Categorical mirror symmetry on cohomology for a complex genus 2 curve
CCATEGORICAL MIRROR SYMMETRY ON COHOMOLOGY FOR ACOMPLEX GENUS 2 CURVE
CATHERINE CANNIZZO
I dedicate this paper to my mother and father, Catherine and John, for being the best role models, pavingthe way, and making everything in my life possible.
Abstract.
Motivated by observations in physics, mirror symmetry is the concept thatcertain manifolds come in pairs X and Y such that the complex geometry on X mirrorsthe symplectic geometry on Y . It allows one to deduce symplectic information about Y from known complex properties of X . Strominger-Yau-Zaslow [SYZ96] described how suchpairs arise geometrically as torus fibrations with the same base and related fibers, knownas SYZ mirror symmetry. Kontsevich [Kon95] conjectured that a complex invariant on X (the bounded derived category of coherent sheaves) should be equivalent to a symplecticinvariant of Y (the Fukaya category, see [Aur14], [FOOO09], [MTFJ19], [cfFc]). This isknown as homological mirror symmetry. In this project, we first use the construction of“generalized SYZ mirrors” for hypersurfaces in toric varieties following Abouzaid-Auroux-Katzarkov [AAK16], in order to obtain X and Y as manifolds. The complex manifold isthe genus 2 curve Σ (so of general type c <
0) as a hypersurface in its Jacobian torus. Itsgeneralized SYZ mirror is a Landau-Ginzburg model (
Y, v ) equipped with a holomorphicfunction v : Y → C which we put the structure of a symplectic fibration on. We thendescribe an embedding of a full subcategory of D b Coh (Σ ) into a cohomological Fukaya-Seidel category of Y as a symplectic fibration. While our fibration is one of the first nonexact,non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea indefining it is the same as in Seidel’s construction for Fukaya categories of Lefschetz fibrationsin [Sei08] and in Abouzaid-Seidel [AS]. Date : March 10, 2020.
Key words and phrases.
Differential geometry, Symplectic geometry, Symplectic aspects of mirror sym-metry, homological mirror symmetry, and Fukaya category. a r X i v : . [ m a t h . S G ] M a r ontents List of Figures 11. Context and main result 21.1. Context 21.2. Main result 21.3. Acknowledgements 42. HMS for abelian varieties 42.1. The symplectic side 42.2. The complex side 112.3. HMS statement 173. Refresher on toric varieties 193.1. Algebro-geometric construction 203.2. Rays define divisors which define line bundles 223.3. Polytope determines ample line bundle 243.4. Sections of O ( D P ) define a K¨ahler potential 244. Construction of symplectic fibration on ( Y, v ) 264.1. Finding Lagrangian torus fibration 264.2. Background needed to define the generalized SYZ mirror 304.3. The definition of ( Y, v ) 334.4. Definition of complex coordinates on ˜ Y / Γ B ω Y, v ) 525.1. Context and definition 525.2. Monodromy 575.3. Moduli spaces definition 635.4. Existence of regular choices to define moduli spaces 705.5. Quasi-invariance of the Fukaya category on regular choices 776. Computing the differential on ( Y, v ) 786.1. Simplify using the Leibniz rule 786.2. Cobordism between generic choice and specific choice for computation 836.3. Count of discs regular for J J D b L Coh ( H ) (cid:44) → H F S ( Y, v ) 928. Appendix A: Enough space to bound derivatives 939. Appendix B: Negligible terms in defining the symplectic form 969.1. Region I 969.2. Region IIA 989.3. Region IIB 1029.4. The remainder of C patch 10410. Notation 106References 108 MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 1
List of Figures V ∨ contributing to µ , viewed in ξ , ξ plane in the universal cover R
82 Moment map gives Lagrangian torus fibration: CP example 273 Base of exterior and interior blow-up on CP
284 Trop(1 + x + x ) = 0 305 Moment polytope in R
306 The (0,0) tile delimited by the tropical curve 327 The ( m , m ) tile delimited by the tropical curve 338 Moment polytope for central fiber of ( Y, v ) when H = Σ . 369 In one dimension lower, the boundary of ∆ ˜ Y is the moment map image of a stringof P ’s. In the polytope, | v | increases in the (0 ,
1) direction. In the fibration v , | v | is the radius of the circle in the base. 3610 Depiction of 3D ∆ ˜ Y . Coordinates respect Γ B -action, see below; magentaparallelogram = fundamental domain. Vertices = C charts. Coordinate transitions,see Lemma 4.13. Expressions in the center of tiles indicate e.g. η ≥ ϕ ( ξ ) = − ξ − (0 , is given by ( ξ , ξ ) ∈ { ( ξ , ξ ) | − ≤ ξ , ξ , − ξ + ξ ≤ } . 3711 L) Regions near a vertex, R) Number of regions interpolated between 4412 How the three angular directions vary for r x r y r z constant on a fiber 4513 Delineating regions in coordinates ( d I , θ I ) and ( d II , θ II ) 4714 Delineated fundamental domain 5215 L i = parallel transport (cid:96) i around U-shaped curve in base of v ξ , ξ ) (cid:55)→ ( f ( ξ , ξ ) , f ( ξ , ξ )). 5817 Example of strip-like end 6418 Leibniz rule 7919 Diagram illustrating Leibniz rule 7920 Simplified diagram on fibers 8021 A triangle in V ∨ contributing to µ , viewed in ξ , ξ plane in the universal cover R ∂ (left) and the count of discs we compute (right) 8323 Gromov compactification 8424 Gromov-Witten theory background for mirror symmetry of toric varieties 8925 Proof of Main Theorem 92 CATHERINE CANNIZZO Context and main result
Context.
Progress in mirror symmetry began with compact Calabi-Yau manifolds( c = 0). In particular, the geometric mirror for those of complex dimension three can beconstructed from T-duality three times [SYZ96] by inverting the radius of each S in a torusfiber to go from the A-model → B-model → A-model → B-model.For Fano manifolds with c >
0, [HKK +
03] describe a physical reason why a mirror shouldbe a
Landau-Ginzburg model , which for mathematicians is a non-compact complex manifold M equipped with a holomorphic function W : M → C called a superpotential . In [CO06],they explicitly compute the superpotential in the case of Fano toric varieties to be a weightedsum of discs according to their intersections with the toric divisors.Homological mirror symmetry (HMS) [Kon95] has been proven in the Calabi-Yau case[She15], [Sei15], [Fuk02]. Proven examples in the Fano case include [Abo09], [Ued06],[AKO08], [She16]. In the case of general type ( c < P ’s identifying their north polesto a point and their south poles to a point. This is known as the “banana manifold.”1.2. Main result.Definition 1.1. A symplectic fibration is a symplectic manifold ( Y, ω ) with a fibration suchthat fibers of the fibration are symplectic with respect to ω . Theorem 1.2 ([Can19]) . Let V be the abelian variety ( C ∗ ) / Γ B where Γ B := Z (cid:104) γ (cid:48) , γ (cid:48)(cid:48) (cid:105) for (1.1) γ (cid:48) := (cid:18) (cid:19) , γ (cid:48)(cid:48) := (cid:18) (cid:19) acts on ( C ∗ ) by (1.2) Z × ( C ∗ ) (cid:51) ( γ , γ ) · ( x , x ) (cid:55)→ ( τ − γ x , τ − γ x ) ∈ ( C ∗ ) . for τ ∈ R + (cid:28) . Let L → V be the ample line bundle ( C ∗ ) × C / Γ B where (1.3) γ · ( x , x , v ) := ( γ · ( x , x ) , x (cid:18) (cid:19) − γ τ − γ t (cid:18) (cid:19) − γ v ) MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 3 and with nonzero section s : V → L . Then H := s − (0) is a complex genus 2 curve and thefollowing diagram commutes, with fully-faithful vertical embeddings corresponding to HMSon cohomological categories. D b L Coh ( V ) ι ∗ (cid:45) D b L Coh ( H ) H F uk ( V ∨ ) HMS on V (cid:63) ∩ ∪ (cid:45) H F S ( Y, v ) HMS on H = Σ (cid:63) ∩ • where V ∨ is the SYZ dual abelian variety to V , • ( Y, v ) is the Γ B -quotient of a toric variety of infinite type and v = xyz is a Γ B -invariantproduct of the local toric coordinates, • ( Y, v ) is a Landau-Ginzburg model which is equipped with the structure of a symplecticfibration via symplectic form ω defined in Definition 4.24, • ( Y, v ) has generic fiber V ∨ degenerating to the singular fiber CP (3) / Γ B , • D b L Coh denotes the full subcategory generated by {L i [ n ] } i,n ∈ Z , • ι ∗ denotes the restriction functor of line bundles to the hypersurface H , • ∪ denotes the functor that parallel transports Lagrangians in fiber V ∨ around U-shapes inthe base of v using the symplectic horizontal distribution ( T V ∨ ) ω , and • FS stands for Fukaya-Seidel category and denotes the Fukaya category of the Landau-Ginzburg model ( Y, v ) . Remark 1.3 (Relevance) . This cohomology-level result already gives a lot of information.The product structure in Floer theory can be computed by counting triangles, and is mirrorto the ring structure on D b L| H Coh ( H ). Then since L| H is the canonical bundle of the genus2 curve, this determines the product structure on the canonical ring (cid:76) i ≥ H (Σ , L i ) offunctions on H = Σ . Once we know this ring structure, we can describe embeddings of thegenus 2 curve into projective space CP N − where N = h (Σ , L k ) for a very ample powerof L , using its sections. As a subvariety in projective space, functions on the genus 2 curvebecome polynomials, i.e. restrictions of homogeneous polynomials on the ambient CP N − .That is, degree m homogeneous polynomials are sections of O ( m ) for positive integers m ,which in turn are identified with sections of L| mk Σ via this embedding. Remark 1.4 (Previous related work) . HMS for abelian varieties of arbitrary dimensionand quotient lattice was proven in [Fuk02] using more advanced machinery. We presenta different argument for this particular case in the left vertical arrow of the Theorem 1.2.Seidel proved HMS with the A-model of the genus 2 surface [Sei11], i.e. the symplecticside. Seidel’s complex mirror is a crepant resolution of C / Z , quotienting by rotation andresolving the orbifold singularity without changing the first Chern class. The critical locusof the superpotential in his paper and of the mirror here are the same. He also speculatedin [Sei12] HMS for the genus 2 curve on the complex side. Remark 1.5 (Future directions) . One future direction is to relate Seidel’s genus 2 mirror toours. Another is enhancing the theorem to A ∞ -functors, namely proving that higher ordercomposition maps match in addition to objects, morphisms and composition. Powers of L split-generate the derived category so an A ∞ -enhancement of the result would allow us to CATHERINE CANNIZZO extend the functors to iterated mapping cones and hence would give a HMS statement forthe whole derived category.
Remark 1.6 (Structure of paper) . In Section 2 we describe the fully-faithful embeddingon abelian varieties, that is, the left vertical arrow of Theorem 1.2. In Section 3 we stateproperties of toric varieties, which are used in constructing (
Y, v ) in the following Section4. The symplectic form on Y is also defined in this section. Next, we equip the symplecticfibration ( Y, v ) with a Fukaya-Seidel category in Section 5 and compute its differential inSection 6. Finally in Section 7 we prove the fully-faithful embedding result for the genus 2curve on the cohomological level, which is the right vertical arrow in Theorem 1.2.1.3. Acknowledgements.
I would first like to thank my thesis advisor Denis Auroux forthe immensely helpful mathematical advice and discussions on this project. This project hada lot of moving parts and I benefited from the expertise of many in discussions during con-ferences and research talks. I thank Mohammed Abouzaid, Melissa Liu, Katrin Wehrheim,Kenji Fukaya, Mark McLean, Charles Doran, Alexander Polishchuk, Sheel Ganatra, HeatherLee, Sara Venkatesh, Haniya Azam, Zack Sylvan, Jingyu Zhao, Roberta Guadagni, WeiweiWu, Wolfgang Schmaltz, Zhengyi Zhou, Benjamin Filippenko, Andrew Hanlon, and Hiro LeeTanaka for fruitful mathematical discussions. I also thank the Fields Institute for hostingme as a short term visitor during their thematic program on Homological Mirror Symmetry,which resulted in several potential collaborations. This work was partially supported by NSFgrants DMS-1264662, DMS-1406274, and DMS-1702049, and by a Simons Foundation grant(
HMS for abelian varieties
The symplectic side.
We define the action-angle coordinates corresponding to a T -action on the abelian surface. All Lagrangians will be expressed in these coordinates. Wewill find that the image of the moment map is the same as the toric polytope, an instance ofDelzant’s theorem e.g. see [MS17]. Note that a usual moment map would land in R n where n is the dimension of the torus, but here the moment map will land in R / Γ B instead. Thisis known as a quasi-Hamiltonian action. Claim 2.1 (Symplectic coordinates) . Let V be the abelian variety ( C ∗ ) / Γ B as above and V ∨ the SYZ mirror abelian variety with complex coordinates x and y . Consider the stan-dard positive rotation T -action i.e. ( e πiα x, e πiα y ) for ( α , α ) ∈ T acting on ( x, y ). Let( ξ , ξ , θ , θ ) denote the action-angle coordinates, so ξ , ξ are the quasi-moment map coor-dinates for the above quasi-Hamiltonian T -action with respect to a symplectic form (whichwill be the restriction to a fiber of a symplectic form on Y , defined below). Then ξ i = log τ | x i | for ( x , x ) ∈ V = ( C ∗ ) / Γ B , hence γ ∈ Γ B acts on ( ξ , ξ ) by translating in the negativedirection − γ . Lastly θ := arg( x ) and θ := arg( y ). Proof.
Since the Lagrangian torus fibration on X is special with respect to the n -form Ω = d log x ∧ d log x ∧ dy for n = 3, we have integral affine structures on the bases of bothfibrations, for X and for Y . The complex affine structure on the B-model corresponds to thesymplectic affine structure on the A-model. In the construction of SYZ mirror symmetry,e.g. cf [Aur07], the complex affine structure on one side (the log |·| of the complex coordinates)is mirror to the symplectic affine structure on the other side (the moment map). Hence MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 5 ξ i = log τ | x i | . Since we rotate the complex coordinates x and y , their angles are the remainderof the action-angle coordinates. So the symplectic form on V ∨ in these coordinates is dξ ∧ dθ .The statement about the Γ B -action follows from the Γ B -action on V , namely γ · ( x , x ) =( τ − γ x , τ − γ x ). Thus taking the logarithm base τ implies γ acts additively in the negativedirection. (cid:3) Definition 2.2 (Setting up notation) . Let T B := R / Γ B where γ acts by negative translation − γ , and T F := R / Z . Thus V ∨ = T B × T F (cid:51) ( ξ , ξ , θ , θ ) with symplectic form dξ ∧ dθ + dξ ∧ dθ . Furthermore, let λ denote the linear map on R in standard bases given by (cid:18) (cid:19) − = (cid:18) − −
13 23 (cid:19) . In particular, λ ( γ (cid:48) ) = (1 ,
0) and λ ( γ (cid:48)(cid:48) ) = (0 , κ ( γ ) := − (cid:104) γ, λ ( γ ) (cid:105) . Remark 2.3 (Intuition for choice of Lagrangians) . HMS for abelian varieties was previouslyknown in more generality by Fukaya [Fuk02]. In his paper, he uses Family Floer theory todefine a line bundle E by requiring E p = Ext( E , O p ) to be defined as the corresponding homset on the mirror side, i.e. C × ( L p ∩ L ) where L is a linear Lagrangian on the torus, L p isvertical (infinite slope), and this intersection has one point. He then constructs a holomor-phic structure on this line bundle.Here we will write explicit linear Lagrangians first, which will be two-dimensional analoguesof those considered in [PZ98]. In [PZ98] they consider a square depicting ( T , (cid:82) T ω = a )as mirror to elliptic curve with complex structure C / Z + ia Z , on which lines of slope k onthe square are mirror to L k powers of a degree 1 line bundle on the mirror elliptic curve.Similarly our linear Lagrangians denoted (cid:96) i will be mirror to L i for the L defined in (1.3). Remark 2.4 (Intuition for choice of line bundles) . We will see how we arrived at thedefinition of the holomorphic line bundle
L → V in Theorem 1.2. Its transition functions ondifferent γ translates are given by: ( γ, x ) (cid:55)→ x λ ( γ ) τ κ ( γ ) where λ ∈ hom(Γ B , Z ) = hom(Γ B , Γ ∗ F ) corresponds to the first Chern class and is (cid:18) (cid:19) − in Theorem 1.2. This is because the first Chern class arises as follows: H ( V ; Z ) = H ( T B × T F ) ∼ = ⊕ i H i ( T B ) ⊗ H − i ( T F ) ∴ c ( L ) ∈ H ( V ; Z ) ∩ H , ( V ) = ⇒ c ( L ) ∈ H ( T B ; Z ) ⊗ H ( T F ; Z )= hom(Γ B , Z ) ⊗ hom(Γ F , Z )= Γ ∗ B ⊗ Γ ∗ F (2.1) = hom(Γ B , Γ ∗ F ∼ = Z ))= ⇒ c ( L ) ∈ hom(Γ B , Z )where the first implication follows from Corollary 2.16. Lemma 2.5.
The following defines a full subcategory of
F uk ( V ∨ ) . The objects are (2.2) (cid:96) k := { ( ξ , ξ , θ , θ ) ∈ V ∨ | ( θ , θ ) ≡ − k (cid:18) (cid:19) − (cid:18) ξ ξ (cid:19) mod Z } CATHERINE CANNIZZO
The morphisms HF ( (cid:96) i , (cid:96) j ) have rank ( i − j ) for i (cid:54) = j and HF ( (cid:96) i , (cid:96) i ) ∼ = H ( T ) . Themultiplicative structure for CF ( (cid:96) j , (cid:96) k ) ⊗ CF ( (cid:96) i , (cid:96) j ) → CF ( (cid:96) i , (cid:96) k ) is (cid:10) µ ( p , p ) , q (cid:11) = (cid:88) γ A ∈ Γ B τ − ll (cid:48) l (cid:48)(cid:48) · κ (cid:16) l (cid:48)(cid:48) l γ e,l + γ A (cid:17) summing over possible intersection points of (cid:96) i ∩ (cid:96) j , where l (cid:48) = j − i , l (cid:48)(cid:48) = k − j , and l = k − i .Proof. Objects.
The definition of (cid:96) k is well-defined because the minus sign ensures that it’swell-defined as a graph modulo group action; Γ B acts negatively on ( ξ , ξ ), so in the thetacoordinates it becomes the standard positive additive Z action in the angular coordinates.Secondly, the (cid:96) k are Lagrangian. Given a path p ( t ) := ( ξ ( t ) , ξ ( t ) , − kλ ( ξ ( t )) , − kλ ( ξ ( t )) ) : ( − (cid:15), (cid:15) ) → V ∨ consider the vector ddt | t =0 p ( t ) tangent to (cid:96) k . It’s of the form( c , c , − kλ ( c ) , − kλ ( c ) ) ≡ c ∂ ξ + c ∂ ξ − kλ ( c ) ∂ θ − kλ ( c ) ∂ θ Hence the tangent bundle
T (cid:96) k is spanned by the vectors with c = (1 ,
0) and c = (0 , T (cid:96) k = R (cid:28) ∂ ξ − k ∂ θ + k ∂ θ , ∂ ξ + k ∂ θ − k ∂ θ (cid:29) =: R (cid:104) X , , X , (cid:105) The symmetry of the matrix representing λ implies that dξ ∧ dθ of these two vectors is zero.(2.3) ω ( X , , X , ) = (cid:88) i =1 , dξ i ( X , ) dθ i ( X , ) − dθ i ( X , ) dξ i ( X , )= (cid:18) · k − − k · (cid:19) + (cid:18) · − k − k · (cid:19) = ( k/ − ( k/
3) = 0and ω ( X , , X , ) = 0 = ω ( X , , X , ) by skew-symmetry of ω . Thus ω | (cid:96) k ≡ (cid:96) k are Lagrangian. Note that an alternative proof would be to show (cid:96) k is Hamilton-ian isotopic to (cid:96) . Morphisms: geometric set-up.
A 4-torus is aspherical and there are no bigons betweentwo of these linear Lagrangians on a 4-torus, which prohibits sphere bubbling and strip-breaking respectively. The latter statement follows because two planes in R which intersectin a finite number of points can only intersect in one point, and strip-breaking occurs onstrips between two Lagrangians. These linear Lagrangians also do not bound discs, whichexcludes the remaining limiting behavior in Gromov compactness for discs with Lagrangianboundary condition, namely disc bubbling. Transversely intersecting Lagrangians.
Thus HF ( (cid:96) i , (cid:96) j ) = CF ( (cid:96) i , (cid:96) j ) since the dif-ferential is zero. So Floer cohomology is freely generated by intersection points of (cid:96) i and MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 7 (cid:96) j . (2.4) (cid:96) i ∩ (cid:96) j = { ( ξ , ξ , θ , θ ) ∈ T B × T F | ( θ , θ ) ≡ − iλ ( ξ ) ≡ − jλ ( ξ ) mod Z }⇐⇒ ξ ∈ ( j − i ) − Γ B / Γ B ∴ | (cid:96) i ∩ (cid:96) j | = ( j − i ) Notation to be used throughout.
Notationally, we write a generic intersection point as (cid:18) γ i ∩ j j − i , − iλ (cid:18) γ i ∩ j j − i (cid:19)(cid:19) ∈ (cid:96) i ∩ (cid:96) j where γ i ∩ j = e γ (cid:48) + e γ (cid:48)(cid:48) for 0 ≤ e , e < ( j − i ) (since the lattice Γ B has index ( j − i ) in( j − i ) − Γ B ). Letting l := j − i and 1 ≤ e ≤ l index the possible choices of ( e , e ) in γ i ∩ j ,we write the collection of all the intersection points as (cid:110)(cid:16) γ e,l l , − iλ (cid:16) γ e,l l (cid:17)(cid:17)(cid:111) ≤ e< ( j − i ) (Note that we could’ve taken j instead of i as the coefficient on the λ , and otherwise use thesame notation.) Non-transversely intersecting Lagrangians.
The above holds when i (cid:54) = j . Since La-grangians are half-dimensional, if they intersect transversely then their intersection is 0-dimensional and we have a discrete set of points. In the case i = j , we perturb (cid:96) i by aHamiltonian as in standard Floer theory. That is, we introduce a Hamiltonian function H and flow one Lagrangian along the symplectically dual vector field X H to dH , until the twoLagrangians intersect transversely, c.f. the introductory paper to Fukaya categories [Aur14].Furthermore, we want to take care how we perturb around the intersection points that arealready transverse. We pick a 1-form φ defined to be zero near boundary punctures on the do-main curve D where there is no problem and the intersection is transverse, and nonzero nearpunctures where the Lagrangians do not intersect transversally. Since we have moved oneLagrangian, we keep track of how this affects the Cauchy-Riemann equation. Holomorphicmaps u should satisfy the modified Cauchy-Riemann equation(2.5) ( du − X H ◦ φ ) , = 0with a modified asymptotic condition: at a puncture (the preimage in D of a perturbedintersection point) the map u converges to a trajectory of X H from (cid:96) i to (cid:96) j . So that theelements of hom sets are again intersection points, we can instead flow the image of u along X H to obtain a Cauchy-Riemann equation ( du H ) , with modified almost complex structureand boundary conditions. If we take the Hamiltonian to be a Morse function, then intersec-tion points of the 0-section of T ∗ (cid:96) i and the graph of df are critical points of f : T → T .By Morse theory, HF ( (cid:96) i , (cid:96) i ) ∼ = H ( T ). Existence of regularity, moduli spaces, independence of choices.
The standard com-plex structure J = i is regular. This is because the linearized ∂ J operator at a holomorphicmap u ∈ π ( V ∨ , ∪ i ∈ I (cid:96) i =: L ) for some index set I is ∂ on (cid:86) (0 , (( D , ∂ D ) , ( u ∗ T V ∨ , u ∗ T L )) ⊂ Ω (0 , ( D ) where smoothness follows from considering smooth maps u , by elliptic regularity.However, a Riemann surface has trivial H , since there are no (2 ,
0) forms (so no (0 , H (1 , ( D ) = ker( ∂ ) / Im( ∂ ) = (cid:86) , ( D ) / Im( ∂ ) hence the image of the CATHERINE CANNIZZO ∂ -operator is surjective.In particular, moduli spaces of k -pointed i -holomorphic discs are smooth orbifolds, whichwe can take the 0-dimensional part of and count. The structure map µ k which inputs k intersection points and outputs one intersection point, counts holomorphic polygons withvertices mapping to those intersection points with boundary on the corresponding intersect-ing Lagrangians. Furthermore: Lemma 2.6.
There exists a dense set J reg ⊂ J ( V ∨ , ω ) of ω -compatible almost complexstructures J such that, for all J -holomorphic maps u : D → V ∨ with suitable Lagrangianboundary condition, the linearized ∂ -operator D u is surjective. We postpone the proof to our discussion of regularity below in more generality in Section 5.5for quasi-invariance of the Fukaya category on regular choices. However here we can makethe stronger statement that the two µ counts are equal on the nose. Lemma 2.7.
The µ for two regular almost-complex structures J and J on V ∨ are equal.Outline. The proof will rely on arguments from the proof of the previous Lemma 2.6. Theargument will be similar, but the Fredholm problem will have an additional [0 ,
1] factor inthe Banach bundle setup. So we will obtain a 1-dimensional manifold. There is no otherboundary expected because 1) sphere bubbling cannot happen as π ( T ) = 0, furthermore 2)strip breaking would break off a bigon but there are no bigons between two linear Lagrangianson a torus (all intersection points have the same index since the Lagrangians have constantslope, whereas a broken strip would have intersection points with indices differing by two) and3) disc bubbling doesn’t occur because Lagrangians don’t bound discs on a torus (since π ispreserved upon taking the universal cover of the torus which is R so has no π ). Since thesigned boundary of a 1-dimensional manifold is zero, we find that M ( p , p , p , [ u ] , J ) = M ( p , p , p , [ u ] , J ) for regular J and J . (These moduli spaces will be defined in Section5.4.) The existence of a dense set of regular paths is similar to the proof for the existence ofregular J . (cid:3) Since we consider H F uk ( V ∨ ) only here, it remains to compute the multiplication µ . Counting triangles.
We compute µ : CF ( (cid:96) j , (cid:96) k ) ⊗ CF ( (cid:96) i , (cid:96) j ) → CF ( (cid:96) i , (cid:96) k ). This willcount J -holomorphic triangles between points p , p , q as in Figure 21, weighted by area. Wecan compute their area as they wrap around the abelian variety, by lifting to the universalcover. q p p ˜ (cid:96) i ˜ (cid:96) j ˜ (cid:96) k Figure 1.
A triangle in V ∨ contributing to µ , viewed in ξ , ξ plane in theuniversal cover R MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 9
Use subscript ξ and θ to denote the ( ξ , ξ ) coordinates and ( θ , θ ) coordinates respectivelyof a point in the universal cover R of V ∨ . Choose k > j > i . Fix q = (cid:10) γ k ∩ i k − i , − kk − i λ ( γ k ∩ i ) (cid:11) tobe one of the ( k − i ) intersection points in the fundamental domain. In particular, the sumof the three vectors around the triangle must be zero. Let the ξ coordinates of these vectorsbe ξ, ξ (cid:48) , ξ (cid:48)(cid:48) respectively. Then setting their sum, and the sum of θ -coordinates equal to zero:(2.6) ξ + ξ (cid:48) + ξ (cid:48)(cid:48) = 0 iξ + jξ (cid:48) − k ( ξ + ξ (cid:48) ) = 0 ∴ ξ (cid:48) = − ll (cid:48)(cid:48) ξξ (cid:48)(cid:48) = − l (cid:48) l (cid:48)(cid:48) ξ Now we apply the constraint that p ∈ (cid:96) i ∩ (cid:96) j and p ∈ (cid:96) j ∩ (cid:96) k . The first constraint isequivalent to − jλ ( p ξ ) ≡ p θ (mod Z ) = p θ + λ ( γ A ) for some γ A ∈ Γ B , where the A indicates we are on the symplectic side. p = ( p ξ , p θ ) = (cid:16) γ e,l l + ξ, − λ (cid:16) k γ e,l l + iξ (cid:17)(cid:17) ∴ − jλ ( p ξ ) ≡ p θ = ⇒ j γ e,l l + jξ = k γ e,l l + iξ + γ A (2.7) = ⇒ ξ = l (cid:48)(cid:48) ll (cid:48) γ e,l + γ A l (cid:48) Let ξ := l (cid:48)(cid:48) ll (cid:48) γ e,l + γ A l (cid:48) . To compute the area of the triangle, recall that the symplectic formin action-angle coordinates ξ, θ is ω = dξ ∧ dθ . In particular, in the plane spanned by (cid:126)u := (cid:104) ξ , (cid:105) and (cid:126)v := (cid:104) , λ ( ξ ) (cid:105) , with respect to these vectors we can write (cid:126)qp = (cid:126)u − i(cid:126)v and (cid:126)qp = l (cid:48) l (cid:48)(cid:48) (cid:126)u − k · l (cid:48) l (cid:48)(cid:48) (cid:126)v so under the parametrization Ψ : ( a, b ) (cid:55)→ a(cid:126)u + b(cid:126)v we find thatparallelogram (cid:126)qp × (cid:126)qp is Ψ((1 , − i ) × ( l (cid:48) l (cid:48)(cid:48) , − k l (cid:48) l (cid:48)(cid:48) )) hence area (∆ p p q ) = 12 (cid:90) (cid:126)qp × (cid:126)qp dξ ∧ dθ = 12 (cid:12)(cid:12)(cid:12)(cid:12) (1 , − i ) × ( l (cid:48) l (cid:48)(cid:48) , − k l (cid:48) l (cid:48)(cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) [0 , Ψ ∗ ( dξ ∧ dθ )= 12 [ − kl (cid:48) /l (cid:48)(cid:48) + il (cid:48) /l (cid:48)(cid:48) ] (cid:90) [0 , d ( a(cid:126)u + b(cid:126)v ) ξ ∧ d ( a(cid:126)u + b(cid:126)v ) θ = − (cid:18) ll (cid:48) l (cid:48)(cid:48) (cid:19) ξ · λ ( ξ ) (cid:90) [0 , da ∧ db (2.8) ξ · λ ( ξ ) = (cid:28) l (cid:48)(cid:48) ll (cid:48) γ e,l + γ A l (cid:48) , λ (cid:18) l (cid:48)(cid:48) ll (cid:48) γ e,l + γ A l (cid:48) (cid:19)(cid:29) = − l (cid:48) κ (cid:18) l (cid:48)(cid:48) l γ e,l + γ A (cid:19) ∴ area (∆ p p q ) = ll (cid:48) l (cid:48)(cid:48) · κ (cid:18) l (cid:48)(cid:48) l γ e,l + γ A (cid:19) = ⇒ (cid:10) µ ( p , p ) , q (cid:11) = (cid:88) γ A ∈ Γ B τ − ll (cid:48) l (cid:48)(cid:48) · κ (cid:16) l (cid:48)(cid:48) l γ e,l + γ A (cid:17) = (cid:10) µ ( p e (cid:48)(cid:48) ,l (cid:48)(cid:48) , p e (cid:48) ,l (cid:48) ) , p e,l (cid:11) Choice of J was regular. This concludes the triangle count computation. It remains toprove that these triangles are holomorphic for some regular J . In the basis (cid:126)u, (cid:126)v above, wecan construct the standard J . Claim 2.8.
Let J := ξ θ ξ − (cid:18) (cid:19) θ (cid:18) (cid:19) − J is a compatible almost complex structure and the triangles described above boundedby (cid:96) i , (cid:96) j , (cid:96) k are J -holomorphic, regular, and are the only J -holomorphic triangles boundedby these Lagrangians. Proof of Claim 2.8.
Let M λ := (cid:18) (cid:19) . Then J = (cid:18) − M λ M − λ (cid:19) (cid:18) − M λ M − λ (cid:19) = (cid:18) − I − I (cid:19) ω ( · , J · ) = ( dξ ∧ dθ )( · , J · )(2.9) = (cid:18) I − I (cid:19) − (cid:18) (cid:19)(cid:18) (cid:19) − = (cid:18) (cid:19) − (cid:18) (cid:19) − > J = − and J is compatible with ω , namely ω ( · , J · ) is a metric since the matrix aboveis positive definite. Linear triangles are J -holomorphic. Recall the discussion before Equation (2.8). Tak-ing the universal cover of V ∨ , we split up the resulting linear space into a product of two2-planes. Let P be the plane spanned by (cid:126)u = (cid:104) ξ , (cid:105) and (cid:126)v = (cid:104) , λ ( ξ ) (cid:105) for the choice of ξ right below Equation (2.7). Let P ω be the symplectic orthogonal complement. In par-ticular, P and P ω are J -holomorphic planes since λ = (cid:18) (cid:19) − . Then the Lagrangians˜ (cid:96) i , ˜ (cid:96) j , ˜ (cid:96) k decompose as products of a straight line of slope − i (respectively − j , − k ) in P , anda straight line of the same slope in P ω . In P we obtain the triangles previously consideredand projected to P ω the straight lines all meet in a single point. Therefore the maps in themoduli spaces for µ , u : D → V ∨ , have a triangle image in P and are constant maps in P ω at the triple intersection point of the three Lagrangians. (Namely, the boundary conditionof the projection of the 3-punctured disc to the latter plane must be constant.) J is regular. Now we can view the universal cover as P × P ω ∼ = R with respect to basisvectors (cid:126)u, (cid:126)v for P and (cid:126)u ω , (cid:126)v ω for P ω . In this basis, we have a J -holomorphic disc in P where J is a complex structure (as every almost complex structure is integrable in two dimensions)with a specified Lagrangian boundary condition. By the Riemann mapping theorem it’sunique. The discs are regular by the same argument on page 7 that J is regular. (cid:3) MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 11
This concludes the proof of Lemma 2.5 that the linear Lagrangians and their morphismsdefine a full subcategory of the cohomological Fukaya category, and that the multiplicativestructure is as stated in the lemma. (cid:3)
The complex side.
We now come to the HMS statement on abelian varieties statedin the main Theorem 1.2.
Lemma 2.9.
We can define a fully-faithful functor D b L Coh ( V ) (cid:44) → H F uk ( V ∨ ) by L i (cid:55)→ (cid:96) i . To prove this we will need to understand line bundles on abelian varieties, and their sections.Specifically, we will be interested in polarized abelian varieties, i.e. ones equipped with anample line bundle. The background on abelian varieties is based on [BL04], [Pol03]. Thetheme is to find Γ B -equivariant properties on ( C ∗ ) . Then we prove that the map in thestatement of this Lemma defines a functor. Here are the steps of the proof:(1) Define cohomology H ( V ) in terms of the lattice.(2) Classify L topologically by c .(3) Define factors of automorphy of the lattice, i.e. transition functions of holomorphicline bundles.(4) Define H , ( V ) in terms of the lattice.(5) Classify P ic ( V ) in terms of the lattice.(6) Express sections of L k .(7) Prove the map is a functor, namely that it respects composition. Prove that thefunctor is a fully-faithful embedding, namely that morphism groups are isomorphicto their image under the functor. Step 1
It will be easier to compute cohomology of V if we take log so the lattice acts additively.Taking the natural logarithm of ( x , x ) ∈ V , which is a locally holomorphic map, where weset | x i | = τ ξ i , we obtain coordinates(2.10) ( ξ log τ + i arg( x ) , ξ log τ + i arg( x )) ∈ C / (log τ )Γ B + 2 πi Z Define Γ F = Z then V + := C / log τ Γ B + 2 πi Γ F . Then the lattice Γ B now acts by subtrac-tion on ξ and Γ F acts by addition on πi arg( x ). Let Γ := (log τ )Γ B + 2 πi Z . Note thattopologically we can express V + as a product of tori: V + = R ξ ,ξ / Γ B × R θ x ,θ x / Γ F = T B × T F where θ x i := π arg( x i ). (However, as an abelian variety, V ∼ = V + are not product ellipticcurves.) Claim 2.10 (Cohomology of abelian variety V ) . H n ( V ; Z ) ∼ = ∧ n Hom(Γ , Z ) Proof outline, c.f. [BL04, § . We have Γ = π ( V + ) = H ( V + ; Z ) ∴ H ( V + ; Z ) = Hom(Γ , Z ).Then the de Rham isomorphism and induction imply (cid:86) n Hom(Γ , Z ) ∼ = (cid:86) n H ( V + ; Z ) ∪ −→ H n ( V + ; Z ). (cid:3) Step 2Corollary 2.11 ([Pol03, § . Complex line bundles on V are topologically classified by theirfirst Chern class, which is equivalent to a skew-symmetric bilinear form E : Γ × Γ → Z . Proof.
For a complex line bundle, c ( L ) ∈ H ( V + ; Z ) ∼ = ∧ Hom(Γ , Z ) by the previous claim,and ∧ Hom(Γ , Z ) ∼ = Hom( ∧ Γ , Z ). (cid:3) Step 3
Next consider a holomorphic line bundle on V . Since log is a holomorphic map, such linebundles pullback to holomorphic line bundles on V + . So first we describe holomorphic linebundles on V + , and then determine which pass under exp to V . One definition of a holomor-phic line bundle is by Cartier divisors { f i } where transition functions are f i /f j and f i locallydefine the divisor corresponding to the line bundle. On a complex torus, we can packagetransition functions into Γ-equivariant factors of automorphy H ( π ( V + ); H ( O ∗ ˜ V + )). On theother hand, holomorphic line bundles are represented by cocycles in the sheaf cohomologygroups H ( V + , O ∗ ). Claim 2.12 ([BL04, Appendix B]) . Holomorphic line bundles on V + are classified by(2.11) H ( π ( V + ); H ( O ∗ ˜ V + )) ∼ = H ( V + , O ∗ )where ˜ V + = C .The requirement that the functions be Γ-invariant implies they form a cocycle, hence rep-resent a class in H ( π ( V + ); H ( O ∗ ˜ V + )). The isomorphism on these cohomology groups isconstructed in [BL04, Proposition B.1] using a description of the transition functions for aholomorphic line bundle in terms of deck transformations corresponding to elements of Γ. Idea of proof following [BL04, Proposition B.1] . Let π : C → V + be the universal covering.The isomorphism of Equation 2.11 is constructed as follows. Line bundles consist of chartsand transition functions satisfying the cocycle condition. Here that condition is equivalentto Γ (cid:121) C on a collection of charts in the universal cover (eg the fundamental domain andtranslates), each one a lift of a chart on V + .More concretely, let { U i } be a covering of V + . A holomorphic line bundle is defined byholomorphic functions g ij : U i ∩ U j → C ∗ by identifying ( p, z ) in one chart with ( p, g ij ( p ) z ) inthe other. Alternatively, lift each U i to W i so that π | W i : W i → U i is a biholomorphism. Inparticular, π − i ( x ) and π − j ( x ) are two lifts of the same point, so by definition of the universalcover they differ by some element γ ij ∈ Γ. Thus we can collect transition functions into afunction f which depends on the choice of lift and the γ ij ∈ Γ deck transformation where π − j ( x ) = γ ij π − i ( x ). f ( γ ij , π − i ( x )) := g ij Γ is a group action so γ ij γ jk = γ ik . Since g ij is a cocycle on a line bundle(2.12) g ij g jk = g ik ∴ f ( γ ij , π − i ( x )) f ( γ jk , π − j ( x )) = f ( γ ik , π − i ( x )) ∴ f ( γ ij , π − i ( x )) f ( γ jk , γ ij π − i ( x ))) = f ( γ ij γ jk , π − i ( x )) ⇐⇒ [ f ] ∈ H (Γ , H ( O ∗ C ))i.e. f is a factor of automorphy. A trivial line bundle has transition functions of the form g ij = h j /h i . Setting g ij = f ( γ ij , π − i ) = h j /h i we can define a global function h on the MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 13 universal cover by h ( π − i ( x )) := h ( π − j ( x )) · h i h j corresponding to a coboundary ( γ, ˜ x ) (cid:55)→ h ( γ · ˜ x ) h (˜ x ) − for some h ∈ H ( O ∗ C ). Thus f respects coboundaries and cocycles, since under g ij (cid:55)→ f ( γ ij , π − i ( · )) we obtain ˇCech coho-mology groups with the discrete topology on Γ and values in the sheaf O ∗ C . Therefore theimage gives a representative of a class in H ( π ( V + ); H ( O ∗ ˜ V + )).For the reverse direction, given a factor of automorphy f , we will describe how to construct aholomorphic line bundle in the proof of the Appell-Humbert Theorem, Theorem 2.15 below. (cid:3) Step 4
Complex line bundles L admitting a holomorphic structure are those for which c ( L ) ∈ H , ( V ), or equivalently the wedge product of c ( L ) with generators of H , ( V ) and H , ( V )equals zero. Furthermore, taking cohomology of the exponential sequence 0 → Z →O exp (cid:45) C ∗ →
1, one can prove e.g. [Huy05] that the coboundary map is: − c : H ( V + , O ∗ ) → H ( V + , Z )Thus holomorphic line bundles have c ∈ H , ( V + ) ∩ H ( V + , Z ), and conversely each suchclass gives rise to torus-worth of holomorphic line bundles ([Huy05]). So we would like todescribe H , ( V + ). Claim 2.13. H , ( V + ) ∼ = Hom C ( C , C ) ⊗ Hom C ( C , C ) Proof.
By [Huy05, Lemma 1.2.5] and [BL04, Theorem 1.4.1], C = ( C ) , ⊕ ( C ) , = ⇒ ( T , C ) ∗ = Hom C ( C , C ) and ( T , C ) ∗ = Hom C ( C , C )Let t v denote translation by v on the torus C / Γ. Then pulling back by dt − v translates C to cover the whole tangent bundle of V + and we can extend ( p, q ) forms to V + .(2.13) Hom C ( C , C ) ⊗ Hom C ( C , C ) ∼ = H , ( V + ) ω (cid:55)→ { dt ∗− v ω } v ∈ V + (cid:3) Corollary 2.14 (C.f [BL04, Proposition 2.1.6]) . A complex line bundle L with first Chernclass c ( L ) = E admits a holomorphic structure if and only if H ( · , · ) := E ( i ( · ) , · ) + iE ( · , · ) is Hermitian.Proof. By the previous claim, it suffices to consider c ( L ) at a point. Again by the previousclaim it arises from a holomorphic line bundle if and only if it is a hermitian matrix (afterchoosing a basis), namely an inner product that is complex linear in the first argument andcomplex anti-linear in the second argument.Given an alternating 2-form E so that E (Λ , Λ) ⊂ Z , consider H ( · , · ) := E ( i ( · ) , · ) + iE ( · , · ).Then H is a Hermitian form if and only if E respects the complex structure i , e.g. see [Huy05,Lemma 1.2.15]. That is i ∗ E = E and E = Im H the imaginary part of H . (cid:3) Step 5
Many holomorphic structures exist on a complex line bundle. Putting everything together,the
Appell-Humbert Theorem classifies holomorphic line bundles on V + in terms of H andan additional piece of data α : Γ → U (1). The first Chern class isn’t sufficient to define theholomorphic structure, since there is a torus’ worth of holomorphic line bundles with thesame c . Theorem 2.15 ([Pol03, Theorem 1.3] and [BL04, Appell-Humbert Theorem 2.2.3]) . ThePicard group of V + can be classified by the following set of pairs. (2.14) P ic ( V + ) ∼ = { ( H, α ) | H : C × C → C Hermitian , E (Γ , Γ) ⊆ Z , α : Γ → U (1) α ( γ + ˜ γ ) = exp( πiE ( γ, ˜ γ )) α ( γ ) α (˜ γ ) , where E = Im H } Proof from [Pol03, Chapter 2] and [BL04, Chapter 2] . We saw that complex structures cor-respond to a choice of E , and here we describe how holomorphic structures on a complex linebundle are parametrized by a semicharacter α : Γ → U (1) ([Pol03, § ⇐ ) Suppose we have a pair ( H, α ) as in the lemma. The corresponding holomorphic linebundle is defined by the factor of automorphy (see [Pol03, Equation (1.2.2)] and followingdiscussion):(2.15) Γ × C f (cid:45) C ∗ ( γ, v ) (cid:55)→ α ( γ ) exp( πH ( v, γ ) + π H ( γ, γ ))Namely, Γ → O ∗ ( ˜ V ) given by γ (cid:55)→ f ( γ, − ) describes the necessarily Γ-periodic gluing func-tions of the line C on different translates under the Γ action, or equivalently as we go aroundelements of π ( V ). This factor of automorphy is a natural definition from the perspectiveof generalized Heisenberg groups . (See [BD16] for an explanation of the connection betweenthe Heisenberg group defined in [Pol03, § U (1) × C with group law( α , v ) · ( α , ˜ v ) = (exp( πiE ( v, ˜ v )) α α , v + ˜ v )These naturally arise in the case of abelian varieties because the condition on α implies thatthere is a representation of the Heisenberg group U (1) × C on the space of L -sections ofthe line bundle we’ve constructed corresponding to ( H, α ), see [Pol03, § Fock representation. ( ⇒ ) Conversely, suppose we have a factor of automorphy f for a holomorphic line bundle L .Because f is nonvanishing we can express f = exp(2 πig ) for some holomorphic function g .Then by [BL04, Theorem 2.1.2] c ( L ) corresponds to the alternating form Γ × Γ → Z givenby:(2.16) E L ( γ, ˜ γ ) = g (˜ γ, v + γ ) + g ( γ, v ) − g ( γ, v + ˜ γ ) − g (˜ γ, v )for all γ, ˜ γ ∈ Γ and v ∈ C (it turns out this expression is independent of v ). Since L isholomorphic, recall E gives rise to a Hermitian form, see Corollary 2.14. The condition that f is a factor of automorphy implies that we can construct a semi-character α satisfying theproperty in the statement of the Theorem, as described at the bottom of [BL04, pg 31],namely α ( γ + ˜ γ ) = exp( πiE ( γ, ˜ γ )) α ( γ ) α (˜ γ ) MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 15
In this way, a holomorphic line bundle defines a pair (
H, α ).We show uniqueness now. Heisenberg groups have unique unitary irreducible representationsby a theorem due to Stone and von Neumann. Furthermore, sections of a holomorphicline bundle uniquely define the line bundle by their transition functions. Thus the Fockrepresentation of U (1) × C (with group law determined by E , i.e. by the factor of automorphy f as in Equation 2.16) on L -sections of L is the only unitary irreducible representation. Arepresentation on the vector space of L -sections by any other L must be isomorphic. Thatis, by uniqueness of the representation, L is the only holomorphic line bundle that cancorrespond to ( H, α ). (cid:3) Finally we arrive at our goal for Step 5, namely classifying
P ic ( V ) by considering its imagethe pullback exp ∗ ( P ic ( V )) ⊆ P ic ( V + ). Corollary 2.16.
Holomorphic line bundles on V are in one-to-one correspondence withpairs ( H, α ) ∈ P ic ( V + ) (see Theorem 2.15) where H can be represented as a real integralsymmetric × matrix, under the pullback by exp .Proof. Recall that for a complex line bundle on an abelian variety, c ( L ) can be representedby an alternating form E : Γ × Γ → Z for Γ ⊂ C . Thus taking R linear combinations wecan extend this to an R -linear form E : C × C → R . With respect to the basis given by thegenerators of Γ B and Γ F in Step 1, this gives a 4 by 4 skew-symmetric matrix with respectto Γ B × Γ F . Then E := Im H for some hermitian form H = (cid:18) a bb a (cid:19) so a ∈ R .(2.17) ( γ + iw γ + iw ) (cid:18) a b + icb − ic d (cid:19) (cid:18) ˜ γ + iv ˜ γ + iv (cid:19) We require that E is trivial in the T F directions, because under exp the T F directions arealready quotiented by in ( C ∗ ) . These are the purely imaginary vectors in C by Equation2.10, so we want E ( iw, iv ) = 0. We find that E ( iw, iv ) = Im H ( iw, iv ) = Im( aw v + bw v + icw v + bw v + dw v − icv w )= c ( w v − v w ) = 0 ∀ w, v ∈ Z ∴ c = 0so H is real hence symmetric. We can express H on R as H = Γ B Γ F (cid:18) (cid:19) Γ B A Γ F A A isreal symmetric. Note that the information of H is that of a homomorphism Γ B → Γ F . (cid:3) Step 6
Since the genus 2 curve is defined as the zero set of a section of a line bundle on V , we nextdescribe sections of the line bundles above. We’ll see the genus 2 curve is a theta divisor. Claim 2.17. s : ( γ ξ , x ) (cid:55)→ x λ ( γ ξ ) τ κ ( γ ξ ) is a factor of automorphy for L where recall λ and κ are defined in Definition 2.2. Steps of proof.
We show its pullback to V + is a factor of automorphy which is trivial in theΓ F directions. L V exp ∗ L V + exp ∗ s s We work in holomorphic coordinates on V and V + . In particular, let v := (log τ ) ξ + 2 πiθ bythe coordinate on V + , where Γ B acts by addition in the negative direction on ξ and Γ F = Z by addition on θ . Furthermore, let γ = γ ξ + iγ θ . Then(2.18) exp ∗ s : ( γ, v ) (cid:55)→ exp(( λ ( γ ξ ) log τ ) ξ + 2 πiλ ( γ ξ ) θ + κ ( γ ξ ) log τ )= exp( λ ( (cid:60) γ ) · v + κ ( (cid:60) γ ) log τ )So we have a linear term in γ and a quadratic term in γ , as expected from Theorem 2.15, ifwe take α ≡
1. (Since α determines the holomorphic structure, if it was non-trivial it wouldcontribute a linear term to κ .) This is because recall from Theorem 2.15 that a pair ( H, α )gives rise to a factor of automorphy on V + by(2.19) ( γ, v ) (cid:55)→ α ( γ ) exp( πH ( v, γ ) + π H ( γ, γ ))By the proof of Corollary 2.16, we found H = iE where E is a real 2 by 2 symmetric matrix.Thus we may define H in terms of the map λ : Γ B → Γ F so that:(2.20) ( γ, v ) (cid:55)→ α ( γ ) exp( πH ( v, γ ) + π H ( γ, γ ))= exp( (cid:104) v, λ ( γ ξ ) (cid:105) + κ ( γ ξ ) log τ )= exp( (cid:104) (log τ ) ξ + 2 πiθ, λ ( γ ξ ) (cid:105) + κ ( γ ξ ) log τ )= x λ ( γ ξ ) τ κ ( γ ξ ) So the condition that the line bundle on V + passes to one on V is the condition that H = iE ,since under exponentiation the Γ F action becomes multiplication by e πn = 1 for some n ∈ Z . (cid:3) We now use γ instead of γ ξ to denote group elements of Γ B . Claim 2.18.
Sections of holomorphic lines bundles on V are functions on ( C ∗ ) with theperiodicity property s ( γ · x ) = τ κ ( γ ) x λ ( γ ) s ( x )so have a Fourier expansion. Proof.
A section s : V → L must have the same transition functions as the line bundle, byconsidering the Cartier data. s ( γ · x ) /s ( x ) = τ κ ( γ ) x λ ( γ ) (cid:3) Corollary 2.19.
Let L be the line bundle defined above in Claim 2.17. Then using thenotation from the proof of Lemma 2.5, H ( V, L ⊗ l ) has the following basis of sections: (2.21) s e,l := (cid:88) γ τ − lκ ( γ + γe,ll ) x − lλ ( γ ) − λ ( γ e,l ) MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 17 where γ e,l = e γ (cid:48) + e γ (cid:48)(cid:48) , ≤ e , e < l . So h ( L l ) = l , e.g. L is a degree 1 line bundle.Proof. Tensoring the line bundle l times means we multiply the transition function of L by l times. In particular, the exponents now add in λ and κ . So equivalently, we could scalethe lattice Γ B to l Γ B and note that the quotient has l lattice points. If we think of theparallelogram in Γ B of length l in the γ (cid:48) and γ (cid:48)(cid:48) directions, then unique lattice points indexthe sections. So the functions in the statement of this Corollary are l linearly independentsections with the same transition functions as L ⊗ l . (cid:3) HMS statement. Step 7
Now we show that D b L Coh ( V ) (cid:51) L ⊗ k (cid:55)→ (cid:96) k ∈ H ∗ F uk ( V ∨ ) is a functor, by showing it respectscomposition on elements of a basis. Basis on complex side.
Recall that D b L Coh ( V ) is defined in Theorem 1.2 to be generatedby powers of L , and for j > i Hom( L i , L j ) ∼ = H ( O , L j − i ) (see [PZ98] for the case of linebundles on an elliptic curve). Set ˜ l := j − i , ˜˜ l := k − j , and l := ˜ l + ˜˜ l = k − i . Recall thatCorollary 2.19 gives a basis of sections of H ( V, L ⊗ l ): s e,l := (cid:88) γ τ − lκ ( γ + γe,ll ) x − lλ ( γ ) − λ ( γ e,l ) where γ e,l = e γ (cid:48) + e γ (cid:48)(cid:48) , ≤ e , e < l . Basis on symplectic side.
On the symplectic side, we consider a basis of Hom V ∨ ( (cid:96) i , (cid:96) j ) = (cid:76) p ∈ (cid:96) i ∩ (cid:96) j C · p given by the ( j − i ) = l intersection points of Equation 2.4:(2.22) p e,l := (cid:16) γ e,l l , − kλ (cid:16) γ e,l l (cid:17)(cid:17) where again γ e,l = e γ (cid:48) + e γ (cid:48)(cid:48) , ≤ e , e < l . Remark 2.20.
This section is a bit notationally heavy so we collect the notations in thisremark: • γ (cid:48) and γ (cid:48)(cid:48) form a basis for Γ B • e indexes the intersection points of two Lagrangians • One tilde corresponds to j − i = ˜ l , two tildes corresponds to k − j = ˜˜ l , and no tilde corre-sponds to the indexing lattice element once we’ve multiplied together and are considering l = k − i . • s denotes sections and p denotes intersection points Example 2.21.
E.g. for i = 0 , j = 1, and k = 2 we have O , L , L and (cid:96) , (cid:96) , (cid:96) with mapsbetween objects and between homs as follows: hom ( O , L ) = H ( V, L ) (cid:51) s , (cid:55)→ p , ∈ hom ( (cid:96) , (cid:96) ) = (cid:77) p ∈ (cid:96) ∩ (cid:96) C · p = C · p , hom ( O , L ) = H ( V, L ) (cid:51) ( s , , s , , s , , s , ) (cid:55)→ ( p , , p , , p , , p , ) ∈ (cid:77) p ∈ (cid:96) ∩ (cid:96) C · p, | (cid:96) ∩ (cid:96) | = 4 hom ( L , L ) ∼ = hom ( O , L ∗ ⊗ L ) ∼ = hom ( O , L ) = H ( V, L ) (cid:51) s , (cid:55)→ p , ∈ hom ( (cid:96) , (cid:96) ) = C · p , In particular we define the map to send units to units. The statement that this map respectscomposition is the following.
Lemma 2.22.
The left vertical map of Theorem 1.2, D b L Coh ( V ) HMS −−−→
F uk ( V ∨ ) , respectscomposition so is a functor. Namely, for s e,l and p e,l bases defined as above: (2.23) HM S ( s e,l ) = p e,l HM S ( s ˜˜ e, ˜˜ l · s ˜ e, ˜ l ) = HM S ( s ˜˜ e, ˜˜ l ) · HM S ( s ˜ e, ˜ l ) = p ˜˜ e, ˜˜ l · p ˜ e, ˜ l ⇐⇒ s ˜˜ e, ˜˜ l · s ˜ e, ˜ l = (cid:88) e ∈ Z /l Z C e · s e,l p ˜˜ e, ˜˜ l · p ˜ e, ˜ l = (cid:88) e ∈ Z /l Z C e · p e,l for the same C e .Proof. First note that C e was computed above in the count of triangles in Equation (2.8).On the other hand, multiplying the theta functions gives(2.24) s ˜˜ e, ˜˜ l · s ˜ e, ˜ l = (cid:88) ˜ γ, ˜˜ γ τ − ˜ lκ (˜ γ + γ ˜ e, ˜ l ˜ l ) − ˜˜ lκ (˜˜ γ + γ ˜˜ e, ˜˜ l ˜˜ l ) x − λ (˜ l ˜ γ + γ ˜ e, ˜ l +˜˜ l ˜˜ γ + γ ˜˜ e, ˜˜ l ) We want to find new variables ( γ, γ A ) to sum over so that we can factor out the bases s e,l .In particular, − λ ( lγ + γ e,l ) must be the exponent on x . So we want this product to equal (cid:80) e C e (cid:80) γ τ − lκ ( γ + γe,ll ) x − λ ( lγ + γ e,l ) = (cid:80) e C e s e,l . The other factor that sums over γ A will arisefrom counting triangles on the A-side, hence the subscript A . Define(2.25) lγ + γ e,l := ˜ l ˜ γ + γ ˜ e, ˜ l + ˜˜ l ˜˜ γ + γ ˜˜ e, ˜˜ l If we sum over γ and 1 ≤ e ≤ l , we will obtain some of the lattice Γ B × Γ B (cid:51) (˜ γ, ˜˜ γ ). Given γ and e , there are multiple corresponding solutions in (˜ γ, ˜˜ γ ). We need another variable. Wedo a weighted version of the change of coordinates ( u, v ) (cid:55)→ (( u + v ) / , ( u − v ) / u := γ + γ e,l l = l (˜ l ˜ γ + γ ˜ e, ˜ l + ˜˜ l ˜˜ γ + γ ˜˜ e, ˜˜ l ). We want to find v such that(2.26) u + c v = ˜ γ + γ ˜ e, ˜ l ˜ lu − c v = ˜˜ γ + γ ˜˜ e, ˜˜ l ˜˜ l for some constant c and c such that the v terms cancel when we multiply the first equationby ˜ l and add it to the second equation multiplied by ˜˜ l . In other words, c ˜ l − c ˜˜ l = 0. So take c = ˜˜ l and c = ˜ l . Then we can simplify the exponent on τ in Equation 2.24:(2.27) ˜ lκ ( u + ˜˜ lv ) + ˜˜ lκ ( u − ˜ lv ) = lκ ( u ) + ˜ l ˜˜ l · lκ ( v )since lv = ˜ γ + γ ˜ e, ˜ l ˜ l − ˜˜ γ − γ ˜˜ e, ˜˜ l ˜˜ l . Thus we now can factor out lκ ( u ) as needed to obtain s e,l whensumming over γ . On the other hand, recall from Equation (2.8) that p ˜ e, ˜ l · p ˜˜ e, ˜˜ l = (cid:88) e (cid:88) γ A ∈ Γ B τ − l ˜ l ˜˜ l · κ (cid:18) ˜˜ ll γ e,l + γ A (cid:19) · p e,l MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 19
That is C e = (cid:80) γ A ∈ Γ B τ − l ˜ l ˜˜ l · κ (cid:18) ˜˜ ll γ e,l + γ A (cid:19) . So in order for the functor to respect composition, wewould like this to be the coefficient on the s e,l as well. Comparing exponents on τ implies:(2.28) l ˜ l ˜˜ l · κ (cid:32) ˜˜ ll γ e,l + γ A (cid:33) = ˜ l ˜˜ l · lκ ( v )In other words, multiplying by ˜ l ˜˜ l and equating the arguments of lκ :(2.29) ˜˜ ll γ e,l + γ A = ˜ l ˜˜ l · l (cid:18) ˜ γ + γ ˜ e, ˜ l ˜ l − ˜˜ γ − γ ˜˜ e, ˜˜ l ˜˜ l (cid:19) ⇐⇒ γ A = ˜ l ˜˜ l · l (cid:18) ˜ γ + γ ˜ e, ˜ l ˜ l − ˜˜ γ − γ ˜˜ e, ˜˜ l ˜˜ l (cid:19) − ˜˜ ll γ e,l Recall that lγ + γ e,l = ˜ l ˜ γ + γ ˜ e, ˜ l + ˜˜ l ˜˜ γ + γ ˜˜ e, ˜˜ l . Thus simplifying we find that:(2.30) γ A = ˜ l ˜˜ ll (cid:18) ˜ γ + γ ˜ e, ˜ l ˜ l − l (cid:16) lγ + γ e,l − ˜ l ˜ γ − γ ˜ e, ˜ l (cid:17)(cid:19) − ˜˜ ll γ e,l = ˜ l ˜˜ ll (cid:16) ˜ γ (1 + ˜ l/ ˜˜ l ) + γ ˜ e, ˜ l (1 / ˜ l + 1 / ˜˜ l ) − ( l/ ˜˜ l ) γ (cid:17) − γ e,l (˜ l + ˜˜ l ) /l = ˜ l (˜ γ − γ ) + γ ˜ e, ˜ l − γ e,l ∈ Γ B so γ A ∈ Γ B as we would like. Hence using Equation (2.28):(2.31) s ˜ e, ˜ l · s ˜˜ e, ˜˜ l = (cid:88) ˜ γ, ˜˜ γ τ − ˜ lκ (˜ γ + γ ˜ e, ˜ l ˜ l ) − ˜˜ lκ (˜˜ γ + γ ˜˜ e, ˜˜ l ˜˜ l ) x − λ (˜ l ˜ γ + γ ˜ e, ˜ l +˜˜ l ˜˜ γ + γ ˜˜ e, ˜˜ l ) = (cid:88) e (cid:88) γ A τ − l ˜ l ˜˜ l κ (cid:18) ˜˜ ll γ e,l + γ A (cid:19) (cid:88) γ τ − lκ ( γ + γe,ll ) x − λ ( lγ + γ e,l ) = (cid:88) e (cid:32) (cid:88) γ A ∈ Γ B τ − l ˜ l ˜˜ l κ (cid:18) ˜˜ ll γ e,l + γ A (cid:19) (cid:33) s e,l So we see that the two coefficients on the basis elements agree between multiplication ofsections and of intersection points, hence composition is respected, and we do indeed have afunctor. (cid:3) Refresher on toric varieties
We collect facts used when defining the SYZ mirror (
Y, v ) to the genus 2 curve. In particular,they will illuminate our choice of definition of symplectic form on Y . Here is the plan ofaction.(1) Describe the algebraic geometry of toric varieties.(2) Describe a natural polarization, i.e. ample line bundle.(3) Sections of that line bundle give rise to a natural K¨ahler potential for defining ω .The background here is based on [CLS11, p 59, p128] and [Ful93, Chapter 1]. Algebro-geometric construction.
The main point here is to illustrate how a lattice M is the “algebra” and dual lattice N is the “geometry” in the algebraic geometry settingof toric varieties. Definition 3.1 (Notation) . • Geometry: torus T N := ( C ∗ ) n is dense in the toric variety, where N := Z n vectorsencode 1-parameter subgroups (1-PS) parametrized by C ∗ . So N ⊗ Z C ∗ = T N andwe can define N R := N ⊗ Z R . • Algebra: weight vectors are elements of M := Hom Z ( N, Z ) and exponentiate to func-tions on the toric variety, called toric monomials or characters. • Define a nondegenerate pairing M R × N R → R by (cid:104) m, n (cid:105) = m ( n ). • Let e i and f i denote the standard basis vectors on N and M . Definition 3.2 (Cone) . A cone σ ⊂ N R is an intersection of half spaces (cid:92) { u }⊂ M H + u := { n ∈ N R | (cid:104) u, n (cid:105) ≥ } over a set of vectors u ∈ M such that u ⊥ ∂H + u . Example 3.3.
For example, taking u = (1 ,
0) and u = (0 ,
1) in M ∼ = Z gives a cone thatis the first quadrant σ = { u ∈ N R | u , u ≥ } . Claim 3.4 (Rays) . A cone σ is a convex linear combination of lattice vectors ρ called rays ,the set of which is denoted σ (1). Example 3.5.
The rays of the cone given by the first quadrant are e , e . Definition 3.6 (Dual cone) . The dual cone σ ∨ ⊂ M R is the intersection of dual half-spaces. σ ∨ := { m ∈ M R | (cid:104) m, n (cid:105) ≥ ∀ n ∈ σ } = (cid:92) n ∈ σ (1) { m ∈ M R | (cid:104) m, n (cid:105) ≥ } So cones give the geometry and dual cones give the algebra.
Example 3.7.
For σ ⊂ N R given by the first quadrant, σ ∨ is the intersection of half-spaceswhere the normals now are the rays e , e . So the dual cone in M R is again the first quadrant. Remark 3.8 (Duality on faces of cones) . If σ consists of convex linear combinations of n , . . . , n s , then these are the normal vectors of the half-spaces in σ ∨ i.e. σ ∨ = H + n ∩ . . . H + n s .A face τ in σ gives rise to a dual cone τ ∨ of complementary dimension in σ ∨ , e.g. in ourexample τ = R + e has dual given by the right half plane. Definition 3.9 (Strongly convex rational polyhedral cone) . Strongly convex means σ doesnot contain generators ± e for any direction e (or equivalently σ ∩ − σ = { } ), and rational means σ has integral generators, namely they are in Z n . Definition 3.10 (Character on affine chart) . Let S σ := σ ∨ ∩ M be the lattice points in thedual cone. Then C [ S σ ] defines local functions on the open affine chart U σ := Spec C [ S σ ] inthe toric variety by m (cid:55)→ χ m , called a character or toric monomial . In particular, because S σ is a semi-group, meaning we can add elements and it contains zero, C [ S σ ] has a naturalring structure. Corollary 3.11.
All local charts contain the dense torus ( C ∗ ) n = Spec C [ χ ± e , . . . , χ ± e n ] . MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 21
Proof.
We know M = Span Z + (cid:104)± e , . . . , ± e n (cid:105) ⊃ { m ∈ σ ∨ (1) } . Furthermore, rays σ (1) (cid:51) ρ (cid:55)→ χ ρ generate the semi-group S σ . Since σ (cid:55)→ σ ∨ reverses inclusions, looking at the ring offunctions reverses again, and taking Spec does a third time, we find that the above inclusionimplies ( C ∗ ) n ⊂ U σ for all cones σ . (cid:3) Claim 3.12.
The elements χ m for m ∈ S σ define local complex toric coordinates if σ isstrongly convex and maximal. Proof.
A character χ m is a function from the dense torus to C ∗ which is also a homomorphism,for m ∈ M : χ m : ( C ∗ ) n → C ∗ , ( t , . . . , t n ) (cid:55)→ t m := t m · . . . · t m n n Sometimes these functions can be extended to the whole toric variety, where the C ∗ coordi-nates tend to 0 or ∞ . For a strongly convex cone, S σ doesn’t contain ± m for any vector m .So fix minimal generators { m , . . . , m n } for S σ which have no relations. These define localcomplex coordinates χ m , . . . , χ m n by:( χ m , . . . , χ m n ) : U σ → C n (cid:3) Example 3.13.
In this paper we will consider the cone generated by (1 , , , , − , − , χ (1 , , =: x , χ (0 , , =: y , and χ ( − , − , =: z . We will also define χ (1 , , χ (0 , , χ ( − , − , = xyz =: v , which will be thesuperpotential on Y . Remark 3.14.
When σ is not maximal, the generators { m i } ≤ i ≤ k>n are not a Z -basis for M and C [ S σ ] ∼ = C [ χ m , . . . , χ m k ] /I where I is a toric ideal that records these relations. Theimage U σ will be an n -dimensional singular variety in C k , otherwise in the maximal case weobtain a smooth variety. In the example of CP below, Example 3.18, and in this paper, wewill not need to quotient by toric ideals. Definition 3.15 (Toric variety) . Recall that cones σ give affine charts. A fan is a collectionof cones σ i ⊂ N R , arranged in a way so that every face of a cone is a cone and τ ij := σ i ∩ σ j is a face of both σ i , and furthermore the transition functions for gluing charts U σ i to U σ j isdone along U τ ij as follows.Suppose σ and σ are two maximal cones so that τ = σ ∩ σ is ( n − τ = H m ∩ σ = H m ∩ σ for m ∈ σ ∨ ∩ ( − σ ) ∨ ∩ M . In particular, the semi-group S τ nowhas the ± m direction whereas σ only had m and σ only had − m . This corresponds toinverting the coordinate χ m in the two charts from C [ S σ ] and C [ S σ ], which will be nonzeroon the overlap. We glue these two charts along the localizations: U τ = ( U σ ) χ m = ( U σ ) χ − m For consistency with [CLS11], we let u denote lattice elements in N . We illustrate how u ∈ N give rise to 1 parameter subgroups. Definition 3.16 (1 parameter-subgroup (1-PS)) . A vector u = ( u , . . . , u n ) ∈ N defines adirection we travel in the dense ( C ∗ ) n , i.e. we have a 1-PS λ u via the multiplicative homo-morphism: λ u : C ∗ → ( C ∗ ) n , t (cid:55)→ ( t u , . . . , t u n ) Using coordinates ( χ m , . . . , χ m n ) on the dense ( C ∗ ) n we obtain a 1-dimensional manifold byfixing u and taking the closure in the toric variety of:(3.1) N ⊗ Z C ∗ → U σ : u ⊗ t (cid:55)→ ( χ m , . . . , χ m n ) ◦ λ u ( t ) = ( t (cid:104) m ,u (cid:105) , . . . , t (cid:104) m n ,u (cid:105) ) Example 3.17.
For example u = (1 ,
1) gives the complex 1-PS ( t, t ) t ∈ C ∗ ⊂ ( C ∗ ) . Example 3.18 ( CP ) . The fan Σ has three cones σ = R + f , σ = R + f , σ = R + ( − f − f ).The dual cones of functionals non-negative on the original cones are σ ∨ = span ( e , e ), σ ∨ = span ( − e , − e + e ), σ ∨ = span ( − e , e − e ). Each original cone is of maximaldimension so the vectors in each dual cone don’t have relations between them. They hencecorrespond to an affine chart isomorphic to C [ x, y ] for suitable variables x and y . The choiceof generator in each case gives a complex coordinate on the chart. U σ = Spec C [ χ , , χ , ] ∼ = C U σ = Spec C [ χ − , , χ − , ] ∼ = C U σ = Spec C [ χ , − , χ , − ] ∼ = C If τ = σ ∩ σ , then this corresponds to inverting χ , and τ ∨ is H +(1 , . U τ sits inside charts U σ and U σ as follows.(3.2) C [ S σ ] = C [ χ (1 , , χ (0 , ] ⊂ C [ χ ± (1 , , χ (0 , ] = C [ S τ ] Spec == ⇒ U τ ⊂ U σ C [ S σ ] = C [ χ − (1 , , χ ( − , ] ⊂ C [ χ ± (1 , , χ ( − , ] = C [ S τ ] Spec == ⇒ U τ ⊂ U σ since C [ S τ ] = C [ χ ± (1 , , χ (0 , ] = C [ χ ± (1 , , χ ( − , ]. Thus U τ has two different sets of coordi-nates based on which U σ i it sits in. Each choice gives us an identification of U τ with C ∗ × C included as the identity map into C in the two charts U σ i . The coordinate change gives usthe transition map. g ( χ (1 , , χ (0 , ) := (cid:18) χ (1 , , χ (0 , χ (1 , (cid:19) This recovers how we think about CP (cid:51) [ z : z : z ] with χ (1 , = z /z and χ (0 , = z /z .3.2. Rays define divisors which define line bundles.
The vanishing of a section of aline bundle defines a divisor, for example the vanishing of a coordinate on the variety. Thedefining functions of a divisor give transition functions for line bundles, [Huy05, § MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 23
Lemma 3.19 (Orbit-Cone correspondence) . There is a 1-1 correspondence between n − k dimensional ( C ∗ ) n -orbits of a point in the toric variety, and k dimensional cones of the fan,e.g. between divisors and rays.Proof. We’ll show that each 1-parameter subgroup defined by some u ∈ N (recall Definition3.16 and Equation (3.1)) in the interior of a cone limits to the same limit point p . Suppose weselect any u ∈ int ( σ ) ∩ N . Then by Equation (3.1) the limit of a function χ m for m ∈ σ ∨ ∩ M on the corresponding 1-PS is:lim t → χ m ( λ u ( t )) = lim t → t (cid:104) u,m (cid:105) = (cid:40) (cid:104) u, m (cid:105) = 00 if (cid:104) u, m (cid:105) > χ m on this 1-PS limits to 1 when m ∈ σ ⊥ ∩ σ ∨ and to 0 if not (it can never benegative by the definition of σ ∨ ). Intuitively, a point is a map Spec C → Spec C [ S σ ] orequivalently a homomorphism C [ S σ ] → C , i.e. if we know the value of all the functions onthe point p then we know the point. Since u ∈ int ( σ ) ∩ N was arbitrary, we get the samelimit point p for all such u . The corresponding orbit is ( C ∗ ) n · p . In complex coordinates( χ m , . . . , χ m n ) we see p is a collection of 0’s and 1’s.In particular, writing each ray ρ i as the convex span of a vector u i ∈ σ (1) ⊂ N in the interiorof its span, its orthogonal space ρ ⊥ i is n − p i has n − n − toric divisor (adivisor invariant under the ( C ∗ ) n action) D ρ := O ( ρ ) = (cid:91) ρ ≤ σ O ( σ ) (cid:3) Definition 3.20 (Equivariant divisor) . An equivariant or toric divisor is a linear combina-tion of closures of toric orbits. Example 3.21 ( CP ) . The fan has rays and dual cones: ρ = (cid:104) (1 , (cid:105) = (cid:104) u (cid:105) , ρ ∨ = (cid:104)± (0 , , (1 , (cid:105) = (cid:104)± m , m (cid:105) ρ = (cid:104) (0 , (cid:105) = (cid:104) u (cid:105) , ρ ∨ = (cid:104)± (1 , , (0 , (cid:105) ρ = (cid:104) ( − , − (cid:105) = (cid:104) u (cid:105) , ρ ∨ = (cid:104)± ( − , , ( − , (cid:105) The limit points γ ρ i send vectors m ∈ σ ⊥ ∩ σ ∨ to 1 and the rest of σ ∨ to 0. We can takeinterior points u i ∈ int ( ρ i ) ∩ N to be the generators given above. Then with respect tocomplex coordinates ( χ m , χ m ) in each chart we input t (cid:104) u i ,m j (cid:105) and take a limit to find p j :(3.3) p = lim t → (cid:0) t (1 , · (0 , , t (1 , · (1 , (cid:1) = (1 , p = lim t → (cid:0) t (0 , · (1 , , t (0 , · (0 , (cid:1) = (1 , p = lim t → (cid:0) t ( − , − · ( − , , t ( − , − · ( − , (cid:1) = (1 , z i = 0. Polytope determines ample line bundle.
A polytope dual to a fan describes atoric variety with the additional information of a line bundle from a toric divisor, where thesize of the polytope determines the coefficients on each D ρ coming from a ray ρ . Definition 3.22 (Polytope) . A polytope P ⊆ M R is P := { m ∈ M R | (cid:104) m, u i (cid:105) ≥ − a i } ⊂ M R for finitely many i , where the u i ∈ N generate rays ρ i of a fan Σ and a i ≥
0. With infinitelymany i we get more generally a polyhedron .The u i are the inward normals to the sides of the polygon, as well as generators for ρ i of thecorresponding fan, and the a i indicate how far away the side is from the origin (e.g. a i = 0if the side contains the origin). Definition 3.23 (Faces, facets) . Codimension 1 faces in the polytope are called facets , (cid:104) m, u i (cid:105) = − a i for some i and (cid:104) m, u j (cid:105) > − a j for j (cid:54) = i . Higher codimension faces have (cid:104) m, u i (cid:105) = − a i for more than one i . Claim 3.24 (Obtaining fan from P ) . We can read off charts of a toric variety from verticesof a full dimensional lattice polytope defining it, c.f. [CLS11, p 76], since charts arise fromdual cones, and the polytope is also dual to the fan.
Proof outline.
Given polytope P with vertices v , we translate each vertex to the originone-by-one and define C v := Cone ( P ∩ M − v ) ⊆ M R . Then σ v := C ∨ v = Cone ( u i | i th face contains v) ⊆ N R gives a cone which will appear in the fan we are constructing.This gives chart U σ v = Spec C [ Cone ( P ∩ M − v ) ∩ M ]. Similarly, for a face Q of the poly-tope, we have σ Q is the cone on the u i of faces containing Q . The collection of all the cones σ Q gives a fan which we define to be Σ P . (cid:3) Remark 3.25.
Note that P is not the polar polytope P . The latter is the analogue ofdualizing σ → σ ∨ on the level of polytopes, see [Ful93, § P in M R , take the dual Cone ( P × { } ) ∨ ⊂ N R × R . Then P is defined by setting this cone to be Cone ( P × { } ). Definition 3.26 (Divisor D P ) . Let P := { m ∈ M R | (cid:104) m, u i (cid:105) ≥ − a i } ⊆ M R be an integralpolytope, namely a i ∈ Z ≥ and the u i have integer coordinates too. The facets F of P arein bijection with rays u i of the corresponding fan, and by the Orbit-Cone correspondence ofLemma 3.19 these correspond to toric divisors D F . Using the index F instead of i , we define D P := (cid:88) F facet a F D F Sections of O ( D P ) define a K¨ahler potential.Claim 3.27 (Sections of O ( D P ), [CLS11][Proposition 4.3.3]) . The sections of O ( D P ), overtoric variety Y P defined by the fan dual to P , areΓ( Y P , O ( D P )) = (cid:77) m ∈ P ∩ M C · χ m where P := { m ∈ M R | (cid:104) m, u F (cid:105) ≥ − a F } ⊂ M R and D P = (cid:80) F facet a F D F . MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 25
Proof.
By definition Γ( Y P , O ( D P )) = { f ∈ C ( Y P ) ∗ | div ( f ) + D P ≥ } ∪ { } , where C ( Y P ) ∗ consists of invertible rational functions on Y P . Note that Supp( D P ) ∩ T N = ∅ because T N is the orbit where all coordinates are nonzero and Supp( D P ) consists of D F defined by thevanishing of a coordinate. Hence D | T N = 0 and div ( f ) | T N ≥ f is a regular function on T N = Spec C [ M ]. We deduce that f ∈ Spec C [ M ] and Γ( Y P , O ( D P )) ⊂ C [ M ].Furthermore, Γ( Y P , O ( D P )) is a T N -invariant subset under the action T N × Γ( Y P , O ( D P )) (cid:51) ( t, f ) (cid:55)→ f ◦ t − ∈ Γ( Y P , O ( D P )) where ( C ∗ ) n acts by coordinate-wise multiplication on points p ∈ Y P . This is well-defined because the D F are T N -invariant and f and f ◦ t − have thesame vanishing set. Since χ m : T N → C ∗ is a homomorphism, we see that T N preserves thespace spanned by χ m : t · χ m ( p ) = χ m ( t − · p ) = χ m ( t − ) χ m ( p ) ∴ t · χ m = χ m ( t − ) χ m There exists a finite subset A ⊂ M and complex coefficients c m (cid:54) = 0 such that f = (cid:80) m ∈ A c m χ m . Let B := span { χ m | m ∈ A } . Then W := B ∩ Γ( Y P , O ( D P )) is also T N -invariant. The vector space W gives a representation of the commutative group ( C ∗ ) n ,hence commuting matrices on W . These can then be simultaneously diagonalized by a basisof eigenvectors. In other words, we can write the representation T N → GL( W ) as a directsum of 1-dimensional representations given by characters χ m .Hence f ∈ W can be written as a sum of characters in C [ M ] only, but also by ones indexedby W only. By uniqueness the two expressions are the same. So χ m ∈ W for m ∈ A and f ∈ (cid:76) χ m ∈ Γ( Y P , O ( D P )) C · χ m and since the reverse inclusion holds by definition we find that:(3.4) Γ( Y P , O ( D P )) = (cid:77) χ m ∈ Γ( Y P , O ( D P )) C · χ m Lastly, χ m is a global section if and only if div ( χ m )+ D P ≥
0, or equivalently (cid:104) m, u F (cid:105) + a F ≥ m ∈ P ∩ M and hence(3.5) Γ( Y P , O ( D P )) = (cid:77) m ∈ P ∩ M C · χ m (cid:3) Claim 3.28 ( O ( D P ) is basepoint free, [CLS11][Proposition 6.1.1]) . There does not exist apoint p ∈ Y P where χ m i ( p ) = 0 for all m i ∈ P ∩ M . Proof.
Recall that faces Q ⊂ P give cones σ Q that describe the fan of the polytope Σ P . Weshow that for each affine piece U σ Q , there is a global section which does not vanish on thatpiece. Since the U σ Q cover the toric variety, that will suffice. Write the face Q = (cid:84) F ⊃ Q F = { m ∈ M R | (cid:104) m, u F (cid:105) = − a F , ∀ F ⊃ Q } ⊂ M R as an intersection of facets. Pick a lattice point m Q ∈ Q , e.g. a vertex. Then χ m Q is a global section of O ( D ) by Claim 3.27. Furthermore (cid:104) m Q , u F (cid:105) + a F = 0 for F ⊃ Q implies( div ( χ m Q ) + D P ) | U σQ = (cid:88) F ⊃ Q ( (cid:104) m Q , u F (cid:105) + a F ) D F = 0In other words, χ m Q ( U σ Q ) (cid:54) = 0. (cid:3) Corollary 3.29 (Definition of a symplectic form, [Huy05][Example 4.1.2]) . In the contextof the previous two claims, we can define a K¨ahler form by ω P := i π ∂∂h, h = 1 (cid:80) si =1 | χ m i | where | · | refers to the standard norm in C and s = | P ∩ M | . Remark 3.30. O ( D P ) is an ample line bundle from combinatorics of the polytope ([CLS11]),which implies that the toric variety Y P can be embedded into projective space via sectionsof a sufficiently high power of O ( D P ). The idea is that some multiple kP of the polytopeis normal (Chapter 2) which implies the polytope is ample (definition in Chapter 2) whichis seen later to be equivalent to the line bundle being ample (Chapter 6). A very amplepolytope intuitively has enough lattice points, corresponding to there being enough sectionsto define an embedding. Example 3.31 ( CP ) . Recalling the fan for CP above in Example 3.18, the toric polytopefor CP is a triangle. Suppose we size the polytope so its edges are described by the twocoordinate axes (say m = 0 , m = 0) and m + m = −
1. There are three toric divisors D , D , D from z = 0, z = 0, and z = 0. Two of the facets go through the origin so a F = 0 for those. The equation of the third facet is (cid:28) m, (cid:18) (cid:19)(cid:29) + 1 = 0. Hence the linebundle described by this polytope is O ( { z = 0 } ) with sections given by the integral verticesof the polytope χ (0 , = 1 , χ ( − , , χ (0 , − . Setting χ (1 , = z /z and χ (0 , = z /z as before,the above construction gives ω = i π ∂∂ log(1 + | z /z | + | z /z | ), which recovers the Fubini-Study form. Note that these coordinates are only defined on z (cid:54) = 0, but | z | is harmonicso we can add its log to the K¨ahler potential without changing the symplectic form.4. Construction of symplectic fibration on ( Y, v )4.1. Finding Lagrangian torus fibration.
We would like to construct an SYZ mirror to H = Σ the genus 2 curve. The required input to do this is a special Lagrangian torus fibra-tion on H . Finding a special Lagrangian torus fibration is a hard problem. Guadagni’s thesis[Gua17] finds Lagrangian torus fibrations on central fibers of toric degenerations, which willbe the setting of the mirror Y in our case. However there is not an obvious Lagrangian torusfibration on, or toric degeneration to, Σ .What one can do, as in Abouzaid-Auroux-Katzarkov [AAK16], is embed Σ in an abelianvariety. We then take the trivial fibration over C with the abelian variety as a fiber, andblow-up the copy of H over 0. The resulting fibration has H as a critical locus in the centralfiber. It also admits another fibration which is a Lagrangian torus fibration, by taking themoment map before the blow-up (a toric Lagrangian torus fibration), and then keeping trackof the blow-up in the base as in [Sym03] to obtain a non-toric
Lagrangian torus fibration.Note that [AAK16] did this process for hypersurfaces of toric varieties, and Seidel speculatedthat this could be done on hypersurfaces of abelian varieties [Sei12], which we do here.The SYZ construction [SYZ96] produces a candidate mirror complex manifold by prescribingdual fibers over the same base. The points of a dual fiber are parametrized by unitary flatconnections on the trivial line bundle on the original fiber. This process is also discussed
MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 27 in [Aur07]. The SYZ construction inverts the radius of each S on a torus fiber, known asT-duality, and passes between the A- and B-models. In particular on a Calabi-Yau 3-fold asin [SYZ96], SYZ mirror symmetry is T -duality three times. Toric Lagrangian torus fibration
A toric variety with its corresponding symplectic formas in Corollary 3.29, has a natural Hamiltonian T n action given by rotation on the dense( C ∗ ) n , which extends to the full toric variety. Here is an example with CP . Example 4.1 ( CP ) . Consider the complex projective plane with the Fubini-Study form:( CP , ω F S = i π ∂∂ log( (cid:80) i =0 | z i | )) where points are denoted [ z : z : z ]. There is a well-defined Hamiltonian T -action where ( θ , θ ) ∈ R / Z = T acts on CP by rotation: ρ ( α , α )[ z : z : z ] = [ z : e πiα z : e πiα z ]. This is Hamiltonian with Hamiltonianfunctions defining the moment map coordinates µ i . For local CP coordinates ( x , x ) wecan define θ i := arg( x i ) and the infinitesimal rotation action is:(4.1) X j := dρ (cid:18) ∂∂α j (cid:19) = 2 πiz j ∂∂z j = ⇒ ι X i ω F S = dµ i where µ i := − | z i | | z | + | z | + | z | = ⇒ ω F S = dµ ∧ dθ + dµ ∧ dθ The last line is true more generally, that in action-angle coordinates ω = dµ ∧ dθ , e.g. see[CdS, Theorem 1.3.4] and set f i = µ i . The contraction with the vector field rotating coor-dinates gives dµ i more generally. Thus the moment map here µ := ( µ , µ ) : CP → R ∼ =( Lie ( T ) , [ , ] = 0) is given by µ = (cid:18) − | z | | z | + | z | + | z | , − | z | | z | + | z | + | z | (cid:19) Its image can be seen in Figure 2, where the diagonal edge follows from adding µ + µ andallowing the coordinates to vary: Figure 2.
Moment map gives Lagrangian torus fibration: CP exampleThe moment map gives a Lagrangian torus fibration because the moment map is a functionof the norms, as is the K¨ahler potential. Thus ω F S | µ − ( pt ) = 0. We can read off the geometryof the fibration from Figure 2. When both local complex coordinates are non-zero, the preimage is T by rotating under the T -action. When one coordinate goes to zero, we onlyhave the other coordinate to rotate, so the fiber is an S . And when both coordinates arezero in each of the three local C charts we obtain a point.The above Lagrangian torus fibration arose from a moment map, so is called toric . If aLagrangian torus fibration is not from a moment map, it’s called non-toric . Note that CP blown up at [1 : 0 : 0] corresponds to removing a small triangle on the base, which gives aquadrilateral. This can still be the base of a Lagrangian torus fibration by taking T ’s aboveinterior points, S on the edges and points at the vertices. Note that we get CP × CP whichis again toric. However, we could have blown up at a point interior in the toric divisor, suchas [1 : 1 : 0]. The result cuts a triangle out of the base again, but we introduce monodromyaround the top vertex by gluing via the Dehn twist. See Figure 3 and also [AAK16, Fig 2]. Figure 3.
Base of exterior and interior blow-up on CP These describe Lagrangian torus fibrations over a base with a symplectic affine structure.Taking the Legendre transform, we obtain a Lagrangian torus fibration over a base with acomplex affine structure, namely log | · | . Indeed SYZ sends the symplectic affine structureto the complex affine structure on the mirror base. In other words, the torus fibration onthe mirror is log | · | over the interior of the base polytope.
Theorem 4.2 (Construction of Lagrangian torus fibration from [AAK16, § . X := Bl H ×{ } ( V × C ) admits a Lagrangian torus fibration. Furthermore, invariants on X are related to H : (4.2) F uk ( X ) ∼ = F uk ( H ) D b Coh ( X ) = (cid:10) D b Coh ( V × C ) , D b Coh ( H ) (cid:11) where the angle brackets denote semi-orthogonal decomposition. An example of this, and a non-toric Lagrangian torus fibration, is when H = (1 , ⊂ CP defined by s ( x ) = x −
1. This gives mirror symmetry for the point H , and the mirror is aLefschetz fibration generated by one thimble. Namely if dim C V = 1, then the zero fiber of y : X → C involves a normal crossings divisor of the form yz = 0, which for dimensionalreasons is a Lefschetz singularity. Hence Seidel’s Fukaya category for Lefschetz fibrations[Sei08] can be used, where H = a point and F uk ( X ) is generated by a thimble. Claim 4.3. H = Crit ( y ) is the critical locus of the Bott-Morse fibration y : X → C . MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 29
Proof.
The zero fiber is the union of the proper transform of V , namely ˜ V := p − ( V \ H × { } )and the exceptional divisor. The normal bundle N V × C /H × = L ⊕ O . Then H = Σ in theblow-up in P ( L ⊕ O ) is the intersection of two divisors in a normal crossing singularity. Thisintersection forms the critical locus of a Morse-Bott fibration given by y : ( x, y, ( u : v )) (cid:55)→ y .Let p be the blow-up map: p : X → V × C , ( x, y, ( u : v )) (cid:55)→ ( x, y )Geometrically ˜ V is a copy of V , i.e. the closure of the part of V away from H in the blow-upwhich fills in the rest of the V copy. Note that the closure adds in only a point for dimensionreasons as we approach each point in H , so the closure adds back in a copy of H . Theexceptional divisor is a P -bundle over H if we consider projection to V × C and then takethe “zero-th” level at V × { } . Let s p be a section of this P -bundle. E := { ( x, , ( u : v )) | x ∈ H } = p − ( H × { } ) p | E : E → H × { } s p : H → E, s p ( x ) := ( x, , (1 : 0))Now we can see H as the critical locus of the y fibration, as the fixed point set of the S -action that rotates y , namely ( x, e iθ y, ( e − iθ u : v )). y − (0) = { ( x, , ( u : v )) | s ( x ) v = 0 · u } = { ( x, , (1 : 0)) } ∪ { ( x, , ( u : v )) | s ( x ) = 0 } = ˜ V ∪ E Thus ˜ V ∩ E = s p ( H ) ∼ = H = Crit ( y )since x ∈ E implies s ( x ) = 0 and x ∈ ˜ V = ⇒ ( u : v ) = (1 : 0). (cid:3) Proof overview of Theorem 4.2 from [AAK16] . Let x = ( x , x ) ∈ V, y ∈ C and s : V → L define the hypersurface which here is the theta divisor. Then, the blow-up projectivizes thenormal bundle to H × { } namely L ⊕ O by the adjunction formula.(4.3) X = Bl H ×{ } V × C = graph[ s ( x ) : y ]= { ( x, y, [ s ( x ) : y ]) ∈ ( P ( L ⊕ O ) → V × C ) } = { ( x, y, [ u : v ]) | s ( x ) v = yu } ⊂ P ( L ⊕ O )a subset of the P -bundle P ( L ⊕ O ) on V × C . The torus fibration on V × C is (log τ | · | , µ S )where µ S is the moment map from the Hamiltonian S -action that rotates the y complexcoordinate. The base of this fibration is B = T B × R + because of quotienting by the Γ B -action in the first two coordinates, which scales | x i | . We keep track of the blow-up in thebase, as above in the case of interior blow-up, to obtain a Lagrangian torus fibration on X for a suitable symplectic form ω constructed in [AAK16]. It is symplectomorphic to thepullback of the canonical toric K¨ahler form p ∗ ω V × C away from E and controls symplecticarea near the exceptional divisor.For the relations between invariants, [AAK16, Corollary 7.8] states that F uk ( H ) is equivalentto a Fukaya category F s ( X, y ), because Lagrangians in H can be parallel transported from the central fiber to obtain non-compact Lagrangians in X admissible with respect to thesuperpotential y . On the complex side, [Orlov, [APS]] implies there is a semi-orthogonaldecomposition of D b Coh ( X ) into D b ( V × C ) and D b Coh ( H ). (cid:3) Background needed to define the generalized SYZ mirror.
We have localcharts for the mirror over open sets in the base B , as described in [Aur07], which are gluedacross walls in the base B . A wall occurs in the base over which the Lagrangian fibers aresingular. The gluing information can be encoded in a polytope by [AAK16]:∆ ˜ Y := { ( ξ , ξ , η ) ∈ R | η ≥ Trop( s )( ξ , ξ ) } where the tropicalization of a function describes how it tends to infinity as its variables tendto infinity, as a function of the direction ξ we let the variable norms | x i | := τ ξ i go to infinity.Mathematically: Definition 4.4 (Tropicalization) . Let f ( x ) = (cid:80) a ∈ A ⊂ Z n c a x a τ ρ ( a ) . Let | x i | = τ ξ i . Then f ( x/ | x | , ξ ) = (cid:88) a c a (cid:18) x | x | (cid:19) a τ ρ ( a )+ (cid:104) a,ξ (cid:105) The tropicalization of f is: Trop( f )( ξ ) := − min a ∈ A (cid:104) a, ξ (cid:105) + ρ ( a )As τ →
0, the leading order term in f has exponent − Trop( f )( ξ ). The vanishing of f limitsto a tropical curve given by those ξ ∈ R n where two terms can cancel in f , namely wheretwo different a ∈ A give the same minimum for ξ . Example 4.5 ([AAK16][ § . Suppose H ⊂ V is the pair of pants f ( x , x ) := 1+ x + x =0 in V = ( C ∗ ) . This is a pair of pants because x ∈ C ∗ \{− } and a cylinder minus a pointis a pair of pants. Then ρ ≡ A = { (0 , , (1 , , (0 , } hence Trop( f )( ξ , ξ ) =max { , ξ , ξ } . If ξ , ξ < ξ > ξ > ξ is the maximumand if ξ > ξ > ξ is the maximum. So the vanishing of Trop( f ) is Figure 4, and themoment polytope is ∆ := { ( ξ, η ) | η ≥ Trop( f )( ξ ) } ⊂ R n +1 depicted in Figure 5 projectedto ( ξ , ξ ) coordinates with the η -coordinate coming out of the page: Figure 4.
Trop(1 + x + x ) = 0 η ≥ ξ η ≥ ξ η ≥ Figure 5.
Moment polytopein R The corresponding toric variety is Spec C [ x, y, z ] = C where x , y and z are the three toriccoordinates arising from the toric monomials with weight vectors given by primitive gener-ators of the three edges. The superpotential is v = xyz , giving the expected pair of pants MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 31 mirror ( C , xyz ).Tropicalizing the infinite series given by the theta function at first sight seems hard. In fact,it satisfies a periodicity property which allows us to see the tropicalization as a honeycombshape when projected to ( ξ , ξ ) coordinates. Claim 4.6.
The tropicalization of the theta function ϕ := Trop s satisfies the followingperiodicity property(4.4) ϕ ( ξ + ˜ γ ) = ϕ ( ξ ) − κ (˜ γ ) + (cid:104) ξ, λ (˜ γ ) (cid:105) Proof.
Recall from the definition in Claim 2.17 s ( x ) = (cid:88) γ ∈ Γ B τ − κ ( γ ) x − λ ( γ ) = (cid:88) γ ∈ Γ B τ (cid:104) γ,λ ( γ ) (cid:105) x − λ ( γ ) Since | x i | = τ ξ i , and letting τ → ϕ ( ξ ) := Trop( s )( ξ ) = max γ κ ( γ ) + (cid:104) ξ, λ ( γ ) (cid:105) Since κ is a negative definite quadratic form of degree 2 and λ is positive of degree 1, thisshould have a maximum. For example, ϕ (0) = 0. We have the following periodicity property.(4.6) ϕ ( ξ + ˜ γ ) = max γ κ ( γ ) + (cid:104) ξ + ˜ γ, λ ( γ ) (cid:105) = max γ κ ( γ ) + (cid:104) ξ, λ ( γ ) (cid:105) + (cid:104) ˜ γ, λ ( γ ) (cid:105) κ ( γ − ˜ γ ) = κ ( γ ) + κ (˜ γ ) + (cid:104) γ, λ (˜ γ ) (cid:105) = ⇒ ϕ ( ξ + ˜ γ ) = max γ (cid:104) ξ, λ ( γ ) (cid:105) + [ κ ( γ − ˜ γ ) − κ (˜ γ )]= (cid:18) max γ κ ( γ − ˜ γ ) + (cid:104) ξ, λ ( γ − ˜ γ ) (cid:105) (cid:19) − κ (˜ γ ) + (cid:104) ξ, λ (˜ γ ) (cid:105) = ⇒ ϕ ( ξ + ˜ γ ) = ϕ ( ξ ) − κ (˜ γ ) + (cid:104) ξ, λ (˜ γ ) (cid:105) (cid:3) Claim 4.7.
The vanishing set V (Trop s ) ⊂ R ξ ,ξ is a honeycomb shape that is a tiling byhexagons. Proof.
Let ξ = 0. ThenTrop( s )(0) = max γ ∈ Γ B κ ( γ ) ≤ ⇒ Trop( s )(0) = 0since κ is negative definite so its maximum is achieved when γ = 0. We know thatTrop( s )( ξ , ξ ) is a piecewise linear function, so let F , denote the piece that is identicallyzero.In order to prove that the projection of ∆ ˜ Y to the first two coordinates is a tiling of hexagons(equivalent to the statement of the claim), we will proceed as follows. We determine whereadjacent hyperplanes intersect, by finding the equations of their lines of intersection. Thiswill produce the hexagonal shape. Fix ( ξ , ξ , η ) ∈ F , . Add i γ (cid:48) + i γ (cid:48)(cid:48) for ( i , i ) ∈ { ( ± , , (0 , ± , ± (1 , − } . These sixchoices will give rise to the hexagonal shape. Recall γ (cid:48) and γ (cid:48)(cid:48) are the generators for Γ B . ByEquation (4.6) and that ϕ ( ξ ) = 0:(4.7) η = ϕ ( ξ + i γ (cid:48) + i γ (cid:48)(cid:48) + 0) = ϕ ( ξ ) + (cid:104) ξ, λ ( i γ (cid:48) + i γ (cid:48)(cid:48) ) (cid:105) − κ ( i γ (cid:48) + i γ (cid:48)(cid:48) )= i ξ + i ξ + i + i + i i = i ξ + i ξ + 1On the other hand, the equation of the plane when ( ξ, η ) ∈ F i ,i is the set of ξ and η = ϕ ( ξ )such that η = i ( ξ − i − i ) + i ( ξ − i − i ) + 1 = i ξ + i ξ − ξ from Equation (4.7) so it lies in F i ,i and not F , .) So points ( ξ , ξ , η ) onthe intersection of the two planes must satisfy both, hence: i ξ + i ξ = 1We get the shape enclosed by the lines ξ i = ± i = 1 ,
2, a box, and ξ − ξ = ± ξ = ξ shifted up and down by 1. This means F , is a hexagon in the η = 0 plane.(0 , ξ = − ξ = ξ − ξ = 1 Figure 6.
The (0,0) tile delimited by the tropical curveNow pick any ( ξ , ξ ) ∈ R . Choose γ such that ( ξ − γ , ξ − γ ) ∈ F , , which we can do sincethe hexagon is the same size as the fundamental domain for the Γ B -action by subtraction.Let λ ( γ ) =: ( m , m ) t . Then again by Equation (4.6) and using that ϕ ( ξ − γ ) = 0:(4.8) η = ϕ ( ξ ) = ϕ ( ξ − γ ) + (cid:104) ξ − γ, λ ( γ ) (cid:105) − κ ( γ )= 0 + ξ m + ξ m − ( m + m m + m ) ⇐⇒ (cid:42) m m − , ξ ξ η (cid:43) − ( m + m m + m ) ⇐⇒ (cid:28)(cid:18) λ ( γ ) − (cid:19) , (cid:18) ξη (cid:19)(cid:29) + κ ( γ )Then using Equation (4.8), we find the intersection of the ( m , m ) plane with the ( m + i , m + i ) plane, i.e. those ( ξ, η ) satisfying both plane equations.∆ η = 0 = ϕ ( ξ + ( i , i ) · ( γ (cid:48) , γ (cid:48)(cid:48) )) − ϕ ( ξ ) MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 33 = (cid:104) λ ( γ ) + ( i , i ) , ξ (cid:105) + κ ( γ + ( i , i ) · ( γ (cid:48) , γ (cid:48)(cid:48) )) − [ (cid:104) λ ( γ ) , ξ (cid:105) + κ ( γ )]= i ξ + i ξ − (cid:104) γ, ( i , i ) (cid:105) − (cid:104) i γ (cid:48) + i γ (cid:48)(cid:48) , ( i , i ) (cid:105) (4.9) = i ξ + i ξ − (cid:104) (2 m + m , m + 2 m ) , ( i , i ) (cid:105) − (cid:104) (2 i + i , i + 2 i ) , ( i , i ) (cid:105) = i ξ + i ξ − i (2 m + m ) − i ( m + 2 m ) − ⇒ ξ = 2 m + m ± ξ = m + 2 m ± ξ − ξ = ± m − m m γ (cid:48) + m γ (cid:48)(cid:48) ξ = m + 2 m − ξ = ξ − − m + m ξ = 2 m + m + 1 Figure 7.
The ( m , m ) tile delimited by the tropical curve (cid:3) The definition of ( Y, v ) . The smooth manifold Y is constructed as a portion ofa toric variety ˜ Y quotiented by Γ B acting properly discontinuously via holomorphic maps.The defining polytope is ∆ ˜ Y := { ( ξ , ξ , η ) ∈ R | η ≥ Trop( s )( ξ ) } ∆ Y := (∆ ˜ Y ) | η ≤ T l / Γ B whereTrop( s )( ξ ) := max γ κ ( γ ) + (cid:104) ξ, λ ( γ ) (cid:105) (4.10) γ · ( ξ , ξ , η ) := ( ξ + γ , ξ + γ , η − κ ( γ ) + (cid:104) ξ, λ ( γ ) (cid:105) ) ∵ Trop( s )( ξ + γ ) = Trop( s )( ξ ) − κ ( γ ) + (cid:104) ξ, λ ( γ ) (cid:105) where the last line is Equation 4.4. The parameter T (cid:28) Y . The polytope ∆ ˜ Y is illustrated in Figure10, where η is bounded below by the expression in the center of the tile, and comes out ofthe page. The superpotential v was defined in Example 3.13; the three toric coordinates( x, y, z ) correspond to the three edges on the polytope ∆ ˜ Y from the lower left vertex of the(0 ,
0) hexagon. In particular, η is the weight vector (0 , ,
1) which gives the toric monomial v = xyz . The fact that v is well-defined under the Γ B -quotient will be proven below. Inparticular, the action on complex coordinates only gives a nontrivial action if we restrict v = xyz to be very small. This is why the condition η ≤ T l is imposed.More specifically, recall that vertices of ∆ ˜ Y correspond to C charts. For example, considerthe ( x, y, z ) coordinates and the green vertices to the right in Figure 10, call them ( x (cid:48) , y (cid:48) , z (cid:48) ).The way in which we glue these two charts is encoded by the edge they are connected by, asfollows. Recall that the normal ν n to the ( n , n ) tile is(4.11) ν n := − n − n The two vertices, taking the normals to the three facets at the vertex, gives us two conesspanned by the convex hull of the following rays: σ = (cid:42) , , (cid:43) = ⇒ U σ = Spec C [ x, y, z ] σ = (cid:42) , − , (cid:43) = ⇒ U σ = Spec C [ x − , xy, zy − ](4.12) τ := σ ∩ σ = (cid:42) , (cid:43) = ⇒ U τ = Spec C [ Z ∩ τ ∨ ] = Spec C [ x ± , y, zy − ]Thus in coordinates on C ∗ × C ∗ × C , which is the overlap of the two charts(4.13) φ = : C (cid:51) ( x, y, z ) (cid:55)→ ( x, y, z ) φ : C (cid:51) ( x, y, z ) (cid:55)→ ( x − , xy, zy − )we find that identifying the U τ ⊂ U σ i for each i , we obtain the transition map:(4.14) φ : C ∗ × C ∗ × C (cid:121) ( x, y, z ) (cid:55)→ ( x − , xy, zy − )Thus this gluing gives P in the first component. In particular, gluing all the toric chartswill contain the dense ( C ∗ ) but infinitely many toric divisors coming from the CP (3) gluedalong P ’s. In the fibration v = xyz , we have this infinite Z -chain of toric divisors over0, and since the coordinates in each chart preserve v , a generic fiber is ( C ∗ ) . So we havenon-compact fibers and a non-compact base for the fibration v . When we take the quotient,the base stays the same but the fibers become compact. Definition 4.8 ([AAK16][Definition 1.2]) . ( Y, v ) is the generalized SYZ mirror of H = Σ . Remark 4.9.
Note that one can apply SYZ in the reverse direction by starting with aLagrangian torus fibration on Y minus a divisor to recover X as its complex mirror, see[AAK16, §
8] or [CLL12].We now discuss the complex coordinates on Y . MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 35
Definition of complex coordinates on ˜ Y / Γ B .Reason for 1-parameter family. We can strengthen our result to be not just betweentwo manifolds, but between two families of manifolds. Namely a family of genus 2 curvesparametrized by τ and a family of symplectic manifolds parametrized by τ . Symmetrically,we can also allow T to parametrize a symplectic structure on the genus 2 curve which ismirror to a complex structure parametrized by T on Y , which is what we define in thissubsection.More specifically, we view τ or T parametrizing the complex structure via scaling the lat-tice we quotient by, so multiplication by i gets scaled in a 1-parameter family. The way inwhich they parametrize the symplectic structure as Novikov parameters is by symplecticallyweighting counts of discs in homology class β by τ − ω ( β ) = e − (log τ ) ω ( β ) , i.e. the symplecticform is scaled to (log τ ) ω as τ → τ is the complex parameter on the genus 2 curve/Novikov parameter on Y , and T willbe the complex parameter on Y /Novikov parameter on the genus 2 curve. Although thecomplex structure on Y won’t affect its symplectic geometry, the K¨ahler potential will bedefined in terms of the complex coordinates, necessarily in a way invariant under T . Thisis why we need to define the complex structure on Y . (So adding in the T doesn’t extendour results to more manifolds in the direction we are considering, because we consider Y asa symplectic manifold and T varies the complex manifold. But if we ever wanted to con-sider mirror symmetry in the other direction, swapping the A- and B-models, we would have1-parameter families in that direction too. So it’s a bit stronger result to include the T here.) Properties defining the symplectic form.
We would like ω to have some nice propertiesthat allow us to compute parallel transport and monodromy later on. (However, there areother symplectic forms one can equip ( Y, v ) with.) These properties will enforce the wayin which the Γ B -lattice acts on the complex coordinates in terms of T . We will define aK¨ahler potential for ω as a function of the norms of the complex coordinates. That way itis a function of the moment map coordinates as well by Legendre transform. In order toadapt Seidel’s Fukaya category for Lefschetz fibrations, we also would like v : Y → C to bea symplectic fibration. That is, ω V ∨ is a symplectic form on the fiber. Over zero we havea singular fiber, so we only require ω to restrict to a symplectic form away from the zerofiber. In a neighborhood of the zero fiber, this corresponds to a neighborhood of the facetsin ∆ ˜ Y . Away from a vertex, we take the limit of the symplectic form as v →
0. Near a C -vertex, we take the standard ω C on C . In between the two, we interpolate with bumpfunctions. Checking non-degeneracy in these regions is the bulk of the calculation for ω , towhich readers may refer to Appendix B. We proceed to describe the symplectic form in afiber so the limit to the zero fiber is well-defined. Properties defining the symplectic form on a v -fiber. The central fiber of v isthe Γ B -quotient of the toric variety with moment polytope given by the hexagon in Figure8. This comes equipped with a symplectic form as described in the theory of Section 3with the toric variety CP (3 points), i.e. blown-up at three points. As we move away from Figure 8.
Moment polytope for central fiber of (
Y, v ) when H = Σ . v = 0 but still near a vertex of the polytope, the toric variety is locally modelled on the( C , xyz ) picture that was the local model of Example 4.5. So we will use bump functionsto interpolate between the toric symplectic form of CP (3 points) and the standard form on C . Away from v = 0 and away from the C vertex we use the toric K¨ahler form ω CP (3) . Illustrative example: symplectic form on mirror fibration to H = pt . We illustratethis in one dimension down, where the polytope is two dimensional. Let H be a point insidean elliptic curve. Then the polytope ∆ ˜ Y is two-dimensional, so we can draw it. Figure 9.
In one dimension lower, the boundary of ∆ ˜ Y is the moment mapimage of a string of P ’s. In the polytope, | v | increases in the (0 ,
1) direction.In the fibration v , | v | is the radius of the circle in the base.A fiber of v : ˜ Y → C is topologically a cylinder with belts increasingly pinched, Z -periodically, as | v | →
0, degenerating to a string of P ’s over the central fiber. Aroundthe widest parts of the cylinder, it looks like a portion of a sphere, and P comes canonicallyequipped with the Fubini-Study form. So on (fiber neighborhood of v -fiber blue circle) × C the symplectic form is a product of that on the base and on the fiber.On the other hand, neighborhoods of the vertices of the polytope give C charts where thelocal picture of v = xy : ˜ Y → C is the Lefschetz fibration C → C , ( x, y ) (cid:55)→ xy where cylin-ders degenerate to a double cone over zero. In particular, C comes canonically equipped MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 37 with the standard form ω std with K¨ahler potential | x | + | y | . When the toric coordinates x, y are very small (these coordinates correspond to the weight vectors (cid:104) , (cid:105) and (cid:104)− , (cid:105) in∆ ˜ Y ), the Fubini-Study form and the standard form are approximately the same by a Taylorexpansion of log. For example when T r x is very small, we have log(1 + ( T r x ) ) ≈ ( T r x ) . Soa symplectic form can be constructed by interpolating between these two K¨ahler forms. Symplectic form on mirror fibration to H = Σ . Now we do the same one dimensionup, for H = Σ inside an abelian surface. xz = v xy y T − v − zy T v xT − v − zT v y xT − y − T v z − = T xyT v x − = T yz T − y − T v z − T − x − T v y − T v z − = T xyT − x − T − z − T v x − T v y − − ξ − − ξ − Figure 10.
Depiction of 3D ∆ ˜ Y . Coordinates respect Γ B -action, see be-low; magenta parallelogram = fundamental domain. Vertices = C charts.Coordinate transitions, see Lemma 4.13. Expressions in the center of tilesindicate e.g. η ≥ ϕ ( ξ ) = − ξ − (0 , is given by( ξ , ξ ) ∈ { ( ξ , ξ ) | − ≤ ξ , ξ , − ξ + ξ ≤ } .Let r x = | x | and the same for y and z , and let 0 < T << C , ω std ) with K¨ahler potential T ( r x + r y + r z ) on C -charts at each vertex in Figure 10,and the toric K¨ahler form for CP (3 points), the blow-up at three points, induced by thehexagon in Figure 10. Recall from Corollary 3.29 that such a potential is the logarithm of the sum of squares of sections corresponding to lattice points.Toric geometry determines the complex coordinates up to scaling. We don’t want to changethe symplectic form so we scale the K¨ahler potential as well to cancel out the scaling of thecomplex coordinates. This reflects the phenomenon that under mirror symmetry, varying thecomplex structure on Y doesn’t change the symplectic structure. In particular, recall thatintroducing T allows us to make a statement about mirror symmetry that is between familiesif we were to flip the A and B sides (analogous to the role of τ in the current direction).Since we are free to scale the sections by a scalar multiple, we may scale in a way that allowsus to factor their sum and write the K¨ahler potential as:(4.15) g xy := log(1 + | T a x | )(1 + | T b y | )(1 + | T c xy | )from a to-be-determined choice of ( a, b, c ). The three dimensional moment polytope ∆ ˜ Y inFigure 10 has a Z / P along each axis is defined to have the same symplecticarea under ω , say equal to 1. This symmetry will enforce how Γ B acts on the local complexcoordinates. Definition 4.10 (Complex coordinates on ˜ Y ) . We define the following coordinate charts onthe polytope ∆ ˜ Y . Let g k index the chart obtained by rotating k vertices clockwise from thelower-left vertex of the 0,0 hexagon. Then the coordinate charts are: • U ,g : ( x, y, z ) • U ,g : ( T v x − , T − y − , T v z − ) =: ( x (cid:48)(cid:48) , y (cid:48)(cid:48) , z (cid:48)(cid:48) ) • U ,g : ( x, T v y, T − v − z ) • U ,g : ( T − x − , T − y − , T v z − ) • U ,g : ( T v x, y, T − v − z ) • U ,g : ( T − x − , T v y − , T v z − ) =: ( x (cid:48) , y (cid:48) , z (cid:48) )Furthermore, we extend this definition to all the coordinate charts on ˜ Y by symmetry.E.g. note that going along the z axis we get to the g − vertex in the ( − ,
0) tile and maydefine U ( − , ,g − : ( T v x − , T v y − , T − z − ) =: ( x (cid:48)(cid:48)(cid:48) , y (cid:48)(cid:48)(cid:48) , z (cid:48)(cid:48)(cid:48) ). In the new coordinate systemdenoted ( x (cid:48)(cid:48) , y (cid:48)(cid:48) , z (cid:48)(cid:48) ) in Figure 10, they will have the same small values as ( x, y, z ) in theoriginal chart. It is the same idea to obtain the coordinate charts at all the other vertices. Definition 4.11 (Definition of complex structure on Y by Γ B -action) . The complex struc-ture on ˜ Y defines a complex structure on Y as follows. We define the group action to be(4.16) ( − γ (cid:48) ) · ( x, y, z ) := ( T v x, y, T − v − z )( − γ (cid:48)(cid:48) ) · ( x, y, z ) := ( x, T v y, T − v − z )The convention is that moving up and right in Figure 10 is negative since the powers of T << γ (cid:48) and γ (cid:48)(cid:48) map thecoordinates ( x, y, z ) to the charts centered at ( − , −
1) and ( − , −
2) respectively in Figure10. Then Γ B acts properly discontinuously on the restriction of ˜ Y to small | v | and Y is awell-defined complex manifold. In particular we obtain a product complex structure.The reason we choose this group action is explained in the proof of Claim 4.12. MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 39
Claim 4.12.
The symplectic forms for CP (3) (i.e. ω on a fiber) and C (i.e. ω in a neigh-borhood of the origin in C ) are invariant under the Γ B action so descend to forms on thecorresponding neighborhoods in Y , (shaded in Figure 4.4 in the one dimension lower case.) Proof.
The interior of each n th hexagonal tile in ∆ ˜ Y is identified under the Γ B -action on thepolytope, so on the K¨ahler potential for CP (3) we would like γ ∗ g xy and g xy to differ by anelement in the kernel of ∂∂ e.g. a harmonic function such as | x | = xx . One way to guaranteethese conditions is if γ (cid:48) , γ (cid:48)(cid:48) arise from a Z / G denote the Z / ˜ Y . Suppose that g acts by(4.17) ( x, y ) (cid:55)→ ( T α y − , T β xy )for some α and β , so that the action on the z coordinate is g · z = T − α − β yz , determined by v = xyz gluing to a global function so must be preserved under G . Thus:(4.18) g ∗ log(1 + | T a x | )(1 + | T b y | )(1 + | T c xy | )= log(1 + | T a + α y − | )(1 + | T b + β xy | )(1 + | T c + α + β x | ) g xy = log(1 + | T a x | )(1 + | T b y | )(1 + | T c xy | )Comparing coefficients on x , y and xy , in order for the g xy and g ∗ g xy to differ by a harmonicfunction we want a = c + α + β , b = − a − α and c = b + β . Since we have five unknownsand only three equations, there are multiple solutions. We make a choice so we will be ableto define a symplectic form, and fix a = b = 1. Then the rest is determined: α = − c = 3 − β = 1 + β hence β = 1 and c = 2. We find that(4.19) g xy := log(1 + | T x | )(1 + | T y | )(1 + | T xy | ) g · ( x, y ) = ( T − y − , T xy )and we see that g and g − do indeed give the group action defined in Definition 4.11. (cid:3) Now we can see where the choice of complex coordinates on ˜ Y arose from. Lemma 4.13.
The action of G defines the complex coordinate charts given above in Lemma4.16.Proof. The g k allows us to index the charts, but the coordinates are a permutation of thecoordinates g k · ( x, y, z ) so the subscript is mainly for indexing. Namely, we re-label thecoordinates in each C chart to match with the ( x, y, z ). Since g · ( x, y ) = ( T − y − , T xy ),applying the group action twice we find( x, y ) ∼ ( T − y − , T xy ) ∼ ( T − ( T − x − y − ) , T ( T − T − )( T xy )) = ( T − x − y − , x )Since Γ B fixes v = xyz so we may rewrite the transformed y -coordinate as T v y to obtainthe γ (cid:48)(cid:48) action, after suitable permutation σ g :(4.20) σ g · g · ( x, y, z ) = ( x, T v y, T − v − z )= ⇒ − γ (cid:48)(cid:48) · ( x, y, z ) := ( x, T v y, T − v − z )(Note that γ (cid:48) and γ (cid:48)(cid:48) increase the coordinate norms, analogous to the Γ B -action increas-ing norms by τ − γ on the mirror side.) The γ (cid:48) calculation is similar; since g − · ( x, y ) = ( T xy, T − x − ) we obtain g − · ( x, y, z ) = ( T v z − , T − x − , T v y − ). Then:(4.21) − γ (cid:48) · ( x, y, z ) = σ g − ◦ g − · ( x, y, z )= σ g − ◦ g − · ( T v z − , T − x − , T v y − )= ( T − v − x, y, T v z )For example, to find the coordinate system ( x (cid:48)(cid:48)(cid:48) , y (cid:48)(cid:48)(cid:48) , z (cid:48)(cid:48)(cid:48) ), we first move down and left by γ (cid:48) action to be in the (0 , −
1) tile, then rotate by g − and finally apply a suitable permutation. (cid:3) Corollary 4.14. ω CP (3) defines a symplectic form in each v fiber on a neighborhood givenby the preimage of an open set in the interior of the hexagon at fixed | v | , (correspondingto the yellow shading of Figure 4.4.) This open set is defined in Figure 13 in terms of thecoordinates introduced beginning with Equation 4.27.Proof. We are justified in defining it just on a fiber, and taking the compactification as v → v (cid:54) = 0. A fiber has two complex coordinates ( x, y ),and in particular | v | is also fixed so in the moment map this corresponds to fixing a certainheight above the base of the infinite bowl. Note that all ξ , ξ , η will vary, but η is a functionof ( ξ , ξ ) since all points lie on a surface. We restrict | v | < T l for some large power of l ,where T <<
1. (In other words, up to rescaling, this A-side is either non-compact from thebase, or compact from boundary on the base of v .) Namely, for | v | = T l with T << l sufficiently small, Γ B acts properly discontinuously and holomorphically so the quotient isa well-defined complex manifold Y . (cid:3) Claim 4.15 (Γ B -action on symplectic coordinates) . γ ∈ Γ B acts by γ · ( ξ, η ) = ( ξ − γ, η − κ ( γ ) + (cid:104) ξ, λ ( γ ) (cid:105) ). Proof.
Recall that the Γ B -action on V was given by x (cid:55)→ τ − γ x . Since τ ξ = | x | , we find thatΓ B acts on ξ by negative addition. For the statement about η , recall that η takes values η ≥ ϕ ( ξ ), so since ϕ ( ξ + γ ) = ϕ ( ξ ) − κ ( γ ) + (cid:104) ξ, λ ( γ ) (cid:105) γ · η = η − κ ( γ ) + (cid:104) ξ, λ ( γ ) (cid:105) when γ ∈ Γ B . (cid:3) The additional piece of information we need to check when taking an SYZ mirror in thissetting of a Γ B -action, is that the group action respects the Lagrangian torus fibration. Lemma 4.16.
The Γ B -action commutes with the moment map, i.e. ∀ γ ∈ Γ B , ( ξ , ξ )( γ · ( x, y, z )) = γ · ( ξ , ξ )( x, y, z ) Proof.
The right-hand side is ξ − γ . In order to determine the left-hand side, we need tocompute how the moment map changes as a function of the complex coordinates. To do this,we let F denote the local K¨ahler potential for the symplectic form. So F is interpolatingbetween the three toric CP (3) potentials around a vertex. The change in moment mapvalue can be calculated by integrating the symplectic form to compute the area of a disc, asfollows. Claim 4.17.
Consider a disc D ⊂ Y , whose lift ˜ D ⊂ ˜ Y is invariant under the action of an S subgroup of T . Denote the corresponding moment map by µ . In particular the boundary of˜ D is an S -orbit S . ( x, y, z ), while its center is a fixed point ( x , y , z ). Then the symplecticarea of the disc D is equal to µ ( x, y, z ) − µ ( x , y , z ). MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 41
Proof.
We claim µ ( x, y, z ) − µ ( x , y , z ) = (cid:82) ˜ D ω . The CP moment map image is a segmentover which we can draw an elongated sphere and map a value in the segment to the area ofthe part traced out in the sphere. The lift ˜ D has boundary component given by an orbit,which we can think of as the integral flow of the vector field generated by the infinitesimalaction, call it X . Then the integral over ˜ D involves integrating over this X and the linefrom ( x, y, z ) to ( x , y , z ). Call this line C . Then we can write the integral as (cid:90) C ι X ω = (cid:90) C dµ = µ ( x, y, z ) − µ ( x , y , z ) (cid:3) So in order to compute the change in moment map coordinates, we go ahead and computearea as follows.
Claim 4.18.
The symplectic area of the P along each of the x , y and z axes is 1. Proof.
In the simplified case of P , if the gluing is normally x ∼ x − , the analogue here wouldbe to define a gluing x ∼ Kx − for some constant K (so a Z -action). We do the computa-tion along the y -axis, and the others will be the same because we will impose x ↔ y ↔ z symmetry on the symplectic form. Recall P has an open covering U , U ∞ and charts φ and φ sending [ z : z ] to z /z and z /z respectively. We want to split up the integration over P into these two charts, but only the portion of the chart up to where they intersect (elsewe integrate over too much). So we have y is the coordinate on φ ( U ) ∼ = C then it is T α y − on φ ∞ ( U ∞ ) and | y | = | T α y − | = ⇒ | y | = T α/ . Let C T α / denote this circle of complex ra-dius T α/ . Then let D be the disc in φ ( U ) and D ∞ the corresponding disc in the other one.Let F denote the local K¨ahler potential for ω and F , F ∞ be the K¨ahler potential in the twocharts U and U ∞ on the y -axis P . Then for K¨ahler potential F given above from the toricK¨ahler form on CP (3): (cid:90) P i π ∂∂F = i π (cid:34)(cid:90) φ − ( D ) d ( ∂F ) + (cid:90) − φ − ∞ ( D ∞ ) d ( ∂F ) (cid:35) = i π (cid:20)(cid:90) ∂D ∂ ( φ − ) ∗ F − (cid:90) ∂D ∞ ∂ ( φ − ∞ ) ∗ F (cid:21) = i π (cid:90) C Tα/ ∂ ( F − F ∞ )= i π (cid:90) C Tα/ ∂ log( | T α/ y | ) = i π (cid:90) C Tα/ T α/ ydy | T α/ y | = i π (cid:90) π e iθ ( − i ) e − iθ dθ = 1 (cid:3) Commutes in ξ , ξ . Thus when gluing across the z -axis the coordinate chart U ( − , ,g − with coordinates ( x (cid:48)(cid:48)(cid:48) , y (cid:48)(cid:48)(cid:48) , z (cid:48)(cid:48)(cid:48) ) to the main coordinate chart U ,g with coordinates ( x, y, z ),the K¨ahler potential transforms by F (cid:48)(cid:48)(cid:48) V = F V − log( | T z | ). Thus this discrepancy betweenthe local K¨ahler potentials implies that the value of ξ is modified by adding the constant −
1, and similarly for ξ . Similarly, when gluing across the x -axis the main chart to the chart U ,g with coordinates ( x (cid:48) , y (cid:48) , z (cid:48) ) the local K¨ahler potentials differ by − log( | T x | ), whichmodifies ( ξ , ξ ) by (1 , γ (cid:48) ,namely changing coordinates from ( x (cid:48)(cid:48)(cid:48) , y (cid:48)(cid:48)(cid:48) , z (cid:48)(cid:48)(cid:48) ) to ( x (cid:48) , y (cid:48) , z (cid:48) ). This action modifies ( ξ , ξ )by adding (2 , γ (cid:48) . Similarly for γ (cid:48)(cid:48) . This completes the proof that the T -moment map is Γ B -equivariant. (cid:3) We will need the following claim later when we define the full symplectic form.
Claim 4.19.
The moment map coordinates ( ξ , ξ ) are monotonic increasing functions of r x and r y in a v = xyz fiber. Proof.
Recall the action-angle coordinates ( ξ, θ ) of Claim 2.1 from the fiber-wise action ρ : ρ : T × V ∨ (cid:51) ( α , α ) · ( x, y, z ) (cid:55)→ ( e iπα x, e iπα y, e iπ ( − α − α ) z ) ∈ V ∨ Let X i := dρ ( ∂ α i ). If ι X i ω | V ∨ were exact, say dH i for some function H i (known as theHamiltonian), then the torus action would be called a Hamiltonian group action and ( H , H )would be the moment map. This leads us to the caveat at the start of Section 2.1. In thesetting here, ω is complicated so computing ι X i ω | V ∨ is involved. However, the 1-forms ι X i ω are closed, and hence locally exact, so there exist locally defined functions ξ , ξ so that ι X i ω = dξ i . Notationally we are assuming ξ η ≡ η . Globally, as seen above, the first two ξ i are Γ B -periodic and their differentials pass to 1-forms on a torus fiber R / Γ. The infinitesimalaction is expressed by the pushforwards dρ ( ∂ α i ) for i ∈ { , } . These vector fields are(4.22) X = ∂∂θ = ( ix, , − iz ) , X = ∂∂θ = (0 , iy, − iz )Then the action coordinates can be expressed locally in terms of the K¨ahler potential: ι X i ω | V ∨ = dξ i = ⇒ ω | V ∨ = dξ ∧ dθ = ⇒ ι X i (cid:18) i π ∂∂F (cid:19) = i π dι X i ∂F = dξ i ∴ ξ := i/ π∂F (2 πix∂ x − πiz∂ z ) + const ∴ ξ = − (cid:18) ∂F∂ log | x | − ∂F∂ log | z | (cid:19) + const ∴ ξ = − (cid:18) ∂F∂ log | y | − ∂F∂ log | z | (cid:19) + const using ∂∂ = − ∂∂ = − d∂ , the conversion to polar coordinates from § F ispreserved by rotating x, y, z as it is a function of their norms, hence the Lie derivative L X i ∂F = 0 and also ∂ θ x F = 0. The calculation for ξ is similar. This calculation of ξ i is up to additive constants. The upshot is that the moment map ( ξ , ξ ) provides periodicaction-coordinates which are monotonic increasing in | x | and | y | for fixed v because of howwe defined F . (But recall the caveat, we are calling it a moment map but it takes values ina torus instead of affine space, so we are expanding the definition of moment map here toallow the ξ i to be periodic multivalued functions for a quasi-Hamiltonian action .) We seethat V ∨ is symplectomorphic to ( T B × T F , ω std ). (cid:3) Note that an orbit is precisely the kernel of ( dξ , dξ ) so that a preimage of a moment mapvalue is a T -orbit. Said another way, the torus action preserves the moment map. Or said MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 43 yet another way, the tangent space to a fiber of the moment map is ∂ θ , ∂ θ . How to view η as a moment map coordinate. Let θ η = arg( v ) = arg( xyz ). We can’textend the above to a T -action. If we were to rotate v too, then x and y would change in theprocess as well, due to monodromy. Because T has trivial bracket on its Lie group, we shouldrotate one variable while fixing the others. When v (cid:54)∈ R + the angle variables θ and θ on V ∨ are only well-defined up to additive constants, as they jump by arg( v ) = θ η under the actionof the generators of Γ B due to the transformation rules for the complex coordinates x and y , see Corollary 4.11. Thus we can only define an infinitesimal T action generated locallyby sufficiently small rotations α i , P ( α , α , α η ) · ( x, y, z ) = ( e iπα x, e iπα y, e iπ ( α η − α − α ) z ).The infinitesimal action on v is expressed by the pushforward dP ( ∂ α η ):(4.23) ∂∂θ η = (0 , , iz )This isn’t global as it transforms non-trivially under Γ B . (Note that, notationally, ξ η is thesame thing as η .) The upshot is that upon integrating we find that η = 12 · ∂F∂ log | z | + const Though the action coordinates ( ξ , ξ , η ) are globally well-defined on the universal cover ˜ Y ,on which the T -action discussed above is well-defined and Hamiltonian, η is only definedlocally on Y . Definition 4.20 (Superpotential) . The superpotential v is the holomorphic function Y → C (4.24) v ( x, y, z ) := xyz which is well-defined as a global function on Y because γ ∗ v = v for all γ ∈ Γ B . Example 4.21.
To see why the fibers are complex tori with complex coordinates x and y , e.g. consider just x ∼ x on C ∗ : if we identify the unit circle with the circle of twicethe radius, we’ve just formed an elliptic curve or 2-torus. Now we compactify: if we have x ∼ T n x for all n then taking n → ±∞ , we see that 0 and ∞ are identified to give thepinched torus. Remark 4.22 (Mirror to non-standard symplectic form on mirror) . Note that the symmetryproperties required for ω told us what complex structure was needed (it produces a productcomplex structure), which by mirror symmetry will correspond to a specific symplectic formon A-model of X . The complex structure is a product here on x and y . In particular, theΓ B -action on complex coordinates here is different than that described in [AAK16, § v m ∼ v (cid:104) λ ( γ ) ,m (cid:105) T (cid:104) γ,m (cid:105) v m where they use complex coordinates v = ( v , v ). Thus setting ( x, y, z ) = ( v − , v − , v v v ),their complex structure arises from the following Γ B -action: γ (cid:48) · ( x, y, z ) = ( T − v − x, T − y, T v z ) γ (cid:48)(cid:48) · ( x, y, z ) = ( T − x, T − v − y, T v z )So our complex structure is mirror to a non-standard K¨ahler form on the genus 2 curve. Figure 11.
L) Regions near a vertex, R) Number of regions interpolated between
Remark 4.23 (Terminology) . The toric variety ˜ Y is referred to as a toric variety of infinite-type by [KL19], because of the infinitely many facets, where the neighborhood of the toricdivisor there is the same as our restriction to | v | small, i.e. η small.4.5. The symplectic form.
Now we define the symplectic form, first on a fiber, as a function of the norms r x , r y , r z viewed in the moment polytope. See Figure 11. We are starting with a polytope, which wewant to be the moment map image with respect to some symplectic form that we construct.In particular, we have found a symplectic form so that the boundary P ’s of the hexagonhave length 1 i.e. symplectic area 1. This follows from the change in K¨ahler potential undergluing across coordinate axes, see Claim 4.15.Now to the definition. The three tiles adjacent to the main corner define toric coordinates x, y and x, z and y, z respectively. Recall the three CP (3 points) potentials are denoted g xy , g xz , g yz by Equations (4.15) and (4.19). In between we interpolate between the two po-tentials on either side. In the Roman-numbered regions, all three potentials are at play andall of r x , r y , r z are very small. They are small because in a fiber we fix v and we’ve restrictedto small v in the definition of ∆ Y in Equation (4.10). So if | v | = T l , then T r ∗ are each ofthe form T l ∗ where l x + l y + l z = l . Furthermore in the Roman-numbered sections aroundthe vertex, because all three toric coordinates go to zero at the vertex, near the vertex theirnorms are still small by continuity. Geometrically, the toric coordinates are small in thecorresponding region of a torus fiber close to the zero fiber. MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 45
We introduce local real radial and angular coordinates d and θ to define delineations on themoment polytope around the vertex. Their subscripts indicate which region of Figure 11they are defined in. This will allow us to interpolate between the three K¨ahler potentials g xy , g xz , g yz as functions of these local coordinates. We fix | v | (= | xyz | ) = T l for T << l a large positive constant, so that in the red regions denoted by Roman numerals of Figure11 we have r x , r y , r z << F = α g xy + α g xz + (1 − α − α ) g yz ; 1 / ≤ α , α ; α + α ≤ α = α = 1 / F = ( g xy + g xz + g yz ) ≈ (( T r x ) + ( T r y ) +( T r z ) ) via the log approximation and in the other regions α , α interpolate between thethree K¨ahler potentials.We want d and θ to locally around a vertex be approximated by Figure 12 where we canread off how r x , r y and r z compare to each other. It is hard to define them as functions ofthe moment map coordinates, but we use that the moment map coordinates are monotonicincreasing functions in the norm coordinates. This is proven in Claim 4.19. In other words, r x increases in the (1 , ,
0) direction, r y increases in the (0 , ,
0) direction on the polytopeand r z increases in the ( − , − ,
1) direction. These motivate the radial variable definition.Hence, as shown in Figure 12 we see that e.g. T ( r y − r z ) increases in the upward verticaldirection. This motivates the angular variable definition. r z (cid:28) r x , r y r x (cid:28) r y , r z r y (cid:28) r x , r z r x r y r z r y − r z increases r z − r x increases r x − r y increases Figure 12.
How the three angular directions vary for r x r y r z constant on a fiber Definition of new radial and angular coordinates d and θ . Recall Claim 4.19 that themoment map coordinates are monotonic increasing in the norms of the complex coordinates.So our choices for d and θ approximate to the expressions in Figure 12. We define functions φ x , φ y , φ z for expressions we will use often in the coming definitions. φ x ( x, y, z ) := log T | T x | | T yz | φ y ( x, y, z ) := log T | T y | | T xz | (4.27) φ z ( x, y, z ) := log T | T z | | T xy | Here are the radial and angular coordinates in regions I and II, as well as their approximationsthere. d I := φ x −
12 ( φ y + φ z )= log T (cid:32) | T x | | T v x − | / (cid:115) | T y | | T v y − | · | T z | | T v z − | (cid:33) d I ≈ ( T r x ) − (cid:0) ( T r y ) + ( T r z ) (cid:1) for r x , r y , r z << θ I := φ y − φ z = log T (cid:18) | T y | | T z | · | T xy | | T xz | (cid:19) ∴ θ I ≈ ( T r y ) − ( T r z ) d IIA := φ x −
12 ( φ y + φ z ) + 32 α ( θ II ) · φ y ≈ T [ r x −
12 ( r y + r z ) + 32 α ( θ II ) · r y ](4.29) d IIB := φ x + φ y − φ z d IIC := φ y −
12 ( φ x + φ z ) + 32 α ( − θ II ) · φ x θ II := log T r y − log T r x where α will be a cut-off function of the angular direction. By symmetry, we define d III := φ y −
12 ( φ x + φ z )(4.30) θ III := φ z − φ x d V := φ z −
12 ( φ x + φ y ) θ V := φ x − φ y In Figure 13, we define regions I and IIA of the polytope in terms of ( d I , θ I ) and ( d II , θ II )coordinates (indicated as ( , ) I or ( , ) II respectively). The picture looks like this using ap-proximations for d and θ . This defines the rest of the regions by symmetry. This will us toestimate how much r x , r y and r z vary in each of the regions and show there is enough spaceto bound the second derivatives of the bump functions used. Namely, in order to go fromthe values of the bump functions at either end of a region, the function has space to growsufficiently gradually that the slope and rate of change of slope can be made small. Thisis proven in Appendix A. Now we define the symplectic form in terms of r x , r y , r z in theseregions. MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 47
Figure 13.
Delineating regions in coordinates ( d I , θ I ) and ( d II , θ II ) Definition 4.24 ( Definition of symplectic form on Y ). We set ω | V ∨ = i π F where F isdefined locally as follows in terms of the coordinates in Equation 4.29 and(4.31) g yz = log(1 + | T y | )(1 + | T z | )(1 + | T yz | ) g xz = log(1 + | T x | )(1 + | T z | )(1 + | T xz | ) g xy = log(1 + | T x | )(1 + | T y | )(1 + | T xy | )We introduce new bump functions α , α , α , α of the new variables ( d, θ ) as follows:23 ≤ α ( d I ) = α + α ≤ , − ≤ α ( θ I ) ≤ , ≤ α ( d I ) ≤ , ≤ α ( θ II ) ≤ α ( θ I ) · α ( d I ) = 12 ( α − α )These bump functions are smooth, increasing as functions of the specified variable, and nearthe ends of their domain of definition they are constant at the bounds given. We also requirethat α is an odd function. See the subsection below entitled Motivation for new bumpfunctions defined in ω for an explanation of the properties of these bump functions. Nowthe definition is as follows, noting that g xz − g yz = φ x − φ y and similarly permuting ( x, y, z ): Regions g ∗• in Fig 11: F = g xy , F = g yz , F = g xz respectivelyRegion I: F = g yz + α ( d I ) d I + α ( θ I ) α ( d I ) θ I Region IIA: F = g yz − α ( θ II ) φ y + α ( d IIA ) d IIA + 12 α ( d IIA )( φ y − φ z − α ( θ II ) φ y )Region IIB: F = ( g yz − φ y ) + α ( d IIB ) d IIB − α ( d IIB ) φ z (4.32) Region IIC: F = g xz − α ( − θ II ) φ x + α ( d IIC ) d IIC + 12 α ( d IIC )( φ x − φ z − α ( − θ II ) φ x )Region VII: F = 13 ( g xy + g xz + g yz ) These formulas match at the boundaries, which allows us to define the rest of the regionsIII – VI similarly to I and II by symmetry, via permuting the coordinates ( x, y, z ). Forexample, one can check that the formula for region IIA agrees with that for region VII when α = and α = 0; with that for region I when α = 0 and α = ; with that for region IIBwhen α = 1; and with g xy when α = α = 1. The calculation is similar for the other regions.Along the coordinate axes (namely the regions shaded red, blue, and black) we interpolatebetween the relevant g ∗• ’s using the same formulas as in regions I, III, and V, with α ≡ r x -axis (blue region) the formula is F = ( g xy + g xz ) + α ( θ I )( g xy − g xz ) andsimilarly for the other edges.Finally by adding a term proportional to | xyz | on the base, with sufficiently large constantof proportionality, we obtain a non-degenerate form ω on Y . This completes the definitionof the symplectic form. Motivation for new bump functions in ω . Note that α , α , − α − α is asymmetricbut all three should be treated symmetrically. In other words, if we rotate ( α , α ) thoughtof as a vector, by π/
4, we get something proportional to (cid:18) −
11 1 (cid:19) (cid:18) α α (cid:19) = (cid:18) α − α α + α (cid:19) We accordingly rearrange terms of the initial expression of F in terms of α − α and α + α . F = α g xy + α g xz + (1 − α − α ) g yz = α log(1 + | T x | )(1 + | T y | )(1 + | T xy | )+ α log(1 + | T z | )(1 + | T x | )(1 + | T xz | )+ (1 − α − α ) log(1 + | T z | )(1 + | T y | )(1 + | T yz | )(4.33) = g yz + ( α + α ) φ x − (cid:18) α + α − α − α (cid:19) · φ y − (cid:18) α + α α − α (cid:19) · φ z = g yz + ( α + α )( φ x −
12 ( φ y + φ z )) + 12 ( α − α )( φ y − φ z )We want bump functions to be multiplied by the variable they are a function of because dαd (log T µ ) = α (cid:48) ( µ ) · µ ≈ ∆ α ∆ log T µ , so we can find estimates on terms containing α (cid:48) ( µ ) · µ when µ is the argument of α . In particular, when µ is d ≈ ( T r ∗ ) or θ ≈ ( T r ∗ ) we can estimate theLog T -derivative of α ( µ ) as O (1) l . Recall from Equation (4.29) that d I := φ x − ( φ y + φ z ) ≈ ( T r y ) − ( T r z ) is a radial direction in region I and θ I := φ y − φ z ≈ ( T r y ) − ( T r z ) anangular direction in region I. So define(4.34) α = α + α , / ≤ α ≤ d I going from 2/3 to 1 in region I, by Equation (4.26).We use ( α − α ) to define a bump function α that varies in the angular direction θ I .The range of α − α depends on d I . At the start of region I we have α = α = 1 / α − α goes from 0 to 0 as we trace out the angle θ I . However at the end of region I we go MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 49 from ( α , α ) = (0 ,
1) at the bottom to ( α , α ) = (1 ,
0) at the top, so that ( α − α ) goesbetween − / /
2. Thus we multiply α ( θ I ) by another bump function α ( d I ) of theradial direction that goes from 0 to 1, so we scale the interval that α varies in. We define α to be an odd function for symmetry reasons. Let(4.35) α ( θ I ) α ( d I ) = 12 ( α − α ) , − / ≤ α ≤ / , ≤ α ≤ F as:(4.36) F = g yz + α ( d I ) · d I + α ( θ I ) · α ( d I ) · θ I We can similarly define F in regions III and IV, and then region II will interpolate betweenregions I and III. This is where the defining equations for ω | V ∨ in Definition 4.24 came from.4.6. Leading order terms in ω . We convert ∂∂F to polar coordinates where calculationsare easier, using the real transformation ( r x , θ x ) ↔ ( x , x ), where x = x + ix = r x e iθ x andsimilarly for y . Recall (e.g. [Huy05])(4.37) ∂ x = 12 ( ∂ x − i∂ x ) , ∂ x = 12 ( ∂ x + i∂ x )By the Chain Rule, since r x = x + x , θ x = tan − ( x /x ), and arctan( t ) (cid:48) = 1 / (1 + t ):(4.38) ∂ x i = ∂ x i ( r x ) ∂ r x + ∂ x i ( θ x ) ∂ θ x = x i r x ∂ r x + ∂ x i ( x /x )1 + ( x /x ) ∂ θ x = x i r x ∂ r x ∓ x i +1 x + x ∂ θ x Hence plugging in for ∂ x i and ∂ x i :(4.39) ∂∂x = 12 (cid:18) ∂∂x − i ∂∂x (cid:19) = 12 (cid:18) x − ix r x ∂∂r x + − x − ix r x ∂∂θ x (cid:19) = 12 (cid:18) e − iθ x ∂ r x − ie − iθ x r x (cid:19) ∂∂x = 12 (cid:18) e iθ x ∂ r x + ie iθ x r x (cid:19) We also need to rewrite the differentials dx = dx + idx and dx in terms of polar coordinates.Since d is complex linear, and using Euler’s formula e iθ = cos θ + i sin θ :(4.40) dx = d ( r x cos θ x ) + id ( r x sin θ x ) = e iθ x dr x + ir x e iθ x dθ x dx = e − iθ x dr x − ir x e − iθ x dθ x Similarly for y . Now we can convert ∂∂F into polar coordinates. i∂∂F = (cid:88) i,j =1 ∂ F∂z i ∂z j dz i ∧ dz j = i (cid:88) i,j ( 12 e − iθ zi (cid:104) ∂ r zi − i/r z i ∂ θ zi (cid:105) )( 12 e iθ zj (cid:104) ∂ r zj + i/r z j ∂ θ zj (cid:105) )( F )( e iθ i ( dr i + ir i dθ i )) ∧ ( e − iθ j ( dr j − ir j dθ j ))= i (cid:88) i,j e − iθ i (cid:20) e iθ j ∂ r i r j F − ir i ∂ r j F δ ij ie iθ j (cid:21) ( e iθ i ( dr i + ir i dθ i )) ∧ ( e − iθ j ( dr j − ir j dθ j )) = i (cid:34)(cid:88) i ( ∂ r i F + 1 r i ∂ r i F )( dr i + ir i dθ i ) ∧ ( dr i − ir i dθ i ) (cid:35) + i (cid:88) i (cid:54) = j ( ∂ r i r j F )( dr i + ir i dθ i ) ∧ ( dr j − ir j dθ j ) A K¨ahler form is compatible with its complex structure by definition (e.g. [Huy05, Definition1.2.13]), so J := i given by multiplication by i in the toric coordinates, is ω -compatible.That is, ω ( · , J · ) is a metric, which we want to express in polar coordinates to facilitatecalculations below. This is a 6 by 6 block diagonal matrix with the r -derivatives blockand the θ -derivatives block. Recall the complex structure acts on real tangent vectors by ∂ x (cid:55)→ ∂ x and ∂ x (cid:55)→ − ∂ x . Again by the Chain rule:(4.41) ∂ r = ∂x ∂r ∂ x + ∂x ∂r ∂ x = cos θ ∂ x + sin θ ∂ x ∂ θ = − r sin θ ∂ x + r cos θ ∂ x J ( ∂ x ) = ∂ x , J ( ∂ x ) = − ∂ x = ⇒ J ( ∂ r ) = 1 r ∂ θ , J ( ∂ θ ) = − r ∂ r Hence the entries along the diagonal in the r block will be g ii = ω ( ∂ r i , J ∂ r i ) = ω ( ∂ r i , r i ∂ θ i ) = 1 r i ω ( ∂ r i , ∂ θ i ) = 12 ( ∂ r i F + 1 r i ∂ r i F )because we pick up the dr i ∧ dθ i term, i.e. r i dr i ∧ dθ i times the F derivative term. Similarly g ij = ω ( ∂ r i , J ∂ r j ) = 1 r j ω ( ∂ r i , ∂ θ j ) = 12 ( ∂ r i r j F ) Dominant terms for metric Region I.
The metric in polar coordinates is: ∂ r x F + r x ∂ r x F ∂ r x r y F ∂ r x r z F∂ r x r y F ∂ r y F + r y ∂ r y F ∂ r y r z F∂ r x r z F ∂ r y r z F ∂ r z F + r z ∂ r z F F = f + α d + α · α θd ≈ T [ r x − (cid:0) r y + r z (cid:1) ] θ ≈ T [ r y − r z ] f ≈ T [ r y + r z ] Note that because everywhere r x appears is as r x , applying ∂ r x r x is the same as applying ∂ r x /r x . Here are the terms that do not involve derivatives of the α i , where for ease of notationsubscript x means ∂ r x : f xx + α d xx + α α θ xx ) f xy + α d xy + α α θ xy f xz + α d xz + α α θ xz “ 2( f yy + α d yy + α α θ yy ) f yz + α d yz + α α θ yz “ “ 2( f zz + α d zz + α α θ zz ) ≈ T α − α + 4 α α
00 0 4 − α − α α The approximation follows from the estimates on bump function derivatives in Appendix B.One of the coordinates does go to zero as the bump functions reach their bounds. However,
MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 51 the one that goes to zero in the xy -plane is the z term and similarly in the xz -plane it’s the y term. Since we add a term | xyz | for the base, this will ensure positive definiteness awayfrom the zero fiber. Dominant terms in Regions III and V.
In region I, r x was the dominating coordinate.In region III, r y will dominate and in region V, r z will dominate. So we take the analogousdata for half regions of III and V, by modeling I. Dominant terms in Region II.
The expressions for the K¨ahler potentials in regions IIIand V thus differ from that in region I by a permutation of the coordinates x, y, z and allthe estimates above carry through under this permutation. Thus we patch together the d coordinate across regions II, IV and VI. This uses another bump function α going from 0to 1 as we increase a suitable θ II -coordinate. In particular, functions of d become functionsof d ◦ α . We have F ≈ T [( r y + r z − α ( θ II ) · r y ) + α ( d IIA ) · ( r x −
12 ( r y + r z ) + 32 α ( θ II ) · r y )+ 12 α ( d IIA ) · ( r y − r z − α ( θ II ) · r y )]since g yz ≈ T [ r y + r z ] d IIA ≈ T [ r x −
12 ( r y + r z )] + 32 α (log r y − log r x ) · ( T r y ) θ II = log( r y /r x )The terms not involving derivatives of the bump functions will form the nondegenerate partof the metric on region II. Off-diagonal terms ∂ r • r (cid:63) for (cid:63) (cid:54) = • are zero because derivativesof non-bump functions means differentiating r ∗ for some ∗ . On the diagonal terms we get r ∗ ∂ r ∗ + ∂ r ∗ r ∗ = 2 ∂ r ∗ r ∗ which, applied to ( T r ∗ ) is 4 T . So in the ( ∗ , ∗ ) entry of the matrix,the leading terms are 4 T times the coefficients on r ∗ . x : T (4 α ) ≥ T / y : T (4 − α − α + 6 α α + 2 α − α α ) = T [4 + (2 − α )( α − α ) + 4 α ( α − ≥ T [4 + 2(0 − /
3) + 4( − / T · z : T (4 − α − α )Note that when α = α = 1 the z term goes to zero. However, it is bounded below in a regionwhere α , α are bounded away from 1. In the region where it goes to 1, we add a term to F from the base, i.e. | xyz | , to maintain nondegeneracy. Again because xyz is bounded belowin the region where we add it, we can take its partial derivatives and the result will be posi-tive definite. Region IIC, IVA, IVC and VIA, VIC are the same after permuting x ↔ y ↔ z .For example, swap r x ↔ r y to get to IIC, and the subscripts I are replaced with subscripts III.The characteristics in region IIB which we did not have in regions IIA and C are 1) r x and r y go from r x >> r y to r y >> r x , passing through r x = r y and 2) α ≡
1. All of r x , r y , r z are still small so we still have an approximation for the K¨ahler potential. The calculation Figure 14.
Delineated fundamental domainfor the negligible terms is given in Appendix B.
Dominant terms in the remainder of C patch. See Figure 14 for the regions left toconsider. In the region between region I and region IIA the only bump functions at play are α and α and they are allowed to vary in the same amount of space described in AppendixA, where nondegeneracy was checked in Appendix B. Here still r x >> r y , r z so those esti-mates still apply.What remains to be checked is that the symplectic form glues positive-definitely along the z axis, then the other axes will follow by symmetry, and in the remaining regions ω agrees withthe standard K¨ahler form of the blowup CP (3). In the black region along the z -axis we nolonger have dependence on the d V coordinate because α , α ≡
1. We still have an angularcoordinate that allows us to interpolate between g xz and g yz in the unprimed coordinates,or g x (cid:48)(cid:48)(cid:48) z (cid:48)(cid:48)(cid:48) and g y (cid:48)(cid:48)(cid:48) z (cid:48)(cid:48)(cid:48) in the tripled primed coordinates at the lower left vertex. So we need tocheck we have positive definiteness when only α is at play and r z is large. The proof thatthe bump function derivative terms can be made arbitrarily small is again in Appendix B.With the Appendices, this proves the following: Lemma 4.25.
The ω defined in Definition 4.24 on Y is non-degenerate and puts the struc-ture of a symplectic fibration on v : Y → C . Donaldson-Fukaya-Seidel type category of linear Lagrangians in ( Y, v )5.1. Context and definition.
Donaldson introduced the pair of pants product and inthis paper we only define and compute the differential and ring structure for the morphism
MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 53 groups. [FOOO09] explain obstructions to defining an A ∞ -category on a symplectic manifoldwhich record how Lagrangians intersect upon perturbations, and is the reason we excludethe infinite-sloped linear Lagrangian parallel transported around a circle in the base in oursubcategory. (Such a Lagrangian bounds i -holomorphic discs.) Seidel adapted the definitionof a Fukaya category to the case of symplectic Lefschetz fibrations, which we adapt to thesymplectic fibration ( Y, v ). Note we’ve constructed ( Y, ω ) so that v : Y → C is a symplecticfibration because we started by constructing ω in a fiber in Chapter 4.5.The notation “Donaldson-Fukaya-Seidel (DFS)- type category of linear Lagrangians in ( Y, v )”means that we only consider a subset of possible objects, and their collection forms a cat-egory that will be a subcategory of the full Fukaya-Seidel category of ( Y, v ) once such acategory is defined. In particular, it is expected this subcategory would split-generate thefull category. The definition of linear Lagrangians in a fiber V ∨ of Y was inspired by therational slope Lagrangians considered in [PZ98] on categorical mirror symmetry for the el-liptic curve. Because of the linearity of Lagrangians considered, they have a lift that allowsfor Maslov grading given by their slope, and they have a Spin structure as well. We obtainLagrangians in the total space by parallel transporting these Lagrangians in the fiber overan arc in the base with respect to the horizontal distribution induced from the symplecticform. Definition 5.1.
The symplectic horizontal distribution of a symplectic fibration π : Y → C to a base manifold C is H ⊂ T Y such that if F is a generic fiber of π then H = T F ω is the ω -complement, i.e. ω ( H, T F ) = 0Given two points p , p ∈ C and a path γ : I → C between them (i.e. γ (0) = p and γ (1) = p ), the parallel transport map is a symplectomorphismΦ : F p → F p defined as follows: given x ∈ F p , we set Φ( x ) to be ˜ γ (1) where ˜ γ : I → Y such that π ◦ ˜ γ = γ , dπ (˜ γ (cid:48) ) = γ (cid:48) and ˜ γ (cid:48) is in the horizontal distribution. Claim 5.2.
By standard theory, this last condition implies Φ is a symplectomorphism.
Proof.
There is a unique horizontal vector field X H on π − ( γ ( I )) with flow φ H by existenceand uniqueness of differential equation solutions and that horizontal implies there is nocomponent of the vector field in the fiber direction. Then since d Φ is the identity on vectorsin H we have ω ( d Φ( X H ) , d Φ( v )) = ω ( X H , v ) for v ∈ H . Also, H is ω -perpendicular to T F ,which is a condition also preserved under parallel transport: when v ∈ T F is transportedinfinitesimally in the parallel direction, it must still be in
T F , otherwise any component in H could be reverse parallel-transported to a vector component in H at the original fiber,contradiction. So ω ( X H , v ) = 0 = ω ( d Φ( X h ) , d Φ( v )) and Φ ∗ ω = ω since T Y = T F ⊕ H inregions where we parallel transport. (cid:3) Corollary 5.3. Φ( (cid:96) i ) is Lagrangian in V ∨ with respect to ω | V ∨ . Claim 5.4.
Φ fixes ξ , ξ . Proof.
Let ρ be the quasi-Hamiltonian T -action rotating coordinates ( x, y ) ∈ V ∨ by angles( α , α ). Let X H be the horizontal vector field with flow φ tH . Then ddt ( ξ i ◦ φ tH ) | t =0 = dξ i ( X H )= ι dρ ( ∂ αi ) ω ( X H )= ω ( ∂ θ i , X H ) = 0 ∵ X H ⊥ ω ∂ θ i ∈ T V ∨ ∴ ξ ◦ φ tH = ξ (cid:3) Corollary 5.5.
Parallel transport Φ is of the form ( ξ , ξ , θ , θ ) (cid:55)→ ( ξ , ξ , θ + f ( ξ ) , θ + f ( ξ )) for some functions f, g depending on ξ but not the angular coordinates.Proof. Let π = v . We know by the previous result that parallel transport does not affect ξ , ξ . We are stating further that f and g are independent of the angles, so we can expressmonodromy as a graph of a function T B → T F . So again let ρ be the T -action on afiber, namely addition on the θ coordinates. Since ω is a function of the norms only, ρ is asymplectomorphism and preserves that T V ∨ ⊥ ω H and acts fiber-wise. Hence0 = ρ ∗ ω ( T V ∨ , H ) = ω ( ρ ∗ T V ∨ , ρ ∗ H ) = ω ( T V ∨ , ρ ∗ H ) ∴ ρ ∗ H = H Furthermore since π = v and ρ preserves fibers, we have π ◦ ρ = π and dπ ( ρ ∗ X hor ) = d ( π ◦ ρ )( X hor ) = dπ ( X hor ) ∴ ρ ∗ ( X hor ) ◦ ρ − = X hor because ρ ∗ X hor ∈ H and has the same horizontal component as X hor ∈ H . Integrating bothsides of the last equality we obtain dρ (cid:18) ddt φ tH (cid:19) ◦ ρ − = ddt (cid:0) ρ ◦ φ tH (cid:1) ◦ ρ − = ddt ( φ tH ) ∴ ρ ◦ φ tH = φ tH ◦ ρ In other words, we get the same answer whether we rotate the coordinates θ (cid:55)→ θ + α andthen transport by adding f ( θ + α ), or transport and then rotate. So parallel transportdoesn’t depend on the angular coordinates. Namely θ i + α i + tf i ( ξ, θ + α ) = θ i + α i + tf i ( ξ, θ ) ∴ f i ( ξ, θ + α ) = f i ( ξ, θ ) ∀ α Thus f , f are independent of θ , θ . (cid:3) Now we can define the DFS-type category we consider in this paper. Objects are U-shapedLagrangians as in the appendix of [AS10] and morphisms are defined via categorical local-ization as in [AS], [Gan16, Chapter 4].
Definition 5.6 (Definition of category on A-side) . Objects.
Recall the definition of (cid:96) k = { ( ξ , ξ , − k (cid:18) (cid:19) − (cid:18) ξ ξ (cid:19) } ( ξ ,ξ ) ∈ T B in Lemma 2.5 for Lagrangians in V ∨ . Also define t x := { (log τ | x | , θ ) } θ ∈ [0 , ⊆ V ∨ . In Y , define these Lagrangians to exist in the v -fiber over −
1. Let (cid:83) γ denote paralleltransport with respect to ( T V ∨ ) ω over a curve γ ⊂ C in the base of v . Let γ : R → C bea smooth curve so that γ (0) = − γ ( R ) traces out a U-shape in the base given outside MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 55
Figure 15. L i = parallel transport (cid:96) i around U-shaped curve in base of v a compact set by rays in the right half plane, i.e. lines re iθ for a fixed θ ∈ [ − π, π ), as inFigure 15. Then define(5.1) L k := ∪ γ (cid:96) k T x := ∪ γ t x to be fibered Lagrangians which generate all objects in the DFS-type category. In particular,we don’t allow the rays to wind around the origin in the compact set. I.e. in the languageof stops we have a stop at −∞ along the real line in C . Floer complexes.
This is based on [Gan16, § K and L be objects. Then first set(5.2) CF directed ( K, L ) = CF ( K, L ) , if K > L C · e L , if K = L , elsewhere the ordering K > L denotes that, outside a compact set, all the rays of K lie aboveall the rays of L and e L is defined as follows. If K = L , then push the rays of K to lieabove the rays of L as in Figure 15. Denoting this deformation of K as L (cid:15) , this definesthe quasi-unit e L ∈ CF ( L (cid:15) , L ) as the count of discs with those Lagrangians as boundaryconditions. We then localize at these e L , in other words we set them to be isomorphisms.Recall that formally inverting morphisms involves taking equivalence classes of roofs K ← K (cid:15) → L This is the definition of an arrow K → L when K < L , namely push up to K (cid:15) > L so thatthe first arrow of the roof is the quasi-unit e K , and then K (cid:15) → L is defined as above. Morphisms.
The morphisms in this category are obtained by quotienting by the differentialon the hom spaces just defined. The differential ( µ ), composition ( µ ), and higher order µ k count bigons, triangles, and ( k + 1)-gons respectively, with the usual Lagrangian boundaryconditions from these L i , and are J -holomorphic for some regular J . This is theory fromChapter 12 of [Sei08], contents of which are listed here.However, for self-Floer cohomology groups, Hamiltonian perturbation is only used in thefiber, as in the Non-transversely intersecting Lagrangians section of the proof of
Lemma 2.5. Not in the total space, because we would need to find a Hamiltonian vectorfield tangent to the fibers of v , so the projection of a Hamiltonian-perturbed holomorphiccurve (i.e. one that satisfies the perturbed Cauchy-Riemann equation (2.5)) under v is stilla holomorphic polygon. Since the symplectic form is not a product of one on the base andfiber, we instead use the categorical localization method of [AS], see also [Gan16]. For eachLagrangian in the fiber, there are many different objects of F S obtained by parallel transportalong U-shapes that go to infinity in slightly different directions, but these will be isomorphicto each other after localizing.
Moduli spaces.
This concludes the definition of the DFS-type category used on the A-side(
Y, v ), modulo the definition of the moduli spaces, which is the remainder of this chapterafter we compute monodromy. Remark 5.7.
If we include t x parallel transported in a circle around the base of v : Y → C as a Lagrangian in the subcategory we are considering, then this Lagrangian boundsnonconstant holomorphic discs. A future direction is to incorporate and define M forthe category containing this Lagrangian. Note that M is a degree 2 operation, and theLagrangian is only Z / J -spheres, but they will show up onlyin the c in M ( c, − ) considered as a map on Floer groups in Section 6.2. Lemma 5.8.
For Lagrangians in correct position, morphisms and compositions in the local-ized Fukaya-Seidel category coincide with those in the directed category. Namely, if
K > L ,then hom
F S ( K, L ) (cid:39) CF ∗ ( K, L ) , and if L > ... > L k , then µ k in F S ( Y, v ) is the same asin the directed category. Namely it is given by counting J -holomorphic discs.Proof references. This is unpublished work of Abouzaid-Seidel, e.g. see [Gan16, Proposition119, 120], and Abouzaid-Auroux. (cid:3)
Lemma 5.9.
The composition of roofs is a roof.Proof references.
This is a consequence of the naturality of quasi-units with respect to con-tinuation maps, see Abouzaid-Seidel and Abouzaid-Auroux. (cid:3)
Definition 5.10.
The
Donaldson-Fukaya category is obtained by passing to cohomology onthe morphism chain complexes.
Remark 5.11 (Not exact) . One difference of our fibration v : Y → C to those in [Sei08] isthat we are in the non-exact case, because fibers are compact tori. In particular, we will havesphere bubbles which would have been excluded. Furthermore, we do not have a horizontalboundary since fibers are tori, but we do have a vertical boundary by taking the preimage ofthe boundary of the base. (Recall that the polytope locally describing Y has the restriction η ≤ T (cid:96) for T (cid:28)
1, so equivalently | v | (cid:28) Remark 5.12 (Not Lefschetz) . In [Sei08] the Fukaya category of a Lefschetz fibration hasLagrangians given by thimbles obtained by parallel transporting a sphere to the singularpoint in the Morse singular fiber where it gets pinched to a point. In our situation, thedegenerate fiber over 0 is not a Lefschetz singularity (i.e. modeled on (cid:80) i z i ) but insteadthe fiber T degenerates by collapsing a family of S ’s, and also collapsing a T in twopoints. This produces singular Lagrangians. So instead, we go around this singular fiber ina U-shape as in [AS19]. MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 57
Remark 5.13 (Not monotone) . Lastly, a compact symplectic manifold (
P, ω ) is monotone if (cid:82) c ( P ) = α (cid:82) ω : π ( P ) → R for some α >
0. Our setting is not monotone: taking P := Y , which is Calabi-Yau, we have c ( Y ) = 0 however [ ω ] (cid:54) = 0 so α would have to bezero, contradiction. Furthermore, the Lagrangians are not monotone, which we now define. Definition 5.14 (Maslov index of a map) . Given u : ( D , ∂ D ) → ( M, L ) we first trivialize u ∗ T M over the contractible disc. Then the
Maslov index counts the rotation of u ∗ T L aroundthe boundary in this trivialized pullback.
Definition 5.15. A monotone Lagrangian L in symplectic manifold ( M, ω ) is such that[ ω ] = k · [ µ ( u )] for all u ∈ π ( M, L ), for some constant k . Namely, the areas of discs areproportional to the Maslov indices of those discs. Example 5.16.
An example can be found in Oh [Oh93], who shows the Clifford torus in( CP n , ω F S ) is a monotone Lagrangian submanifold. In this thesis, all discs considered haveMaslov index 2, but the areas vary as prescribed by a formula of [CO06] which we willelaborate on later.Now that we’ve indicated how this set-up differs from those currently in the literature, we givean outline of the subsections of this chapter. In Section 5.2 we discuss monodromy aroundthe central fiber in v : Y → C because it is used to find intersections of the Lagrangians L k we defined in the total space. Then Section 5.3 sets up the background needed to define thestructure maps and proves the moduli spaces involved have the required conditions to be putinto the definition. Finally we show the definition of the DFS-type category is independentof choices, in Section 5.4.5.2. Monodromy.Definition 5.17.
Monodromy is the parallel transport map Φ of Definition 5.1 around aloop which goes once around a singularity in the base.
Claim 5.18. If ψ ( t )( v ) = e πit v then the symplectic horizontal lift of dψ ( d/dt ) is of theform X hor = ∂/∂θ η + f ( ξ, η ) ∂/∂θ + f ( ξ, η ) ∂/∂θ where f i are now functions of ξ , ξ , and η . Proof of Claim 5.18.
We saw above in Corollary 5.5 that X hor does not involve ∂/∂ξ i . Wenow show it also preserves η because we are considering parallel transport around a circlewith fixed | v | . Let w be the coordinate on the base C of π = v . In particular, X hor isdefined by the property(5.3) dπ ( X hor ) = ∂/∂θ w With respect to the action-angle coordinates π : ( ξ , ξ , η, θ , θ , θ η ) → ( | w | , θ w ) := ( | v | , θ η )and noting that | v | is a function of ( ξ , ξ , η ) only, Equation (5.3) can be expressed as: (cid:18) ∂ | v | /∂ξ ∂ | v | /∂ξ ∂ | v | /∂η (cid:19) af f b = (cid:18) a∂ | v | /∂ηb (cid:19) = (cid:18) (cid:19) (0,1)(1,0) (0,0)(1,1) ξ ξ Figure 16.
Monodromy in fiber, thought of as a section over the parallelo-gram ( ξ , ξ ) (cid:55)→ ( f ( ξ , ξ ) , f ( ξ , ξ )).thus b = 1 and a must be zero as η depends on | v | . So the horizontal lift is of the form: X hor = ∂/∂θ η + f ∂/∂θ + f ∂/∂θ Also we saw above in Claim 5.5 that for a fixed fiber, the f i do not depend on θ and θ . Asimilar argument shows they are independent of θ η , i.e. ρ λ ◦ φ tH = φ tH ◦ ρ λ for λ ∈ S definingthe rotation action on the v coordinate. This can be seen by replacing ρ with ρ λ in theproof of Claim 5.5. (cid:3) Example 5.19 (One dimension down, 2D local case) . The case of C with symplectic fi-bration ( x, y ) (cid:55)→ xy is the setting of a Lefschetz fibration, with singular fiber given by twocopies of C from x = 0 or y = 0. The monodromy is a Dehn twist about the S given by thebelt of the cylindrical fibers. f ( x, y ) = xy and S -action is ( e iθ x, e − iθ y ). The holomorphic vector field corresponding tothis is iz ∂ z − iz ∂ z , whose contraction with ω = i ( dz ∧ dz + dz ∧ dz ) gives Hamiltonianvector field − ( | z | − | z | ). We have a new set of coordinates on the two dimensional fiber:the moment map coordinate µ and the angle coordinate of the action θ . As we approach xy = 0, the orbit at µ -height 0, namely | x | = | y | , goes to zero. That’s one way to see howwe get the picture of a cylinder with the belt pinching to zero.Recall from Claim 5.18 that we can describe parallel transport by the graph of a function T B → T F by adding ( f ( ξ, η ) , f ( ξ, η ) ,
1) to the angular coordinates ( θ , θ , θ η ). The mainresult of this section is the following Lemma. Lemma 5.20 (Monodromy, see Figure 16) . Over the parallelogram-shaped fundamental do-main of the torus T B , the monodromy for a loop around the base is given by ( f ( ξ ) , f ( ξ )) asfollows. They equal (0 , for ξ in the upper right corner where r − z >> r − x , r − y , then (0 , on the right where r − y is the largest, (1 , on the left where r − x largest and thus (1 , in thebottom left corner. MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 59
Proof of Lemma 5.20.
First we show the result holds for ξ in the C patch. Then we showit holds in the middle region of the hexagon where the K¨ahler potential is that of CP (3).Then we conclude the result by showing the contributions in between are negligible. Parallel transport in C -patch with standard metric. We want to find a horizontal liftof the angular vector field ∂/∂θ w , or with respect to complex coordinates ∂/∂w this wouldbe the vector field on the complex plane whose value at w is iw . Namely dπ ( X hor ) = iw . Wecan explicitly compute real vectors in the horizontal distribution H in the C -patch. Thenwe can find a vector parallel to iw∂/∂w and scale it suitably so it projects to iw∂/∂w andis not just parallel to it. Claim 5.21 (Finding the horizontal subspace) . X (cid:60) ( v ) , X (cid:61) ( v ) generate the horizontal dis-tribution H . Proof.
The kernel of d p v when v = v is also the tangent space of a v -fiber. This allowsus to conclude the following. Let p ∈ v − ( c ) for some c ∈ C and | c | (cid:28) T l by Definition 4.10.Then ker( d p v ) = T p ( v − ( c )). On the other hand ker( dv ) = ker( d (cid:60) ( v )) ∩ ker( d (cid:61) ( v )). Thisnow allows us to find generators for H . H = ker( dv ) ω = [ker( d (cid:60) ( v )) ∩ ker( d (cid:61) ( v ))] ω = ker( d (cid:60) ( v )) ω + ker( d (cid:61) ( v )) ω ⊇ R · X (cid:60) ( v ) + R · X (cid:61) ( v ) where the last line follows from the general notation of X f as ω -dual to df and that ω ( X f , X f ) = 0 for alternating form ω . Thus since H is rank two, we have equality inthe last line. (cid:3) Claim 5.22.
The horizontal distribution H is a complex subspace, i.e. invariant undermultiplication by i . Proof.
Note v = (cid:60) ( v ) + i (cid:61) ( v ). If we consider v as a coordinate (rather than a function)then since J = i on vector fields sends ∂ (cid:60) ( v ) (cid:55)→ ∂ (cid:61) ( v ) and ∂ (cid:61) ( v ) (cid:55)→ − ∂ (cid:60) ( v ) , it does thetranspose on the dual differential forms: J ◦ d (cid:60) ( v ) = − d (cid:61) ( v ). Thus using the duality fromthe metric and that the gradient is related to the Hamiltonian vector field (thinking of v now as a function) by − J :(5.4) J ∇ ( (cid:60) ( v )) = − ∇ ( (cid:61) ( v )) ∴ − J X (cid:60) ( v ) = ∇ ( (cid:60) ( v )) = J ∇ ( (cid:61) ( v )) = − X (cid:61) ( v ) ∴ J X (cid:60) ( v ) = X (cid:61) ( v ) This concludes the proof by Claim 5.21. (cid:3)
Corollary 5.23. X hor = ig ∇ g | v | up to some scalar function g to be determined.Proof. First, ∇| v | = ∇ ( (cid:60) ( v ) + (cid:61) ( v ) ) ∈ H by Claim 5.22. In particular, from calculuswe know that the gradient of a real function in two variables is perpendicular to its levelsets in the plane. Here, that means that dπ ( ∇| v | ) ⊥ ψ ( t )( | v | ) (in the notation of Claim5.18) for 0 ≤ t < π . Multiplication by i turns a vector perpendicular to a circle to avector tangent to the circle. Mathematically, since π = v (now thought of as a function) is holomorphic, its derivative commutes with J and we obtain the following statement of twovectors being parallel:(5.5) J dπ ( ∇| v | ) = dπ ( J ∇| v | ) (cid:107) dψ ( d/dt )while X hor has the property that dπ ( X hor ) = dψ ( d/dt ). Hence since both X hor and J ∇| v | are in H , they must be proportional to each other by some scalar function g . (cid:3) Now we can compute the monodromy in the C -patch. dv = yzdx + xzdy + xydz ∴ ∇ g | v | = 2 (cid:10) x | yz | , y | xz | , z | xy | (cid:11) ∴ X H = 2 ig (cid:10) x | yz | , y | xz | , z | xy | (cid:11) s.t. dv ( X H ) = iv ∂ η This last condition allows us to solve for the scalar function g . ∴ ig xyz ( | yz | + | xz | + | xy | ) = i ( xyz ) = ⇒ g = 2( | yz | + | xz | + | xy | ) ∴ X H = i | yz | + | xz | + | xy | (cid:10) x | yz | , y | xz | , z | xy | (cid:11) So integrating, we obtain the following parallel transport map over the | v | circle in the base. ∴ φ Hθ ( x, y, z, θ ) = ( e iθ | y | | z | | y | | z | | x | | z | | x | | y | x, e iθ | x | | z | | y | | z | | x | | z | | x | | y | y, e iθ | x | | y | | y | | z | | x | | z | | x | | y | z )= ( e iθ | x |− | x |− | y |− | z |− x, e iθ | y |− | x |− | y |− | z |− y, e iθ | z |− | x |− | y |− | z |− z )(5.6)In the last step we divide by | v | to more easily see the Dehn twisting behavior. This givesthe monodromy result, Lemma 5.20 at the start of this section, in the local model near theorigin of C . Namely, as we move counterclockwise across the x axis of the hexagon then | y | (cid:28) | z | becomes | z | (cid:28) | y | and we see that θ stays fixed at a small angle but θ rotatesfrom large to small, by 1. Hence ( θ , θ ) changes from (0 ,
0) to (0 ,
1) as we move from the(0 ,
0) tile then down to the (0 , −
1) tile. The other regions are similar.Note that at first glance, there is more dependence on x once we divide by g , which maymake a harder ODE to solve for finding the flow. However, recall that symplectic paralleltransport preserves the moment map coordinates ξ , ξ , and η . Thus, since these are mono-tonic increasing functions in r x , r y , r x r y r z , we find that norms must also remain constantalong the circle. So since we have that g = | v | / |∇| v | | is a function of the norms, we findthat this is constant and doesn’t contribute to the flow when we integrate. Away from the C -patch. Elsewhere in Y , we may not be able to directly compute ∇ g | v | .All we know from Appendix B is that g = g + g (cid:15) where g is diagonal in polar coordinatesand g (cid:15) can be made to have arbitrarily small entries. Let V be d | v | with respect to polarcoordinates, so it will be zero in the last three angular coordinates, let G + G (cid:15) denote themetric, and V the vector ∇ g | v | , all of them with respect to polar coordinates as in Section4.6. Then if O ( · ) denotes a matrix with entries on the order given in parentheses: d | v | = ( g + g (cid:15) )( ∇ g | v | , · ) MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 61 = ⇒ V t = V t ( G + G (cid:15) )= ⇒ V = ( G t + G t(cid:15) ) − V = ( O (1) + O (1 /l )) − V ∴ V ≈ ( G t ) − V where ( G t ) − is the leading order terms of the metric defined in Definition 4.24. It is diago-nal by Section 4.6 and consists of constants because all bump functions are functions of thenorms r x , r y , r z which are fixed for ( ξ , ξ , η ) fixed. Thus the parallel transport map will besimilar to that in Equation 5.6, but | x | − , | y | − , | z | − will be scaled by certain constants. In the CP (3) hexagons. Recall that in the upper right corner of the fundamental domainparallelogram of Figure 10 where r z (cid:28) r x , r y , the K¨ahler potential is a sum of the CP (3)toric potential g xy , which only involves the complex coordinates x and y and does not involve z , and a term proportional to | xyz | . The latter is constant on the v -circle. Hence horizontalvectors are those whose x and y components vanish, namely H = span (cid:18) ∂∂ (cid:60) z , ∂∂ (cid:61) z (cid:19) Thus in this upper right corner region, parallel transport varies z while fixing x and y . Itfollows that the monodromy preserves θ = arg( x ) and θ = arg( y ), so ( f ( ξ, η ) , f ( ξ, η )) =(0 ,
0) for ( ξ, η ) in the upper right corner, remain the same as near the C -patch.By the same argument, in the region where the fiberwise potential is g yz , the horizontaldistribution is parallel to the x coordinate axis, so parallel transport varies x while keeping y and z constant. In particular the angular parallel transport vector field is ( ix, ,
0) sothe monodromy increases θ at unit rate while keeping θ constant, and ( f , f ) = (1 , g xz , angular parallel transport is (0 , iy, θ while keeping θ constant, and ( f , f ) = (0 , f , f ) are againinteger constants, determined by using the change of coordinate transformations describedin Definition 4.10. In particular at the lower-left corner of the parallelogram, in terms ofthe coordinates ( x (cid:48)(cid:48)(cid:48) , y (cid:48)(cid:48)(cid:48) , z (cid:48)(cid:48)(cid:48) ) the parallel transport only varies z (cid:48)(cid:48)(cid:48) while keeping arg( x (cid:48)(cid:48)(cid:48) ) andarg( y (cid:48)(cid:48)(cid:48) ) constant. However, because x (cid:48)(cid:48)(cid:48) = T v x − , fixing arg( x (cid:48)(cid:48)(cid:48) ) implies arg( v ) − arg( x )remains constant. Thus varying arg( v ) around the unit circle also varies arg( x ) and so θ increases by 1. Similarly for y , and we find ( f , f ) = (1 , B -invariance of ω as proven in Claim 4.12. (cid:3) We need the following main result in order to compute the differential of the Fukaya category.Now that we know the monodromy we can prove it. The place where the monodromy isused is bold-faced in the proof below.
Lemma 5.24.
The parallel transported φ H π ( (cid:96) i ) is Hamiltonian isotopic to (cid:96) i +1 .Proof. We construct an isotopy h t : V ∨ → V ∨ in the fiber in coordinates ( ξ , ξ , θ , θ ). Ona fiber, η is a function of ξ , ξ so doesn’t show up in the notation until the end of the proof,when we consider maps on the total space Y . We want h t to map (cid:96) to φ πH ( (cid:96) ) where φ πH is the monodromy. To prove h t is a Hamiltonian isotopy, i.e. ι ddt h t ω = dH t for some smooth function H t , a classical result of Banyaga (cf [MS17, Theorem 3.3.2] or [Pas14, Equation (6)])implies that it suffices to show that the flux of ω through cylinders traced out by generatorsof H ( V ∨ ) under h t , is zero. That is: (cid:28)(cid:90) t ι X t ω, [ γ ] (cid:29) =: (cid:104) Flux( h t ) , [ γ ] (cid:105) = (cid:90) h t ( γ ) ω = 0where X t := ddt h t and the second equality follows by an argument similar to the proof ofClaim 4.17, where area corresponds to integrating over the height direction X t and angulardirection γ (cid:48) ( t ) on the cylinder. The isotopy h t interpolates linearly between φ πH ( (cid:96) ) and (cid:96) .Recall that the angular coordinates on (cid:96) are given by − λ ( ξ ) which we denote in componentsas − ( λ ( ξ ) , λ ( ξ )). Then define the isotopy on the fiber as: h t ( ξ , ξ , θ , θ ) := ( ξ , ξ , θ + t ( f ( ξ ) + λ ( ξ )) , θ + t ( f ( ξ ) + λ ( ξ ))) : V ∨ → V ∨ This is well-defined modulo Γ B in the first two coordinates. The isotopy is also well-defined modulo Z in the second two coordinates by our monodromy computa-tion. Lemma 5.20 implies that( f , f )( ξ + γ ) = ( f , f )( ξ ) − λ ( γ )therefore( f , f )( ξ + γ ) + λ ( ξ + γ ) = ( f , f )( ξ ) − λ ( γ ) + λ ( ξ + γ ) = ( f , f )( ξ ) + λ ( ξ )for ∀ γ ∈ Γ B so h t indeed descends to an isotopy on V ∨ .Note that H ( V ∨ ) has rank four since V ∨ ∼ = T B × T F . If we let γ be the loop generated by oneof the two angular directions (0 , , , γ ( s ) = (0 , , s,
0) and h t ( γ ( s )) = (0 , , s, t because ξ = 0 and f i (0) = 0 as illustrated in Figure 16. So the integral of ω overthis cylinder of height zero is zero. Now we let γ ( s ) = (2 s, s, , , − ≤ s ≤ γ ( s ) = ( s, s, ,
0) will be similar. (cid:90) h t ( γ ) ω = (cid:90) γ ι X t ω = (cid:90) γ ( dξ ∧ dθ + dξ ∧ dθ )(( f − λ ) ∂ θ + ( f − λ ) ∂ θ , − )= (cid:90) γ ( f − λ ) dξ + ( f − λ ) dξ = (cid:90) / s = − / f (2 s, s ) ds + g (2 s, s ) ds where we used that λ being a linear map implies anti-symmetry across zero, so the integralfrom 0 to 1 / − / f and f . It suffices to show that f i ( − ξ ) = − f i ( ξ ). We do this as follows. Consider the map on Y given by: φ − : ( ξ, η, θ, θ η ) (cid:55)→ ( − ξ, η, − θ, θ η ) MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 63
It is a symplectomorphism because dξ ∧ dθ + dη ∧ dθ η (cid:55)→ d ( − ξ ) ∧ d ( − θ ) + dη ∧ dθ η = dξ ∧ dθ + dη ∧ dθ η . It remains to prove that it is fiber-preserving. By monotonicity of ξ , ξ , η on the coordinate norms, the map φ − on complex coordinates is( x, y, z ) → ( T − x − , T − y − , T v z − )because arg( x − ) = − arg( x ) and log | x | becomes − log | x | up to a constant in T , and similarlyfor | y | and | z | . That this gives the map φ − up to additive constants then follows by Claim4.19. It preserves the polytope ∆ ˜ Y as seen from the coordinates in Figure 10. Since Claim4.19 describes ξ , ξ , η only up to additive constants we have a bit more work to do to show φ − preserves a fiber. Suppose φ − is defined by:( ξ , ξ , η ) (cid:55)→ ( − ξ + c , − ξ + c , η + c )Since the polytope ∆ ˜ Y is preserved, this must map the origin to itself. Hence the constantsare zero. Recall from Claim 5.18 that X hor = ∂∂θ η + f ( ξ, η ) ∂∂θ + f ( ξ, η ) ∂∂θ . Also X hor ispreserved by any fiber-preserving symplectomorphism of Y as in the proof of Corollary 5.5.Since φ − is one such, we have: X hor = ( φ − ) ∗ ( X hor ) = X hor ◦ φ − = ⇒ ( φ − ) ∗ (cid:18) ∂∂θ η + f ( ξ, η ) ∂∂θ + f ( ξ, η ) ∂∂θ (cid:19) = ∂∂θ η − f ◦ φ − ( ξ, η ) ∂∂θ − f ◦ φ − ( ξ, η ) ∂∂θ = ⇒ ∂∂θ η + f ( ξ, η ) ∂∂θ + f ( ξ, η ) ∂∂θ = ∂∂θ η − f ( − ξ, η ) ∂∂θ − f ( − ξ, η ) ∂∂θ ∴ f i ( − ξ, η ) = − f i ( ξ, η )for i = 1 ,
2. So the flux of ω through h t ( γ ) is zero on generators of H ( V ∨ ). By runningthis argument repeatedly, we can see that ( φ πH ) k ( (cid:96) ) is Hamiltonian isotopic to (cid:96) k +1 for any k . Hence φ πH ( (cid:96) i ) is Hamiltonian isotopic to ( φ πH ) i +1 ( (cid:96) ) which is Hamiltonian isotopic to (cid:96) i +1 . (cid:3) Moduli spaces definition.
Now that we have computed the monodromy, the nextstep will be computing the differential between two intersection points in the base. This willinvolve moduli space considerations. More generally, the Fukaya category is an A ∞ -categorymeaning in addition to objects and morphisms, there are structure maps on k morphismsfor any natural number k that satisfy A ∞ -relations. These can be thought of as higher orderassociativity relations on the morphisms, hence the A in A ∞ , and in the case of symplecticfibrations they involve counting pseudo-holomorphic discs which project to polygons in thebase. The remainder of this chapter will set up the theory to define these counts. The word“moduli” indicates we look at a set of objects “modulo” an equivalence relation, and the“space” refers to equipping this set with a topology. We start by defining the structure onthe domains of the pseudo-holomorphic curves we will want to count. These are the sourcecurves . The following background was learned from [Sei08]. Definition 5.25 (Domains) . A punctured boundary Riemann surface S is the data of acompact, connected, Riemann surface with boundary and with punctures on its boundary,as well as the assignment of a Lagrangian L i to the i th component of ∂S . We further “rigidify” by adding extra structure to S ; denote punctures as “positive” or “negative” anddefine strip-like ends via embeddings (cid:15) : ( −∞ , s × [0 , t → D or (cid:15) : [0 , ∞ ) s × [0 , t → D for Figure 17.
Example of strip-like endthe negative and positive punctures ζ ± respectively, such that lim s →±∞ (cid:15) ( s, t ) = ζ ± . This iscalled a Riemann surface with strip-like ends . Remark 5.26.
This “rigidifies” because any operation on S must preserve the additionaldata of strip-like ends, thus placing further restrictions. The strips provide a nice set ofcoordinates near the punctures (namely s and t ) and give a straightforward way to pre-gluetwo sections by identifying linearly in the ( s, t ) coordinates. See [Sei08, § (8i) and § (9k)]. Example 5.27.
In this thesis, S will be one of the unit disc D , two discs glued together at apoint on their boundary with two punctures on one disc, or a disc with a marked boundarypoint identified at one point to a configuration of spheres in the central fiber. An exampleof a strip-like end we use later on to glue two discs is ( −∞ , × [0 , (cid:51) ( s, t ) (cid:55)→ (cid:15) − ( s, t ) := e − π ( s + it ) + ie − π ( s + it ) − i = z + iz − i ◦ e − z ◦ π · ( s + it ) ∈ D \{ } . Note that −∞ is the puncture which would mapto ζ − := 1 in the disc. See Figure 17.Next we find the 2-homology of Y . This is where the pseudo-holomorphic discs map to inthe target. Lemma 5.28 (Homology of Y ) . H ( Y ) ∼ = H ( CP (3) / Γ B ) ∼ = Z where all homology classes will be over Z .Proof. Note that Y deformation retracts onto the central fiber over the contractible base,determined by parallel transport in the inward radial direction. Therefore the homology of Y is the homology of the central fiber, a degeneration of a T fiber.To get H ( CP (3) / ∼ ) we may use Mayer-Vietoris as follows. Cut out an open set projectingto a disc at the center of the hexagon in the hexagonal picture, which then retracts onto theboundary. This gives two open sets, which in the moment map picture are a disc and itscomplement, intersecting in an S .Let B := D × T correspond to the disc in the moment polytope. Let ˜ A be the complement,a string of 6 CP ’s in a circle and A := ˜ A/ Γ B the banana manifold. In particular, ˜ A ∩ B retracts onto S × T = T . Then we have0 = H ( ˜ A ) ⊕ H ( B ) → H ( Z ) = 0 → H ( ˜ A ∩ B ) = Z ∗ ) −→ H ( ˜ A ) ⊕ H ( B ) = Z ⊕ Z → H ( CP (3)) → H ( ˜ A ∩ B ) = Z ∼ = −→ H ( ˜ A ) ⊕ H ( B ) = Z ⊕ Z → H ( CP (3)) = 0 MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 65 where ( ∗ ) = − − − − and the second to last map is the 3-by-3 identity matrix withrespect to the three generators of H ( ˜ A ∩ B ) = H ( S × T ) (call them b for the S in themoment polytope and T , T for the two loops in the 2-torus fiber). Then passing to thequotient CP (3) / ∼ we find that0 = H ( A ) ⊕ H ( B ) → H ( CP (3) / ∼ ) → H ( A ∩ B ) = Z ∗∗ ) −−→ H ( A ) ⊕ H ( B ) = Z ⊕ Z → H ( CP (3) / ∼ ) = Z → Z · b ⊂ H ( A ∩ B ) = Z ∗∗∗ ) −−−→ H ( A ) ⊕ H ( B ) = Z ⊕ Z → H ( CP (3) / ∼ ) = Z where ( ∗∗ ) = and ( ∗ ∗ ∗ ) = . Note that the final term is consistentwith Y being the quotient of its universal cover ˜ Y by Γ B = Z , where π ( Y ) = π ( CP (3) / ∼ )should be Z . So ultimately we look at the projection of the 6 P ’s mapping to the threeglued P ’s. Again we find that the homology is Z . (cid:3) Now that we know what the second homology class of Y is, we are better equipped toclassify the possible choices for β = [ u ] for pseudo-holomorphic maps u . We proceed todefine the class of almost complex structures. Then we can define J -holomorphic curves.The almost complex structures we consider are compatible with the ω defined above andequal the standard J induced from the complex toric coordinates near the boundary. Say J = J outside of the open set U := v − ( B ( )), the preimage of a disc radius 1 / U . We will denote this set as J ω ( Y, U ). It is non-empty because it contains J , and contractible by the same argument asfor the set of all ω -compatible almost complex structures. Definition 5.29 (Maps) . Let J ∈ J ω ( Y, U ). A pseudo-holomorphic map is a J -holomorphicmap u : ( S, ∂S ) → ( Y, (cid:116) i ∈ π ( ∂S ) L i ). A J /pseudo-holomorphic curve is the image of such amap. We require lim s →±∞ u ( (cid:15) ij ( s, t )) ∈ L i ∩ L j where (cid:15) ij is a strip-like end attached at an intersection point L i ∩ L j on the boundary. Moduli spaces for a fibration.
We can think of the above as a section of a trivial fibra-tion with fixed Lagrangian boundary condition. For non-trivial fibrations, the Lagrangianboundary condition now consists of fibered Lagrangians. These are the ones described aboveobtained from parallel transporting a linear Lagrangian in a fiber over a U-shaped arc.
Let J be multiplication by i in the toric coordinates. Given a J -holomorphic map to thetotal space u : S → Y , we can compose with the holomorphic projection v ◦ u : S → C toobtain a biholomorphism onto its image by the Riemann mapping theorem. In particular, ifwe identify S with v ◦ u ( S ), then v ◦ u = S and u is actually a holomorphic section of v which projects to an embedded holomorphic polygon in the base.On the other hand for generic J close to J , whose existence we prove in the next section,pseudo-holomorphic maps u are sections only outside U , where J = J . They still havealgebraic intersection number 1 with fibers of v in the region where J has been perturbedto J . We will refer to these as section-like maps . Example 5.30.
Let t x be the preimage of a moment map value ( c , c ) ∈ T B , intersectedwith a fiber. So t x = { c , c , θ , θ } θ i ∈ [0 , π ) which in particular is invariant under paralleltransport because that map rotates the angles. Since (cid:96) i ∩ t x is the one point of (cid:96) i with( ξ , ξ ) = ( c , c ), any pseudo-holomorphic section u must limit to that one point over thecorresponding puncture in the base. Stable maps.
We have now discussed the Lagrangian boundary condition, the homologyclass, and the almost complex structure. So we next quotient by automorphisms of the do-main. In this thesis, we will be concerned with stable maps to Y where the domain is a discwith one boundary marked point, or a disc union sphere with one boundary marked pointon the disc and one interior marked point on the disc and sphere each where they meet.These are stable maps (in particular they have finite reparametrization action) but not sta-ble domains. So stabilization plays the role of quotienting by the automorphism group. Onereference for the topology of the moduli space of pseudo-holomorphic maps is [Fuk20].A stable map is given by a tree of pseudo-holomorphic maps, where each vertex α is a spherebubble, except for the vertex corresponding to original disc curve. The stable refers to the ab-sence of continuous families of nontrivial automorphisms; in particular if we fix three pointson every constant component there are no nontrivial automorphisms. A tree encodes infor-mation for how to glue, where interior edges are assigned a gluing length, and semi-infiniteedges at either end give the resulting marked points of the final glued disc. The idea is thatwe can have a family of discs, parametrized in the base by possible cyclic configurations. Wealso have a gluing parameter for each interior edge where a sphere bubble is glued. More inte-rior edges increases the codimension of the moduli space of that configuration in the modulispace of a disc. We can encode the possible degenerations in an associahedron, see [Sei08, § Topology and compactification.
The compactification of the moduli space of unparametrizedcurves (i.e. equivalence classes) is a union over possible bubble trees of unparametrized curves,c.f. [MS12, Equation (5.1.5)]. The reason is that these are the possible limit configurations.Lecture5–7 of [Fuk20] discuss the topology, Hausdorfness, and that sequential compactnessimplies compactness when the limit configurations are included.The moduli space of stable maps exhibits
Gromov convergence . A limit of a sequence of J -holomorphic curves is either in the original moduli space or not. If it is not, then in the limitwe could have disc bubbling, sphere bubbling, or strip-breaking. This is done by pregluingthe maps by pasting them together, which may not give something J -holomorphic. Hence MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 67 it is called pregluing. However it is suitable as a compactification of a topological space.Following [MS12, Chapter 5] let u.c.s. := uniformly on compact subsets. If the derivativesin a fixed homology class are bounded in W ,p for some p >
2, then we have a Bolzano-Weierstrass type result that any sequence has a convergent subsequence. However, in theborderline p = 2 case, as is the case here, the limit may not be in original class of curves.Gromov convergence to a stable map means: for each vertex α of the tree we have a familyof reparametrizations φ να .To achieve a compact moduli space we consider limites of pseudo-holomorphic curves. Finiteenergy curves, determined by the homology class β by pseudo-holomorphicity, can still havebubbles because energy is invariant under rescaling. However, the derivative may blow upas in the following example: CP → CP given by [ x : y ] (cid:55)→ [ x : (cid:15)y : xy ]. This parametrizesthe holomorphic curve in CP given by ab = (cid:15)c . As (cid:15) →
0, we only see the curve [ x : 0 : y ].However we can reparametrize and obtain a different limit. The image is actually two spheres, ab = 0 and not just b = 0, because of rescaling in CP . In ab = (cid:15)c , let (cid:15) → ab = 0.If we remove the bubbling points, then the sequence of curves converges u.c.s. to the maincomponent. To get C ∞ loc convergence to the main component we use the Bolzano-Weierstrasstheorem on the energy. And the reason the main component is fully defined on S (eventhough we removed the bubble points) is because of removable singularity property of pseudo-holomorphic maps. To find the other bubbled-off components, reparametrize the source curveby z/R n , where the R n goes to infinity, and include a sequence of points tending to the bubblepoint. Energies of the main component plus that of the bubbles should add up to the originalenergy since energy remains the same in the limit. Example 5.31 (Gromov compactness in our setting) . For fixed β and Lagrangian boundarycondition, there is no disc bubbling or strip breaking under reparametrization. Only spherebubbling cannot be excluded. Also, there are no multiply covered discs. Proof of example.
The projection of a hypothetical disc bubble under v , if non-constant,would satisfy the open mapping principle outside of U because it is a J -holomorphic sectionthere. A disc bubble would have boundary in a single Lagrangian. However, all Lagrangiansconsidered project to a U-shaped curve which does not enclose any bounded region of thecomplex plane. So any such disc would have to lie entirely in a fiber. However, linear La-grangians in tori have zero relative π i.e. they don’t bound discs. So there can be no discbubbling in a fiber and hence no disc bubbling in general.Similarly if a strip breaks, any component that breaks off must either lie entirely within afiber of v , or the degeneration must be visible in the projection to v . Namely, the polygonalregion of the complex plane over which the section-like map projects also decomposes into aunion of polygons with boundary on the given arcs. Since Maslov indices add, and all discshave intersection 1 with the central fiber, there is no room for a Maslov zero disc since itwould have to be a bigon in a fiber (as it can’t pass through the central fiber) and linearLagrangian in tori cannot bound bigons.If we did have multiple covers, we may require that J depend on z ∈ S , in which case wereplace J with J z . The text [MS12] lays this foundation for Lagrangian boundary conditions case which is described in [Aur14] and will involve an R -family of J ’s on a strip. We don’tneed to do that here since we are considering section-like discs, which are injective near theirboundary. They are somewhere injective and only wrap once around the boundary. (cid:3) Dimension.
The virtual dimension of the moduli space of section-like maps is given by theMaslov index of the chosen β homology class, minus twice the number of edges in the stabletree. That is, each bubbled off sphere reduces dimension by 2. Thus we expect to have a pseudocycle , i.e. the image of the boundary of the moduli space has codimension at leasttwo in the total space. This can be achieved in the semipositive case, which means thereare no J -holomorphic spheres of negative Chern number for generic J – this is our settingbecause Y is Calabi-Yau. We define the resulting moduli space by taking a collection ofmaps, quotient by reparametrization, and then compactify. Remark 5.32 (Analogue of dimension in Morse theory) . With Morse-Smale data we knowthe dimension of the moduli space by counting eigenvalues. We require transverse intersectionof the unstable manifold of x − with the stable manifold of x + . We have a projection mapfrom the unstable to the perpendicular of the stable. On the other hand, the index of aFredholm section is computed from the spectral flow of a loop of symmetric matrices, see[Wen16, § D ). A spin structure on the Lagrangian determines an ori-entation on the moduli space of parametrized curves, so quotienting by Aut( D ) induces anorientation on the moduli space of unparametrized curves. Example 5.33 (Deligne-Mumford moduli space of domains) . For references see [MS12, § §
6] and [Fuk20, Lectures1–4] A stable disc is a stable map to a point. The Deligne-Mumfordspace M ,d +1 arises from considering degenerations of stable discs with the additional dataof d + 1 boundary marked points. E.g. possible limits under reparametrization in Gromovcompactness, see [MS12, § g curves with l marked points is6 g − l . Role of Maslov class in dimension.
The following is based on [Aur07] and [CO06]. Wegive a more detailed definition of Maslov class.
Definition 5.34 (Maslov class of a Lagrangian and of a disc) . Suppose 2 c ( M ) = 0, so thesquare of the anticanonical bundle is trivializable by some section s . We have a map from LGr to the unit bundle of K − by taking det of a basis for each Lagrangian. We can identifythat unit bundle with S using the trivializing section s mentioned above. The upshot is thatwe get a map from LGr → S . The “Maslov class” of the Lagrangian is the pullback of [ S ],i.e. a homology class in LGr . If it is zero (meaning we can lift to R ), then we can define µ ( β )as the evaluation of this homology class on β . This measures the obstruction to extendingthe square of this normalized section on L to one on a disc representing 2-homology class β . Definition 5.35.
A special Lagrangian L is defined to be one such that if Ω trivializes K X \ D on the complement of an anticanonical divisor D , then Ω | L = e − iφ dvol L for some constantphase φ . Lemma 5.36 ([Aur07, § . Let ( X, ω, J ) be a smooth, compact and K¨ahler manifold. Let Ω ∈ M ( X, ( T ∗ (1 , X ) n ) be a global meromorphic n -form, with poles along an anti-canonical MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 69 divisor D , e.g. using log coordinates. In other words, Ω − is a nonzero holomorphic sectionof the anti-canonical bundle on X \ D . Let L be a special Lagrangian submanifold in X \ D .Let β ∈ π ( X, L ) be nonzero. Then µ ( β ) is twice the algebraic intersection number β · [ D ] ,where µ ( β ) denotes the Maslov class.Proof from [Aur07, § . The tangent space to L is real since being Lagrangian is defined by( T L ) ⊥ = J T L with respect to ω ( − , J − ). Taking a real basis gives a nonvanishing section of K − X | L which we can scale to unit length. Since we’ve normalized, this section is independentof choice of basis. In particular, its square also trivializes the square of the anticanonicalbundle over L . Since L is special Lagrangian, Ω − which defines the divisor of D is equal tothis volume form on L , up to a constant phase factor, by Definition 5.35. The Maslov classof β is then deg(Ω − | β ) = 2 D · β . (cid:3) Example 5.37 ([CO06], [Aur07]) . Consider the moduli space of J -holomorphic discs withboundary in L = (cid:83) circle t x for a circle around the origin in the base of v , i.e. a product torusin a toric variety, and β is a class of Maslov index 2, i.e. by Lemma 5.36 the class of a discintersecting the central fiber once. Dimension.
Let n = 3 be the real dimension of the Lagrangian. For µ ( β ) = 2, the virtualdimension of the moduli space of holomorphic curves in class β is n − µ ( β ) = n − D u is nχ ( D ) + µ ( β ) = n + µ ( β ) where χ denotes the Euler characteristic. Then we subtract out 3 because of thereparametrization action, since only the identity automorphism on a disc fixes 3 boundarypoints. So when n = 3 this is real dimension 2.Using Lemma 5.36, we can interpret the dimension geometrically from intersection numbers.Since we’re considering J -sections u , they pass once through the central fiber at 0, which isalso the divisor D in our setting. D is the union of toric divisors in v − (0). They intersect D transversely once, hence they have Maslov index 2 by the above result. There are nonontrivial Maslov zero discs by the fact that linear Lagrangians in the fiber do not bounddiscs. If we require one boundary marked point to map to a given point of L , this cuts downthe dimension of the moduli space of holomorphic curves by n , but we add one dimensionfrom the choice of marked point. (The collection of discs with boundary on (cid:83) circle t x anda point constraint on the boundary is a zero dimensional family, otherwise we could rotatethe disc by the T action and obtain a family of discs.) So the expected dimension is then 2(from the Maslov index) plus 1 for the choice of marked point on the domain, minus 3 fromquotienting by reparametrization. This results in an expected dimension of zero. Moduli space computation.
The linear fiber Lagrangian t x can be thought of as corre-sponding to the skyscraper sheaf under mirror symmetry. Recall that the definition of (cid:96) k involved rotating an amount 2 πk along the two angle directions as we traverse one loop ineach of the base ξ i moment map directions. Rotating only the angle directions and not inthe base defines t x , namely, we fix the moment map coordinates and let the angles vary.This gives the preimage of a moment map coordinate A := ( a , a , a ) = ( ξ , ξ , η ) and welet | z | := τ A denote the exponentiated coordinates. The reason for choosing the letter A isthat the formula for counting such discs is discussed in a paper of [CLL12] and that notationfollows theirs. Because each disc considered intersects the central fiber only once, its lift to the universalcover ˜ Y can intersect only one of the toric divisors of ˜ Y . Which divisor it intersects is deter-mined by the class β . The image of the disc is contained in the union of the open stratum of˜ Y and the open stratum of the component of D that the disc intersects, which by standardtoric geometry is the image of a toric chart C ∗ × C ∗ × C inside ˜ Y .Thus we think of the disc as mapping to the chart C ∗ × C ∗ × C . Then a disc with boundary in S ( r ) × S ( r ) × S ( r ) implies it is constant in the first two components (by the maximumprinciple) and we obtain a disc in the last v coordinate. And this is the only disc by theRiemann mapping theorem. It cannot be multiply covered because it must be Maslov index2 by the previous paragraph.So the discs we count correspond to selecting a point A in the moment polytope and drawinga line to a facet in that polytope. Geometrically, fixing A implies each disc has the same T ∼ = S ( r ) × S ( r ) × S ( r ) Lagrangian boundary condition, namely the moment mappreimage of A . As we allow | v | →
0, we find that r , r remain constant and the thirdcoordinate goes to zero along the disc. (In the moment polytope this corresponds to a path,depending on β , from A to a facet. And ξ , ξ , η may vary along the path.) We count eachdisc over the possible β , weighted by area, in Theorem 6.4. Here we’ve shown there is onlyone J -holomorphic disc in each β class.5.4. Existence of regular choices to define moduli spaces.Definition 5.38.
We say that an almost complex structure J ∈ End(
T Y ) is regular if, forall J -holomorphic maps u : (Σ , j ) → ( Y, J ) on the complex curve Σ, the linearization (orderivative) of the Cauchy-Riemann operator ∂ J is surjective. Remark 5.39. “Regularization” refers to perturbing the ∂ J operator to be equivariantlytransverse to the zero section of a Fredholm bundle which we can build so that the operatoris a section of the bundle. “Geometric regularization” means the perturbations are obtainedby perturbing the almost complex structure J , so are geometric in nature. Namely theperturbations of ∂ J are ∂ J (cid:48) − ∂ J as J (cid:48) varies. In this setting equivariance will be automatic,as described below. Note that later on, we will need to use a non-regular J for computationsand in that case we will use “abstract regularization” by adding abstract perturbations p which are sections of an “obstruction” bundle built from the non-surjectivity of D u . Theyare not necessarily of the form ∂ J (cid:48) − ∂ J . Remark 5.40 (Intuition) . We claim that a choice of regular J gives rise to a smoothstructure on a moduli space. The concept is analogous to why, e.g. , the sphere x + y + z = 1is smooth when thought of as a regular value of f ( x ) = x + y + z . Namely in standardcoordinates df = (2 x, y, z ) : R → R is surjective whenever x , y and z are not all zero. Forsuch a choice of ( x, y, z ), f ( x ) (cid:54) = 0 is the radius of the sphere. So a regular choice of ( x, y, z ) (cid:32) f − f ( x, y, z ) is playing the role of a regular choice of J (cid:32) ∂ J , and ( f − f ( x, y, z )) − (0) is asmooth manifold analogous to how ∂ − J (0) will be a smooth manifold. The difference is that f is a function between finite-dimensional spaces, whereas J and ∂ J live in infinite-dimensionalfunction spaces. MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 71
Finite rank case: geometric regularization introduction from [Weh14, Lecture 1A] . For a section of a finite rank bundle over a finite dimensional manifold, its zero set is au-tomatically compact. However, we don’t necessarily have that over an infinite-dimensionalmanifold. In the finite dimensional case, there exists a space of perturbations P so that s + p is transverse to the zero section for all p ∈ P , i.e. the derivative D u ( s + p ) is surjective forall u ∈ ( s + p ) − (0). We can guarantee that P is non-empty, the perturbed s is transverseto the zero section, ( s + p ) − (0) is compact, and we can construct cobordisms between zerosets for different choices of p .These properties allow us to define a fundamental class for M := s − (0), denoted [ M ], evenif it is singular from lack of transversality to the zero section (which would have ensured it isa manifold). We view the fundamental class of the singular moduli space as the intersectionof nested open sets W k , which are subsets of points in the base of the bundle that are 1 /k away from s − (0). In particular, by non-triviality of P , we can take a sequence of pertur-bations so ( s + p k ) − (0) ⊂ W k stays the same. A fundamental class is necessary to countthe zero set of s , i.e. compute an intersection of s with the zero section. We build a funda-mental class using that ˇCech homology of the singular manifold is isomorphic to an inverselimit of rational ˇCech homology of the nested open sets. Each W k has a fundamental class inthe top degree of ˇCech homology, which will limit to the fundamental class we are looking for.Fortunately, Banach spaces still come equipped with all the necessary tools to obtain smoothstructures analogous to the finite-dimensional case. We follow the arguments of [MS12,Chapters 3–10], adapting their S = CP setting to our S = D setting with Lagrangianboundary conditions; this is also discussed in [Sei08] and [Gan16]. Example 5.41.
An example of the geometric regularization theory is implemented in [Weh13]for Gromov non-squeezing, which involves illustrating how to show a family of moduli spacesvarying J t is 1 dimensional (Fredholm), a manifold (transversality/regularity), compact (Gro-mov compactness) and has boundary. Remark 5.42. J -curves have some analogous properties as complex curves, such as theCarlemann similarity principle and unique continuation (if two J -holomorphic maps agreeon an open set, or all derivatives agree at a point, then the maps are equal). Other propertiesinclude that there are only finitely many points in the preimage of a point and only finitelymany critical points. Furthermore simple curves (or their analogue in the Lagrangian bound-ary condition case, somewhere injective curves) have an open dense set of injective points. Existence of regular J . Let U be the open neighborhood of the singular fiber of v definedin the paragraph before Definition 5.29. Recall(5.7) J ω ( Y, U ) := { J ∈ Ω ( Y, End(
T Y )) | J = − , ω ( · , J · )is a metric , J | Y \ U ≡ J } In particular, this set is non-empty because it contains J , and is contractible by the sameargument as in the case of no boundary. This set of J is what’s needed to prove geometricregularization in the boundary case, see [MS12, Remark 3.2.3]. Furthermore, let γ i denotecurves in the base of v and (cid:83) γ i (cid:96) i for parallel transport of (cid:96) i over γ i . Recall from Definition5.6 that L i is the Lagrangian given by parallel transporting (cid:96) i from the − Then define the notation(5.8) L | γ := (cid:91) i ∈ I ⊂ Z (cid:91) γ i (cid:96) i Since all maps to Y we consider are polygons when projected to the base of v , all discs passthrough the zero fiber. Remark 5.43 (Notation) . The notation ˙ J does not mean we can only vary J in one directionas is usually the case with the dot notation. We use the notation as a symbolic way to denotetangent vectors to the space of complex structures; it’s denoted Y in [MS12]. Lemma 5.44 (Geometric regularization) . There exists a dense set J ∈ J reg ⊂ J ω ( Y, U ) such that, for all J -holomorphic maps u : ( D , ∂ D ) \{ z , . . . , z | I | } → ( Y, L | γ ) as in Definition5.6, the linearized ∂ -operator D u is surjective. Remark 5.45.
We use the superscript 1 because later we will want existence of the slightlysmaller set J reg of J regular for a disc attached to a sphere with similar Lagrangian bound-ary conditions. These will require not only surjectivity of the linearized operator but alsocompatible behavior when evaluating at the intersection point. Roadmap adapting the 2nd edition book [MS12, pg 55, proof of Theorem 3.1.6 (ii)] . We adaptMcDuff-Salamon to the setting with Lagrangian boundary conditions. Note Theorem C.1.10in [MS12] already proves we have a Fredholm problem for the case of boundary. The back-ground for this was also learned from [Weh14, Lecture 9].Take a homology class β ∈ π ( Y, L | γ ). We make the following definitions(5.9) B ,pβ := { u ∈ W ,p (( D , ∂ D ) , ( Y, L | γ )) | [ u ] = β, lim z → z i u ( z ) = p i }J (cid:96)ω ( Y, U ) := { J ∈ C (cid:96) ( Y, End(
T Y )) | J = − , ω ( · , J · ) is a metric , J | Y \ U ≡ J } ˆ˚ M J := { ( u, J ) | J ∈ J (cid:96)ω ( Y, U ) , u ∈ B ,pβ , ∂ J ( u ) = 0 } where the hat indicates we haven’t yet quotiented by automorphisms of the source curve andthe ring indicates we haven’t compactified yet. Then we claim that(5.10) B ,pβ × J (cid:96)ω ( Y, U ) (cid:51) ( u, J ) (cid:55)→ ∂ J ( u ) ∈ L p ( D , Λ , ⊗ u ∗ T Y )is a Fredholm section s of a Banach bundle, with surjective derivative, hence it also has aright inverse and we can invoke the Inverse Function Theorem to deduce that ˆ˚ M J = s − (0)is a C (cid:96) − Banach submanifold of the base. We will now justify this. We use L to denote L | γ for ease of notation. Banach manifold structure on B ,p . A map u ∈ B ,p has a local Banach chart fromexponentiating T Γ( u ∗ T Y, u ∗ T L ) (cid:51) ξ via u (cid:55)→ exp u ξ . The map exp u ξ still has the correctboundary condition; there exists a metric so that one Lagrangian is totally geodesic [MS12,Lemma 4.3.4] which can be adapted to the argument for two transversely-intersecting La-grangians as is done in [Mil65, Lemma 6.8], [Fra10] for submanifolds of complementarydimensions. Namely, take a convex combination of the two metrics defined for each La-grangian separately, and by uniqueness of geodesics given a starting point and direction, aswell as the definition of totally geodesic, we see that the result still holds in a neighborhood MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 73 of an intersection point. And at any point considered there are at most two Lagrangiansintersecting, so this suffices. Once we have the metric for which both Lagrangians are totallygeodesic, then for p ∈ ∂ D we see that exp u ( p ) ξ is a point on the geodesic which starts in L and travels in the direction of a vector tangent to L , so it must remain in L . Banach manifold structure on J (cid:96)ω ( Y, U ) . The second factor on the base of the Banachbundle we are constructing, J (cid:96)ω ( Y, U ), has Banach charts around elements J as describedin [MS12, § J to obtain the conditions for vectors ˙ J inthe tangent space, and local charts can then be recovered from the tangent space by expo-nentiating. Three conditions on J become linearized: 1) J | Y \ U ≡ J implies ˙ J | Y \ U ≡
0, 2) J = − implies ˙ J J + J ˙ J = 0, and 3) ω ( ˙ J · , · ) + ω ( · , ˙ J · ) = 0 arises from ω -compatibility.Equivalently, the second and third conditions impose that ˙ J = J ˙ J J and ˙ J is self-adjointwith respect to metric ω ( · , J · ). So a chart centered at J is constructed via ˙ J (cid:55)→ J exp( − ˙ J J ).A fiber of the bundle will not have boundary conditions because it is given by the spacewhere ( du ) , = ( du + J ◦ du ◦ j ) lands in, namely L p ( D , Λ , ⊗ u ∗ T Y ), and that doesn’tconcern the boundary. Note that J ◦ du ◦ j does not a priori have the same behavior as du in the direction tangent to the boundary, so ( du ) , does not satisfy any particular boundarycondition. Hence the structure of the Banach bundle over open sets in the base will mimicthe case of [MS12, Proof of Proposition 3.2.1] of no Lagrangian boundary conditions. Weagain use the exponential map to trivialize the bundle over a neighborhood N ( u ) in the firstfactor of B ,p × J (cid:96)ω ( Y, U ), and we use parallel transport to trivialize over a neighborhood N ( J ). This trivializes the bundle over a neighborhood in the base, and a composition ofthese gives transition maps that satisfy conditions for a Banach bundle, [MS12, Proof ofProposition 3.2.1] and [Sei08, § (9k)]. Key regularity argument.
We have that s ( u, J ) := ∂ J ( u ) is a Fredholm section of thisBanach bundle by [MS12, Theorem C.1.10]. Furthermore, its derivative at any ( u, J ) suchthat s ( u, J ) = 0 and u is somewhere injective (guaranteed by u being J -holomorphic outsideof U , so a section of v ), is surjective, as follows. See [MS12, Proof of Proposition 3.2.1].(5.11)( d ( u,J ) s )( ξ, ˙ J ) = D u ξ + 12 ˙ J duj : W ,p ( D , u ∗ T Y ) × C (cid:96) ( Y, End(
T Y, J, ω )) → L p ( D , Λ , ⊗ j u ∗ T Y )where(5.12) D u := d F u (0) : W ,p ( D , ∂ D ; u ∗ T Y, u ∗ T L ) → L p ( D , Λ , ⊗ j u ∗ T Y ) F u : W ,p ( D , ∂ D ; u ∗ T Y, u ∗ T L ) → L p ( D , Λ , ⊗ j u ∗ T Y ) ξ (cid:55)→ ∂ J (exp u ξ )see [MS12, Proposition 3.1.1]. In words, D u is defined by, for a nearby u (cid:48) in the exp neigh-borhood of u , parallel transport back to the origin of the chart at u , take ∂ J , then mapforward again on the fiber under parallel transport; the linearized operator D u ξ will be thederivative of this operation at the point 0. This is well-defined because of the totally geodesiccondition above. The D u term is only from varying u . Varying J as well we get (e.g. see [Weh14, Lecture 9]):(5.13) ( d ( u,J ) s )( ξ, ˙ J ) = D u ξ + 12 ˙ J duj
Given that a J -holomorphic map u must be a section of v outside of U (since J is J there),we see that u cannot wrap more than once around the boundary. Hence the curve is some-where injective. We claim that the operator ds is surjective with continuous right inverse,so is regular. Suppose by contradiction the image is not dense. Then we can constructa nonzero linear functional that is orthogonal to im ( ds ) and locally lies in a fiber of theBanach bundle, namely a non-zero L q form η that annihilates D u ξ + ˙ J duj over all W ,p tangent vectors ξ and ˙ J , in particular ˙ J = 0. So D u ξ = 0 for all ξ by Equation 5.13. Thisimplies that η has W ,ploc regularity by the elliptic bootstrapping result of [MS12], proven forLagrangian boundary conditions.Once we have regularity, we can integrate and prove D ∗ u η = 0. Via integration by parts(with evaluation on the 1-form and inner product on the bundles) we have D ∗ η = 0, becausetaking the adjoint leaves a boundary term d (cid:104) η, J ξ (cid:105) ω . Then using Stokes’ theorem and thatthe test vectors ξ are tangent to the Lagrangian at the boundary, this equals zero.However since η (cid:54) = 0, using bump functions we may construct a perturbation ˙ J as in [MS12,page 65] so η integrated on ˙ J is nonzero, contradicting that D ∗ u η = 0. The construction in[MS12, page 65] still works in the Lagrangian boundary setting because the constructed ˙ J is supported in a small neighborhood around a somewhere injective point, so will be zeronear the boundary as required. It is as follows. We’ve assumed η (cid:54) = 0 so pick point p so that η | p (cid:54) = 0. Somewhere injective points are dense so find a neighborhood of them around p .Use bump functions to construct a ˙ J so that (cid:82) D η ( ˙ J duj ) >
0. This is a contradiction. So η vanishes on the open set of injective points, hence vanishes identically by unique continuation[MS12, Theorem 2.3.2]. This is again a contradiction since η (cid:54) = 0. So the annihilator of ds iszero and the Hahn-Banach theorem implies that the image of ds is dense. So combining thatproperty with the image being closed from the Fredholm property of the operator, we findthat the operator surjects onto L p . This will allow us enough freedom to find the vectors ˙ J . Implicit and inverse function theorem find dense set of regular J . Since D u is Fred-holm by [MS12, Theorem C.1.10] and ds = D u ⊕ B for bounded linear operator B = ˙ J duj ,is surjective, [MS12, Lemma A.3.6] implies that ds has a right inverse. Thus 0 is a regu-lar value of s ( u, J ) = ∂ J ( u ) and by the Implicit Function Theorem [MS12, Theorem A.3.3] s − (0) = ˆ˚ M J is a C (cid:96) − -Banach submanifold of B ,pβ × J (cid:96)ω ( Y, U ). Separability of ˆ˚ M J isinherited.Now consider the projection π : ˆ˚ M J → J (cid:96) given by ( u, J ) (cid:55)→ J . This is Fredholm becauseit has the same kernel and cokernel as D u from [MS12, Lemma A.3.6]. Also its linearizationis surjective since it’s a projection. So we have regularity of π at ( u, J ) whenever J isregular. Hence we have the hypothesis of [MS12, Theorem A.5.1 (Sard-Smale Theorem)](which relies on the infinite-dimensional inverse function theorem, [MS12, Theorem A.3.1(Inverse Function Theorem)]). The result of Sard-Smale implies that these regular J values MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 75 are dense, i.e. J reg is dense in J ω ( Y, U ) as we wanted. So in particular we have existence ofregular J . (cid:3) This type of problem shows up often so it has a name.
Definition 5.46. A Fredholm problem concerns the zero set of a
Fredholm section of aBanach bundle. I.e. a section whose linearization is a Fredholm operator, namely dim ker -dim coker is finite and whose image is closed. It’s the set-up for the moduli spaces in question(the main part or fiber products of moduli spaces that show up when Gromov compactifying)as the zero sets of Fredholm sections. Regular values of a Fredholm problem put additionalstructure on the moduli spaces, enabling us to count them.
Lemma 5.47.
The set of parametrized (i.e. before quotienting) J -holomorphic discs u :( D , ∂ D ) → ( Y, L | γ ) for J ∈ J reg is a manifold of finite dimension given by index ( D u ) .Proof from [MS12, Theorem 3.1.6 (i)] . Charts are given by F u : W ,p ( D , ∂ D ; u ∗ T Y, u ∗ T L ) → L p ( D , Λ , ⊗ j u ∗ T Y ) ξ (cid:55)→ ∂ J (exp u ξ )using the exponential map to obtain a diffeomorphism of an open set around 0 in F − u (0)to a neighborhood of u in the space of parametrized J -holomorphic discs. Regularity of J implies d F u (0) = D u is surjective. The implicit function theorem [MS12, Theorem A.3.3]implies these are smooth manifold charts after restricting to potentially smaller open sets.This does not depend on p because J -holomorphic maps u are smooth by elliptic regularity[MS12, Proposition 3.1.10], when J is smooth. Note that [MS12, Appendix B (EllipticRegularity)] covers the necessary background with totally real boundary conditions e.g. theLagrangian boundary condition case here. See also [Weh13, Lectures 4–6]. The dimensionstatement follows from the tangent space of the moduli space being given by ker( D u ), andalso that coker( D u ) = 0, so their difference i.e. the Fredholm index, is also the dimension ofthe moduli space. (cid:3) Remark 5.48.
Even for non-regular J , we can still construct a Fredholm problem by [MS12,Theorem C.1.10 (Riemann-Roch)], which is proven in the case of Lagrangian boundarycondition. How we get the smooth structure will be a different matter though, because J isnot regular. This is where abstract perturbations of the ∂ J -operator are used. Compactification.
We will show that the moduli space, which now has a smooth manifoldstructure, is already compact. Then we take the zero dimensional part, which as a compact0-manifold is now something we can count. The more general case of compactifying andputting on a smooth structure is by gluing, e.g. [MS12, § J ’ we mean that transversality should hold for evaluation maps at marked points on discsand spheres. So we excluded disc bubbling and strip breaking. We now show that, for regular J , themoduli space of a somewhere injective disc union a simple sphere is a manifold of negativedimension, meaning it is empty and can be excluded. Lemma 5.49 (Excluding bubbling in our setting) . There exists a dense set J reg ( Y, ∂Y ; D ∪ P ) of J regular for the moduli space of maps with domain a simple sphere attached to asomewhere injective disc with one boundary marked point. The maps are section-like hencesomewhere injective. Corollary 5.50.
The moduli space of stable configurations consisting of a disc with onemarked boundary point identified at its center to one or more sphere bubbles for regular J has negative dimension, which is empty. In particular, the moduli space of any somewhereinjective disc passing through the open set U union any configuration of multiply-covered andsimple spheres can be excluded.Proof of Corollary 5.50. The Riemann-Roch theorem [MS12, Appendix] implies the dimen-sion of the manifold cut out by the regular J is of negative dimension, specifically dimension −
2. Lazzarini’s result [Laz11] implies any disc can be decomposed into simple discs andhis paper [Laz00] shows that any J -holomorphic disc contains a simple J -holomorphic disc.Thus if we had a nonempty configuration as in the statement of the corollary, we wouldhave a non-constant map in the case of a simple disc union a simple sphere, by factoringthrough the multiple covers and taking one simple disc that goes through the sphere. Butthis is a contradiction, so there couldn’t have been any such nonempty moduli spaces tobegin with. (cid:3) Proof of Lemma 5.49.
The proof is similar to the previous proof of existence of regular J ,however we have an additional constraint which is the point of attachment between the discand the sphere. We have dense sets of J regular for each component (the disc and thesphere); the disc was described earlier and the sphere situation is done in [MS12, Chapter3]. This proof will involve checking that there is still a dense set of J in the intersection ofthese two dense sets which interact well at the point where the disc and sphere intersect.Intersect the dense sets of regular J for the sphere and disc separately. Consider U ⊂ (cid:83) J ∈ ˆ˚ M ( Y,∂Y ) M ( A D ; J ) × ˆ˚ M ( A P ; J ) where A denotes the respective homology classes and u D ( D ) (cid:42) u P ( P ) (in contrast with the case of just spheres where we require that the imagesnot be equal). Namely, U is the subset of pairs of maps where the disc image isn’t containedin the CP image. We have the pointwise constraint that the sphere and disc are attachedat a point. So we need transversality of the evaluation map U → Y × Y . More specifically,the disc and sphere must intersect at a marked point which we place at 0 in the domain. Wealso fix a point on the boundary of the disc so there are no nontrivial automorphisms.Using Sard-Smale we can deduce that U is a manifold after an additional check at the in-tersection point. This is from [MS12, Chapter 6]. In order to show the evaluation map at0 is transverse, we want the linearized evaluation map to be surjective. So we select anytwo tangent vectors in the codomain at (0 D , P ), and then construct two ˙ J supported intwo disjoint small balls, one on each component, to ensure surjectivity. Each ball should notintersect the other component. This is possible because the Lagrangian boundary conditionimplies ˙ J is zero near the boundary and so if it’s only supported on a small ball in the MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 77 interior it is of this form; then we can extend each ˙ J by zero and simply add them.We now construct the ˙ J . We follow [MS12, § §
6] for transversality of the evaluation map. Note that the reference considers thecase of a sphere, and the argument works for the case of a disc because we have less totest due to more geometric constraints. Recall the construction of η on page 5.4 in KeyRegularity argument.
We can construct such an η in both cases of sphere or disc, andalso require that ξ now vanish on the point of intersection of the sphere and disc. We arestill working on a single component, P or D . This surjection tells us that we can find a ξ pointing in a specified direction at a specified point and tangent to its respective disc orsphere moduli space. Moreover one can do this in a small neighborhood around a specifiedpoint that is not the intersection point. Hence we can add vectors that work on differentcomponents using bump functions to extend each one by zero.The implicit function theorem puts the structure of a Banach submanifold on (cid:91) J ∈J ( Y,∂Y ) ˆ˚ M ( A D ; J ) × ˆ˚ M ( A P ; J )We want to prove that U (the preimage under the evaluation map at 0 of the diagonal)is a submanifold. Elliptic bootstrapping and the implicit function theorem in Appendicesof [MS12] apply because they are proven for totally real boundary conditions, such as La-grangian boundary conditions. Since the linearized evaluation map is surjective on vectorsas shown in the previous paragraphs, by Sard-Smale the subset of the universal space wheremaps respect the pointwise constraint is a manifold and we have existence of a dense set ofregular J for the disc and sphere so that the evaluation map at their intersection is trans-verse. Note also that we have omitted discussion of the asymptotic behavior at strip-likeends in our sketch of the functional analysis setup; the function spaces we consider and theirtopology need to be appropriately modified in the strip-like ends to enforce the asymptoticconditions, see [Sei08]. This concludes the proof of Lemma 5.49. (cid:3) Now that we have a regular J for the disc union sphere configuration, we have that themoduli space of such configurations is a manifold. In particular, the manifold has dimen-sion given by the Fredholm index, which is < M J (( D , ∂ D ) , ( Y, L ); J, β, pt ) of discs with one marked boundary point is already rigid i.e. di-mension zero by the point constraint so we don’t need to quotient by automorphisms, andit is also already compact as we’ve excluded limit behavior lying outside this moduli space.Hence we can write ˆ˚ M J (( D , ∂ D ) , ( Y, L ); J, β, pt ) as M J (( D , ∂ D ) , ( Y, L ); J, β, pt ).5.5.
Quasi-invariance of the Fukaya category on regular choices.
We pre-facedescriptions with “quasi” when the descriptions hold on the cohomology level.
Lemma 5.51.
Let J and J be two regular almost complex structures. Then they definequasi-equivalent Fukaya A ∞ -categories, i.e. isomorphic Donaldson-Fukaya categories.References for proof. We need to show that the Donaldson-Fukaya categories have isomor-phic object and morphism spaces. We also need to show that there exists a functor between them, namely that it respects composition. The Lagrangians depend only on the symplecticform, so remain unchanged upon changing the almost complex structure. Likewise for theFloer complexes, which are generated by their intersection points. Note that Seidel in [Sei08, § (10c)] discusses upgrading this equivalence to the A ∞ -category, in particular for the productor composition map. See [Sei08, § § (10c)].To show the morphism groups are isomorphic, we show that the differential ∂ is the samefor each J and J . This will follow from the use of a continuation map. (In general thisargument only shows that the Floer complexes with the Floer differentials for J and J are quasi-isomorphic, hence have isomorphic cohomology.) This is the moduli space fromsolutions of a single PDE that is the usual Cauchy-Riemann equation however instead of J we use J t where J t at time 0 is J and J t at time 1 is J . This defines what the functor doeson morphisms. In particular, this will require the existence of a path of regular J in thespace of all almost complex structures. This is discussed in Lectures on Floer Homology byD. Salamon.That existence of J t holds follows from a Sard-Smale argument, as in [MS12, Theorem 3.1.8]in their second edition book. The difference is that our Riemann surface has boundary (adisc with k punctures on the boundary corresponding to the moduli space in defining thestructure map µ k − ). So the base and fiber of the Banach bundle will be the same as in[MS12, pg 55] however the spaces of almost complex structures will restrict to ones that areidentically J outside of the open set U around the origin and moduli spaces will consist ofmaps on discs instead of spheres. The Sard-Smale theorem and elliptic regularity proven inthe Appendices of [MS12] already incorporate Lagrangian boundary conditions since theyassume totally real boundary conditions. (cid:3) Computing the differential on ( Y, v )6.1. Simplify using the Leibniz rule.
Figure 18 will be used to compute the differential ∂ , so we explain the geometry and notation of the picture first. Remark 6.1 (Notation) . Capital M k denotes structure maps on the total space of Y andlowercase µ k denotes structure maps on the torus fiber, in particular ∂ := µ . As illustrated inFigure 18, hom Y ( L i , L j ) decomposes into two hom groups on the fiber, hom right ( (cid:96) i +1 , (cid:96) j )[ − ⊕ hom left ( (cid:96) i , (cid:96) j ) = CF ( (cid:96) i +1 , (cid:96) j )[ − ⊕ CF ( (cid:96) i , (cid:96) j ). In particular, M will map from hom right tohom left in the Floer differential.We would like to use t x as a boundary condition and not an (cid:96) j because this will allow us tocount discs with boundary in the preimage of a moment map, as in [CO06]. Note that M : CF ( (cid:96) i , t x ) → CF ( φ H π ( (cid:96) i ) , t x ) CF (( φ H π ) − ) −−−−−−−→ CF ( (cid:96) i , ( φ H π ) − ( t x )) = CF ( (cid:96) i , t x ) ∼ = C where φ H π is the monodromy from Section 5.2. We’ve used that applying the diffeomorphism( φ H π ) − gives a bijection between intersection points, and t x = { ( ξ , ξ , θ , θ ) } ) θ ,θ ∈ [0 , π ) isinvariant under parallel transport because it only rotates angles. So we get CF ( (cid:96) i , t x ), whichhas only one intersection point. Lemma 6.2. M : CF ( (cid:96) i +1 , (cid:96) j ) → CF ( (cid:96) i , (cid:96) j ) can be computed from the data of M : CF ( (cid:96) i +1 , t x ) → CF ( (cid:96) i , t x ) over all x ∈ V . MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 79
Figure 18.
Leibniz rule(6.1) hom Y ( L j , T x ) ⊗ hom Y ( L i , L j ) M (cid:45) hom Y ( L i , T x )hom Y ( L j , T x ) ⊗ hom Y ( L i , L j ) M ⊗ + ⊗ M (cid:63) M (cid:45) hom Y ( L i , T x ) M (cid:63) Figure 19.
Diagram illustrating Leibniz rule
Proof.
First note that although M is an automorphism of CF ( L i , L j ) = CF ( (cid:96) i +1 , (cid:96) j )[ − ⊕ CF ( (cid:96) i , (cid:96) j ), for degree reasons and geometric considerations the only nonzero contribution isfrom the the first summand to the second summand.Let t j denote the intersection point of (cid:96) j and t x , and let p ki,j denote the k th intersection pointof (cid:96) i ∩ (cid:96) j , where 0 ≤ k < ( i − j ) as described in the definition of the Fukaya-Seidel categoryabove. We will use the diagram in Diagram 6.1 below. The crux of this argument relies onthe Leibniz rule, which is one of the associativity relations in an A ∞ -category. The homs inthis diagram are in the total space Y . We can reduce some calculations to the fiber, becauseeach hom on the total space is a direct sum of Floer groups over two fibers, namely over thepoints of intersection of the curves given by their projection to the base. For the geometricreasoning behind the diagram, see Figure 18.We can simplify this diagram to involving homs only in the fiber, as done in Diagram 6.2below. The domains for M and M have more terms, but below we only list the ones givingthe nonzero contributions. The rest are zero because there are no bigons on two Lagrangians (6.2) hom left ( (cid:96) j , t x ) ⊗ hom right ( (cid:96) i +1 , (cid:96) j ) (cid:51) t j ⊗ p ki +1 ,j M (cid:45) hom middle ( (cid:96) i +1 , t x )hom( L j , T x ) ⊗ hom( L i , L j ) M ⊗ + ⊗ M (cid:63) M (cid:45) hom left ( (cid:96) i , t x ) M (cid:63) Figure 20.
Simplified diagram on fibersin the fiber, hence the only bigons must be over one in the base. Also t x is invariant underthe monodromy hence we don’t need to include monodromy of how the fiber Lagrangianchanges in these maps.Subscripts in the simplified diagram indicate which fiber the homs are referring to from thethree intersection points of Figure 18, which we refer to as left, middle, and right respectively,going from left to right.Differentiating the product M , the Leibniz rule implies (where l = j − i and so p ki,j indexedover k can instead be written as p e,l indexed over e )(6.3) M ( M ( t j , p e,l − )) = M ( M ( t j ) , p e,l − ) + M ( t j , M ( p e,l − )) = M ( t j , M ( p e,l − ))because M ( t j ) = 0, since t j is of degree 0 and there is nothing in degree 1 at the otherintersection points of the two Lagrangians; note that t i is of degree − • RHS of Equation (6.3): M ( p e,l − ) := (cid:80) ˜ e α ˜ e ( e, l ) p ˜ e,l where α ˜ e ( e, l ) is the count of bigonsbetween p e,l − and p ˜ e,l weighted by their area. (In particular, since τ was the complexparameter on the B -side, it is the Novikov parameter here on the symplectic side.) Thisis the differential we are looking for. • Then: M ( t j , M ( p e,l − )) = (cid:80) ˜ e α ˜ e ( e, l ) M ( t j , p ˜ e,l ) = ( (cid:80) ˜ e α ˜ e ( e, l ) n ˜ e ( l )) t i where n ˜ e ( l ) is theweighted count of triangles (in a fiber, for degree reasons) with vertices at t i , p ˜ e,l , and t j .Since it’s in the fiber, we can compute n ˜ e ( l ) directly as in Lemma 2.22, as follows. p k (cid:48) ij t i t j (cid:96) i t x (cid:96) j Figure 21.
A triangle in V ∨ contributing to µ , viewed in ξ , ξ plane in theuniversal cover R The three vertices of the triangle are on lifts of (cid:96) i ∩ t x (cid:51) t i , (cid:96) j ∩ t x (cid:51) t j , and (cid:96) i ∩ (cid:96) j (cid:51) p ˜ e,l .Translate p ˜ e,l so that it lies in the fundamental domain for the Γ B -action. p ˜ e,l = (cid:18) γ i ∩ j j − i , − iλ (cid:18) γ i ∩ j j − i (cid:19)(cid:19) MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 81
Along t x , the ξ stay constant at some translate of a , say a + γ . Thus the sum of the ξ going around the triangle equaling zero implies the amount we add to the ξ coordinate,moving along the (cid:96) i and (cid:96) j directions from p ˜ e,l , are equal. Thus:(6.4) a + γ = γ j ∩ i j − i + ξt i = (cid:18) γ i ∩ j j − i , − iλ (cid:18) γ i ∩ j j − i (cid:19)(cid:19) + ( ξ, − iλ ( ξ )) t j = (cid:18) γ i ∩ j j − i , − iλ (cid:18) γ i ∩ j j − i (cid:19)(cid:19) + ( ξ, − jλ ( ξ ))Thus the triangle is half of the parallelogram spanned by:(6.5) (cid:28) a + γ − γ i ∩ j j − i =: ξ, − iλ ( ξ ) (cid:29) and (cid:104) ξ, − jλ ( ξ ) (cid:105) This is in the 2-plane spanned by (cid:104) ξ, (cid:105) and (cid:104) , λ ( ξ ) (cid:105) . As before, the area of the triangleis half of (cid:104) ξ, λ ( ξ ) (cid:105) times det (cid:18) − i − j (cid:19) = ( j − i ) . That is n ˜ e ( l, x ) = (cid:88) γ ∈ Γ B τ − ( j − i ) κ ( log τ | x | + γ − γi ∩ jj − i ) = s ˜ e,l [log τ | x | ](1)where [log τ | x | ] denotes a shift in the Γ B -lattice by ξ = log τ | x | . • LHS of Equation (6.3): M ( t j , p e,l − = n e ( l − · t (cid:48) i +1 , noting that the fibration is trivial-izable in the beige region of Figure 18. • Then: M ( t (cid:48) i +1 ) := C ( x ) s ( x ) · t i will be computed by a homotopy argument where C ( x ) isan infinite series and by Theorem 6.4 the disc count is s ( x ), our original theta function interms of the Novikov parameter τ . • Thus Equation (6.3) becomes C ( x ) s ( x ) · n e ( l − · t i = M ( M ( p e,l − , t j ) = (cid:32)(cid:88) ˜ e α ˜ e ( e, l ) n ˜ e ( l ) (cid:33) t i therefore comparing the coefficient on t i we find that C ( x ) s ( x ) · n e ( l −
1) = (cid:32)(cid:88) ˜ e α ˜ e ( e, l ) n ˜ e ( l ) (cid:33) Back to the proof of Lemma 6.2. We can compute the ∂ on (cid:96) i ∩ (cid:96) j given the informationabove. Consider the following two maps:(6.6) Hom( L i +1 , L j ) s ⊗ −→ Hom( L i , L j )Hom V ∨ ( (cid:96) i +1 , (cid:96) j ) ∂ −→ Hom V ∨ ( (cid:96) i , (cid:96) j )These are linear maps on vector spaces. Let B denote the basis s e,l and p e,l above for ( i, j ).Then consider s ⊗ s e,l − = s , · s e,l − = (cid:88) ˜ e (cid:32)(cid:88) η τ − ( l − ) κ (cid:16) η − ( l − γ ˜ el (cid:17) (cid:33) s ˜ e,l We would like to prove the equalities in the ? = below:(6.7) ∂ ( p e,l − ) = (cid:88) ˜ e α ˜ e ( e, l ) p ˜ e,l ? = C (cid:88) ˜ e (cid:32)(cid:88) η τ − ( l − ) κ (cid:16) η − ( l − γ ˜ el (cid:17) (cid:33) p ˜ e,l ⇐⇒ α ˜ e ( e, l ) ? = C (cid:32)(cid:88) η τ − ( l − ) κ (cid:16) η − ( l − γ ˜ el (cid:17) (cid:33) What we know is that n ˜ e ( l, x ) = s ˜ e,l [log τ | x | ](1) and for all x ∈ ( C ∗ ) / Γ B C ( x ) s ( x ) · s l − ,e [log τ | x | ](1) = (cid:32)(cid:88) ˜ e α ˜ e ( e, l ) s ˜ e,l [log τ | x | ](1) (cid:33) So we have the desired equality pointwise at each x , and using the formula for multiplicationof theta functions we see that:(6.8) C ( x ) s ( x ) · s l − ,e [log τ | x | ](1) = C ( x ) (cid:88) ˜ e (cid:32)(cid:88) η τ − ( l − ) κ (cid:16) η − ( l − γ ˜ el (cid:17) (cid:33) s ˜ e,l Since this holds for all x , we get the desired equality as follows. s e,l [log τ | x | ](1) = (cid:88) γ ∈ Γ B τ − lκ ( γ +log τ | x |− γ e,l /l ) = (cid:88) γ ∈ Γ B τ − lκ ( γ − γ e,l /l ) − lκ (log τ | x | )+ l (cid:104) log τ | x | ,λ ( γ − γ e,l /l ) (cid:105) = τ − lκ (log τ | x | ) (cid:88) − γ ∈ Γ B τ − lκ ( − γ + γ e,l /l ) | x | − lλ ( − γ + γ e,l /l ) (6.9) = τ − lκ (log τ | x | ) (cid:88) γ ∈ Γ B τ − lκ ( γ + γ e,l /l ) | x | − lλ ( γ + γ e,l /l ) = τ − lκ (log τ | x | ) s e,l ( | x | )Hence for x positive real we obtain:(6.10) s ( x ) · s l − ,e [log τ x ](1) = s , ( x ) · τ − ( l − κ (log τ x ) s l − ,e ( x )= τ − ( l − κ (log τ x ) (cid:88) ˜ e (cid:32)(cid:88) η τ − ( l − ) κ (cid:16) η − ( l − γ ˜ el (cid:17) (cid:33) s ˜ e,l and multiplying by C ( x ) this also equals(6.11) (cid:88) ˜ e α ˜ e ( e, l ) s ˜ e,l [log τ x ](1)= τ − lκ (log τ x ) (cid:88) ˜ e α ˜ e ( e, l ) s ˜ e,l ( x )Thus comparing coefficients we see that ∂ ( p e,l ) = (cid:88) ˜ e α ˜ e ( e, l ) p ˜ e,l MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 83
Figure 22.
The homotopy between ∂ (left) and the count of discs we compute (right)= C ( x ) τ κ (log τ x ) (cid:88) ˜ e (cid:32)(cid:88) η τ − ( l − ) κ (cid:16) η − ( l − γ ˜ el (cid:17) (cid:33) p ˜ e,l (6.12) s ⊗ s e,l = (cid:88) ˜ e (cid:32)(cid:88) η τ − ( l − ) κ (cid:16) η − ( l − γ ˜ el (cid:17) (cid:33) s ˜ e,l and not only have we shown the statement of Lemma 6.2 but we’ve also shown ∂ ∝ s ⊗ aslinear maps on vector spaces, pending the proof of Theorem 6.4. (cid:3) Cobordism between generic choice and specific choice for computation.
Inthis section we will discuss obstruction bundles, whose definition can be found in [MS12, § v , depicted in Figure 22. The left side, with a generic regular J , indi-cates the M ( t (cid:48) i +1 ) we want to calculate. The right side with the standard J = i indicatessomething we can compute. Note that since the set of J regular for all configurations (discsand discs union spheres) is nonempty and dense, and the maps are J sections away from aneighborhood of zero we can claim that J is a limit of such J by the denseness of the regular J . I.e. we have a path of J ’s limiting to J , so these J ’s perturb J near the zero fiber. Thenwith the construction of a cobordism, we can compute the left by computing the right.There is existing theory for computing open Gromov-Witten invariants of J -holomorphicdiscs with boundary on a moment map fiber, one marked point, and passing through thesingular fiber of Y once (as is the case here because they are 1-1 in general and v -sectionswith J ). This is the setting with (cid:83) circle t x that we see on the right side of Figure 22 when weallow J to vary to J at the right end. However, M as is counts bigons through the singularfiber with boundary over a bigon as in the left of Figure 22, instead of a circle. So following[Sei08, § M to M ( cp, · ) where c counts J -holomorphic discs with boundaryon (cid:83) circle t x and marked point p . This deformation constructs the homotopy. Note that inthe book, he deforms the fibrations. However in this setting, the fibration stays the same,while the Lagrangian boundary conditions are deformed via an automorphism of Y relativeto the boundary. So we will need a gluing argument. Figure 23.
Gromov compactificationSee Figure 23 for a pictorial depiction of the analytic and algebraic steps involved. Inparticular, we must abstractly perturb ∂ J in order to see the configurations we want tocount on the right hand side as sitting in a moduli space. At the moment, the moduli spacewith J has too many elements, a two-dimensional family of elements for each configurationthat we only want to count once. Let (cid:83) γ r (cid:96) denote the Lagrangian boundary condition attime r depicted in Figure 22. Lemma 6.3.
Choose β and let β r = φ r ∗ ( β ) where φ r : ( Y, (cid:83) γ (cid:96) ) ∼ = −→ ( Y, (cid:83) γ r (cid:96) ) is a dif-feomorpism inducing an isomorphism φ r ∗ on homology for < r ≤ . Then for a suitablefamily J r described in the proof, (cid:91) r ∈ (0 , ˆ M (( Y, (cid:91) γ r (cid:96) ); β r ; J r ) /Aut where Aut denotes strip-translation, has the structure of a compact topological 1-manifold.In particular, the signed count of its boundary is 0.Proof.
The (0 , manifold structure. We use background from [Weh14, Lecture 9],[Weh13, Lecture 14], and [MS12, § B k,p := { ( φ − r ◦ u, r ) | u : R × [0 , → ( Y, (cid:91) γ r (cid:96) ) ∈ W k,p } MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 85
We can classify the tangent space to a path in ˜ B as one where the derivative of the path atthe boundary is a vector that is a sum of a vector in the tangent space to the Lagrangianboundary and a vector corresponding to the flow of the isotopy φ r . As in [MS12], this willgive us a 1-manifold structure on the set of maps u so that ∂ J r ( φ − r ◦ u ) = 0. Namely, givena Fredholm problem E → B describing the r = 1 moduli space, we can pull back:( φ r ◦ pr ) ∗ E (cid:45) E ˜ B k,p (cid:63) φ r ◦ pr (cid:45) B (cid:63) The bundle of the right vertical arrow admits a regular J by Lemma 5.44. We can choosea path of perturbations given by sections of the obstruction bundle, since at J the cok-ernel of D u is greater than zero but has constant dimension, see Claim 6.5. See [MS12, § r = 1 the perturbation is 0 and at r = 0 the perturbation p issuch that ( ∂ J + p ) − (0) counts the limiting curves on the right in Figure 23. We thenconsider the Fredholm problem in the left downward arrow and find a perturbation of thesection determined by the family in the previous sentence, which vanishes on the bound-ary of ˜ B k,p . Since there isn’t disc bubbling for each r , the boundary of this is just the r = 0 end (and r = 1 end once we compactify). So we obtain a 1-manifold structure on (cid:83) r ∈ (0 , ˆ M (( Y, (cid:83) γ r (cid:96) ); β r ; J r ) /Aut . Topological 1-manifold structure.
The next step will be to Gromov compactify at the r = 0 end. Note that φ r does not have a limit at r = 0, as it becomes very degenerate andis not a diffeomorphism. So instead we consider elements as u in this moduli space insteadof φ − r ◦ u . In order to preglue, which gives the topology at the r = 0 end, we trivialize thenormal bundle in a neighborhood of where we want to glue, and then interpolate linearlybetween the two maps. See [Weh14, lec 3, 1 hr]. Note that we take the gluing parameter tobe e − l which goes to zero as the gluing length l goes to infinity, where we have the configu-ration of two discs in the right side of Figure 22. (Note that even without trivializing, thereare scaling functions on the normal bundle. E.g. the gluing parameter for the two discs inthe base could be a cross ratio of four points around the belt that is getting pinched to apoint.) Preglue the domains.
We remove a neighborhood of the puncture first. In the ( s, t )coordinates on strip-like ends, we glue ( s − l, t ) to ( s, t ). That is, we place an amount l inthe R direction on one strip overlapping onto the other strip. The two parts separately givethe r = 0 case and the two parts glued together is the r = (cid:15) > −∞ , × [0 , → ( −∞ , × [0 , π ]by · π . Then map to the lower half of an annulus by e − z , and then lastly to the right halfof a disc with a puncture at 1 by z + iz − i . See Figure 17. The reason why the preglued map isclose to the J r -holomorphic glued map one would obtain by Newton iteration is because bycontinuity ∂ J r of the glued map is still small; if it were constant on the glued part then itwould actually be holomorphic. We interpolate slowly so it is still close to constant. Preglue the maps.
We define a new preglued map u R u ∞ on the preglued domain de-fined above, where u and u ∞ denote the two maps on discs at the r = 0 end. We knowhow to interpolate in the base v ∈ C coordinate using (1 − ρ ) u + ρu ∞ − i . Then we applythis same linear interpolation in the moment map coordinates ( ξ , ξ , η, θ , θ , θ η ) to pregluemaps u to the total space. This choice of interpolation for the pregluing ensures that, whenpregluing the disc bounded by (cid:83) circle t x and a strip with boundary on (cid:83) γ t x and (cid:83) γ (cid:96) i , theresulting preglued map has boundary on (cid:83) γ r (cid:96) i ∪ t x as in Figure 22. For in the fiber direction,the values of ( ξ , ξ ) on the two components agree along the boundary and the interpolationpreserves them. In the base direction, we can choose the family of paths γ r to be the familyof paths obtained from γ and the circle centered at the origin by our interpolation procedure. Gromov compactness, maps limit to preglued maps.
With J and no α spheres inthe homology class β , we know the moduli space has one disc, by Theorem 5.37. All discsby themselves are regular for J . Then we look at a limit of J r -holomorphic discs u r as r goes to 0, namely they solve the Cauchy-Riemann equation with J r . After possibly passingto a subsequence, then lim r → u r =: u R u ∞ because of the exclusion of disc bubbling andstrip breaking for a fixed Lagrangian by the geometry of ( Y, v ). This is also discussed in[CLL12, Proposition 4.30]. Note that there are more pseudo-holomorphic discs with J than J . The latter only has one in each homology class. This is because with J some of the discsmust converge to a disc union bubbles as r → J := J to J . See Figure23. Deducing result from cobordism.
We have constructed a cobordism between ∂ − J (0)and ( ∂ J + p ) − (0) for an admissible perturbation given by e.g. a section of the obstructionbundle (equivalently, a Kuranishi structure with one chart since the cokernel has constantdimension). So their counts are equal. In particular, one can count the moduli space( ∂ J + p ) − (0) by taking the Euler number of a suitable section of the obstruction bundle.The curve count is done in [KL19] where they use the result of [Cha11], who shows thatone can add in an additional ray to compactify these configurations of disc union sphere toa configuration of all spheres, and the Kuranishi structure on this closed Gromov-Witteninvariant is isomorphic to that on the original open Gromov-Witten invariant. Then theclosed curves can be counted by the Picard-Fuchs equation from algebraic geometry. Thiscan now be computed using the mirror theorem of Givental. See Figure 6.4 for an outline ofthese steps. (cid:3) Count of discs regular for J . In this section we take J = J and consider modulispaces of discs only, for which J is regular. In other words, we only consider homologyclasses β that arise from discs. The homology classes in π ( Y, L ) that we consider cover adisc in the base of v around 0, and pass through the central fiber in one point. Equivalently,taking their real part, they can be depicted in ∆ ˜ Y as a line from a fixed point on the interiorof the polytope to a facet. Varying the facet allows one to enumerate all the homologyclasses, done in [CO06]. Theorem 6.4.
The count of discs equals the defining theta function, up to a change ofcoordinates.Proof.
The count in this setting of discs in a toric variety is given in [CO06]. Recall thatwe weight by τ − (cid:82) ω in the count. (Note that τ corresponds to the complex structure on MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 87 the genus 2 curve, so it corresponds to the symplectic structure on the mirror. The com-plex structure on the mirror was in terms of T , see Figure 10.) If x i are coordinates onthe complex side on V and | x i | := τ ξ i where the point in the polytope we measure from is( a , a , a ) = ( ξ , ξ , η ), then from [CO06] the area of the disc intersecting the ( m , m ) facetis 2 π ( (cid:104) a, ν ( F m ,m (cid:105) + α ( F m ,m )).The facet equations are determined from Equation (4.10). In particular, ( ξ , ξ , η ) = (0 , , η = 0 plane. Denote the facets by F m ,m and let ν ( F m ,m )and α ( F m ,m ) denote the normal and constant defining the plane the facet lies in. Recallthat η ≥ ϕ ( ξ ) where ϕ ( ξ + γ ) = ϕ ( ξ ) − κ ( γ ) + (cid:104) λ ( γ ) , ξ (cid:105) . Suppose ( ξ , ξ , η ) ∈ F m ,m . Weknow from Claim (4.15) that Γ B acts on the moment map coordinates in the following way:(6.13)( − m γ (cid:48) − m γ (cid:48)(cid:48) ) · ( ξ , ξ , η ) = ( ξ − m − m , ξ − m − m , η − κ ( m γ (cid:48) + m γ (cid:48)(cid:48) ) − m · ξ )= ( ξ − m − m , ξ − m − m , η + m + m m + m − m ξ − m ξ )In particular, this point must be in F , . We also know ( ξ , ξ , η ) ∈ F m ,m . Plugging eachpoint into the equation of the corresponding facet, we find that: (cid:42) ν ( F m ,m ) , ξ ξ η (cid:43) + α ( F m ,m ) = 0= ⇒ (cid:42) ν ( F , ) , ξ − m − m ξ − m − m η + m + m m + m − m ξ − m ξ (cid:43) + α ( F , )(6.14) = η + m + m m + m − m ξ − m ξ = 0= ⇒ ν ( F m ,m ) = ( − m , − m , t , α ( F m ,m ) = m + m m + m So comparing the series from counting discs weighted by area for the differential, and thatof the theta function, we find that θ -function = (cid:88) n ∈ Z x − n x − n τ n t n (6.15) disc count by area = τ η (cid:88) n ∈ Z | x | n | x | n τ n t n which agree up to a change of coordinates. (cid:3) Count of spheres not regular for J .Claim 6.5. J is not a regular almost complex structure for disc + sphere configurations. Proof.
The standard J is multiplication by i in the toric coordinates. Recall that H ( v − (0)) (cid:54) =0 and v is holomorphic with respect to J , so submanifolds representing classes in H of thecentral fiber are holomorphic. Then nonzero Dolbeault cohomology implies the spheres arenot regular; this follows by the Riemann-Roch theorem and the fact that the cokernel of the ∂ J operator is Dolbeault cohomology. Since the cokernel is nonzero we find that D u is notsurjective for maps u arising from these holomorphic spheres. (cid:3) Remark 6.6 (Terminology) . Closed Gromov-Witten theory counts spheres, for which wecan use the algebraic geometry of stacks. Chapter 10 of [CK99] gives the stack definition ofmoduli spaces. A reference for an introduction to stacks is [Fan01].
Open Gromov-Witten theory involves counting discs with a Lagrangian boundary condition, and this boundarycondition is why we introduce analysis into definitions and use Fredholm problems to countthe moduli spaces.We will denote sphere classes in the central v -fiber by α and the class of the disc passingthrough the divisor D ij corresponding to the I := ( i, j ) facet by β ij . Let n β I + α denote thecount of the following moduli space: M β I + α ( J ) := { ( u, v ) : ( D , ( S ) k ) → Y | k ∈ N ∪ { } , u ( ∂ D ) ⊂ T x | circle , [ u v ] = β I + α, ( u, v ) ∈ C ∞ , ev ( u ) = ev ( v ) , µ ([ D I ]) = 2 , ∂ J ( u, v ) = 0 } × { p } / Aut(
Y, p ) Theorem 6.7 (Open mirror theorem proved in [KL19, Theorem 3.10]) . ˜ Y is a toric Calabi-Yau manifold of infinite-type. Then (cid:88) α n β I + α q α (ˇ q ) = exp( g I (ˇ q )) where q denotes the K¨ahler parameters, ˇ q the complex parameters, q (ˇ q ) the mirror map and g I (ˇ q ) := (cid:88) d ( − ( D I · d ) ( − ( D I · d ) − (cid:81) I (cid:48) (cid:54) = I ( D I (cid:48) · d )! ˇ q d where the summation over d is taken over all d ∈ H eff ( ˜ Y , Z ) such that − K ˜ Y · d = 0 , D I · d < ,and D I (cid:48) · d ≥ for all I (cid:48) (cid:54) = I . Remark 6.8.
Note that here we consider only a one-parameter family of values of K¨ahlerparameters, because we’ve fixed the symplectic form so that the three toric divisors x = 0, y = 0, and z = 0 have symplectic area 1 (these form the “banana manifold”). Namely, q = τ ∈ R and q α , in our notation, is τ ω ( α ) .Below in Figure 6.4 is a flow chart indicating the necessary background for understandingthe sphere count in Kanazawa-Lau [KL19]. Note that they use J = J as we are using here. Flow chart detailsGivental: [Giv98] . The Picard-Fuchs differential equation describes the behavior of peri-ods arising from Hodge structures on the complex side. Givental introduced the I and J functions, where I computes solutions of the Picard-Fuchs equation and J computes theGromov-Witten invariants. He proved a relation between these two functions, i.e. a mirrortheorem. Closed mirror theorem: [CCIT15] . The closed mirror theorem relates the I and J func-tion (defined in e.g. [CK99, § J function MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 89 [KL19]: infinite toric open mirror thmlimit argument [CCLT16]: open mirror thm[Cha11]: oGW = cGW [CCIT15]: closed mirror thm[Giv98]: I , J fns Figure 24.
Gromov-Witten theory background for mirror symmetry of toric varietieson the symplectic manifold corresponds to the I function on the complex manifold. Theseare functions of the K¨ahler and complex moduli, which are isomorphic. The mirror mapgoes between neighborhoods of a K¨ahler large limit point and a complex large limit point(maximally unipotent monodromy), see [CK99, § J -function and the GWpotential.In particular, [CK99, Equation (10.4)] gives the relation between differentials, intersectiontheory, and Gromov-Witten theory. The Picard-Fuchs equation [CK99, § q near maximally unipotent monodromy (denoted y k in [CK99]). The K¨ahler moduli q is denoted q k in the same reference.Givental’s mirror theorem for toric complete intersections is described in [CK99, § § The result of [Cha11] . In the Fukaya category one would like to compute open GWinvariants, i.e. discs with Lagrangian boundary conditions, so we would like to be able tocount these as well. There is a notion of “capping off” introduced in [Cha11] where, for atoric variety X Σ , one adds an additional ray to the fan Σ to define a partial compactification X Σ . This is done so the discs in the open GW count are “capped off” to become spheres. See[CCLT16, § Definition 6.9 (Kuranishi chart) . A Kuranishi neighborhood of p on moduli space X is thefollowing data: • V p smooth finite-dimensional manifold , possibly with corners • V p × E p → V p is the obstruction bundle , where E p is a finite-dimensional real vector space. • Γ p is a finite group which acts smoothly and effectively (no non-trivial element acts triv-ially) on V p , and E p linearly represents the group. • Kuranishi map s p is a smooth section of V p × E p (smooth map V p → Γ p ), and is Γ p equivariant. • ψ p is a topological chart which is a homeomorphism from the local model s − p (0) / Γ p to aneighborhood of p in X . • V p / Γ p or V p may also be referred to as a Kuranishi neighborhood (rather than the collectionof all these pieces of data). • o p is a point which the Kuranishi map sends to zero and the chart maps to p . • For references in the literature on gluing such charts, see [MTFJ19, Fukaya, Tehrani].
Kanazawa-Lau apply [CCLT16] to the infinite toric setting.
In [KL19] there is anotion of taking a limit to arrive at the infinite toric case, which is our setting as well beforewe quotient by the Γ B -action. They build on the open mirror theorem of [CCLT16] andcompute the sphere count as the coefficient of 1 /z in the mirror map. This concludes theoutline for the flow chart.6.5. Computation of the differential.Lemma 6.10.
Recall our notation above in the second paragraph of Remark 6.6. We makethe following definitions: • Fix A ∈ ∆ ˜ Y . • Set z = τ A . • Lagrangian boundary condition is (cid:83) circle t x over circle around origin in the base of v . • Fix a point pt constraint on this Lagrangian. • Fix disc homology class β ij = [ D ij ] in H ( ˜ Y , T x | circ ) .Then the differential can be computed from the c defined in Remark 5.7, which equals: (6.16) c = C · (cid:32)(cid:88) γ τ ω ( γ ∗ β ) (cid:33) where C := (cid:0)(cid:80) α n β + α τ ω ( α ) (cid:1) for n β + α defined in the proof and where (cid:80) γ τ ω ( γ ∗ β ) was com-puted in the disc count in Theorem 6.4. Lemma 6.2 explains why this suffices to computethe differential, which counts bigons bounded by L i ∪ L j .Proof. Define a Γ B × ( C ∗ ) -action on ˜ Y where ( γ, c γ ) acts by c γ ◦ γ and c γ is complexmultiplication. Specifically, γ ∈ Γ B sends D ij to some D i (cid:48) j (cid:48) , and c γ is defined by requiringthe point γ ∗ ( A ) in the moment polytope to map back to A via the ( C ∗ ) toric action (whilefixing the divisors). Thus Γ B × ( C ∗ ) acts on moduli spaces of curves with a fixed Lagrangianboundary condition, varying homology classes, and a marked point pt , by post-composition.Then we have an isomorphism of moduli spaces: (6.17) { ( u, pt ) , u : D → Y, pt ∈ ∂ D | u ( ∂ D ) ⊂ T x | circ , u ( pt ) = pt constraint , [ u ] = β ij , ∂ J reg ( u ) = 0 } ∼ = { ( u, pt ) , u : D → Y, pt ∈ ∂ D | u ( ∂ D ) ⊂ T x | circ , u ( pt ) = c γ ◦ γ ( pt constraint ) , [ u ] = ( c γ ◦ γ ) ∗ β ij , ∂ ( c γ ◦ γ ) ∗ J reg ( u ) = 0 } In particular, for a regular J reg as exists by Lemma 5.44, (so moduli spaces are manifolds)and introducing the point constraint (so they are zero dimensional), counting points in these MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 91 moduli spaces produces an infinite series of discs and no sphere bubbles (because we ex-cluded them). By denseness, we can choose J reg sufficiently close to J so a limit of regular J ’s limits to J . The disc will either converge to a disc or to a disc with a sphere bubbleconfiguration as we saw in Figure 23. This count of discs for J reg is hence proportional tothe differential in the Fukaya category.By [CLL12, Proposition 4.30], we know that the only homology clases that can appear in thecompactification are stable trees of the form β ij + (cid:80) i n i α i for some integers n i and spheres α i . Note that ( c γ ◦ γ ) ∗ J = J since multiplication by scalars is a holomorphic map. Theclaim we want to prove is that the defined moduli spaces are isomorphic as we vary thehomology classes. Applying these group actions should produce isomorphic moduli spaces,and we know then that the curve count for a particular homology class D ij + α will be thesame for all others and the counts will be the same so we can factor out the common factor.Namely the counts n β + α do not depend on the disc class β since there is a 1-1 bijectionbetween moduli spaces of sphere configurations showing up with D ij and with any other D i (cid:48) j (cid:48) , via the map c γ ◦ γ . That is because it has an inverse ( c γ ◦ γ ) − given by multiplicationby the inverse scalars.Suppose we write for an arbitrary disc and sphere configuration β + α (cid:48) =: γ ∗ ( β + α ) for fixed β with a suitable γ which then determines α . Then we can denote all n β + α (cid:48) independentof β and only depending on γ, α as n β + α (cid:48) = n γ ∗ ( β + α ) = n β + α . The last equality is true asfollows. We streamline notation below and use γ to incorporate both actions of γ and c γ .We choose J reg to also be regular for the homology class γ ∗ ( β I + α ) so γ ∗ J reg is regular forthe class β + α . Invariance on regular J by a continuation map argument (see Section 5.5)then implies M ( β I + α, J reg ) ∼ = M ( γ ( β I + α ) , γ ∗ J reg ) ∼ = M ( γ ( β I + α ) , J reg ) c = (cid:88) β,α n β + α (cid:48) τ ω ( β )+ ω ( α (cid:48) ) = (cid:88) α,γ ∈ Γ B n γ ∗ ( β + α ) τ ω ( γ ∗ β )+ ω ( α ) = (cid:88) γ ∈ Γ B (cid:32)(cid:88) α n β + α τ ω ( α ) (cid:33) · τ ω ( γ ∗ β ) = (cid:32)(cid:88) α n β + α τ ω ( α ) (cid:33) · (cid:32) (cid:88) γ ∈ Γ B τ ω ( γ ∗ β ) (cid:33) (6.18) = ⇒ C ( x ) = (cid:32)(cid:88) α n β + α τ ω ( α ) (cid:33) The first factor we can put in front, and the second is the multi-theta function describedabove in the computation of J -discs. Note that n β = 1 by Theorem 5.37, hence C = 1+(higher order terms) and is invertible.Now we put everything together. Now that we’ve proven Theorem 6.4, the item on page81 regarding M ( t (cid:48) i +1 ) is true so by Equation (6.12), we computed the differential up tothe function C ( x ). Then by Equation (6.18) we find C ( x ) = (cid:80) α n β + α τ ω ( α ) where n β + α are defined by the Open mirror theorem [KL19, Theorem 3.10] cited above in Theorem 6.7: (cid:80) α n β I + α q α (ˇ q ) = exp( g I (ˇ q )), where q denotes the K¨ahler parameters i.e. | q | α = τ ω ( α ) , ˇ q thecomplex parameters on V up to coordinate change, q (ˇ q ) the mirror map and g I (ˇ q ) := (cid:88) d ( − ( D I · d ) ( − ( D I · d ) − (cid:81) I (cid:48) (cid:54) = I ( D I (cid:48) · d )! ˇ q d where the summation over d is taken over all d ∈ H eff ( ˜ Y , Z ) such that − K ˜ Y · d = 0, D I · d < D I (cid:48) · d ≥ I (cid:48) (cid:54) = I . Thus(6.19) ∂ ( p e,l ) = (cid:88) ˜ e (cid:32)(cid:88) α n β + α τ ω ( α ) (cid:33) τ κ (log τ x ) (cid:32)(cid:88) η τ − ( l − ) κ (cid:16) η − ( l − γ ˜ el (cid:17) (cid:33) p ˜ e,l (cid:3) Fully-faithful cohomological embedding D b L Coh ( H ) (cid:44) → H F S ( Y, v )In this chapter we prove that the right vertical arrow of Theorem 1.2 is a fully faithful em-bedding, namely that the arrow is indeed a functor and that the morphism groups betweenobjects and images of those objects are isomorphic.On objects we map L| ⊗ iH (cid:55)→ L i . If φ H π is the monodromy of the symplectic fibration v : Y → C around the origin, then the symmetry of our definition of ω ensures that φ H π ( (cid:96) i )is Hamiltonian isotopic to (cid:96) i +1 by Lemma 5.24. Since Floer cohomology is invariant underHamiltonian isotopy, we can consider linear Lagrangians in the fibers. This allows us toobtain the bottom row of the following diagram, whenever j ≥ i + 2. When j < i + 2 thereare also Ext groups to consider and we get a long exact sequence instead, namely the lasthorizontal map is not surjective anymore. Hom ( L i +1 , L j ) ⊗ s −→ Hom ( L i , L j ) → Hom ( L i , L j ⊗ ι ∗ O H ) → CF ( (cid:96) i +1 , (cid:96) j ) ∼ = · (cid:63) C (cid:48)− ∂ −→ CF ( (cid:96) i , (cid:96) j ) ∼ = (cid:63) → HF ( L i , L j ) (cid:63) → Figure 25.
Proof of Main TheoremExt groups here are computed from injective resolutions:(0 (cid:45) L − (cid:45) O V (cid:45) O H (cid:45) ⊗ L j − i (cid:45) L j − i − (cid:45) L j − i (cid:45) L j − iH (cid:45) → H ( L j − i − ) → H ( L j − i ) → H ( L j − i | H ) → H ( L j − i − ) → H ( L j − i ) → H ( L j − i | H ) → H ( L j − i − ) → H ( L j − i ) → H ( L j − i | H ) MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 93
Our main result is that the left-side square in the diagram commutes, which then impliesthat the rightmost vertical arrow is an isomorphism as well. More precisely, we’ve shownin Lemma 6.2, Equation (6.12), and Lemma 6.10 that under the chosen isomorphisms ofLemma 2.22, the Floer differential ∂ agrees with multiplication by s ∈ H ( V, L ) up to themultiplicative factor C (cid:48) := τ κ (log τ x ) C ( x ), the open Gromov-Witten invariant in [KL19] fora particular choice of K¨ahler parameter. So scaling the first arrow on morphisms by C (cid:48)− gives a commutative diagram on the left, which we can do since C = 1 + . . . .Furthermore, this map on objects induces a functor because it respects composition. This isbecause the product structures on the groups in the right-hand vertical arrow of the diagramin Figure 25 are those naturally induced on the quotients, and the left two vertical maps arefunctorial by Lemma 2.22. So the induced isomorphisms on the cokernels of the horizontalmaps are also functorial. I.e. composition on the complex side versus Floer product on HF match under the constructed isomorphisms.Hence this provides the desired isomorphism between the morphisms groups for the functor D b L Coh ( H ) → H ( F S ( Y, v )) and the proof of Theorem 1.2.8. Appendix A: Enough space to bound derivatives
Lemma 8.1 (Estimates on bump function derivatives) . The definition corresponding toFigure 13 provides enough space to make log T -derivatives small.Proof. Region VII contains the point in its center where r x = r y = r z = T l/ and itsboundary circle is d = T l/ for whichever region d we are using. So ∆ log T r x = O ( l ) is oforder l . Region I is delineated by a rectangle in ( d I , θ I ) coordinates of length 2 T − l/p by T − l/p for avariable p which we constrain below. Namely,(8.1) Region I := { ( d I , θ I ) | ( d I , θ I ) ∈ [ T l/ , T l/ − l/p ] × [ − T l/p , T l/p ] } Thus ∆ log T d I = O ( l/p ). The angular coordinate requires a bit more to analyze since itgoes to zero when r y = r z .The curves delineating region II are θ II constant for the radially outward lines, and d IIA or d IIB constant along angular curves. Note that d IIA constant here is approximately r x constant.Label the points of Figure 13 as follows. Region II
Radial lines are d IIx constant, for x ∈ { A, B, C } . Two curves around the originthrough P , Q and P , Q respectively are θ II constant. • P = B ≈ ( l/ , l/ − l/p, l/ l/p ) • P ≈ ( l/ − l/p, l/ − l/p, l/ l/p ).This follows because the sliver from P to P is θ II = log T r y − log T r x constant, while P is obtained by moving E up along constant d I ≈ ( T r x ) so r x ( P ) ≈ T l/ − l/p . Then constant θ II implies: θ II ( P ) ≈ l/ − l/p − l/ l/ − l/p = θ II ( P ) ≈ log T r y − ( l/ − l/p )= ⇒ log T r y ≈ l/ − l/p • Q ≈ ( l/ , l/ − l/p, l/ l/p )From P to Q along d IIA constant we increase r y by a factor of T − l/p keeping r x approximately constant. • Q ≈ ( l/ − l/p, l/ − l/p, l/ l/p )From Q to Q we have θ II is constant and at Q we have θ II ( Q ) ≈ l/ − l/p − l/ l/ − l/p . From P to Q we have r x approximately constant hence log T r x at Q isapproximately l/ − l/p solog T r y ≈ ( l/ − l/p ) + ( l/ − l/p ) = 3 l/ − l/p The r z coordinate is determined by r x r y r z = T l .Now finally we get a condition on p . We want r x >> r y everywhere in region IIA so thatcontour lines for d IIA look roughly as they are drawn and approximations for d IIA are valid.Looking at Q this means T l/ >> T l/ − l/p and Q gives the same constraint. Hence weneed 1 / − /p > p >
16. E.g. take p = 17.Recall Figure 13 and that from A to B r y moves through l/p orders of T magnitude whilefrom D to E it moves through 3 l/ p which is a lot more for small T . Region I: α , α . Since α is a function of d I ≈ T [ r x − ( r y + r z )] ≈ ( T r x ) in regionI because r x is many orders of magnitude bigger than r y and r z get in that region. Weneed to see how log T ( T r x ) changes. Recall that r x changed by l/p orders of magnitude, solog T ( T r x ) changes by approximately 2 l/p . Thus the log derivative can be made as small aspossible, as explained above.For α (cid:48)(cid:48) log3 we want to know the change in slope over log d time. The derivative goes from 0to l in approximately l log time. Think of a bump function from 2 / α are bounded by a constant times T /l . Region I: α . However, in the calculation for α , we end up dividing by θ . So α will needto be constant for a short while in the middle, so we don’t divide by zero. In the calculationsabove, we have cut out a region where T < ( r y /r z ) < T − . In this region α ≡
0. To dothis, we need to make sure that we have at least one order of magnitude difference between r y and r z . This is fine in region I because we have r y and r z many orders of magnitude apartat B and C , with even more discrepancy as we move out to E and F . (Note however, thereverse would have happened in region II. In other words, r z gets smaller as we move out,so r x and r y get bigger, and constant r y − r x means they will get closer and closer togetheras we move out.)This one is a function of θ I ≈ T [ r y − r z ] or its negative. Furthermore, we’re taking out asliver around the axis where T < ( r y /r z ) < /T . So we need to check there’s enough spaceleft. At the bottom where C and F are, θ I ≈ − ( T r z ) which is on the order of − T ( l − lp +1 ) MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 95 as seen above. Then we stop when r z /r y = 1 / √ T . At this point r z and r y are basicallyequal, since they are only about one T -order of magnitude apart and both really small. Atsome fixed d I , we have r x ≈ T l/ − tl/p where t is fixed at some number between 0 and 1. So r z = √ T − r y and: r x r y r z = T l = ⇒ T l/ − tl/p · r y · √ T − r y = T l = ⇒ r y = √ T · T l/ tl/p Hence θ I ≈ T [ r y − r z ] = T r y (cid:32) − (cid:18) r z r y (cid:19) (cid:33) ≈ T √ T · T l/ tl/p (1 − T − ) ≈ − T l/ tl/p +3 / ∵ T << θ I changes when we cut out this sliver to make sure α has enoughspace for the log derivatives. We find θ I decreases from order of magnitude 3 l/ − l/p + 2to order of magnitude 3 l/ tl/p + 3 /
2. This means a net change of(3 l/ − l/p + 2) − (3 l/ tl/p + 3 /
2) = − l (cid:18) p + tp (cid:19) + 1 / l , so we’re still okay for the log derivative of α because this is the de-nominator of the log derivative which can be made very large because of the l . Note that weonly care about α in region I since it’s constant at 1 / Region II.
Also, there is enough space in θ II and log d IIA for α and α , α respectively.Recall d IIA ≈ T [ r x −
12 ( r y + r z ) + 32 α · r y ]Since everywhere in region IIA we have r x >> r y , the leading order term in d II is ( T r x ) .Likewise since θ II is linear in log T r y − log T r x , for it to stay constant we find that both log T r y and log T r x increase the same amount along contour lines. At P and P , θ II ≈ log T ( T l/ − l/p ).At Q and Q , θ II ≈ log T ( T l/ − l/p ). Thus the change is l/p and α (cid:48) ≈ ∆ α / ∆ θ II ≈ p/l .This can be made small by taking l large.Now we check d IIA for α and α . At the smaller value we have d IIA ≈ ( T r x ) at P where r x has exponent l/
2. At the larger value we get l/ − l/p . So the change is 2 l/p and thederivatives in α and α are approximately p/ l times whatever the full range of α i is. Againthis can be made small.Second-order derivatives also have enough space. In the bump functions α , . . . , α the vari-able we’re taking the derivative with respect to has space k · l for some k > / l really big,we can ensure that while this is happening, the second derivative doesn’t get too big. Thegraph of the bump functions won’t be linear because they have to level off at the endpointsof their support. But with enough space, we can make sure they don’t turn too quickly fromhorizontal to linear. Other regions covered by symmetry.
The argument that there is enough space inIIC follows from IIA by swapping r x and r y in the calculations. The only variable in re-gion IIB is d IIB and we take the two radial curves to be where d IIB is constant, with thesame amount of change in the variable as in d IIA . Note that when r x >> r y we see that d IIB ≈ T [ r x + r y − r z ] ≈ ( T r x ) . So initially, constant d IIB is approximately the samething as constant r x . At some point we increase r y enough to equal r x . Likewise, comingfrom region IIC we have that constant d IIB means, initially, approximately the same thingas constant r y . So in the middle the curve interpolates between vertical (constant r x ) andhorizontal (constant r y ). The derivatives for functions of d IIB have enough space because d IIB goes through the same amount as d IIA and d IIC . Outside C patch. Note that we will have enough space for log derivatives because θ ≈ (1 + | T z | ) T ( r x − r y ) for r x , r y very small and this was already checked earlier when θ ≈ T ( r x − r y ). (cid:3) Appendix B: Negligible terms in defining the symplectic form
Region I. Negligible terms.
The terms that produce derivatives of the bump func-tions are α ( d I ) d I and α ( θ I ) α ( d I ) θ I . Note that d in region I close to where r x = r y = r z is approximately linear in ( T r x ) , ( T r y ) , and ( T r z ) .Some notation: Let d ≡ d I in this section. We want to allow the variable we’re taking thederivative with respect to to vary among { r x , r y , r z } . So we denote those variables to be { r (cid:63) , r • } ∈ { r x , r y , r z } . Furthermore, (cid:48) log means we take the log derivative dα /d (log( d I )) = α (cid:48) · d . The following calculation for the second derivative applies to α as well, and α if wereplace d with θ . d α d (log d ) = ( α (cid:48) · d ) (cid:48) · d = ( α (cid:48)(cid:48) · d + α (cid:48) ) d = ⇒ α (cid:48)(cid:48) · d + α (cid:48) = 1 d · d α d (log d ) = ⇒ α (cid:48)(cid:48) = 1 d · (cid:18) d α d (log d ) − dα d (log d ) (cid:19) Diagonal terms for α ( d ) · d : d ≈ T [ r x − (cid:0) r y + r z (cid:1) ] = ⇒ d r ∗ ≈ λT r ∗ , d r ∗ r ∗ ≈ λT , λ ∈ { , − } (cid:18) r ∗ ∂ r ∗ + ∂ r ∗ r ∗ (cid:19) ( α ( d ) · d ) = 1 r ∗ ( α (cid:48) d r ∗ d + α ( d ) · d r ∗ )+ ( α (cid:48)(cid:48) d r ∗ d + α (cid:48) d r ∗ r ∗ d + 2 α (cid:48) ( d ) d r ∗ + α ( d ) · d r ∗ r ∗ ) ≈ r ∗ ( α (cid:48) λT r ∗ d + α ( d ) · λT r ∗ )+ ( α (cid:48)(cid:48) ( λT r ∗ ) d + α (cid:48) λT d + 2 α (cid:48) ( d )( λT r ∗ ) + α ( d ) · λT )= λT [ α (cid:48) d + α + λT α (cid:48)(cid:48) r ∗ · d + α (cid:48) d + 2 λT α (cid:48) r ∗ + α ( d )]= λT [ α (cid:48) (2 d + λT r ∗ ) + λT r ∗ ( α (cid:48)(cid:48) d + α (cid:48) )] + 2 λT α MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 97 = λT [ α (cid:48) log3 (2 + λT r ∗ d ) + λ T r ∗ d α (cid:48)(cid:48) log3 ] + 2 λT α α (cid:48) log3 ≈ ∆ α ∆ log d ≈∝ lT r ∗ d ≈ T r ∗ T [ r x − (cid:0) r y + r z (cid:1) ] = r ∗ r x − (cid:0) r y + r z (cid:1) = r ∗ /r x − (cid:16) r y r x + r z r x (cid:17) ≈ ∵ r x >> r y , r z α (cid:48)(cid:48) log3 ≈∝ l Thus all terms involving bump function derivatives can be made much smaller than 2 λT α for l sufficiently large. Off-diagonal terms for α ( d ) · d where ∗ (cid:54) = (cid:63) : d r ∗ r (cid:63) = 0 ∂ r ∗ r (cid:63) ( α · d ) = ∂ r ∗ ( α (cid:48) d r (cid:63) d + α ( d ) · d r (cid:63) )= α (cid:48)(cid:48) d r ∗ d r (cid:63) d + α (cid:48) d r (cid:63) r ∗ d + 2 α (cid:48) d r (cid:63) d r ∗ + α · d r (cid:63) r ∗ ≈ T λµr ∗ r (cid:63) α (cid:48)(cid:48) d + 0 + 2 T λµr ∗ r (cid:63) α (cid:48) + 0 , λ, µ ∈ { , − } = T λµ r ∗ r (cid:63) d d ( α (cid:48)(cid:48) d + α (cid:48) + α (cid:48) )= T λµ r ∗ r (cid:63) d ( α (cid:48)(cid:48) log3 + α (cid:48) log3 ) T r ∗ r (cid:63) d ≈ r ∗ r (cid:63) /r x − (( r y /r x ) + ( r z /r x ) ) ≈ T /l . Diagonal terms r • ∂ r • + ∂ r • r • and off-diagonal terms ∂ r • r (cid:63) for α α θ I . θ ≈ T ( r y − r z ) d ≈ T ( r x −
12 ( r y + r z )) θ r • ≈ λ • T r • , λ • ∈ { , ± } d r • ≈ µ • T r • , µ • ∈ { , − } r • ∂ r • ( α α θ ) = 1 r • ( α (cid:48) θ r • α θ + α α (cid:48) d r • θ + α α θ r • ) ≈ T ( λ • α (cid:48) log4 α + µ • α α (cid:48) log5 θd + λ • α α ) ∂ r (cid:63) r • ( α α θ ) ≈ T ∂ r (cid:63) [ r • λ • α (cid:48) θα + r • µ • α α (cid:48) θ + r • λ • α α ]= T [ ∂ r (cid:63) ( r • ) λ • α (cid:48) θα + r • λ • α (cid:48)(cid:48) θ r (cid:63) θα + r • λ • α (cid:48) θ r (cid:63) α + r • λ • α (cid:48) θα (cid:48) d r (cid:63) + ∂ r (cid:63) ( r • ) α α (cid:48) θ + r • µ • α (cid:48) θ r (cid:63) α (cid:48) θ + r • µ • α α (cid:48)(cid:48) d r (cid:63) θ + r • µ • α α (cid:48) θ r (cid:63) + ∂ r (cid:63) ( r • ) λ • α α + r • λ • α (cid:48) θ r (cid:63) α + r • λ • α α (cid:48) d r (cid:63) ] ≈ T [ δ • (cid:63) λ • α (cid:48) log4 α + T [ λ • λ (cid:63) r • r (cid:63) θ ( α (cid:48)(cid:48) log4 − α (cid:48) log4 ) α + λ • λ (cid:63) r • r (cid:63) θ α (cid:48) log4 α + λ • µ (cid:63) r • r (cid:63) d α (cid:48) log4 α (cid:48) log5 ]+ δ (cid:63) • θd α α (cid:48) log5 + µ • λ (cid:63) T r • r (cid:63) d α (cid:48) log4 α (cid:48) log5 + µ • µ ∗ T r • r (cid:63) θd α ( α (cid:48)(cid:48) log5 − α (cid:48) log5 ) + µ • λ (cid:63) T r • r (cid:63) d α (cid:48) log5 α + δ (cid:63) • λ • α α + λ • λ (cid:63) T r • r (cid:63) θ α (cid:48) log4 α + λ • µ (cid:63) T r • r (cid:63) d α α (cid:48) log5 ] We’ve already seen above that T r (cid:63) r • d and hence θd are bounded. So it remains to check that T r (cid:63) r • θ is bounded. Also, we only need to consider the cases where the numerator does notinvolve r x by the comment above that we get zero otherwise and that second-order partialderivatives are symmetric. So we have to bound the following expressions: r y r y − r z , r z r y − r z , r y r z r y − r z We are considering the top half of region I, where r y > r z . We declare that α is constant inthe region 1 < (cid:16) r y r z (cid:17) < T . So the support of α is where (cid:16) r y r z (cid:17) > T . In particular, we seethat(9.1) 1 (cid:16) r y r z (cid:17) − < T − − T − T ≈ T So these terms are bounded, which can be seen dividing top and bottom by r z . So we’ve seen λ • λ ∗ T r • r ∗ θ , T r • r (cid:63) d , θd are bounded, and multiply log derivatives which can be made sufficientlysmall for l large.9.2. Region IIA. Negligible terms.
Bump function terms in F are: α · ( T r y ) , α · d IIA , { α ( T r ∗ ) } ∗ = y,z , α α · ( T r y ) α (log( r y /r x )) · ( T r y ) First derivative divided by r (cid:63) :1 r x ∂ r x [ α (log r y − log r x ))( T r y ) ] = − r x α (cid:48) · ( T r y ) = − T · r y r x α (cid:48) , (cid:12)(cid:12)(cid:12)(cid:12) r y r x (cid:12)(cid:12)(cid:12)(cid:12) < r y ∂ r y [ α (log r y − log r x ))( T r y ) ] = 1 r y α (cid:48) · ( T r y ) + α · (2 T ) = T α (cid:48) + 2 T α r z ∂ r z [ α (log r y − log r x )( T r y ) ] = 0 α (cid:48) ≈∝ l MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 99
So all bump function derivative terms from first order derivatives of this term can be madesmall.
Second derivative ∂ (cid:63) • : ∂ r x r x ( α ( T r y ) ) = ∂ r x ( − ( T r y ) r x α (cid:48) ) = ( T r y ) r x ( α (cid:48) + α (cid:48)(cid:48) ) < T ( α (cid:48) + α (cid:48)(cid:48) ) ≈∝ T ( 1 l + 1 l ) , small ∂ r x r y ( α ( T r y ) ) = ∂ r y ( − ( T r y ) r x α (cid:48) ) = − T r y r x (2 α (cid:48) + α (cid:48)(cid:48) ) , norm < T (2 α (cid:48) + α (cid:48)(cid:48) ) ∂ r y r y ( α ( T r y ) ) = ∂ r y ( T α (cid:48) r y + 2 T α r y ) = T (3 α (cid:48) + α (cid:48)(cid:48) + 2 α ) Note the first derivative α (cid:48) goes from 0 to a maximum of 1 /l + (cid:15) at the half way point of l/ /l , still small. (Strictly speaking, (1 /l + (cid:15) )(2 /l ).) Thusthe derivatives of α r y can be made small by taking l sufficiently large. α ( d IIA ) d IIA
First derivative. r (cid:63) ∂ r (cid:63) ( α d IIA ) = ( α (cid:48) d IIA + α ) · d IIA(cid:63) r (cid:63) .Here are the partial derivatives of d IIA .(9.2) d IIA ≈ T [ r x −
12 ( r y + r z )] + 32 α (log r y − log r x ) · ( T r y ) ( d IIA ) x r x = T r x [2 r x + 32 ( α (cid:48) · − r x · r y )] = T [2 + 32 ( α (cid:48) · − r y r x )]( d IIA ) y r y = T r y [ − r y + 32 ( α (cid:48) · r y · r y + 2 r y α )] = T [ − α (cid:48) + 2 α )]( d IIA ) z r z = T r z [ − r z ] = − T Thus derivative terms are either α (cid:48) log3 = α (cid:48) d IIA or a regular derivative of α , which may bemultiplied by ( r y /r x ) but r x > r y in region IIA. So derivative terms of α ( d IIA ) d IIA can bemade small for l sufficiently large. Second derivative terms.
We differentiate each of the first derivative terms. Let’s say P is a term in Equation 9.2 above. Then we want to differentiate r (cid:63) P because above wedivided by r (cid:63) . Thus using the product rule with a differential operator D = ∂ r • we get D ( r (cid:63) ) P + r (cid:63) D ( P ). The first term gives 0 or 1 times P , which we already know is small. Sowe’ll only need to consider r (cid:63) D ( P ) for 8 choices of P .(1) P = α (cid:48) d IIA : r (cid:63) ∂ r • ( α (cid:48) d IIA ) = r (cid:63) α (cid:48)(cid:48) · ( d IIA ) • · d IIA + r (cid:63) α (cid:48) ( d IIA ) • = r (cid:63) ( d IIA ) • d IIA α (cid:48)(cid:48) log3 ( d IIA ) • terms : { T r • , T α r y , T α (cid:48) r y , T α (cid:48) r y r x } < T r • , ∵ α ≤ , α (cid:48) ≈∝ l , r y r x < T r (cid:63) r • d IIA ≈ r (cid:63) r • r x − ( r y + r z ) + α r y = r (cid:63) r • /r x − (( r y /r x ) + ( r z /r x ) ) + α ( r y /r x ) (9.3) ≈ r (cid:63) r • r x ≈∈ { , } ∵ r x >> r y , r z in region IIA
00 CATHERINE CANNIZZO ∴ r (cid:63) ∂ r • ( P ) = r (cid:63) ( d IIA ) • d IIA α (cid:48)(cid:48) log3 ≈ (bounded) · l (2) P = α (cid:48) d IIA α term from differentiating F first wrt y , i.e. ∂ r y ( F ) (same calculationworks replacing α with α (cid:48) ) r y ∂ r • ( α (cid:48) d IIA α ) = r y ∂ r • ( α (cid:48) d IIA ) · α + r y ( α (cid:48) d IIA ) · ∂ r • ( α )= (above case) · α ± α (cid:48) log3 α (cid:48) · cr y , c ∈ { r x , r y , } = small + small(3) P = α (cid:48) d II α (cid:48) · (cid:16) r y r x (cid:17) from differentiating first wrt x (same argument for α α (cid:48) ( r y /r x ) replacing α (cid:48) d II α (cid:48) with α α (cid:48) ) r x ∂ r • ( α (cid:48) d II α (cid:48) · (cid:18) r y r x (cid:19) ) = r x ∂ r • ( α (cid:48) d II α (cid:48) ) · (cid:18) r y r x (cid:19) + r x ( α (cid:48) d II α (cid:48) ) · ∂ r • (cid:18) r y r x (cid:19) = r y r x · r y ∂ r • ( α (cid:48) d II α (cid:48) ) ± ( α (cid:48) log3 α (cid:48) ) · cr x , c ∈ { r y /r x , r y /r x , } = (small)(previous case) + (small)( r y /r x ) i , i ∈ { , } = small ∵ r x >> r y (4) P = α : shows up in first derivative of F wrt any variable, r (cid:63) ∂ r • α = r (cid:63) α (cid:48) ( d IIA ) • = α (cid:48) log3 · ( r (cid:63) ( d IIA ) • ) /d IIA , and ( r (cid:63) ( d IIA ) • ) /d IIA bounded by Equation (9.3).(5) P = α α (same argument for α α (cid:48) : shows up in first derivatives of F wrt y , r y ∂ r • ( α α ) = r y (( α ) • α + α ( α ) • ). First term with α derivatives ok by previ-ous item, second term gives α α (cid:48) times one or zero, since r y /r x (cid:28) α ( d IIA ) d IIA . α · ( T r ∗ ) for ∗ ∈ { y, z } We run through the same argument as with α · d IIA above, replacing d IIA with (
T r ∗ ) inthe second term. First derivative. r (cid:63) ∂ r (cid:63) ( α ( T r ∗ ) ) = r (cid:63) α (cid:48) ( d IIA ) (cid:63) ( T r ∗ ) + α · (( T r ∗ ) ) (cid:63) r (cid:63) .The bump function derivative term is r (cid:63) α (cid:48) ( d IIA ) (cid:63) ( T r ∗ ) = α (cid:48) log5 T r ∗ ( d IIA ) (cid:63) d IIA r (cid:63) . Note that T r ∗ d IIA and ( d IIA ) (cid:63) r (cid:63) are bounded, the latter by Equation (9.2) and the former since: (9.4) r ∗ r x − ( r y + r z ) + α r y = r ∗ /r x − (( r y /r x ) + ( r z /r x ) ) + α ( r y /r x ) (cid:28) ∵ r x >> r y , r z Note that ∗ ∈ { y, z } . So first derivatives are bounded for sufficiently large l in α · ( T r ∗ ) since they are either non-bump function derivatives or a bounded quantity multiplied by asmall log derivative. Second derivatives.
Differentiating first derivatives α (cid:48) ( d IIA ) (cid:63) ( T r ∗ ) and 2 T r (cid:63) α :1) MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 101 r (cid:63) ∂ r • ( α (cid:48) ( T r ∗ ) ) = r (cid:63) α (cid:48)(cid:48) · ( d IIA ) • · ( T r ∗ ) + r (cid:63) α (cid:48) (( T r ∗ ) ) • = ( α (cid:48)(cid:48) log5 − α (cid:48) log5 ) r (cid:63) · ( d IIA ) • · ( T r ∗ ) d IIA + α (cid:48) log5 T r • r (cid:63) d IIA
Already checked bounded: r (cid:63) ( d IIA ) • d IIA ∵ (9 . , ( T r ∗ ) d IIA ∵ (9 . , T r • r (cid:63) d IIA ∵ (9 . ∴ r (cid:63) ∂ r • ( α (cid:48) ( T r ∗ ) ) = (small)(bounded) + (small)(bounded)2) α (cid:48) ( T r ∗ ) α (same for α (cid:48) ( T r ∗ ) α (cid:48) ): term from differentiating F first wrt y , i.e. ∂ r y ( F ) r y ∂ r • ( α (cid:48) ( T r ∗ ) α ) = r y ∂ r • ( α (cid:48) ( T r ∗ ) ) · α + r y ( α (cid:48) ( T r ∗ ) ) · ∂ r • ( α )= (above case) · α ± α (cid:48) log5 ( T r ∗ ) d IIA α (cid:48) · cr y , c ∈ { r x , r y , } = (small)(bounded) + (small)(bounded)3) α (cid:48) ( T r ∗ ) α (cid:48) · (cid:16) r y r x (cid:17) : from differentiating first wrt x r x ∂ r • ( α (cid:48) ( T r ∗ ) α (cid:48) (cid:18) r y r x (cid:19) ) = r x ∂ r • ( α (cid:48) ( T r ∗ ) α (cid:48) ) · (cid:18) r y r x (cid:19) + r x ( α (cid:48) ( T r ∗ ) α (cid:48) ) · ∂ r • (cid:18) r y r x (cid:19) = r y r x · r y ∂ r • ( α (cid:48) ( T r ∗ ) α (cid:48) ) ± ( α (cid:48) log5 ( T r ∗ ) d IIA α (cid:48) ) · cr x , c ∈ { r y r x , r y r x , } = (small)(previous case) + (small)( r y /r x ) i , i ∈ { , } = small ∵ r x >> r y
4) 2 T α : shows up in first derivative of F wrt any variable. Taking another derivative gives r (cid:63) ∂ r • T α = r (cid:63) T α (cid:48) ( d IIA ) • = 2 T α (cid:48) log5 · ( r (cid:63) ( d IIA ) • ) /d IIA . So we want ( r (cid:63) ( d IIA ) • ) /d IIA tobe bounded. This was checked in Equation (9.3).This concludes our check of the first and second order derivatives of α ( d IIA )( T r ∗ ) for ∗ ∈ { y, z } . We have one more remaining type of term showing up in F to check. α α · ( T r y ) r (cid:63) ∂ r (cid:63) ( α ( T r y ) ) · α + α ( T r y ) · r (cid:63) ∂ r (cid:63) α = (previous) · α ± α ( T r y ) · cr (cid:63) α (cid:48) (cid:63) = z = ⇒ c = 0 , (cid:63) = x = ⇒ ≈ ∵ r x >> r y , (cid:63) = y = ⇒ r y r (cid:63) = 1So again first derivatives may be made small by taking l sufficiently large. Finally, we checksecond order derivatives. ∂ r • [ ∂ r (cid:63) ( α ( T r y ) ) · α + α ( T r y ) · ∂ r (cid:63) ( α )] = ∂ r • r (cid:63) ( α ( T r y ) ) · α + ∂ r (cid:63) ( α ( T r y ) ) ∂ r • α + ∂ r • ( α ( T r y ) ) ∂ r (cid:63) α + α ( T r y ) · ∂ r • ∂ r (cid:63) ( α )
02 CATHERINE CANNIZZO α ( T r y ) . Terms 2 & 3: ∂ r (cid:63) ( α ( T r y ) ) · ∂ r • α = ( α (cid:48) log5 ( d IIA ) (cid:63) d IIA ( T r y ) + 2 α T δ y(cid:63) r y ) · ± r x or r y · α (cid:48) = ± [ T α (cid:48) log5 ( d IIA ) (cid:63) r y d IIA · r y r x or r y · α (cid:48) + 2 α T r y r x or r y · α (cid:48) ] α ( T r y ) ∂ r • ( α (cid:48) r x or r y ) = α ( T r y ) (cid:20) α (cid:48)(cid:48) · ± r x or r y · r x or r y − α (cid:48) r x or r y (cid:21) = T α α (cid:48)(cid:48) ± r y r x , r x r y , or r y − T α α (cid:48) r y r x or r y = small , ∵ r x >> r y = ⇒ ∂ r • ∂ r (cid:63) ( α α · ( T r y ) ) = (small)where the final line follows from the calculations for the 1st term of region IIA, which was α ( T r y ) , and the third term of region IIA, which was α ( T r y ) . So the upshot is: allterms involving derivatives of bump functions can be made arbitrarily small because theyare multiplied by expressions which are bounded (taking either log derivatives of α , α orregular derivatives of α .) So we get a positive-definite form for l sufficiently large, becausethe terms not involving derivatives of bump functions are O (1) so they dominate, and wealready showed they give something positive-definite.9.3. Region IIB.
The characteristics in region IIB which we did not have in regions IIAand C are 1) r x and r y go from r x >> r y to r y >> r x , passing through r x = r y and 2) α ≡ r x , r y , r z are still small so we still have an approximation for the K¨ahler potential: F ≈ T r z + α ( d IIB ) d IIB + 12 α ( d IIB ) · ( − ( T r z ) ) d IIB ≈ T [ r x + r y − r z ]Let’s repeat the calculations above for α ( d IIA ) · d IIA and α · ( T r z ) with region IIB, andsee if they relied on r x >> r y . What we know in region IIB is that r x , r y >> r z . α ( d IIB ) d IIB
First derivative. r (cid:63) ∂ r (cid:63) ( α d IIB ) = ( α (cid:48) d IIB + α ) · d IIB(cid:63) r (cid:63) .Here are the partial derivatives of d IIB . d IIB ≈ T [ r x + r y − r z ]( d IIB ) x r x ≈ T r x (2 r x ) = T d IIB ) y r y ≈ T d IIB ) z r z ≈ − T r (cid:63) ∂ r (cid:63) ( α d IIB ) = ( α (cid:48) d IIB + α ) · d IIB(cid:63) r (cid:63) are proportional to α (cid:48) d IIB (a logderivative, so small) and α (not a derivative). So first derivatives of α ( d IIB ) d IIB may be
MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 103 made small for l sufficiently large. Second derivative terms.
We differentiate each of the first derivative terms. Let’s say P is a term in the list above. Then we want to differentiate r (cid:63) P because above we divided by r (cid:63) .Thus using the product rule with a differential operator D = ∂ r • we get D ( r (cid:63) ) P + r (cid:63) D ( P ).The first term gives 0 or 1 times P , which we already know is small by the above item foreach P on the list. So we’ll only need to consider r (cid:63) D ( P ) for the 2 choices of P listed above.(1) P = α (cid:48) d IIB . Then this term contributes to ∂ r (cid:63) r • ( F ) via r (cid:63) ∂ r • ( P ) i.e. r (cid:63) ∂ r • ( α (cid:48) d IIB ) = r (cid:63) α (cid:48)(cid:48) · ( d IIB ) • · d IIB + r (cid:63) α (cid:48) ( d IIB ) • = r (cid:63) ( d IIB ) • d IIB α (cid:48)(cid:48) log3 ( d IIB ) • terms : T r • T r (cid:63) r • d IIB ≈ r (cid:63) r • r x + r y − r z = r (cid:63) r • /r x r y /r x ) − ( r z /r x ) ≈ r (cid:63) r • /r x r y /r x ) ( (cid:63), • ) ∈ { ( r x , r x ) , ( r x , r z ) , ( r z , r z ) } = ⇒ r (cid:63) r • r x ∈ { , small } ∵ r x >> r z ( (cid:63), • ) = ( r y , r z ) = ⇒ r y r z r x < r y r x ( (cid:63), • ) ∈ { ( r x , r y ) , ( r y , r y ) } suffices to bound: a a , a a , a = r y /r x a a ≤ , a < ⇒ a a < , a ≥ ⇒ a a ≤ a a ≤ ∴ r (cid:63) ∂ r • ( α (cid:48) d IIB ) = (bounded)(small)(2) P = α . Taking another derivative gives r (cid:63) ∂ r • α = r (cid:63) α (cid:48) ( d IIB ) • = α (cid:48) log3 · ( r (cid:63) ( d IIB ) • ) /d IIB .So we want ( r (cid:63) ( d IIB ) • ) /d IIB to be bounded. This was just checked above. α · ( T r z ) We run through the same argument as with α · d IIB above, replacing d IIB with (
T r z ) inthe second term. First derivative. r (cid:63) ∂ r (cid:63) ( α ( T r z ) ) = r (cid:63) α (cid:48) ( d IIB ) (cid:63) ( T r z ) + α · (( T r z ) ) (cid:63) r (cid:63) .1 r (cid:63) α (cid:48) ( d IIB ) (cid:63) ( T r z ) = α (cid:48) log5 T r z ( d IIB ) (cid:63) d IIB · r (cid:63) Note that T r z d IIB and ( d IIB ) (cid:63) r (cid:63) are bounded. The latter is approximately constant because d IIB is approximately linear in r x , r y and r z . For the former: r z r x + r y − r z = r z /r x r y r x − (( r z /r x ) ) ≈ r z /r x r y r x ≈ ∵ r x >> r z So first derivatives of α · ( T r z ) may be made small by taking l sufficiently large. Second derivative terms.
We differentiate each of the first derivative terms. They are α (cid:48) ( d IIB ) (cid:63) ( T r z ) and 2 T r (cid:63) α . P is defined as in the second derivative calculation on page103.
04 CATHERINE CANNIZZO (1) P = α (cid:48) ( T r z ) d IIB ) (cid:63) r (cid:63) . It suffices to consider α (cid:48) ( T r z ) because ( d IIB ) (cid:63) r (cid:63) is a constant. r (cid:63) ∂ r • ( α (cid:48) ( T r z ) ) = r (cid:63) α (cid:48)(cid:48) · ( d IIB ) • · ( T r z ) + r (cid:63) α (cid:48) (( T r z ) ) • = ( α (cid:48)(cid:48) log5 − α (cid:48) log5 ) r (cid:63) · ( d IIB ) • · ( T r z ) d IIB + α (cid:48) log5 T r z r (cid:63) d IIB
Already checked bounded: r (cid:63) ( d IIB ) • d IIB , ( T r z ) d IIB ∵ (above) T r z r (cid:63) d IIB ≈ r z r (cid:63) /r x r y /r x ) − ( r z /r x ) ≈ r z r (cid:63) /r x r y /r x ) < r (cid:63) /r x r y /r x ) (cid:63) = x = ⇒ bounded (cid:63) = y = ⇒ a/ (1 + a ) bounded as above (cid:63) = z = ⇒ small ∴ r (cid:63) ∂ r • ( α (cid:48) ( T r z ) ) = (small)(bounded) + (small)(bounded)(2) P = 2 T α : shows up in first derivative of F wrt any variable. Taking anotherderivative gives r (cid:63) ∂ r • T α = r (cid:63) T α (cid:48) ( d IIB ) • = 2 T α (cid:48) log5 · ( r (cid:63) ( d IIB ) • ) /d IIB . So we want( r (cid:63) ( d IIB ) • ) /d IIB to be bounded, which was already checked above.This completes the calculation of positive definiteness in region IIB.9.4.
The remainder of C patch. Recall from the construction of coordinate chartson Y (Definition 4.10 and Lemma 4.13) that the charts U ,g with coordinates ( x, y, z )and U ( − , ,g − with coordinates ( x (cid:48)(cid:48)(cid:48) , y (cid:48)(cid:48)(cid:48) , z (cid:48)(cid:48)(cid:48) ) are glued to each other via ( x (cid:48)(cid:48)(cid:48) , y (cid:48)(cid:48)(cid:48) , z (cid:48)(cid:48)(cid:48) ) =( T v x − , T v y − , T − z − ). Also recall from Equation 4.32 that along the z -axis, where α = α = 1, ω is defined by the K¨ahler potentials F = 12 ( g xz + g yz ) + α ( θ V ) θ V , where θ V = φ x − φ y , and F (cid:48)(cid:48)(cid:48) = 12 ( g (cid:48)(cid:48)(cid:48) xz + g (cid:48)(cid:48)(cid:48) yz ) + α ( θ (cid:48)(cid:48)(cid:48) V ) θ (cid:48)(cid:48)(cid:48) V , which differs from F by a harmonic term (see proof of Claim 4.18).By symmetry it suffices to check that ω is positive definite on just half of the z -axis. Namely,we can assume that | z | ≤ T − , since otherwise | z (cid:48)(cid:48)(cid:48) | = | T − z − | ≤ T − . Although | z | ≤ T − ,we may assume that | x | and | y | are very small. They are at most of the order of T l/ − l/p ,see Figure 13; otherwise α is constant and equal to ± /
2, so F agrees with either g xz or g yz . Thus | T xz | ≤ | T x | and | T yz | ≤ | T y | hence these quantities are very small and wehave again small scale approximations for their logarithms. Differentiating α ( θ ) θ once andtwice: θ = φ x − φ y = log T (1 + | T x | ) − log T (1 + | T yz | ) − log T (1 + | T y | ) + log T (1 + | T xz | ) ≈ (1 + | T z | )( | T x | − | T y | ) MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 105 r ∗ ∂∂r ∗ ( α θ ) = ( α (cid:48) θ + α ) θ ∗ r ∗ ∂∂r • ( α (cid:48) θθ ∗ ) = α (cid:48)(cid:48) θ • θθ ∗ + α (cid:48) θ • θ ∗ + α (cid:48) θθ ∗• = ( α (cid:48)(cid:48) θ + α (cid:48) θ ) · θ • θ ∗ θ + α (cid:48) log4 θ ∗• ∂∂r • ( α θ ∗ ) = α (cid:48) θ • θ ∗ + α θ ∗• = α (cid:48) log4 θ • θ ∗ θ + α θ ∗• we see we’ll need to bound terms as follows: • θ ∗ r ∗ has estimates 2(1 + | T z | ) T for ∗ ∈ { x, y } and for ∗ = z we have the estimate θ z r z ≈ T ( r x − r y ) ≈ r x , r y (cid:28) r z .) • θ • θ ∗ θ implies we’ll need to consider r • r (cid:63) r x − r y , which is approximately zero unless both ∗ and • are not z , because θ is approximately (1+ T r z ) T ( r x − r y ) and r x , r y (cid:28) r z . Otherwise recallfrom Equation 9.1 that we make α constant equal to zero on a sliver T < ( r x /r y ) < /T .We divide top and bottom by r x or r y depending on which variable is larger, and thenthe numerator is at most 1 while the denominator is bounded below from the constraint T < ( r x /r y ) < /T . • θ ∗• is approximately zero unless ∗ = • in which case it’s a constant, so bounded.
06 CATHERINE CANNIZZO
Notation
Chapter 1: Main theorem • ( γ , γ ) = Z -coordinates of γ • γ (cid:48) := (cid:18) (cid:19) , γ (cid:48)(cid:48) := (cid:18) (cid:19) • Γ B = Z (cid:104) γ (cid:48) , γ (cid:48)(cid:48) (cid:105)• ( n , n ) = coordinates of γ with respect to γ (cid:48) , γ (cid:48)(cid:48) basis • V = ( C ∗ ) / Γ B • V ∨ generic fiber of ( Y, v ) and SYZ mirror abelian variety of V • τ ∈ R parametrizes family of complex structures on the complex genus 2 curve • Σ = genus 2 curve • H hypersurface defined by • D b L Coh is generated by powers and shifts of L Chapter 2: HMS for abelian variety • x , x are complex coordinates on V • T B := R / Γ B • λ ( n γ (cid:48) + n γ (cid:48)(cid:48) ) := (cid:18) n n (cid:19) • κ ( γ ) := − λ ( γ ) t M λ ( γ ) • (cid:96) k are linear T -Lagrangians in V ∨ of slope k • t x := { (log τ | x | , θ ) } θ ∈ [0 , ⊆ V ∨ • s e,l denotes a basis of sections for H ( L l ) of size l • p e,l denotes a basis of intersection points for HF V ∨ ( (cid:96) i , (cid:96) j ) where j − i = l , and thereare l such points • γ i ∩ j denotes the remainder modulo Γ B for a given such intersection point • ( ξ , ξ , η ) are the moment map coordinates where | x i | = τ ξ i • θ i = arg( x i ) for i = 1 , Chapter 3: Toric geometry refresher • σ denotes a cone • Σ denotes a fan • M lattice gives the algebra on the toric variety • N lattice gives the 1-parameter subgroups of toric variety • P is the polytope that defines a toric variety Chapter 4: Definition of ω • T is the complex parameter on Y /Novikov parameter on genus 2 curve • τ is the Novikov parameter on Y /complex parameter on the genus 2 curve • ∆ ˜ Y defines the universal cover of Y and a toric variety of infinite type • x, y, z are the complex toric coordinates on Y • r x , r y , r z denotes their norms • ρ denotes T -action • P denotes infinitesimal T -action • v = xyz is the superpotential • F denotes the K¨ahler potential • ( d I , θ I ) denotes coordinates on delineated region I MS ON COORDINATE RINGS FOR A COMPLEX GENUS 2 CURVE 107
Chapter 5: Definition of the DFS-type category • DFS - Donaldson-Fukaya-Seidel • H is the symplectic horizontal distribution • F briefly at the start denotes a fiber V ∨ • Φ is the parallel transport map • X hor denotes the horizontal vector field over a given curve in the base of v • φ tH is the flow of X hor • (cid:83) γ (cid:96) k or (cid:83) γ t x denotes parallel transport of the fiber Lagrangian over γ in ( Y, v ) • L k is (cid:83) γ (cid:96) k over U-shaped γ • ( f , f ) denote amount we add to ( θ , θ ) from monodromy • π denotes v when we think of it as a fibration in Section 5.2. • φ H π denotes monodromy • J ω ( Y, U ) denotes the class of compatible almost complex structures which are iden-tically J outside open set U about the origin in v base • D anti-canonical divisor • A = ( a , a , a ) is fixed value for ( ξ , ξ , η ) in the polytope ∆ ˜ Y • p is either an abstract perturbation or a fixed point in a Lagrangian • J reg is J regular for disc configurations • In Key Regularity argument: η used briefly as annihilator to image of linearizedoperator on universal Fredholm problem, ξ for tangent vectors on space of maps, and˙ J for tangent vectors on space of almost complex structures • J reg denotes J regular for disc attached to sphere configuration • B base of a Fredholm problem Chapter 6: Computing the differential • M k are structure maps on Y • µ k are structure maps on fiber V ∨ , ∂ = µ • p ki,j , p e,l both denote intersection points in (cid:96) i ∩ (cid:96) j where l = j − i • t i ∈ (cid:96) i ∩ t x • α ˜ e ( e, l ) is weighted count of bigons between p e,l − and p ˜ e,l • n ˜ e ( l ) is weighted count of triangles between t i , p ˜ e,l and t j where l = j − i • C ( x ) is the sphere bubble count • c is the disc count times the sphere bubble count • β i,j + α denotes ( i, j )th disc class plus a sphere configuration class α • n β + α denote curve counts
08 CATHERINE CANNIZZO
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