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Lagrangian fibrations and theta functions

In this thesis we study asymptotic behavior of projective embeddings of abelian varieties and their amoebas. The projective embeddings are given by theta functions. It is known that a Lagrangian fibration of the abelian variety determines a basis of theta functions. After reviewing the relation from the viewpoint of geometric quantization and mirror symmetry, we prove that the Lagrangian fibration of the abelian variety can be approximated by the moment maps of natural torus action in a suitable sense. In the final section we discuss a part of the result in more general setting.


Introduction
The purpose of this thesis is to discuss some relations between theta functions and Lagrangian fibrations of abelian varieties.
Lagrangian fibration is a map (M, ω) → B from a symplectic manifold such that each (smooth) fiber is a Lagrangian submanifold. Lagrangian fibrations appear in several areas of mathematics and mathematical physics such as completely integrable systems (classical mechanics), geometric quantization and mirror symmetry. In particular, the picture of mirror symmetry via special Lagrangian fibrations proposed by Strominger-Yau-Zaslow [39] has attracted much attention and a lot of works relating to this program have been carried on (e.g. [12,13], [18], [35,36,37]).
However, little is known about (special) Lagrangian submanifolds and fibrations at the moment. In general, it is more difficult to deal with submanifolds compared to functions or connections. In the case of (special) Lagrangian fibrations, the main difficulties seems to be the following two points. The first one is the fact that general Lagrangian fibrations have singular fibers. Singularities appearing in Lagrangian fibrations are not completely understood yet. The other one concerns the case of special Lagrangian fibrations. To define the special Lagrangian submanifolds, we need a Ricci-flat metric. Ricci-flat metrics are difficult objects to analyze.
The cases of Abelian varieties and toric varieties are typical examples of Lagrangian fibrations which are well understood. Mirror symmetry via special Lagrangian fibrations works well for abelian varieties and theta functions play an important role in the theory( [34], [11]).
Theta functions, or more generally holomorphic sections of ample line bundles L on a smooth projective variety X, are the other main characters in this paper. Of course, the space H 0 (X, L k ) of holomorphic sections does not have natural basis in general. However, some projective varieties have natural basis. For example, in the case of toric varieties, monomials are natural basis of holomorphic sections. Also in the case of abelian varieties, there are natural basis of theta functions(one can find such basis in the Mumford's book [29]). We can think that such basis are determined by Lagrangian fibrations by using the notion of geometric quantization (the case of abelian varieties is discussed in [9], [42], [44] and other cases are, for example, in [1], [4], [15], [16], [20]). The case of abelian varieties can be also interpreted as a special case of mirror symmetry.
Note that, for large k, each basis of H 0 (X, L k ) defines an embedding into the projective space CP N k = PH 0 (X, L k ) * : ι k : X ֒→ CP N k , z → (s 0 (z) : · · · : s N k (z)) We see the relation between Lagrangian fibration of X and projective embeddings defined by the basis corresponding to the Lagrangian fibration. Namely, we compare the Lagrangian fibration with the standard one of projective spaces (i.e. the moment maps of natural torus actions) We denote the restriction of the moment map to X by where B k is the image of X under the moment map.
The simplest example is the case of toric varieties. If we take monomials as a basis of H 0 (X, L k ), the corresponding embeddings are torus equivariant. This means that π k : X → B k coincides with the moment map of X.
The main theorem of this paper deal with the case of abelian varieties. In this case, the situation is not so trivial. The rough statement is the following. Theorem 1.1. Let X be an abelian variety. We consider projective embeddings oof X by using theta functions determined by a Lagrangian fibration of X. Then the sequence {π k : X → B k } converges to the original Lagrangian fibration as k → ∞ in the "Gromov-Hausdorff topology".
The precise statement is given in Section 6. This theorem is closely related to the theory of approximation of Kähler metrics by Fubini-Study metrics. Tian [40] and Zelditch [46] proved that every Kähler metric in a fixed Kähler class c 1 (L) can be approximated by restrictions of (normalized) Fubini-Study metrics on CP N k = PH 0 (X, L k ) * for large k by choosing appropriate embeddings. There is another theory on this subject and it is related to the notion of stability in the sense of geometric invariant theory. This is a possible approach to prove "Hitchin-Kobayashi correspondence for manifolds" which claims the equivalence of stability of a projective variety and the existence of special Kähler metrics such as Kähler metric of constant scalar curvature. In this approach, we relate the choice of projective embeddings with the stability condition and study the asymptotic behavior of the restrictions of Fubini-Study metrics. In particular, this theory could be effective for the the study of Ricci-flat metrics.
The projective embeddings defined by theta functions give examples of both of the above theories. The main theorem can be regarded as a prototype of approximations of Kähler metrics and Lagrangian fibrations at the same time. Our method might be useful for the study of special Lagrangian fibrations. This paper is organized as follows. In Section 2, we recall basic facts on symplectic geometry used in this paper. In Section 3, we review the theory of geometric quantization, especially, the part related to Lagrangian fibration in detail. Section 4 is devoted to a brief summary of mirror symmetry for abelian varieties via special Lagrangian fibrations. We recall the theorem of Tian and Zelditch stated above in Section 5. In this section, we also recall the relation between projective embeddings, stability of polarized manifolds and canonical metrics. The precise statement and proof of the main theorem is given in Section 6. Finally we discuss a generalization of the above result in Section 7.

Basic facts on symplectic geometry
In this section, we summarize basic facts on symplectic geometry which we need later. This proposition follows from dω = 0 and the following formula

Symplectic manifolds and automorphisms
where L ξ is the Lie derivative.
Definition 2.3. For a smooth function f ∈ C ∞ (M ), its Hamilton vector field ξ f is defined by A Hamilton diffeomorphism is a diffeomorphism generated by Hamilton vector fields. We denote the group of Hamilton diffeomorphisms by Ham (M, ω).
From the definition, the Lie algebra of Ham (M, ω) is identified with the space B 1 (M ) of exact 1-forms. In particular, every Hamilton diffeomorphism preserves ω. Proposition 2.4. Assume that ω represents an integral cohomology class: Then there exists a Hermitian line bundle L → M with a unitary connection ∇ such that c 1 (L, ∇) = ω.
Such (L, ∇) is called a prequantum bundle of (M, ω). This is a fundamental ingredient in geometric quantization.
Proof. Let A (resp. A * ) be the sheaf of smooth functions (resp. non-zero smooth functions) and consider the following exact sequence: Recall that H 1 (M, A * ) parameterizes the isomorphism classes of smooth complex line bundles on M . Since A is a fine sheaf, the long exact sequence becomes This implies that there exists a line bundle L → M such that c 1 (L) = [ω]. Now we take an arbitrary Hermitian metric and a unitary connection ∇ ′ on L. Note that its curvature ω ′ represents c 1 (L). There exists a smooth 1-form α such that ω = ω ′ + dα .
gives a unitary connection such that its curvature coincides ω.
We consider the automorphism group G of a prequantum bundle (L, ∇) → (M, ω). G consists of isomorphisms such thatF preserves the Hermitian metric and the connection ∇ of L (hence F preserves ω).
Proposition 2.5. The Lie algebra of G is formally identified with the space of smooth functions C ∞ (M ) on M , here the Lie algebra structure of C ∞ (M ) is given by the Poisson bracket.
We can see this as follows. Consider the natural projection and denote its image by G 0 . The kernel consists of automorphisms which preserve the base space. It is easy to see that this is isomorphic to S 1 . Then we have the following exact sequence: Proposition 2.5 is a consequence of the fact that G 0 ∼ = Ham (M, ω). In fact, the corresponding exact sequence of Lie algebras is given by Let Γ(L) = Γ(M, L) be the space of smooth sections of L. Then, for each f ∈ C ∞ (M ) ∼ = LieG, its actionf : Γ(L) → Γ(L) is given bŷ

Action-angle coordinate
In this and the next subsection, we recall basic properties of Lagrangian fibrations.
Definition 2.6. A submanifold S ⊂ M is called a Lagrangian submanifold if it satisfies ω| S = 0 and dim S = 1 2 dim M . We call a fibration π : M → B a Lagrangian fibration if general fibers are Lagrangian.
Remark 2.7. It is natural to allow Lagrangian fibrations to have degenerate fibers. However we do not care about singular fibers in this section.
Example 2.8. Let B be an n-dimensional manifold and T * B its cotangent bundle. Then the natural projection T * B → B is a Lagrangian fibration with respect to the standard symplectic structure ω 0 on T * B. Now we recall the notion of moment maps. Definition 2.9. Let G be a Lie group acting on a symplectic manifold (M, ω) as symplectomophisms. A moment map where , is the natural pairing and we identified ξ ∈ g with the vector field on M defined by its action.
1. Every smooth fiber has a natural affine structure. In particular, compact fiber is a torus T n .
Outline of the proof. 1. Let U be a neighborhood of b 0 ∈ B. We may assume that U ⊂ R n is a small ball. Denote the coordinate of R n by (x 1 , . . . ,x n ) and Proof. Since f i is constant along fibers, for each ξ ∈ T π −1 (b) . In particular, ξ fi ∈ T π −1 (b) and Note that ξ f1 , . . . , ξ fn are linearly independent. Furthermore, we have [ξ fi , ξ fj ] = ξ {fi,fj } = 0 from the above claim. Then ξ f1 , . . . , ξ fn generates an R n -action on π −1 (U ) and each fiber π −1 (b) is preserved by this action. In particular, if fibers are compact, we have π −1 (b) ∼ = T n .
Fix a Lagrangian section λ : U → π −1 (U ) and define a map We define the coordinate of exp(ξ y i ξ x i ) · λ by (y 1 , . . . , y n ). Note that the zero section of T * U corresponds to λ. Then the coordinate (y 1 , . . . , y n ) has period 1.
Remark 2.11. If there exists a quantum line bundle (L, ∇), we can take θ to be a connection 1-form of ∇. In this case, exp(2π √ −1x i ) is the holonomy of (L, ∇) on each fiber.

Global structure of Lagrangian torus fibrations
Next we discuss global structures of Lagrangian fibrations. Here we assume that the Lagrangian fibration π : M → B has no degenerate fiber. In the proof of theorem 2.10, we constructed a map ψ : In other words, we have This means that Λ U defines a global Z n -bundle Λ ⊂ T * B. Since Λ is spanned locally by dx 1 , . . . , dx n , Λ is a Lagrangian submanifold in (T * B, ω 0 ). Hence ω 0 induces a symplectic structure on T * B/Λ and the natural projection T * B/Λ → B is a Lagrangian torus fibration. We remark that T * B/Λ has a Lagrangian section (i.e. the zero section). This is called the Jacobian fibration of the Lagrangian fibration π : M → B ( [12]). Now assume that the Lagrangian fibration π : (M, ω) → B has a Lagrangian section λ : B → M . Then the map ψ extends to a global map Hence we have an isomorphism T * B/Λ → M of Lagrangian fibrations.
Next we consider the case that π : M → B does not necessarily have a global Lagrangian section. Take a open cover M = i U i of M such that a Lagrangian section λ i : U i → π −1 (U i ) and an action-angle coordinate are defined on each U i . Then we have can be reconstructed by gluing T * U i /Λ| Ui → U i . The gluing is determined by Lagrangian sections µ ij : We summarize as follows:

Geometric Quantization
Geometric quantization is a method to construct representations geometrically.
In the case of abelian varieties, we obtain the representations of finite Heisenberg groups. This view point gives an interpretation of the relation between Lagrangian fibrations of abelian varieties and theta functions.

Brief review of quantum mechanics
First we recall a formulation of quantum mechanics on Euclidean spaces. Let M = T * R n = R 2n be a symplectic vector space with the standard symplectic structure ω = 1 n i=1 dq i ∧ dp i , where (q 1 , . . . , q n ) is a coordinate on R n , (p 1 , . . . , p n ) the canonical coordinate on the cotangent spaces and is the Planck's constant.
The goal is to correspond a representation of a suitable Lie subalgebra of C ∞ (M ). For each function f , we denote the corresponding operator byf : H → H, where H is a Hilbert space. From the requirement that f →f is a Lie algebra homomorphism, we obtain the canonical commutation relation This is realized as follows: Let H = L 2 (R n ) be the space of L 2 -functions on R n with respect to the Lebesgue measure and definê for ϕ(q) ∈ L 2 (R n ). Then the relation (2) are satisfied.
Remark 3.1. From (2),q i andp i get commutative when → 0. Such a limit is called the semi-classical limit.
Definition 3.2. The Heisenberg algebra heis(R 2n ) is the Lie algebra generated byq 1 , . . . ,q n ,p 1 , . . . ,p n (and a center) satisfying the Heisenberg relations. In other words, heis(R 2n ) is given by the central extension where the extension class is given by ω.  [30])). L 2 (R n ) is the unique (up to isomorphisms) irreducible representation of heis(R 2n ).

Prequantization
Let (M, ω) be a symplectic manifold of dimension 2n and assume that there exists a prequantum line bundle (L, ∇) → M .
As we saw in the previous section, the Lie algebra of the automorphism group G of the prequantum bundle (L, ∇) is identified with the space C ∞ (M ) of smooth functions on M . Hence we have a representation Γ(L) of G and Lie G ∼ = C ∞ (M ). However this representation is not the desired one. This is "too large" as we see in the following example (This is why we call this "prequantization").
Example 3.4. We consider the case of symplectic vector space M = T * R n = R 2n with the symplectic form ω = 1 n i=1 dq i ∧dp i , as in the previous subsection. In this case, the prequantum bundle L is the trivial line bundle with connection . This representation is different from the one discussed in the previous subsection.
To get the desired representation, we have to eliminate half of the parameters (q 1 , . . . , q n , p 1 , . . . , p n ). This is done by choosing a "polarization".
Remark 3.5. We can take kω as a symplectic form instead of ω. In this case, L k is a prequantum bundle. 1 k play the role of the Planck's constant . We also call the limit k → ∞ the semi-classical limit.

Polarizations
Definition 3.6. Extend ω on T M ⊗ C complex bilinearly. A polarization P is an integrable Lagrangian subbundle in T M ⊗ C, i.e. a complex subbundle P ⊂ T M ⊗ C of rank n satisfying [P, P ] ⊂ P and ω| P ≡ 0.
For a polarization P , we "define" the space of polarized sections by (This definition is a temporary one. For some class of polarization, we need to modify the definition. We will discuss this point later.) Remark 3.7. The conditions of polarization are necessary for the integrability condition of ∇ ξ s = 0. Since the curvature of (L, ∇) coincides with ω, the integrabiliy condition can be written as for every ξ, η ∈ P . The integlability of P implies [ξ, η] ∈ P and the Lagrangian condition implies ω(ξ, η) = 0.
There are two important classes of polarizations.
Then the complexified relative tangent bundle gives a polarization. This polarization satisfiesP = P . We call polarizations satisfying this condition real polarizations. 1 In this case, the space Γ P (L) of polarized sections consists of sections of L which are covariantly constant along fibers. Note that the restriction of L on each fiber is flat since each fiber is Lagrangian. We discuss this case later. Giving a Kähler polarization is equivalent to fixing a compatible complex structure on (M, ω). In this case, Γ P (L) is nothing but the space H 0 (M, L) of holomorphic sections.
Remark 3.10. The action of G (or C ∞ (M )) does not preserve Γ P (L) ⊂ Γ(L). In general, the subgroup of G which preserves the polarization is very small. Example 3.11. We consider the case of symplectic vector space again. The natural projection π : T R n → R n is a Lagrangian fibration. Let P = ker dπ. Then Γ P (L) is identified with L 2 (R n ). Furthermore, the actions of q i and p i ∈ C ∞ (T R n ) preserve this subspace. In fact, its action is given by for ϕ(q) ∈ L 2 (R n ), as desired.
We can take a Kähler polarization on T R n ∼ = C n . In this case, we also have an irreducible representation of Heisenberg group on a space of holomorphic functions on C n . This is called the Bergman-Fock representation. (See [30].) Example 3.12 (Borel-Weil theory). Let G be a compact Lie group, T a maximal torus in G, G C the complexification of G, and B ⊂ G C a Borel subgroup. We denote the flag variety by X = G/T = G C /B. Then every G C -equivariant line bundle on X can be determined by a holomorphic character B → C * by 1 In general, Lagrangian fibrations have degenerate fibers. Thus it is natural to allow real polarizations to be degenerate. However we ignore singular fibers in this section.
On the other hand, every character of B descend to a character of Hence there is a one to one correspondence For each character λ : T C → C * , we denote the corresponding line bundle by Any finite dimensional irreducible representation of G is given in this way.
Next we discuss the case of real polarizations with compact Lagrangian fibers. Let π : M → B be a Lagrangian torus fibration. Since ω is the curvature of L, the restriction L| π −1 (b) is a flat line bundle on π −1 (b) for each b ∈ B. However L| π −1 (b) has non-trivial holonomy in general. Therefore, there is no nontrivial smooth section of L which is constant along fibers. Hence we must change the definition of Γ P (L).
For each k ∈ N, we can take kω to be a symplectic form instead of ω. In this case, L k is a prequantum bundle.
From Proposition 2.10 and Remark 2.11, π −1 (b) is a Bohr-Sommerfeld fiber of level k if and only if the action coordinate (q 1 , . . . , q n ) of b takes its value in 1 k Z n . In particular, Bohr-Sommerfeld fibers appear discretely.
Instead of smooth sections, we consider distributional sections of L supported only on Bohr-Sommerfeld fibers and are covariantly constant on the support: Since each fiber is connected, covariantly constant sections on a Bohr-Sommerfeld fiber is unique up to constants. In particular, the dimension of Γ P (L) is equal to the number of Bohr-Sommerfeld fibers.
Remark 3.16. There exists a cohomological definition of Γ P (L). Consider the following complex where d ∇ P is the restriction of the connection to P . This is in fact a complex since (d ∇ P ) 2 = ω| P = 0 . Then Theorem 3.17 (Śniatycki [38]).

Comparison of the spaces of wave functions for Kähler and real polarizations
Assume that we have a Lagrangian fibration π : (X, ω) → B of a compact Kähler manifold and a prequantum bundle L → X. Then we have two spaces of wave functions, i.e. the space H 0 (X, L) of holomorphic sections and Γ T X/B (L). It is natural to ask whether these spaces are isomorphic or not. In general, it is difficult to define and compute Γ T X/B (L) since there exist degenerate fibers for general Lagrangian fibrations. However it is shown that these spaces are isomorphic for several cases. The moment map of a natural S 1 -action is a Lagrangian fibration. This can be written explicitly as follows. We identify CP 1 with the unit sphere Then the moment map is given by It is easy to see that π −1 (b) satisfies the Bohr-Sommerfeld condition of level k if and only if b = 2i−k k , i = 0, 1, . . . , k is a lattice point. Note that In this case, the isomorphism between two spaces of wave functions is given by just the restrictions: . The same is true for general projective toric manifolds (with a suitable Kähler form).
Guillemin-Sternberg [15] proved the case of flag varieties. This is also true for the case of abelian varieties. We will see this in detail in the next subsection. Theorem 3.19 (Andersen [1]). Let (X, ω) be a compact Kähler manifold, L → X a prequantum bundle. Assume that we have a Lagrangian fibration π : X → B with no degenerate fiber. Then Remark 3.20. In general, the restriction of holomorphic sections to BS fibers does not give covariantly constant sections.
Remark 3.21. Very few compact Kähler manifolds admit Lagrangian fibration without degenerate fibers. However it is expected that this theorem holds for more general cases such as K3 surfaces. We discuss this point in section 4 again.
The dimension of H 0 (X, L k ) is given by Riemann-Roch formula and vanishing theorem of cohomologies. Thus what we need to do is to count the number of Bohr-Sommerfeld fibers. The idea is to use its "mirror", i.e. a dual torus fibration.
Proof. From Riemann-Roch theorem, By using the symplectic form, T X/B can be identified with π * T * B. Since T * B and T B contains lattices of maximal rank, these are flat. Then
Next we calculate the number of Bohr-Sommerfeld fibers. First we define a dual torus fibration of π : X → B. Let T * B/Λ → B be the Lagrangian fibration associated to π, as in the previous section. We denote the dual lattice of Λ by Recall that the dual torusŤ n of a torus T n parameterizes flat line bundles E → T n . In our situation, for each b ∈ B, we can associate a flat line bundle L k | π −1 (b) on π −1 (b). This defines a section λ k : B → T B/Λ * of the dual torus fibration. On the other hand, the zero section λ 0 corresponds to the trivial bundle on X. By definition, λ k can be written explicitly as follows. Let (x 1 , . . . , x n , y 1 , . . . , y n ) be the action-angle coordinate constructed in section 2 and take the dual coordinate (x 1 , . . . , x n , v 1 , . . . , v n ). Then λ k is given by From this expression, we see that λ k and λ 0 intersect transversely and positively (under a suitable orientation). Therefore the number of Bohr-Sommerfeld fibers of level k coincides with the intersection number of λ k (B) and λ 0 (B). Put α = dv 1 ∧ · · · ∧ dv n ∈ Ω n (T B/P ). This is the Poincaré dual of λ 0 (B).

Geometric quantization for abelian varieties
Let X = C n /(ΩZ n + Z n ) be an abelian variety of complex dimension n, where Ω is an n × n complex symmetric matrix whose imaginary part Im Ω is positive definite, and L an ample line bundle on X. We assume that L is symmetric of degree 1, for simplicity. Recall that L is said to be symmetric if (−1 X ) * L ∼ = L, here −1 X : X → X is the inverse morphism of X. Explicitly, L is given by where the action of ΩZ n + Z n on C n × C is for λ ∈ ΩZ n + Z n . Any ample bundle of degree 1 is a pull back of L by some translation of X. From the definition of L, defines a Hermitian metric on L whose first Chern form gives a flat Kähler metric on X, here we put (h ij ) = (Im Ω) −1 . Take the following two Lagrangian tori and consider the natural projection Then we have two polarizations and hence two spaces of wave functions. Next we define finite Heisenberg groups in a similar way as in the case of vector space R 2n . For each k ∈ N, let X k (resp. X ± k ) be the subgroup of points of X (resp. X ± ) of order k. Then It is easy to see that X k can be characterized by where C * acts on L k by multiplications on each fiber. The finite Heisenberg group G k is defined by where µ k is the group of k-th roots of 1. The group law is Kähler polarization G k acts naturally on H 0 (X, L k ). Note that the Hermitian metric h 0 is invariant under the G k -action. This means that the induced G k -action on H 0 (X, L k ) is unitary.
is a unique (up to isomorphism) irreducible representation of G k , called the Heisenberg representation.
Holomorphic sections of L k can be explicitly written by using theta functions. We take a basis of H 0 (X, L k ) defined by where we denote ..,k n and By definition, the action of G k on H 0 (X, L k ) is given by From (7), we can show that H 0 (X, L k ) is an irreducible representation of G k .

Real polarizations
Next we consider the real polarization defined by (5). In this case, Bohr-Somerfeld fibers are characterized as follows: In particular the number of Bohr-Somerfeld fibers of level k coincides with dim H 0 (X, L k ) = k n .
The basis s i ∈ H 0 (X, L k ) can be reconstructed from σ i 's by using the Bergman kernel. The Bergman kernel Π k (z, w) is the integral kernel of the orthogonal projection Proposition 3.24. Let Π k (z, w) be the Bergman kernel of the Hermitian line bundle (L k , h 0 ). Then Proof. We will show that s 1 , . . . , s k n define an orthonormal basis of H 0 (X, L k ) in the next subsection. Then the Bergman kernel is given by From this expression, it suffices to show that By direct computation, we have and we obtain (8) by integrating this.

Special Lagrangian fibrations and mirror symmetry
Mirror symmetry is a duality in string theory. Mathematically, this is regarded as a duality between symplectic geometry on a Calabi-Yau manifold M and complex geometry on another Calabi-Yau manifold W . Homological mirror symmetry conjectured by Kontsevich [23] claims the equivalence of the Fukaya category on M and the derived category of coherent sheaves on W . Strominger-Yau-Zaslow [39] proposed mirror symetry via special Lagrangian fibrations. This picture gives a geometric construction of mirror manifolds and correspondence of two categories. This works successfully for abelian varieties and theta functions play an important role again.

Special Lagrangian submanifolds
An important fact is: [45]). If X is a compact Kähler manifold with c 1 (X) = c 1 (K −1 X ) = 0, then each Kähler class contains a unique Ricci-flat Kähler metric. Let X be a compact Calabi-Yau manifold of complex dimension n. By definition, X carries a non-vanishing holomorphic n-form Ω ∈ H 0 (X, K X ). Recall that the Ricci form of a Kähler form ω is given by Ric(ω) = −∂∂ log ω n . Hence ω is Ricci-flat if and only if ω n = cΩ ∧Ω for some constant c.
Now we fix a Ricci-flat metric ω and normalize Ω so that This condition is satisfied for ω = An important property of special Lagrangian submanifolds is the following: Theorem 4.4 (Harvey-Lawson [17]). Every special Lagrangian submanifold is volume minimizing in its homology class.
Note that the special Lagrangian condition is a first order differential equation. On the other hand, the condition of minimal submanifolds is second order. In addition, minimal submanifolds are not neccesarily volume minimizing. Such a situation is similar to that of ASD connections and Yang-Mills connections in gauge theory.
Recall that the normal bundle of a Lagrangian submanifold S is identified with the cotangent bundle T * S by using the symplectic structure. Under this identification, every small deformations of S is given as a graph of a 1-form on S. Then the graph is also Lagrangian if and only if the 1-forms is closed. Furthermore, every infinitesimal deformation is unobstructed. In particular, the tangent space of special Lagrangian submanifolds is identified with H 1 (S, R).
Definition 4.6. A fibration π : X → B is called a special Lagrangian fibration if each smooth fiber is special Lagrangian.
Since each fiber π −1 (b) is a Lagrangian torus, . This implies that B can be considered as (a component of) the moduli space of special Lagrangian tori in X. h ij dz i ∧ dz j , where τ is a n × n matrix with positive definite imaginary part and (h ij ) = (Im τ ) −1 . Then π : X −→ T n , τ x + y −→ x is a special Lagrangian fibration.
Example 4.8. Let X be a K3 surface and assume that π : X → CP 1 is an elliptic fibration with respect to a complex structure I. We fix a Ricciflat Kähler metric g. Since X is hyperKähler, there exist compatible complex structures J and K satisfying I 2 = J 2 = K 2 = −1 and IJ = K. We denote the corresponding Kähler form by ω I , ω J and ω K respectively. Then Since π is holomprphic with respect to I, we have Ω I | fiber = 0. This implies that π : X → S 2 is a special Lagrangian fibration with respect to J.
In this case, we know the structure of singular fibers from the result for elliptic surfaces by Kodaira [22].

A local model of mirror symmetry
Homological mirror symmetry conjecture states the equivalence of the Fukaya category of M and the derived category of coherent sheaves on W . Roughly, this means that each pair (S, L) of a special Lagrangian submanifold in M and a flat line bundle on S corresponds to a coherent sheaf on W . In this subsection, we see this correspondence from the point of view of special Lagrangian fibrations, following Leung-Yau-Zaslow [25] and Leung [24] (see also Hitchin [18]). This picture gives a concrete description of the correspondence of (special) Lagrangian submanifolds in M and coherent sheaves on W . In this subsection, we see the correspondence of holomrphic condition and special Lagrangian condition in a local model.
Let B be a small ball in R n with coordinate (x 1 , . . . , x n ). Then every Lagrangian torus fibration has the form for some Z n -bundle Λ ⊂ T * B. We denote the standard symplectic form by ω = dx i ∧ dy i . As in the previous section, we construct the dual torus fibration byπ where Λ * = Hom(Λ, Z) is the dual lattice bundle of Λ. Then W has a complex structure induced from where y i is the dual coordinate of x i . Note that Ω = dz 1 ∧ · · · ∧ dz n is a holomorphic n-form on X.
Remark 4.10. Here we assume that the B-field is zero. If we consider non-zero B-fields, we obtain more general complex structures.
Next we introduce Kähler metrics on M and W . From the definition of the complex structure on W , every Kähler metric and its Käher form have the form The induced metric on M is written as where (g ij ) = (g ij ) −1 . Then we have a compatible almost complex structure J on M defined by ω(ξ, η) = g(Jξ, η) .
Recall that each fiber of π andπ have a natural affine structure. Hereafter we assume that g is constant along fibers with respect to the natural affine structure (semi-flat condition). Then the Kähler condition (dω = 0) implies that Remark that the Jacobian is given by ∂xi ∂x j = g ij . We denote the Legendre transform of φ by Then Hence we have where z i = x i + √ −1y i . In particular J is integrable. The Ricci-flat condition for g is equivalent to the Monge-Ampère equation Namely, in this setting, mirror of M is nothing but its Legendre transform. We summarize the above discussion. M = T * B/Λ has a standard symplectic structure while W = T B/Λ * has a natural complex structure. Each compatible almost complex structure J (or, equivalently, a metric g) on M determines a metricǧ (orω) on W . Under the semi-flat condition, closedness ofω corresponds to the integrability condition of J. Moreover, Ricci-flat condition for g is equivalent to that forǧ.
Remark 4.11. The semi-flat condition is quite strong. In fact, few compact Calabi-Yau manifolds admit semi-flat Ricci-flat metrics. Nevertheless this local model is still important because it is expected that Calabi-Yau manifolds posses structures close to this semi-flat model near the large complex structure limit (i.e when it is close to the most degenerate Calabi-Yau's). Gross-Wilson [14] analyzed the case of K3 surfaces.
Next we consider the correspondence of special Lagrangian submanifolds in M and coherent sheaves on W .  First we consider the case that S ⊂ M is a special Lagrangian section of π : M → B. S can be given as a graph of 1-form α = A i dx i on B. The Lagrangian condition is equivalent to dα = 0. We set Since Hence The special Lagrangian condition Im e √ −1θ Ω = 0 becomes Then the corresponding supersymmetric B-cycle is given by C = W and E is a line trivial bundle with the connection This follows from (9) (i.e. the closedness of α). Furthermore the generalized Hermitian-Yang-Mills condition is equivalent to the special Lagrangian condition (10).
Next we consider the case that S is a multisection, i.e. the restriction π| S : S → B is a r-fold covering. In this case, the corresponding supersmmetric Bcycle is a holomorphic vector bundle E → W of rank r. To see this, we consider the following r-fold covering: Then the problem is reduced to the case of single sections.S corresponds to a line bundle L → W . E is defined by Since f : W → W is a r-fold covering, E is a vector bundle of rank r.

Mirror symmetry for abelian varieties
The correspondence discussed in the previous subsection works in the case of abelian varieties. For elliptic curves, Polishchuk-Zaslow [34] proved an equivalence of the derived category D b (W ) of coherent sheaves on W and a modified version F 0 (M ) of Fukaya category from this view point. For higher dimensional case, Fukaya [11] constructed a functor from the A ∞ category of affine Lagrangian submanifolds in M to D b (W ) for abelian varieties. We recall these theory very briefly. Let M = R 2n /Z 2n be a 2n diemsional torus with a constant complexified and consider the following Lagrangian fibration Then the mirror is given by The construction in the previous subsection can be rephrased by using "the Poincaré bundle". Let p : M × B W → W be the natural projection. Then the Poincaré bundle is a line bundle is the flat line bundle corresponding to ξ ∈π −1 (π(ξ)) = (π −1 (π(ξ))) ∨ . Let S ⊂ M be an affine Lagrangian submanifold and L → S a line bundle. Assume that π| S : S → B is an unramified covering. Let ("Fourier-Mukai transform" of (S, L)). Newt we see the correspondence of morphisms. Morphisms in Fukaya category are given by Floer homologies HF (S 1 , L 1 ), (S 2 , L 2 ) . Roughly speaking, Flore homologies are generated by intersection points of Lagrangian submanifolds. Here we are interested in the case that Lagrangian submanifolds intersect transversally.  (Fukaya). If S 1 and S 2 are transverse, where Hom(L 1 , L 2 ) is homomorphisms of vector spaces underlying the local systems at S 1 ∩ S 2 .
Theorem 4.17 (Fukaya [11]). Let S 1 and S 2 be affine Lagrangian submanifolds in M (not necessarily transverse) and L 1 , L 2 flat line bundles on S 1 and S 2 respectively. Then These Lagrangian sections S k are just the section λ k appeared in the proof of Theorem 3.19. In other words, HF (S 0 , S k ) can be identified with the space of wave functions defined by the real polarization π : M → B. In this case, the isomorphism of the spaces of wave functions can be regarded as a part of mirror symmetry.
The next theorem states the correspondence of the product structure (compositions) of morphisms.
The matrix elements are given by theta functions. We see this in the following example.
Example 4.20 ). Let M = T 2 = R 2 /Z 2 be a 2-torus with a complexified Kähler form ω C = − √ −1τ dx ∧ dy. In this case, its mirror is given by W = C/τ Z + Z. We consider the affine Lagrangian submanifolds S k of M corresponding to the lines y = kx in R 2 for k = 0, 1, 2. Then where b 0 and b 1 are the points corresponding to (0, 0), ( 1 2 , 0) ∈ R 2 respectively.
Then HF (S 0 , S 1 ) = Cb 0 , HF (S 1 , S 2 ) = Cb 0 , HF (S 0 , S 2 ) = Cb 0 ⊕ Cb 1 and the product m 2 : HF (S 0 , S 1 ) ⊗ HF (S 1 , S 2 ) → HF (S 0 , S 2 ) is defined by counting triangles bounded by S 0 , S 1 , S 2 . More precisely, where the summation is taken over all triangles with vertices b 0 , b 0 , b 0 , etc. Consequently we have On the other hand, S k corresponds to a holmorphic line bundle on W of degree k: In this case, the product is the natural product ("the addition formula").

Remark 4.21.
For more general Calabi-Yau manifolds with (special) Lagrangian fibrations satisfying suitable conditions, and its "topological mirror" W → B, we can generalize some parts of the above discussion.
It is not known whether the Floer homology HF (S 0 , S) can be defined, and dim HF (S 0 , S) coincides with the number of the intersection S 0 ∩ S.
By definition, every projective manifold X can be embedded into projective spaces. A way to equip a Kähler metric on X is to restrict the Fubini-Study metric on the projective space. On the other hand, some projective manifolds have standard Kähler metrics such as Ricci-flat metrics and Kähler-Einstein metrics. The restriction of the Fubini-Study metric is not a desired one in general. For example, the restriction of the Fubini-Study metric to cubic curves in CP 2 is not flat. Kähler metrics obtained from the Fubini-Study metric is restricted.
A way to deal with more general Kähler metrics from the view point of projective embeddings is to consider all tensor power of the polarization L → X. We can think of this as an analogy of the approximation of smooth functions by polynomials. In this section, we recall two theories of approximations of Kähler metrics.

Bergman kernels and projective embeddings
Let X be a smooth projective variety and L → X an ample line bundle. In this subsection, we fix a Kähler metric ω in the class c 1 (L) and a Hermitian metric on L whose first Chern form coincides with ω.
For each k, we take a basis s 0 , . . . , s N k of the space H 0 (X, L k ) of holomorphic sections, and consider the embeddings defined by ι k : X ֒→ CP N k , z → (s 0 (z) : · · · : s N k (z)) .
We denote the pull-back of the Fubini-Study metric by ω k : here we normalize the metric so that ω k represents c 1 (L). Tian and Zelditch proved the following theorem: Theorem 5.1 (Tian [40], Zelditch [46]). If s 0 , . . . , s N k are orthonormal for each k ≫ 1, then ω k converges to the original metric ω as k → ∞ in the C ∞ -topology. More precisely, for each r.
Namely, any Kähler metric representing c 1 (L) can be approximated by pullback of Fubini-Study metrics on CP N k = PH 0 (X, L k ) for large k.
Remark 5.2. The case of abelian varieties is discussed by Ji [19] and Kempf [21]. This theorem follows from the asymptotic behavior of the Bergman kernels Π k (z, w). Recall that Π k (z, w) is given by for orthonormal basis s 0 , . . . , s N k of H 0 (X, L k ). Its diagonal part gives a distortion function, i.e.
Theorem 5.1 follows from the following theorem.

Stability and Kähler metrics
The notion of stability is introduced by Mumford to construct moduli spaces. Moduli spaces are constructed as quotient spaces. In general, quotient spaces are not Hausdorff. To obtain a Hausdorff space, we consider moduli space of (semi-)stable objects. On the other hand, stability is closely related to existences of differential geometric structures such as metrics and connections. The following theorem is a typical example, known as "Hitchin-Kobayashi correspondence". Theorem 5.4 (Narasimhan-Seshadri [31], Donaldson [5,6], Uhlenbeck-Yau [43]). Let (X, L) be a polarized manifold and E → X a holomorphic vector bundle. Then E is stable in the sense of Mumford-Takemoto if and only if E admits an irreducible Hermitian-Yang-Mills connection.
It is expected that such a theorem also holds for the case of manifolds. There are several notion of stability (these are closely related each other but not equivalent) and it is not clear which stability is the right one at the moment.
Recently, Donaldson proved a part of the above conjecture. Here we review the result .
Definition 5.6. For a basis s 0 , . . . , s N k of H 0 (X, L k ), put and M = (M ij ) ∈ su(N k + 1). The basis s 0 , . . . , s N k is said to be balanced if M is a constant multiple of the identity matrix.
Since ω k changes as s 0 , . . . , s N k , it is not clear whether balanced basis exist or not. The existence of balanced basis is closely related to the stability of the polarized manifold (X, L).
Theorem 5.8 (Donaldson [7]). Assume that X admits a Kähler metric ω ∞ in the class c 1 (L) of constant scalar curvature. Further we assume that the automorphism group Aut(X, L) of the polarized variety (X, L) is discrete. Then, for each k ≫ 1, we can take a balanced embeddings. In particular, (X, L) is stable. Furthermore, the pull-backs of the Fubini-Study metrics ω k = 1 k ω FS converges to the constant scalar curvature metric ω ∞ .

Abelian varieties and theta functions
In section 3 and 4, we saw some relations between theta functions and Lagrangian fibrations on abelian varieties from the point of view of geometric quantization and mirror symmetry. It is natural to ask what happen if we embed the abelian varieties into projective spaces by these theta functions. In this section, we prove that the Lagrangian fibration of the abelian variety can be approximated by the restrictions of the moment maps of projective spaces in a certain sense.

Statement of the main theorem
We consider the same situation as in section 3.5. Let X = C n /(ΩZ n + Z n ) be an abelian variety of complex dimension n, where Ω is an n × n complex symmetric matrix whose imaginary part Im Ω is positive definite, and take an ample symmetric line bundle L on X of degree 1 defined by We fix a Hermitian metric on L and a flat metric on X, where we denote (h ij ) = (Im Ω) −1 . We consider the following Lagrangian torus fibration as before: π : X = X + × X − −→ X − = T n , z = Ωx + y −→ y .
Write ω k = 1 k ι * k ω FS . We compare the Lagrangian fibration π : X → X − and the restriction of a natural Lagrangian fibration (the moment map of the natural torus action) on projective spaces.
Let T k n −1 ⊂ SU (k n ) be the maximal torus which consists of diagonal matrices and consider its action on CP k n −1 given by Then this action is Hamiltonian and its moment map is given by here we identify the dual of the Lie algebra of T k n −1 with Denote the image of ι k (X) ⊂ CP N under µ T k n −1 by B k and consider the following map Note that π k has many degenerate fibers. In fact the most degenerate one is 0-dimensional. We also remark that each fiber is isotropic with respect to ω k (at smooth points).
Next we introduce a distance on B k . For that purpose, we define a distance on the polytope ∆ k = µ T k n −1 (CP k n −1 ). Note that µ T k n −1 has no degenerate fiber on the interior • ∆ k of ∆ k . We take a metric on • ∆ k so that the moment map µ T k n −1 : (CP k n −1 , ω FS ) → ∆ k is a Riemannian submersion on the interior. This is equivalent to the following definition. Consider the restriction µ T k n −1 : RP k n −1 → ∆ k of the moment map to RP k n −1 ⊂ CP k n −1 . This is a 2 k n −1sheeted covering which branches on the boundary of ∆ k . By identifying • ∆ k with a sheet of RP k n −1 , we have the restriction of the Fubini-Study metric on • ∆ k . We define a distance on B k induced by the normalized metric on • ∆ k with normalized factor 1 k . On the other hand, We equip the metric on X − so that π : (X, ω) → X − is a Riemannian submersion.
Under the above preparation, we can state the main theorem Theorem 6.1.
2. B k converge to X − in Gromov-Hausdorff topology.
3. The sequence of maps π k : X → B k between metric spaces converge to π : X → X − .
Before the proof of this theorem, we recall the definition of Gromov-Hausdorff distances. If Z is a metric space and X, Y ⊂ Z, then the Hausdorff distance of X and Y is defined by where we denote B(X, ε) the ε-neighborhood of X in Z.
For metric spaces X and Y , the Gromov-Hausdorff distance is defined by This is equivalent to the following definition: where the infimum is taken over all metrics on X Y compatible with these on X and Y .
Next we define convergence of maps. Let X i , Y i , X, Y be metric spaces and consider the maps f i : Suppose that X i and Y i converge to X and Y respectively in Gromov-Hausdorff topology. Then, from the definition of Gromov-Hausdorff distance, there exist metrics on X ( i X i ) such that X i converge to X in Hausdorff topology in X ( i X i ) (and the same is true for Y i and Y ). In this case, we say that {f i } converges to f if for every sequence

Proof of Theorem 6.1
Let {s 1 , . . . , s k n } be the basis of H 0 (X, L k ) as above.
Proposition 6.2. The basis {s 1 , . . . , s k n } of H 0 (X, L k ) are both balanced and orthonormal with respect to h 0 and ω 0 .
Proof. Recall that the G k -action on H 0 (X, L k ) preserves the L 2 -inner product defined by ω 0 and h 0 . From the formula (7), i.e. all s i 's have the same L 2 -norm, here we identify b i with (1, 0, b i ) ∈ G k . Furthermore, we have for all a ∈ X + k . This implies that (s bi , s bj ) L 2 = 0 if i = j. In other words, s 1 , . . . , s k n are orthonormal basis up to a constant.
To show that each s i has unit norm, we consider the following function From the definition of h 0 and s i , By using z = Ωx + y, we have From the definition of theta functions, Therefore we have From the fact that {b i } = 1 k Z n /Z n , we have the following identity: e( t lb i ) = k n , l ∈ kZ n , 0, otherwise.
By using this, we obtain Therefore, C = s i 2 L 2 (h0) = 1. Next we prove that these basis satisfies the balanced condition. Let h k be the pull back of the standard Hermitian metric (the Fubini-Study metric) on the hyperplane bundle (7) we have This means that h k and ω k are invariant under the G k -action. Hence this action on H 0 (X, L k ) is also unitary with respect to h k and ω k . We can prove that s 1 , . . . , s k n are balanced by the same argument as above.
The next lemma is a key of the proof of Theorem 6.1. for some distance dist on X − .

Proof.
For z = Ωx + y ∈ X, we write From (14), we have converges to R n e 1 2 t uΩu du uniformly on a fundamental domain of ΩZ n + Z n as k → ∞, this part can be bounded uniformly from above and below. Furthermore exp 2πk t (y − b i )(Im Ω −1 )(y − b i ) is of the form e −kCdist(y,bi) 2 , since Im Ω −1 is negative definite. Hence we have the desired estimate.
The convergence of Kähler forms ω k follows from Proposition 6.2 and Theorem 5.1 or 5.8. Lemma 6.4. Let d 0 and d k denote the distance on X defined by ω 0 and ω k respectively. Then there exists a constant C independent of k such that holds for each p, q ∈ X.
From the estimate of Γ, for some constant C > 0. Hereafter we denote constants independent of k be the same C. Therefore d dt This implies that 1 From this, we have a uniform bound |ξ 0 +η| ≤ C ′ for large k. Hence we have |η| = |Γ(ξ 0 +η, ξ 0 +η)| ≤ C k .
Sinceη(0) = 0, we have |η| = O 1 k . In particular Therefore the length of γ k is given by Lemma follows from this estimate.
Let {x ij = Ωa i + b j } i,j=1,...,k n = X k with the distance induced by d 0 and denote the same set with distance d k by {y ij }. Then Define a distance on {x ij } {y ij } by where C is the constant in Lemma 6.4. Then By combining this with (16), we have This prove (1) of the theorem.
Definition 6.5. Let (X, d X ), (Y, d Y ) be two compact metric spaces. A map ϕ : X → Y is said to be an ε-Hausdorff approximation if the following two conditions are satisfied.

2.
For each x, y ∈ X, We identify X − with a section {0} × X − ⊂ X and put We prove that ϕ k is a C/ √ k-Hausdorff approximation.
Lemma 6.6. There exist a constant C > 0 such that the C/k-neighborhood of Proof. Take any point π k (z) ∈ B k and put y = π(z) ∈ X − . Since π k : (X, ω k ) → B k is invariant under the action of X + k , there exists z ′ ∈ π −1 (y) such that π k (y) = π k (z ′ ) and d k (z, z ′ ) ≤ C/K. Then The proof in [32] is not correct.
For each ξ ∈ T p CP N k , we denote its vertical and horizontal parts by where T CP N k /∆ k ,p = ker dµ k is the tangent space of the fiber of µ k : CP N k → ∆ k and (T CP N k /∆ k ,p ) ⊥ is its orthogonal complement with respect to the Fubini-Study metric. Let (Z 0 : · · · : Z N k ) be the homogeneous coordinate and write Then T CP N k /∆ k and (T CP N k /∆ k ) ⊥ are spanned by ∂ ∂v i 's and ∂ ∂u i 's respectively. Let γ : [0, l] → B k be a curve and take a liftγ : [0, l] → X of γ. Then the length of γ is given by Proof. From the definition of the theta function, we have Similarly we have Lemma 6.7 follows from this.
From this Lemma we have for some constant C j . Let γ : [0, l] → X − be a curve and take a horizontal liftγ : [0, l] → X, i.e.
Then the length of γ is given by Note that (T X/X − ) ⊥ is spanned by J ∂ ∂x i 's, where J is the complex structure on X. (Recall that ∂ ∂x i tangents to fibers.) By direct computation, we have From this and (18), we have This implies that From this, we obtain Lemma 6.6 and 6.8 prove the second statement of Theorem 6.1.
From the second statement of Theorem 6.1, there is a distance on X − B k compatible with those on X − and B k such that for all y ∈ X − . It suffices to show that π k (p) converges to π(p) for each p ∈ X. For p ∈ X, we denote the corresponding point in X/X + k byp. We regard that π(p) ∈ B ⊂ X/X + k . Then the distance between π(p) andp with respect to ω k is

Lagrangian fibrations and holomorphic sections
In the previous section, we proved that theta functions can be reconstructed from a Lagrangian fibration by using the Bergman kernels. This construction can be applied to more general settings. In this section, we study the asymptotic behavior of the holomorphic sections constructed in this way by using the asymptotic behavior of the Bergman kernels ( [3], [26], [2]).

Asymptotic behavior of the Bergman kernels
The asymptotic behavior of the Bergman kernel Π k (z, w) is studied in detail.
for some constants C, c > 0, where d(z, w) is the distance of z, w ∈ X.
The asymptotic behavior of Π k (z, w) near the diagonal can be written as follows. Fix a point z 0 ∈ X and take a neighborhood U of z 0 with local holomorphic coordinate (z 1 , . . . , z n ) around z 0 . We write ω = √ −1 g ij dz i ∧ dz j and set G = (g ij ). Then we can take a holomorphic local frame e L of L satisfying ∂h ∂z i = ∂ 2 h ∂z i ∂z j = 0 at z 0 for each i, j = 1, . . . , n. Here we put h(z) = h (e L (z), e L (z)). In other words, From now on, we fix a trivialization of L over U defined by and express sections of L k with respect to this trivialization. Note that this trivialization is not holomorphic but unitary. Under this setting, Π k (z, w) can be write down as follows: Theorem 7.2 (Bleher-Shiffman-Zelditch [3]).
We study the asymptotic behavior of s i 's by using the results quoted in the previous subsection. All the proofs of the results in this subsection are given in the next subsection.
Hereafter we fix an arbitrary constant ε > 0. Let ρ > 0 be a constant such that local holomorphic coordinates considered in the previous subsection can be defined in the ball of radius ρ. We take a constant R > 0 such that Furthermore we take r > 0 such that for each z ∈ X and b ∈ B with d(π(z), b) ≤ r, there exists y 0 ∈ π −1 (b) with d(z, y 0 ) = d(z, π −1 (b)) and for all y ∈ π −1 (b). Let k be large enough so that √ kd(z,π −1 (bj )) .

If
Next we give the behavior of s j near π −1 (b j ). Fix z 0 ∈ π −1 (b 0 ) and take an action-angle coordinate (x 1 , . . . , x n , y 1 , . . . , y n ) around π −1 (π(z 0 )) constructed in section 2. Then we can take a holomorphic coordinate around z 0 of the form z = y + Ωx + O(|z| 2 ) with some symmetric matrix Ω with Im Ω > 0. In this case, ω = √ −1 g ij dz i ∧ dz j = − dx i ∧ dy i and Im Ω = 1 2 G −1 at z 0 . Note that b j ∈ 1 k Z n with respect to this action angle coordinate.
By using the above results, we compute L 2 -inner products of s i 's.
Theorem 7.7. Let b = b 0 or b 1 and assume that d here we take an action-angle coordinate around π −1 (b 0 ) as above. In particular, Corollary 7.8. Assume that the complex structure is invariant under the T naction along fibers (i.e. G is constant along each fiber), then i.e. s i 's are "asymptotically orthonormal".
Remark 7.9. We cannot conclude that s i 's are linearly independent, because dim H 0 (X, L k ) = O(k n ) is much bigger compared to the error term.
The next theorem is a weaker version of Theorem 5.3. for k ≥ k 0 . In particular, s i 's are base point free.
Lemma 7.11. For b = b j ∈ 1 k Z n with |b| ≤ ε, Proof. From the choice of local trivialization of L, the Hermitian metric h(z) has the form h(z) = exp − t zGz + O(|z| 3 ) .
From this, the solutions of ∇σ = 0 is of the form σ = C exp k 2 t (y +Ωb)G(y +Ωb) + kO(|y| 3 ) e L (y) for some constant C. This constant is determined by the normalization condition |σ j | 2 h ≡ 1 Vj .
Proof of Theorem 7.5 We put s j (z) = k 2π proof of Proposition 7.6. Let Proof of Theorem 7.7. We put 4c log k) = I + II + III .
Since d(z, π −1 (b 0 )) ≥ R/ √ k, |s 0 (z)| h ≤ Ck Finally, we consider I. From Theorem 7.5, We take an action-angle coordinate and write z = Ωx + y around a point z 0 as above. Then Hereafter we omit the error terms. Consequently,
Next we compute (I). From Theorem 7.5, we have and (I) = k 2π Similarly,