; see as pictured in Figure 2. From this we see that the differential µ onthe complexes is given by µ ( y i ) = ( e πiθ − ) z i for 0 ≤ i ≤ k and we conclude thatHom H ∗ F τ ( A ) ( L O , L ξ ) = ⎧⎪⎪⎨⎪⎪⎩ H ∗ ( S ; C ) ⊗ H ∗ ( P , O ( k )) ξ = q k , k ∈ Z . When k ≥
0, the mirror functor Φ A sends z n z n to the action-corrected generator y n . To check that thefunctor intertwines the Floer and Yoneda products, we note that computation of the Floer product reducesto a calculation in T ∗ S , and is given by µ ( Φ A ( z n z n ) , Φ A ( z n ′ z n ′ )) = µ ( y n , y n ′ ) = y n + n ′ = Φ A ( z n + n ′ z n + n ′ ) . . 10. Diagrams of categories
The following theorem shows that our mirror correspondences commute with localization.
Theorem 10.0.1.
For any pair ( A ) = (( m , k ) , ( m ∞ , k ∞ )) such that ( m , k ) ≠ ( m ∞ , − k ∞ ) , the followingdiagram commutes: (104) Pic S τ ( A ) C τ ( m , k ) H F τ ( A ) H F τ ( m , k ) C τ ( m ∞ , k ∞ ) C τ H F τ ( m ∞ , k ∞ ) H F τ, W r
192 3 1156 7410 128 where: ● the arrows 1, 2, 3, and 4 are restriction functors induced by the covering of S ( A ) by D τ ( m , k ) and D τ ( m ∞ , k ∞ ) ; ● the arrows 5 and 7 are the localization functors of Section 6.2; ● the arrows 6 and 8 are the localization functors composed with the symplectomorphism Ψ ∶ Y → Y given by ( s, t, ϕ s , ϕ t ) ↦ ( s, − t, ϕ s , − ϕ t ) ; ● the arrows 9, 10, 11, and 12 are the mirror functors..Proof. The commutativity of the ( , , , ) and ( , , , ) squares is immediate; the commutativity of the ( , , , ) and ( , , , ) squares follows from Theorem 8 and Remark 6.3.1. Thus it remains only to checkthat the ( , , , ) and ( , , , ) squares commute. Note that the maps 3 , , , and 8 are embeddings ofcategories; thus to show that these remaining squares commute it suffices to show that the diagramPic S τ ( A ) H F τ ( A ) C τ H F τ, W r . Φ A i ∗ A ι ( m ,k ) ○ ι A Φ W r commutes.Let L i = L ( γ ) = L ( f i , g i , ξ i ) i = ,
1, be line bundles in Pic S τ ( A ) . Let L = L − L = L ( f, g, ξ ) . Let L i = L ( γ i ) denote the corresponding type- A Lagrangians. Recall that ι ∗ A ( L i ) = L ( m , k , , ξe − πig ) ; let L W ri = L ( m i ,k i , ,ξ i e − πigi ) . Let H and H be type- A Hamiltonians which satisfy the constraints above, and H ′ and H ′ type- W r Hamiltonians with ( ψ H i ( L i )) ∩ ([ − , ]) = ( ψ H ′ i ( L W ri )) ∩ ([ − , ]) . Let G i ∶ R × S be small functionswith support within [ − , ] × S such that the functions H ′ i + G i and H ′ i − H i + G i are smooth and such that ψ H ′ i − H i + G i ( ψ H i ( L i )) and ψ H i ( L i ) intersect transversely; note that this implies that H ′ i + G i and H ′ i − H i + G i are type- W r perturbing Hamiltonians. Then note that ψ H ′ i − H i + G i ( ψ H i ( L i )) = ψ H ′ i + G i ( L W ri ) ;thus L i and L W ri represent the same object in H F W r . We conclude that on objects, the diagram reads L i [ L i ] i ∗ A L i [ L i ] . Now let z n z n ∈ Ext i ( L , L ) ≅ H ( S τ ( A ) , L ) . Then there exists N ∈ N such that ξe − πig = e πiτN .Using the change of coordinates formula of Equation (79), we know that i ∗ A ( z n z n ) = z ( k ∞ n − k n ) x N = σ i ∗ A ( L ) , ( j,N ) . OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 65
Let [ s n ,n ] ∈ Hom H F τ ( A ) be represented by s H ,H n ,n ∈ CF τ ( ψ H ( L ) , ψ H ( L )) . By Remark 9.0.1, weknow that s H ,H n ,n ∈ CF τ ( ψ H ( L ) , ψ H ( L )) ( j,N ) . Using the same methods as in the proof of Lemma 8.2.1, we can show that µ ( c H i → H ′ i − H i + G i , s H ,H n ,n ) = y H ,H ( j,N ) , as both are the action-corrected generators of degree zero in the ( j, N ) graded piece of the Floer complex.Thus on morphisms, the diagram reads z n z n [ s n ,n ] σ L , ( j,N ) [ y L ,L , ( j,N ) ] . We conclude that the diagram in the statement of the theorem commutes. (cid:3)
Appendix A. Lagrangians mirror to torsion sheaves
The heuristics that guide our mirror construction can also be used to find Lagrangians mirror to sheaveswhich are supported on finitely many elliptic fibers on the Hopf surfaces we consider. These novel Lagrangiansare objects of an enlargement of the Fukaya category constructed in the main section of this paper.Recall that the mirror space Y is the product of the SYZ mirrors to C ∗ and the elliptic curve, anddecompose Y as Y = ( R × S , dt ∧ dϕ t ) × ( S × S , ds ∧ dϕ s ) . Consider a Fukaya category consisting of Lagrangians equipped with higher rank bundles with flat connec-tions, like those in the work of Polischuk and Zaslow on mirror symmetry for the elliptic curve ([22]). Weexpect that the product decomposition of Y commutes with the mirror functor in the following sense: Givena pair ( L , V , ∇ ) and ( L , V , ∇ ) of Lagrangians equipped with vector bundles with flat connections, suchthat L is a Lagrangian submanifold T ∗ S and L is a Lagrangian submanifold of T , we can form a productLagrangian ( L × L , V ⊠ V , ∇ + ∇ ) in Y . We expect that given two coherent sheaves F and G on C ∗ and E respectively, with mirror Lagrangians L F and L G , the sheaf F ⊠ G on C ∗ × E is mirror to the Lagrangian L F × L G . This is made precise in the remainder of this section. Remark A.0.1 (Notation) . Let E τ denote the torus with complex K¨ahler parameter τ . Let CF ∗ E τ ( L, L ′ ) denote the Floer complexes over C associated to an elliptic curve with complex K¨ahler parameter τ thatappear in [22]. We use “Lagrangian” to mean the data of a Lagrangian equipped with a vector bundle anda flat connection. Let F ( E τ ) denote the Fukaya category of such Lagrangians, as in [22]. Let Φ E τ denotethe mirror functor of [22] that interchanges coherent sheaves on E τ with Lagrangians on E τ , with the signconventions reversed (see Footnote 1).A.0.1. Sheaves supported on interior fibers.
Let z = exp ( R + πiθ ) ∈ C ∗ . The SYZ mirror to the skyscrapersheaf O z is the Lagrangian L z = ({( R , ϕ t ) ∣ ϕ t ∈ S } , V , d + πiθ d ϕ t ) consisting of the radial circle with radius R and the flat connection with monodromy e πiθ on the trivialrank-one vector bundle V . Let F be a coherent sheaf on E τ and let Φ E τ ( F ) = L F . Let i z ∶ E τ → C ∗ × E τ denote the inclusion map which sends E τ to { z } × E τ . Then i z ∗ ( F ) = O z ⊠ F so we expect that i z ∗ F is mirror to L z × L F . Let L z × L F = I z ( L F ) .We provide partial verification of this expectation with the following proposition: Proposition A.0.1.
Let G be a line bundle on on a surface S τ ( A ) mirror to a Lagrangian L G . Then forall such z ∈ C ∗ and F ∈ Coh ( E τ ) , there is an isomorphism Ext ∗ ( i z ∗ F , G ) ≅ HF ∗ τ ( I z ( L F ) , L G ) . We allow the case where S τ ( A ) is algebraic. We need the following lemma:
Lemma A.0.2 ([22]) . (a) Let π m ∶ E τ → E mτ denote the isogeny of elliptic curves, and let Π m ∶ E mτ → E τ denote the m -fold covering of tori which sends ( x, ϕ x ) to ( x, mϕ x ) . The induced functor Π ∗ m ∶ F ( E τ ) →F ( E mτ ) that sends a Lagrangian to its preimage satisfies Φ mτ ○ π m ∗ = Π ∗ m ○ Φ τ . (b) Write E τ as ( S ) with coordinates ( x, ϕ x ) . Let T Be πi ( τη + ν ) denote the translation functor on the Fukayacategory of E τ : ( L, V, ∇ ) ↦ ( L + η ˆ e x , V, ∇ + πiν Id V dϕ x ) . Let T Aξ denote the translation functor on E τ given by z ↦ ξz . Then T Bξ ○ Φ E τ = Φ E τ ○ T A ∗ ξ . Part (a) is adapted from Proposition 4 of [22] (with π m denoting the dual isogeny to that which appearsthere); part (b) is evident from the mirror correspondence in Section 5.1 of the same paper. Proof of Proposition A.0.3.
By Corollary 7.1.7, we can write i ∗ A G = L τ ( m , k , d, ξ ) for some d ∈ Z , ξ ∈ C .Let i ∗ z G = G z ∈ Pic dm E τ , and note G z = i ∗ z L τ ( m , k , d, ξ ) = T ∗ ξ − z kd ( L τ, ) ⋅ L m d − τ, . Let G = i ∗ G ∈ Pic dE m τ , and note that π ∗ m ( T B ∗ z k / m G ) = G z .Consider the map f ∶ Y → R × S given by y ↦ ( t ( y ) , m ϕ t ( y ) + k ϕ s ( y )) ; note that f ( L G ) = γ ; f ( I z ( L F )) = { R } × S , where γ ⊂ R × S is a curve which intersects the circle f ( I z ( L F )) at one point p . Let Y p = f − ( p ) . The map ̃ Y p → ( R / Z ) × ( R / m Z ) defined by y ↦ ( s ( y ) , ϕ s ( y )) descends to a symplectomorphism ( Y p , ω ∣ Y p ) → (( R / Z ) × ( R / m Z ) , dx ∧ dϕ x ) . Then one can verify the following identities for the intersections of the Lagrangians with Y p : I z ( L F ) ∩ Y p = T e − πkiθ ○ Π ∗ m ( Φ E τ ( F )) ; L G ∩ Y p = T e − πkiθ ○ Φ E m τ ( T B ∗ z k / m G ) . By choosing an almost-complex structure J for which f is holomorphic, we can guarantee that all holomor-phic curves contributing to the differential on the Floer complex are contained in the fiber Y p . Then thereis a canonical isomorphism CF ∗ τ ( I z ( L F ) , L G ) ≅ CF ∗ E m τ ( I z ( L F ) ∩ Y p , L G ∩ Y p ) . Then the Hom space is given by HF ∗ E m τ ( I z ( L F ) ∩ Y p , L G ∩ Y p ) ≅ HF ∗ E m τ ( T e − πk iθ ○ Π ∗ m ( Φ E τ ( F )) , T e − πk iθ ○ Φ E m τ ( T B ∗ z k / m G )) ≅ HF ∗ E m τ ( Π ∗ m ( Φ E τ ( F )) , Φ E m τ ( T B ∗ z k / m G )) ≅ Ext ∗ ( π m ∗ F , T B ∗ z k / m G ) ≅ Ext ∗ ( F , G z ) ≅ Ext ∗ ( i z ∗ F , G ) . The second isomorphism is the mirror functor and Lemma A.0.2; the third follows from the ( π m ∗ , π ∗ m ) adjunction. (cid:3) OMOLOGICAL MIRROR SYMMETRY FOR ELLIPTIC HOPF SURFACES 67
Figure 3
A.0.2.
Sheaves supported on boundary fibers.
We now consider sheaves supported on the boundary fiber π − A ( ) . Recall that π − A ( ) is the genus-one curve E m τ isogenous to the interior fibers E τ . Let F bea coherent sheaf on E m τ , and consider the sheaf i ∗ F . We claim that i ∗ F is mirror to a “U-shaped”Lagrangian constructed in the following way: Let f ∶ Y → R × S be the map given above. Choose a U-shaped arc γ ∈ R × S which extends to infinity in the negative direction as pictured in Figure 3, with theends of the arc asymptotic to the line m ϕ t + k ϕ s =
0. For a point p ∈ γ , parametrize f − ( p ) = Y p usingcoordinates adapted to the ( m , k ) boundary conditions: map ̃ Y p → ( R / Z ) × ( R / m Z ) by y ↦ ( s − ( k / m ) t, ϕ s ) . Let L F = Φ E m τ ( F ) be mirror to F , and construct a Lagrangian I ( L F ) in Y which fibers over γ and hasthe property that I ( L F ) ∩ Y p = L F by symplectic parallel transport. In the chosen coordinates, the paralleltransport is the identity map. Remark A.0.2.
The Lagrangians of the form I ( L F ) , and their cousins of the form I ∞ ( L F ) , are objects ofa Fukaya-Seidel category consisting of Lagrangians that, at each end, fiber over curves that asymptoticallyapproach the lines m ϕ t + k ϕ s = m ∞ ϕ t − k ∞ ϕ s =
0. The morphisms in this proposed category arecomputed via the same procedure as in the main text, and in particular the Floer complex CF ∗ ( ψ H ( L ) , L ′ ) for H ∶ Y → R a type- A Hamiltonian should be quasi-isomorphic to the Hom space between two Lagrangians L and L ′ ; this motivates the following proposition. Proposition A.0.3.
Let G be a line bundle on on a surface S τ ( A ) mirror to a Lagrangian L G . Then forall F ∈ Coh E m τ , there is an isomorphism Ext ∗ ( i ∗ F , G ) ≅ HF ∗ τ ( ψ H ( I ( L F )) , L G ) for H a type- A Hamiltonian such that ψ H ( I ( L F )) and L G intersect transversely.Proof. After possibly composing with a compactly supported Hamiltonian isotopy, we can ensure that thereis a unique p ∈ ψ H ( γ ) ∩ f ( L G ) , and ψ H ( I ( L F )) ∩ Y p = L F ; L G ∩ Y p = Φ E m τ ( G ) . Thus again there is a sequence of isomorphisms: CF ∗ τ ( ψ H ( I ( L F )) , L G ) ≅ CF ∗ E m τ ( Φ E m τ ( F ) , Φ E m τ ( G )) ≅ Ext ∗ ( F , G ) ≅ Ext ∗ ( i ∗ F , G ) . The situation is pictured in Figure 3. (cid:3)
Remark A.0.3.
The functor I ∶ F ( E m τ ) → F ( A ) is the Orlov functor of Abouzaid and Ganatra ([3],[28]), which in general produces Lagrangians mirror to Lagrangians supported on the fiber of a Landau-Ginzburg model ([14]). In fact this is a very simple instance of this phenomenon. Consider the case where ( m , k ) = ( , ) , so D τ ( m , k ) = C × E τ . The mirror to C is the manifold T ∗ S stopped at one point in theboundary at infinity, and the linking arc of the stop is mirror to the structure sheaf of the point { } ([10]).Denote the arc linking the stop by L . Then note that for a Lagrangian L F mirror to F ∈ Coh ( E τ ) , theconjectured mirror to i ∗ F is the product I ( L F ) = L × L F as expected.All U -shaped Lagrangians mirror to sheaves supported on the fiber over 0 take this form after a differentchoice of symplectic splitting of (a finite cover of) Y , which corresponds to a choice of algebraic coordinateson the surface D ( m , k ) .A.0.3. Further work.
With some further work, one could prove that the inclusion functors constructed aboveintertwine the Floer and Yoneda products, and so the domain of the mirror functors that appear in thiswork could be expanded to a subcategory of Coh an S τ ( A ) which includes torsion sheaves. However, fornon-algebraic S τ ( A ) the subcategory of D b Coh an S τ ( A ) generated by torsion sheaves and line bundles is astrict subcategory: for example, the rank-two vector bundles on the Hopf surface constructed in [21] are notgenerated by these objects. (One can show that the objects of the subcategory in question restrict to thesame object of D b Coh E τ on all but finitely many fibers on the open subset C ∗ × E τ ⊂ S τ fibering over C ∗ ,a property not enjoyed by these rank-two bundles.) We therefore expect that any proof of Conjecture 1.0.1,i.e., of an equivalence of triangulated categories, would take the form sketched in Subsection 1.3 and proceedvia Theorem 10.0.1. References
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