Local structure of principally polarized stable Lagrangian fibrations
aa r X i v : . [ m a t h . AG ] J u l LOCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLELAGRANGIAN FIBRATIONS
JUN-MUK HWANG, KEIJI OGUISO
Abstract.
A holomorphic Lagrangian fibration is stable if the characteristic cycles ofthe singular fibers are of type I m , ≤ m < ∞ , or A ∞ . We will give a complete descriptionof the local structure of a stable Lagrangian fibration when it is principally polarized. Inparticular, we give an explicit form of the period map of such a fibration and conversely, fora period map of the described type, we construct a principally polarized stable Lagrangianfibration with the given period map. This enables us to give a number of examplesexhibiting interesting behavior of the characteristic cycles. Introduction
For a holomorphic symplectic manifold (
M, ω ), i.e., a 2 n -dimensional complex manifoldwith a holomorphic symplectic form ω ∈ H ( M, Ω M ), a proper flat morphism f : M → B over an n -dimensional complex manifold B is called a (holomorphic) Lagrangian fibrationif all smooth fibers are Lagrangian submanifolds of M . The discriminant D ⊂ B , i.e., theset of critical values of f , is a hypersurface if it is non-empty. In [HO1], the structure ofthe singular fiber of f at a general point b ∈ D was studied. By introducing the notionof characteristic cycles, [HO1] shows that the structure of such a singular fiber can bedescribed in a manner completely parallel to Kodaira’s classification ([Kd], see also V. 7 in[BHPV]) of singular fibers of elliptic fibrations. Furthermore, to study the multiplicity ofthe singular fibers, [HO2] generalized the stable reduction theory of elliptic fibrations (cf.V.10 in [BHPV]), explicitly describing how arbitrary singular fiber over a general point of D can be transformed to a stable singular fiber, a singular fiber of particularly simple type.These results exhibit that the theory of general singular fibers of a holomorphic Lagrangianfibration gives a very natural generalization of Kodaira’s theory of elliptic fibrations.The current work is yet another manifestation of this principle. An important part ofKodaira’s theory is the study of the asymptotic behavior of the elliptic modular functionof a given elliptic fibration near a singular fiber. As a generalization of this we will studythe asymptotic behavior of the periods of the abelian fibers near a general singular fiber ofa holomorphic Lagrangian fibration. Here we need to make two additional assumptions onthe Lagrangian fibration.First, we will assume that the singular fibers are of stable type, i.e., its characteristiccycles are of type I k , ≤ k ≤ ∞ ( I ∞ meaning A ∞ ). As explained above, any generalsingular fiber can be transformed into this form by the stable reduction ([HO2], Section 4). Jun-Muk Hwang is supported by National Researcher Program 2010-0020413 of NRF and MEST, andKeiji Oguiso is supported by JSPS Program 22340009 and by KIAS Scholar Program.
The second assumption we will make is that the Lagrangian fibration is principally po-larized , in the sense explained in Definition 3.1. This condition is satisfied if there existsan f -ample line bundle on M \ f − ( D ) whose restriction on smooth fibers give principalpolarizations on the abelian varieties. This assumption is rather restrictive compared withthe setting of [HO1] and [HO2], where the only assumption was that the fibers of f are ofFujiki class.We believe that understanding the structure of Lagrangian fibration under these as-sumptions is essential for the study of general cases. Since this special case already requiressubstantial care and already provides many interesting examples (see Section 5), we restrictour attention to it in this paper and leave the general cases to future study.The main result of this paper is the following. Theorem 1.1.
Let f : M → B be a principally polarized stable Lagrangian fibration (cf.Definition 2.1 and Definition 3.1). Then at a general point b ∈ D of the discriminant, thereexists a coordinate system ( z , . . . , z n ) with D defined by z n = 0 , such that for a suitablechoice of an integral frame of the local system R f ∗ Z on B \ D , the period matrices havethe form θ ij = ∂ Ψ ∂z i ∂z j for ( i, j ) = ( n, n ) and θ nn = ∂ Ψ ∂z n ∂z n + ℓ π √− z n where Ψ is a holomorphic function in z , . . . , z n . Here ℓ is the number of irreducible com-ponents of a general singular fiber.Conversely, given any germ of holomorphic function Ψ( z , . . . , z n ) such that Im( θ ij ) > ,there exists a principally polarized stable Lagrangian fibration whose period matrices are ofthe above form. That the period matrix of Lagrangian fibration is the Hessian of a potential function is awell-known consequence of the action-angle variables (cf. [DM]). The logarithmic behaviorof the multi-valued part reflects the stability assumption on the singular fiber. The noveltyin Theorem 1.1 lies in the choice of the variable z n through which these two aspects areintertwined. The existence of z n follows from the fact proved in Proposition 3.13 that thecharacteristic foliation accounts for the degenerate part of the polarization restricted to thefixed part of the monodromy. The proof of this uses a version of the Monodromy Theoremfrom the theory of the degeneration of Hodge structures and the topological property ofthe stable singular fiber.The converse direction in Theorem 1.1 is shown by explicitly constructing a principallypolarized stable Lagrangian fibration from a given potential function Ψ( z ) imaginary part ofwhose Hessian matrix is positive definite. This part is a sort of generalization of Nakamura’sconstruction [Na] of toroidal degeneration of principally polarized abelian varieties over 1-dimensional small disk. Using our construction, we shall give a concrete 4-dimensionalexample of principally polarized stable Lagrangian fibration in which the types of char-acteristic cycles of singular fibers change fiber by fiber, too. To our knowledge, such anexample has not been noticed previously. In fact, most of the previous constructions ofsingular fibers of Lagrangian fibrations have used product construction from elliptic fibra-tions. OCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 3 Stable Lagrangian fibrations
Definition 2.1. A Lagrangian fibration is a proper flat morphism f : M → B from aholomorphic symplectic manifold ( M, ω ) of dimension 2 n to a complex manifold B of di-mension n such that the smooth locus of each fiber is a Lagrangian submanifold of M . The discriminant D ⊂ B is the set of the critical values of f , which is a hypersurface in B if itis non-empty. Throughout this paper, we assume that D is non-empty. We say that f is a stable Lagrangian fibration if D ⊂ B is a submanifold and each singular fiber f − ( b ) , b ∈ D, is stable, i.e., it is reduced and the characteristic cycle in the sense of [HO1] is of type I k , ≤ k ≤ ∞ . By the description in [HO1], this is equivalent to saying that f − ( b ) isreduced and its normalization is a disjoint union of a finite number of compact complexmanifolds Y , . . . , Y ℓ for some positive integer ℓ such that(i) each Y i is a P -bundle over an ( n − A i whose fibersare sent to characteristic leaves of f − ( b ) in the sense of [HO1], i.e., for a definingfunction h ∈ O ( B ) of the divisor D , the Hamiltonian vector field ι ω ( f ∗ dh ), where ι ω : Ω M → T ( M ) is the vector bundle isomorphism induced by ω , is tangent to theimage of the fibers in M ;(ii) there exist submanifolds S i , S i ⊂ Y i , with S i = S i except possibly when ℓ = 1 , S i ∪ S i is a 2-to-1 unramified cover of A i under the P -bundle projection;(iii) the normalization ν : S Y i → f − ( b ) is obtained by the identification via a collectionof biholomorphic morphisms g i : S i → S i +11 for 1 ≤ i ≤ ℓ − g ℓ : S ℓ → S withthe additional requirement g = g − if S = S and S = S for ℓ = 2.A maximal connected union of the P -fibers in (i) under the identification in (iii) iscalled a characteristic cycle . A characteristic cycle can be either of finite type ( I m -type,1 ≤ m < ∞ ) or of infinite type A ∞ , which we also denote by I ∞ .Recall (cf. [HO2] Section 4) that in a neighborhood of a general singular fiber, any La-grangian fibration whose fibers are of Fujiki class can be transformed to a stable Lagrangianfibration by certain explicitly given bimeromorphic modifications and branched covering.We will be interested in the local property of the fibration at a point of D . Thus we willmake the following( Assumption ) D ⊂ B is the germ of a smooth hypersurface in an n -dimensional complexmanifold and the fundamental group π ( B \ D ) is cyclic.The following is immediate from Proposition 2.2 of [HO1]. Proposition 2.2.
Given a stable Lagrangian fibration, we can assume that there exists anaction of the complex Lie group C n − on M preserving the fibers and the symplectic formsuch that S i , S i are orbits of this action for all ≤ i ≤ ℓ . This action of C n − on Y i descends to the translation action on A i . The patching biholomorphisms g i in Definition2.1 (iii) as well as the P -bundle structure in (i) are equivariant under this action. Inparticular, if S = S (resp. S = S ), the Galois action of the double cover S → A (resp. S → A ) is by a translation on the torus S (resp. S ). Regarding the topology of the singular fiber f − ( b ) , b ∈ D , we have the following. JUN-MUK HWANG, KEIJI OGUISO
Proposition 2.3.
In the setting of Definition 2.1, fix a component Y of the normalizationof f − ( b ) and set Y o := Y \ ( S ∪ S ) , which is equipped with a C ∗ -bundle structure ̺ : Y o → A over a complex torus A of dimen-sion n − coming from Definition 2.1 (i). Then there exists a (not necessarily holomorphic)continuous map µ : f − ( b ) → A ′ to a complex torus A ′ isogenous to A such that when j : Y → f − ( b ) is the natural inclusion and ρ : Y → A → A ′ is the composition of ̺ andan isogeny, µ ◦ j is homotopic to ρ .Proof. Let us use the notation introduced in Definition 2.1 (iii) for the description of thenormalization morphism ν : S Y i → f − ( b ).First, we consider the case S i = S i for all 1 ≤ i ≤ ℓ . Define ^ f − ( b ) as the variety obtainedfrom S Y i with all the patching identification g , . . . , g ℓ − such that the normalizationfactors through ν : [ Y i → ^ f − ( b ) → f − ( b )with the second arrow given by the identification via g ℓ . When ℓ = 1, ^ f − ( b ) = Y . The C n − -action of Proposition 2.2 lifts to a C n − -action on ^ f − ( b ). The connected unionsof the images of the P -fibers of Y i define finite chains of quasi-transversally intersecting P ’s in ^ f − ( b ) , which we call characteristic chains. Each characteristic chain intersectseach S i (resp. S i ), 1 ≤ i ≤ ℓ at exactly one point, inducing a morphism ^ f − ( b ) → A ′′ to acomplex torus of dimension n − A i ’s. This determines a biholomorphism ζ : S ℓ → S . Fix a point α ∈ S ℓ and let β = ζ ( α ) ∈ S . For t ∈ [0 , ⊂ R , let τ t : S → S be the translation by t ( β − g ℓ ( α )) . Define a new family of biholomorphic morphisms g tℓ : S ℓ → S by g tℓ = τ t ◦ g ℓ . Clearly, g ℓ = g ℓ . We claim that g ℓ = ζ . In fact, ζ − ◦ g ℓ is an automorphism of S ℓ which fixes thepoint α. But both g ℓ and ζ must be equivariant under the C n − -action of Proposition 2.2.Thus ζ − ◦ g ℓ must be the identity map of S ℓ , proving the claim.Let f − ( b ) t be the variety obtained from ^ f − ( b ) by identifying S and S ℓ via g tℓ . Then f − ( b ) = f − ( b )and f − ( b ) is homeomorphic to f − ( b ). The C n − -action descends to f − ( b ) t for each t as τ t commutes with the C n − -action. By abuse of terminology, we call the maximal connectedunions of the images in f − ( b ) t of the characteristic chains as characteristic cycles of f − ( b ) t .By the C n − -action, we know that all characteristic cycles in f − ( b ) t are isomorphic. Byour choice of α and β , there exists one finite characteristic cycle in f − ( b ) . Thus weget a morphism µ ′ : f − ( b ) → A ′ to some complex torus A ′ isogenous to A ′′ . Define µ : f − ( b ) → A ′ as the composition of µ ′ with the homeomorphism f − ( b ) → f − ( b ) . Itcertainly satisfies the required property.Now consider the case when ℓ = 1 and S = S . Set ^ f − ( b ) = Y and define ζ : S → S as the Galois action of the double covering S → A in Definition 2.1 (ii). Then the sameargument as in the previous case applies. OCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 5
Finally, consider the case when ℓ = 2, S = S and S = S . By Proposition 2.2,the Galois action on S (resp. S ) of the double cover over A (resp. A ) is given bya translation, say, by γ ∈ C n − (resp. γ ∈ C n − ). By the equivariance of g = g − ,for each α ∈ S , we have g ( γ · α ) = γ · g ( α ). Thus by the normalization morphism ν : Y ∪ Y → f − ( b ), a point α ∈ S is identified with g ( α ) ∈ S , and the point γ · α ∈ S , which lies in the P -fiber through α , is identified with γ · g ( α ), which liesin the P -fiber through g ( α ). Thus we get a morphism µ : f − ( b ) → A ′ to an ( n − A ′ whose fiber is a union of two P ’s identified at two points. This µ satisfies the required property. (cid:3) We have the generalization of the classical action-angle correspondence as follows.
Proposition 2.4.
Given a stable Lagrangian fibration f : M → B , choose a Lagrangiansection Σ ⊂ M of f . Then we have a natural surjective unramified morphism Φ : T ∗ B → M \ E where E is the union of the irreducible components of the fibers of f disjoint from Σ such that (1) f ◦ Φ agrees with the natural projection g : T ∗ B → B , (2) Φ sends the zero section of T ∗ B to Σ and (3) Φ ∗ ω coincides with the standard symplectic form on T ∗ B .In particular, Γ := Φ − (Σ) is a Lagrangian submanifold (with many connected components)in T ∗ B . For each b ∈ B, Φ b := Φ | T ∗ b ( B ) : T ∗ b ( B ) → f − ( b ) \ E is the universal covering and Γ b := Γ ∩ T ∗ b ( B ) is naturally isomorphic to H ( f − ( b ) \ E, Z ) .Proof. Over B \ D , this is just a holomorphic version (cf. Proposition 3.5 in [Hw]) of theclassical action-angle correspondence as described in Section 44 of [GS]. The statementover D follows by the same argument as for the smooth fibers. In fact, for each b ∈ B , thevector group T ∗ b ( B ) acts on the fiber f − ( b ) with n -dimensional orbits on the smooth locusof f − ( b ) (cf. Proposition 3.3 in [Hw]). The morphism Φ b is defined by taking the orbitmap of the point Σ ∩ f − ( b ) under this action, which is a universal covering map for thesmooth locus of the component of f − ( b ) containing Σ ∩ f − ( b ). This shows (1) and (2).The proof of (3) is the same as that of Theorem 44.2 of [GS]. (cid:3) Proposition 2.5.
Let b ∈ D . In the notation of Proposition 2.4, we can assume that theconnected component of Γ containing each point of Γ ∩ T ∗ b ( B ) is a Lagrangian section of T ∗ ( B ) → B , i.e., a closed 1-form on B . Let Γ ′ ⊂ Γ be the union of such sections of Γ over B . Then for each s ∈ B \ D , Γ ′ s := Γ ′ ∩ T ∗ s ( B ) is a sublattice of Γ s satisfying Γ s / Γ ′ s ∼ = Z .Proof. Since f − ( b ) \ E is a C ∗ -bundle over an ( n − ∩ T ∗ b ( B ) has rank 2 n −
1. Thus Γ ′ s has rank 2 n −
1. It remains to show that Γ s / Γ ′ s istorsion-free. Suppose it has a k -torsion, 0 < k ∈ Z , i.e., there exists a point α ∈ Γ s \ Γ ′ s such that kα ∈ Γ ′ s . Let f kα be a closed 1-form given by the component of Γ ′ containing kα .Then the closed 1-form ˜ α = k f kα is also a component of Γ ′ containing α , which implies α ∈ Γ ′ s , a contradiction. (cid:3) Proposition 2.6.
In the notation of Proposition 2.4, let Y o be the fiber of M \ E at a point b ∈ D . Let Φ b : T ∗ b ( B ) → Y o be the universal covering map and ̺ : Y o → A be the C ∗ -bundle JUN-MUK HWANG, KEIJI OGUISO over an ( n − -dimensional torus. Let Υ ⊂ Γ b = Γ ′ b be the rank-1 sublattice correspondingto the kernel of ̺ ∗ : H ( Y , Z ) → H ( A, Z ) . Then for any v ∈ Γ ′ b \ Υ , there exists ̟ ∈ H ( f − ( b ) , Z ) such that h ̟, j ∗ v i 6 = 0 where j ∗ : H ( Y o , Z ) → H ( f − ( b ) , Z ) is induced by the inclusion j : Y o ⊂ f − ( b ) .Proof. Since ̺ ∗ ( v ) ∈ H ( A, Z ) is non-zero, there exists ϕ ∈ H ( A, Z ) such that h ϕ, ̺ ∗ ( v ) i 6 =0. Let ̟ = µ ∗ ϕ where the map µ : f − ( b ) → A is as defined in Proposition 2.3 satisfying µ ◦ j = ̺. Then h ̟, j ∗ ( v ) i = h µ ∗ ϕ, j ∗ ( v ) i = h j ∗ µ ∗ ϕ, v i = h ̺ ∗ ϕ, v i = h ϕ, ̺ ∗ ( v ) i 6 = 0 . (cid:3) For a stable Lagrangian fibration f : M → B , denote by Λ the local system on B \ D defined by the lattice Λ s := H ( f − ( s ) , Z ) for s ∈ B \ D. Proposition 2.7.
For a stable Lagrangian fibration f : M → B and s ∈ B \ D , fix agenerator of the cyclic fundamental group of π ( B \ D, s ) and denote by τ s : Λ s → Λ s themonodromy operator of the generator. Then the fixed part Λ ′ s ⊂ Λ s of τ s at s ∈ B \ D hascorank 1.Proof. For any s ∈ B \ D , we can identify each fiber Λ s = H ( M s , Z ) with the fiber Γ s ofProposition 2.4. Thus the result follows from Proposition 2.5. (cid:3) Principally polarized stable Lagrangian fibration
Definition 3.1.
Let Λ be as in Proposition 2.7. A principal polarization on a stableLagrangian fibration f : M → B is a unimodular anti-symmetric form Q : ∧ Λ → Z B \ D where Z B \ D denotes the constant sheaf of integers on B \ D , which induces a principalpolarization on each smooth fibers of f . A stable Lagrangian fibration with a choice ofprincipal polarization is called a principally polarized stable Lagrangian fibration . Remark 3.2.
In Definition 3.1, the polarization on M \ f − ( D ) may not extend to an f -ample class of the whole M . In fact, f need not be projective. This definition is usefulbecause there are many situations where the polarization exists a priori only on the smoothfibers, e.g., in Kodaira’s study of elliptic fibrations and also in our construction in Section5. Proposition 3.3.
Let f : M → B be a principally polarized stable Lagrangian fibration.Then the monodromy operator in Proposition 2.7 satisfies τ s = Id and τ s ◦ τ s = Id . Proof. If τ s = Id, then we see that f is a smooth fibration, as in the proof of Proposition 3.2in [Hw]. In fact, since there is no monodromy and f is polarized over B \ D , we can extendthe period map of the abelian family on B \ D to the whole B ([Gr], Theorem 9.5). Thus,we obtain a smooth abelian fibration f ′ : M ′ → B such that f and f ′ are bimeromorphicoutside D . Since M ′ contains no rational curves and both M and M ′ have trivial canonicalbundles, this implies M and M ′ are biholomorphic, a contradiction to the non-emptinessof the discriminant D of f . OCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 7 If τ s ◦ τ s = Id , take a double cover g : B ′ → B branched along D and let D ′ = g − ( D ).Denote by ˆ f : ˆ M → B ′ the fiber product of f and g , which has no monodromy on B ′ \ D ′ . By the C n − -action of Proposition 2.2 which lifts to ˆ M , the following property ofˆ M can be seen from the corresponding properties in the case of n = 1 (cf. Proof ofProposition 9.2 in [BHPV]): ˆ M is normal, Gorenstein with singularities of type A × ( germ of (2 n − f ′ : M ′ → B ′ , which is a family with trivial canonical bundle and nomonodromy. Then we get a contradiction as in the previous case. (cid:3) Lemma 3.4.
Let τ : Λ → Λ be an automorphism of a lattice such that Λ ′ := { v ∈ Λ , τ ( v ) = v } is a sublattice of corank 1, i.e., Λ / Λ ′ ∼ = Z . If τ ◦ τ = Id , then η := τ − Id satisfies η ◦ η = 0 . Proof.
Note that Λ ′ ⊂ Ker( η ). The induced automrophism ¯ τ : Λ / Λ ′ → Λ / Λ ′ is either Idor − Id. If ¯ τ = Id, then for a non-zero v ∈ Λ \ Λ ′ , we have τ ( v ) = v + λ for some λ ∈ Λ ′ .Then η ( v ) = λ ∈ Λ ′ ⊂ Ker( η ). This proves that η ◦ η = 0. If ¯ τ = − Id, then for a non-zero v ∈ Λ \ Λ ′ , we have τ ( v ) = − v + λ for some λ ∈ Λ ′ . Then τ ◦ τ ( v ) = − τ ( v ) + τ ( λ ) = − ( − v + λ ) + λ = v. Thus τ ◦ τ = Id , a contradiction. (cid:3) Proposition 3.5.
In the setting of Proposition 3.3, let η := τ s − Id . Then for any β ∈ Im( η ) and an element ϕ ∈ H ( M, Z ) , h i ∗ ϕ, β i = h ϕ, i ∗ β i = 0 where i ∗ : H ( M, Z ) → H ( M s , Z ) and i ∗ : H ( M s , Z ) → H ( M, Z ) are the homomorphismsinduced by the inclusion i : M s := f − ( s ) ⊂ M .Proof. Let H be the local system on B \ D given by H ( M s , Z ) , t ∈ B \ D . Denote by τ ∗ : H s → H s the transformation dual to τ , i.e., for any ̟ ∈ H ( M s , Z ) and u ∈ H ( M s , Z ), h τ ∗ ( ̟ ) , u i = h ̟, τ ( u ) i . By Proposition 2.7, Proposition 3.3 and Lemma 3.4, we have η = 0 and η ◦ η = 0, i.e.,0 = Im( η ) ⊂ Ker( η ) = Λ ′ s . Similarly, η ∗ := τ ∗ − Id is an endomorphism of H s with η ∗ = 0 and η ∗ ◦ η ∗ = 0. Since i ∗ ϕ ∈ Ker( η ∗ ) by (the easy half of) the global invariant cycles theorem (cf. Theorem 4.24of [Vo]), for any ψ ∈ Ker( η ∗ ) and u ∈ Λ s , h ψ, η ( u ) i = h η ∗ ( ψ ) , u i = 0 . It follows that h i ∗ ϕ, Im( η ) i = 0 . (cid:3) Remark 3.6.
If the family f : M → B is projective, we could have used the MonodromyTheorem (cf. Theorem 3.15 in [Vo]) in place of Proposition 3.3 and Lemma 3.4 in theabove proof. We have used the above approach because we do not want to assume that f is projective. Proposition 3.7.
For a principally polarized stable Lagrangian fibration f : M → B and s ∈ B \ D , let τ s : Λ s → Λ s be the monodromy operator of Proposition 2.7, which should JUN-MUK HWANG, KEIJI OGUISO preserve the polarization Q s : ∧ Λ s → Z . Setting η := τ s − Id as in Proposition 3.5, wehave Λ ′ s = Ker( η ) . Then Im( η ) ⊂ Λ s is contained in Ξ s := { v ∈ Λ ′ s | Q ( v, w ) = 0 for all w ∈ Λ ′ s } . Proof.
Since τ s preserves the polarization Q s and η ◦ η = 0 by Lemma 3.4, Q s ( η ( v ) , u ) + Q s ( v, η ( u )) = 0 for all v, u ∈ Λ s . Thus for any v ∈ Λ s and u ∈ Ker( η ) = Λ ′ s , we have Q s ( η ( v ) , u ) = − Q s ( v, η ( u )) = 0 , whichmeans η ( v ) ∈ Ξ s . (cid:3) Definition 3.8.
Let Λ be a free abelian group of rank 2 n . Given a unimodular non-degenerate anti-symmetric form Q : ∧ Λ → Z , a basis { p , . . . , p n , q , . . . , q n } of Λ is calleda symplectic basis of Λ with respect to Q if, in terms of the dual basis { p , . . . , p n , q , . . . , q n } of Hom(Λ , Z ), Q = p ∧ q + p ∧ q + · · · + p n ∧ q n . Lemma 3.9.
In the setting of Definition 3.8, let τ : Λ → Λ be a group automorphism pre-serving Q . Assume that the subgroup Λ ′ ⊂ Λ of elements fixed under τ has corank 1. Thenthere exists a symplectic basis { p , . . . , p n , q , . . . , q n } such that { p , . . . , p n , q , . . . , q n − } ⊂ Λ ′ . Proof.
Fix a symplectic basis { a , . . . , a n , b , . . . , b n } such that Q = a ∧ b + · · · + a n ∧ b n . The anti-symmetric form Q | Λ ′ must have a kernel of rank 1, i.e.,Ξ := { v ∈ Λ ′ , Q ( v, u ) = 0 for all u ∈ Λ ′ } has rank 1. Pick a generator p of Ξ. Since Ξ is primitive, i.e., Λ / Ξ has no torsion, we canwrite p = α a + · · · + α n a n + β b + · · · + β n b n with some integers α i , β i satisfying gcd ( α , . . . , α n , β , . . . , β n ) = 1 . Thus there exists inte-gers α ′ , . . . , α ′ n , β ′ , . . . , β ′ n such that α ′ · α + · · · + α ′ n · α n + β ′ · β + · · · + β ′ n · β n = 1 . Let q := − β ′ a − · · · − β ′ n a n + α ′ b + · · · + α ′ n b n . Then Q ( p , q ) = 1 . DefineΛ ′′ := { v ∈ Λ , Q ( p , v ) = 0 = Q ( q , v ) } . Then Λ ′′ ⊂ Λ ′ is a lattice of rank 2 n − Q | Λ ′′ is unimodular and non-degenerate(cf. [GH], the proof of Lemma in p.304). Let { p , . . . , p n , q , . . . , q n } be a symplectic basis ofΛ ′′ . Then { p , . . . , p n , q , . . . , q n } is a symplectic basis of Λ with the required property. (cid:3) Proposition 3.10.
In the setting of Proposition 2.5, identify Λ s = H ( M s , Z ) with Γ s for s ∈ B \ D as in the proof of Proposition 2.7. Assume that we have a principal polarization Q . Then we can find a collection of components { p , . . . , p n , q , . . . , q n − } of Γ ′ such that foreach x ∈ B \ D , there exists q n,x ∈ Γ x such that { p ,x , . . . , p n,x , q ,x , . . . , q n,x } is a symplecticbasis of Λ x = Γ x with respect to Q x . OCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 9
Proof.
Fix a point s ∈ B \ D . The monodromy τ s : Λ s → Λ s preserves the polariza-tion Q s on Λ s and fixes Λ ′ s = Γ ′ s . Applying Lemma 3.9, we have a symplectic basis { p ,s , . . . , p n,s , q ,s , . . . , q n,s } with p ,s , . . . , p n,s , q ,s , . . . , q n − ,s ∈ Λ ′ s . Since Γ ′ consists of sec-tions of g : T ∗ B → B , the vectors p ,s , . . . , p n,s , q ,s , . . . , q n − ,s uniquely determine compo-nents p , . . . , p n , q , . . . , q n − of Λ ′ . To check the existence of q n,x for any x ∈ B \ D, justpick q n,x as any vector in Λ x contained in the component of Λ containing q n,s . (cid:3) Proposition 3.11.
In the notation of Proposition 3.10, when b ∈ D , the vector p n,b ∈ Γ ′ b regarded as an element of H ( Y o , Z ) in the notation of Proposition 2.6, lies in the lattice Υ of Proposition 2.6.Proof. Suppose not. By our (Assumption) after Definition 2.1, we may assume that M is topologically retractable to f − ( b ) and identify H ( f − ( b ) , Z ) with H ( M, Z ). Then byProposition 2.6, there exists ̟ ∈ H ( f − ( b ) , Z ) = H ( M, Z ) such that h ̟, j ∗ p n,b i 6 = 0. Fora point s ∈ B \ D , the choice in Proposition 3.10 implies that p n,s ∈ Ξ s of Proposition 3.7.Denote by ̟ s the element in H ( M s , Z ) induced by ̟ ∈ H ( M, Z ) under the identification H ( f − ( b ) , Z ) = H ( M, Z ) . Since j ∗ p n,b ∈ H ( f − ( b ) , Z ) = H ( M, Z ) and the image of p n,s ∈ H ( M s , Z ) in H ( M, Z ) belongs to the same class, Proposition 3.5 and Proposition3.7 say that h ̟, j ∗ p n,b i = h ̟ s , p n,s i = 0 . This is a contradiction. (cid:3)
Proposition 3.12.
In Proposition 3.11, the C -linear span of Υ in T ∗ b ( B ) is exactly C · dh where h ∈ O ( B ) is a defining equation of the divisor D .Proof. From the definition of Υ in Proposition 2.6, the linear span of Υ is sent to a fiber ofthe C ∗ -bundle. By Definition 2.1 (i), this fiber is a leaf of the characteristic foliation, whichis given by the Hamiltonian vector field ι ω ( f ∗ dh ) on M . Under the symplecto-morphism Φin Proposition 2.4, this corresponds to C · dh . (cid:3) Proposition 3.13.
Let { p , . . . , p n , q , . . . , q n − } be as in Proposition 3.10. Then thereexists a holomorphic coordinate system { z , . . . , z n } on B such that, regarded as sections of T ∗ ( B ) , p = dz , . . . , p n = dz n and D is given by z n = 0 . Proof.
Since p , . . . , p n are closed 1-forms which are point-wise linearly independent at everypoint of B , we can find coordinates z , . . . , z n with p i = dz i . By Proposition 3.12, we maychoose z n to be a defining equation of D . (cid:3) Let us recall the classical Riemann condition (e.g. [GH], p.306).
Proposition 3.14.
Let V be a complex vector space of dimension n and let Λ ⊂ V be alattice of rank n such that V / Λ is an abelian variety with a principal polarization. Fora symplectic basis { p , . . . , p n , q , . . . , q n } of Λ with respect to the principal polarization Q : ∧ Λ → Z , { p , . . . , p n } becomes a C -basis of V and the period matrix ( θ ji ) defined by q i = n X j =1 θ ji p j in V is symmetric in ( i, j ) and Im( θ ji ) > . Theorem 3.15.
Given a principally polarized stable Lagrangian fibration f : M → B witha Lagrangian section Σ ⊂ M , there exists a holomorphic coordinate system ( z , . . . , z n ) on B such that (i) z n = 0 is a local defining equation of D ; (ii) on B \ D , dz , . . . , dz n − , dz n belong to Γ ′ in the notation of Proposition 2.5; (iii) there exists a symplectic basis { p ,s , . . . , p n,s , q ,s , . . . , q ,n } on each Λ s = Γ s , s ∈ B \ D satisfying p ,s = ( dz ) s , . . . , p n,s = ( dz n ) s and the associated period matrix in the sense of Proposition 3.14 is given by θ ji = ∂ Ψ ∂z i ∂z j + ℓ π √− z n for some holomorphic function Ψ on B , which we call a potential function of theLagrangian fibration, and some integer ℓ .Proof. Let { p , . . . , p n , q , . . . , q n − } be as in Proposition 3.10 and Proposition 3.13. At apoint s ∈ B \ D , we add q n,s to get a symplectic basis of Λ s . By analytic continuation, weget a multi-valued 1-form q n over B \ D such that any choice of a value q n,t of q n at a point t ∈ B \ D , together with p ,t , . . . , p n,t , q ,t , . . . , q n − ,t , gives a symplectic basis of Λ t . Usingthe coordinate system in Proposition 3.13, we can write q i = n X j =1 θ ji dz j , where θ ji is a (univalent) holomorphic function on B for each 1 ≤ i ≤ n − ≤ j ≤ n, while θ jn is a multi-valued holomorphic function on B \ D for each 1 ≤ j ≤ n . By Proposition3.14, θ ij = θ ji for each 1 ≤ i, j ≤ n . It follows that θ jn is univalent holomorphic function on B for each 1 ≤ j ≤ n −
1. By the choice of p n,s ∈ Ξ s and Proposition 3.7, the monodromyoperator τ s : Λ s → Λ s is of the form τ s ( q n,s ) = q n,s + ℓp n,s for some integer ℓ . Thus ˜ θ nn := θ nn − ℓ π √− z n is also univalent. Set ˜ θ ji := θ ji if ( i, j ) = ( n, n ). Then ˜ θ ji is a univalent holomorphic functionon B for all values of 1 ≤ i, j ≤ n and q i = n X j =1 ˜ θ ji dz j for 1 ≤ i ≤ n − q n = n X j =1 ˜ θ jn dz j + ℓ π √− z n dz n . OCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 11
Since q i ’s are closed 1-forms on B , we have ∂ ˜ θ ji ∂z k = ∂ ˜ θ ki ∂z j = ∂ ˜ θ ij ∂z k for any 1 ≤ i, j, k ≤ n . By Poincar´e’s lemma, there exists a holomorphic function Ψ suchthat ˜ θ ji = ∂ Ψ ∂z i ∂z j . (cid:3) Construction of principally polarized stable Lagrangian fibrations withgiven potential functions
In this section, for a sufficiently small n -dimensional polydisk B with coordinate ( z , . . . , z n ),we shall construct a principally polarized stable Lagrangian fibration f : ( M, ω M ) → B witha given potential function Ψ( z ). Our construction closely follows Nakamura’s toroidal con-struction [Na]. However, main differences are the following:(i) the base space B is of dimension n (rather than 1).(ii) the total space should be not only smooth but also symplectic. (I) Construction of a non-proper Lagrangian fibration ˜ M → B . For each integer k ∈ Z , let E k be a copy of C × C equipped with linear coordinates( x k , y k ). We define a complex manifold E by identifying points in ∪ k ∈ Z E k by the followingrule: a point ( x k , y k ) of E k with x k = 0 and y k = 0 is identified with a point ( x k +1 , y k +1 )of E k +1 with x k +1 = 0 and y k +1 = 0, if and only if x k +1 = x k y k , and y k +1 = 1 x k . On E , z n := x k y k is a well-defined holomorphic function independent of k and w n := x − k +1 k y − kk is a meromorphic function independent of k , with zeros and poles supported on ∪ k ∈ Z ( x k y k = 0) . Moreover, the 2-forms dy k ∧ dx k glue together yielding a holomorphic symplectic form ω E on E , satisfying ω E = dz n ∧ dw n w n . Fix coordinates ( z , . . . , z n − , w , . . . , w n − )on C n − × C n − and regard them as functions on the open subset C n − × ( C × ) n − definedby w = 0 , . . . , w n − = 0 . Define ˜ X := C n − × ( C × ) n − × E. On ˜ X , we have the holomorphic functions z , . . . , z n , w , . . . , w n − and the meromorphicfunction w n . Define a morphism ˜ p : ˜ X → C n by ( z , . . . , z n ). The fiber of ˜ p over b with z n ( b ) = 0 isisomorphic to ( C × ) n − × C × with coordinates ( w , . . . , w n − , w n ) and the fiber over b with z n ( b ) = 0 is isomorphic to( C × ) n − × ∪ k ∈ Z P k where P k is a copy of the projective line P with affine coordinate y k .We have a holomorphic symplectic 2-form ω ˜ X := n − X i =1 dz i ∧ dw i w i + ω E = n X i =1 dz i ∧ dw i w i on ˜ X . From now, we regard ˜ X as a symplectic manifold by this symplectic form. Fromthe coordinate expression of ω ˜ X and ˜ p , it is immediate that ˜ p is a non-proper Lagrangianfibration.We denote ˜ M = ˜ X × C n B where B = { ( z , . . . , z n − , z n ) | | z i | < ǫ ( ∀ i ) } and ǫ is a sufficiently small positive real number. We denote the natural projection ˜ M → B induced from ˜ p by ˜ f : ˜ M → B .
Note that the restriction ω ˜ M of ω ˜ X is a symplectic 2-form on ˜ M and ˜ f is a non-properLagrangian fibration. (II) Group action of Γ = Z n on ˜ M . Let Ψ( z , z , · · · , z n ) be a holomorphic function on B such that the imaginary partIm ˜ θ ( z ) of the Hessian matrix ˜ θ ( z ) = ( ∂ Ψ ∂z i ∂z j )is positive definite and ℓ be a positive integer. We define the period matrix θ ( z ) by θ ( z ) = ˜ θ ( z ) + log z n π √− (cid:18) O n − ℓ (cid:19) . We will write ˜ θ ( z ) = (cid:18) ˜Θ ( z ) ˜Θ ( z )˜Θ t ( z ) ˜ θ nn ( z ) (cid:19) , where ˜Θ ( z ) is the ( n − × ( n −
1) matrix, ˜Θ ( z ) is the ( n − × t ( z ) is thetranspose of ˜Θ ( z ) and ˜ θ nn ( z ) is 1 × Z n − ⊕ Z . We define a group action of Γ on ˜ M as follows. Let γ = ( j, m ) ∈ Γ. Then the action T γ : ˜ M → ˜ M is defined in terms of the coordinate functions on C n − × ( C × ) n − × E k ⊂ ˜ M by T ∗ γ z i = z i for i = 1 , . . . , n − OCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 13 T ∗ γ (Π n − i =1 w b i i ) = exp (2 π √− j ˜Θ ( z ) b + m ˜Θ t ( z ) b )Π n − i =1 w b i i where b = ( b i ) n − i =1 is ( n − × T ∗ γ x k = (exp (2 π √− j ˜Θ ( z ) + m ˜ θ nn ( z ))) − x k − mℓ T ∗ γ y k = exp (2 π √− j ˜Θ ( z ) + m ˜ θ nn ( z )) y k − mℓ . It is immediate that T ∗ γ T ∗ γ ′ = T ∗ γ + γ ′ . Then T γ ∈ Aut ( ˜ X/ C n ) and γ T γ defines aninjective group homomorphism from Γ to Aut ( ˜ M / C n ). Here Aut ( ˜ M / C n ) is the group ofautomorphisms of ˜ M over C n , i.e., the group of automorphisms g of ˜ M such that ˜ f ◦ g = ˜ f . Proposition 4.1.
The action Γ on ˜ M is properly discontinuous, free and symplectic, inthe sense that T ∗ γ ω ˜ M = ω ˜ M for each γ ∈ Γ .Proof. Freeness of the action is clear from the description of the action. The proof of properdiscontinuity is essentially the same as the proof of [Na], Theorem 2.6. This can be alsoseen from the concrete description of fibers below in (III), at least fiberwisely.Let us show that the action is symplectic, i.e., ω ˜ M = T ∗ γ ω ˜ M for each γ = ( j, m ). This isa new part not considered by [Na]. We have T ∗ γ dz i = dz i , T ∗ γ dw i = exp (2 π √− f i ( z )) w i forall i , where, in terms of the standard basis h e i i n − i =1 of C n − ,f i ( z ) = j ˜Θ ( z ) e i + m ˜Θ ( z ) e i for 1 ≤ i ≤ n − f n ( z ) = j ˜Θ ( z ) + m ˜ θ nn ( z ) . Thus, for i with 1 ≤ i ≤ n , we have T ∗ γ dz i = dz i ,T ∗ γ dw i w i = T ∗ γ ( d log w i )= d log( T ∗ γ w i ) = d (2 π √− f i ( z )) + d (log w i )= dw i w i + 2 π √− n X k =1 ∂f i ∂z k dz k . Using these identities, we can compute T ∗ γ ω ˜ M = n X k =1 T ∗ γ ( dz i ) ∧ T ∗ γ ( dw i w i )= ω ˜ M − π √− n X i =1 n X k =1 ∂f i ∂z k dz k ∧ dz i = ω ˜ M − π √− X ≤ i Γ. By Proposition 4.1, M is a smooth symplectic manifold with symplecticform ω M induced by ω ˜ M and M admits a fibration f : M → B induced by ˜ f . We denotethe (scheme theoretic) fiber f − ( b ) over b ∈ B by M b . Let us describe the fibers M b . (III-1) Smooth fibers M b First consider the case where z n ( b ) = 0, i.e., the case where M b is smooth. We have˜ M b = ( C × ) n − × { ( x k , y k ) | x k y k = b n } ≃ ( C × ) n ( w ,...,w n − ,w n ) . and w n = z n ( b ) k y k . Let h e i i ni =1 be the ordered standard basis of Γ. From the descriptionin (II), the action of Γ is given by: T ∗ e i w j = exp(2 π √− θ ji ( b )) w j T ∗ e i w n = exp(2 π √− θ ni ( b )) w n for 1 ≤ i ≤ n − θ ji = ˜ θ ji and T ∗ e n w j = exp(2 π √− θ jn ( b )) w j T ∗ e n w n = exp(2 π √− θ nn ( b )) z n ( b ) ℓ w n = exp(2 π √− θ nn ( b )) w n . Hence M b = ˜ M b / Γ is an n -dimensional principally polarized abelian variety of period θ ( b ), as desired. By the description of ω M , the fibers M b , z n ( b ) = 0 , are also Lagrangiansubmanifolds. Proposition 4.2. For b ∈ B, z n ( b ) = 0 , choose the basis p ,b , . . . , p n,b q ,b , . . . , q n,b of H ( M b , Z ) such that ˜ M b = C n / h p j,b i nj =1 and q i,b = n X j =1 θ ji ( b ) p j,b OCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 15 for each i ( ≤ i ≤ n ). Let p b , . . . , p nb q b , . . . , q nb be the dual basis of H ( M b , Z ) . Then the integral -form L b := n X i =1 p ib ∧ q ib give a monodromy invariant principal polarization of M over B \ D where D = ( z n = 0) .Proof. When z n ( b ) = 0, the fiber ˜ M b of ˜ p : ˜ M → B is ( C × ) n and this family has nomonodromy over B \ ( z n = 0). Thus we can fix a basis p ,b , . . . , p n,b of H ( ˜ M b , Z ) uniformlyin b, z n ( b ) = 0 . To get a basis of H ( M b , Z ) , we choose additional elements q ,b , . . . , q n,b ∈ H x ( M b , Z )determined by the deck-transformation of ˜ M b induced by the action T e , . . . , T e n . From thedescription of T ∗ e i on w j , they satisfy the relation q i,b = n X j =1 θ ji ( b ) p j,b . We see that q ,b , . . . , q n − ,b are invariant under the monodromy, while q n,b q n,b + ℓp n,b under the monodromy of the generator γ of π ( B \ D ), i.e., the circle around discriminantdivisor z n = 0. The 2-form L b is a principal polarization on M b . It remains to show that L b is invariant under the monodromy. By definition of θ ( b ), we compute that γ ∗ ( L b ) = γ ∗ ( n − X i =1 p i,b ∧ q i,b ) + γ ∗ ( p n,b ∧ q n,b )= n − X i =1 p ib ∧ q ib + p nb ∧ ( q nb − ℓp nb ) = L b . This implies the invariance. (cid:3) Remark 4.3. As in [Na], one can also describe ˜ f : ˜ M → B in terms of toric geometry.Following an argument similar to [Na], Section 4, it seems possible to give a relativelyprincipally polarized divisor (the relative theta divsor) which is defined globally over B \ D .However, its closure is not necessarily f -ample even if total space is of dimension 4 (casesof stable principally polarized Lagrangian 4-folds). In fact, a failure of f -ampleness of theclosure already happens when the fiber dimension 2 and the base dimension 1 as explicitlydescribed in [Na] Section 4, Page 219. See also Remark 3.6. (III-2) Singular fibers M b Next consider the singular fibers of f . They are M b with z n ( b ) = 0. Recall from (I)that ˜ M b is the product of ( C × ) n − with coordinate ( w i ) n − i =1 and the infinite tree ∪ k ∈ Z P k ofprojective lines P k with affine coordinate y k , and M b = ˜ M b / Γ. Let us denote by (0) k , ( ∞ ) k ∈ P k the two points on the projective line P k such that (0) k is identified with ( ∞ ) k − in thetree. From (II), the action of Γ is given by: T ∗ e i w j = exp(2 π √− θ ji ( b )) w j T ∗ e i y k = exp(2 π √− θ ni ( b )) y k for 1 ≤ i ≤ n − T ∗ e n w j = exp(2 π √− θ jn ( b )) w j T ∗ e n y k = exp(2 π √− θ nn ( b )) y k − ℓ , where h e i i ni =1 is the ordered standard basis of Γ. Here we note that the last equality showsthat the monodromy operation corresponds to the shift of the components of the infinitetree ∪ k ∈ Z P k . Thus ˜ M / < e n > can be described as the variety obtained from( C × ) n − × ∪ ℓ − k =0 P k by identifying the point ( w , . . . , w n − ) × (0) ∈ ( C × ) n − × (0) with the point(exp(2 π √− θ n ) w , . . . , exp(2 π √− θ n − n ) w n − ) ∈ ( C × ) n − × ( ∞ ) ℓ − . From this description, M b consists of ℓ irreducible components, each of whose normal-ization is isomorphic to a P -bundle over ( n − C × ) n − / h e i i n − i =1 , where the action of h e i i n − i =1 is given by the coordinate action T ∗ e i (1 ≤ i ≤ n − 1) on w j (1 ≤ j ≤ n − 1) described above. Note that the quotient( C × ) n − / h e i i n − i =1 is compact because the imaginary part of ˜Θ ( z ) is positive definite fromthe assumption that the imaginary part of ˜ θ is positive definite. We also note that thecharacteristic cycles are of type I m for some 1 ≤ m ≤ ∞ .From this description, the following is now clear: Theorem 4.4. The fibration f : M → B constructed above is a proper, flat, principallypolarized stable Lagraingian fibration with a potential function Ψ( z ) . Moreover, ℓ is thenumber of components of the singular fiber and S i = S i for all i in the notation of Definition2.1. Given a principally polarized Lagrangian fibration, we can find a potential function Ψon B as in Theorem 3.15. Starting from Ψ we can construct a principally polarized stableLagrangian fibration by Theorem 4.4. These two Lagrangian fibrations must agree outsidethe discriminant set. Thus they must be biholomorphic by the following. Proposition 4.5. Let f : M → B and f ′ : M ′ → B be two Lagrangian fibrations with thesame discriminant D ⊂ B , having Lagrangian sections Σ ⊂ M and Σ ′ ⊂ M ′ . Suppose thereexists a biholomorphic morphism Φ : M \ f − ( D ) → M ′ \ f ′− ( D ) such that Φ(Σ) = Σ ′ and Φ is symplectomorphic, i.e., Φ ∗ ω M ′ = ω M . Then Φ extends to a biholomorphic morphism M ∼ = M ′ .Proof. The proof is essentially given in the proof of Proposition 5.1 in [HO2]. Let us sketchthe argument. One can see that Φ is a bimeromorphic map between M and M ′ . Chooseholomorphic coordinates ( z , . . . , z n ) on B such that the discriminant D is defined by z n = 0.Then the Hamiltonian vector fields induced by dz , . . . , dz n − determine C n − -actions on OCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 17 M and M ′ such that Φ is equivariant with respect to them. These C n − -actions are freeand all orbits have dimension n − 1. Since both M and M ′ have trivial canonical bundle,the exceptional loci of the bimeromorphic map must be of codimension ≥ 2. Since theyare invariant under the C n − -actions, they must be union of finitely many orbits of C n − -actions. But then each component of the exceptional loci in M must be transformed to acomponent of the exceptional loci in M ′ biholomorphically. This implies that there are noexceptional loci and Φ is biholomorphic. (cid:3) As a corollary of Theorem 4.4 and Proposition 4.5, we obtain Corollary 4.6. For any principally polarized stable Lagrangian fibration, the positive inte-ger | ℓ | in Theorem 3.15 is the number of components of the singular fiber and S i = S i foreach i in the notation of Definition 2.1. Theorem 3.15, Theorem 4.4 and Corollary 4.6 complete the proof of Theorem 1.1.5. Periods and the characteristic cycles In this section, we will examine the relation between types of the characteristic cyclesand the periods. For simplicity, we will restrict our discussion to the case of ℓ = 1. Thegeneralization to arbitrary ℓ is straightforward. Explicit constructions (Constructions I-III) will be given when n = 2, i.e., constructions of 4-dimensional principally polarizedstable Lagrangian fibrations. Construction I gives an explicit example in which the typesof characteristic cycles change fiber by fiber. Construction II gives an explicit example inwhich the types of characteristic cycles are constant type I n ( n < ∞ ) and Construction IIIgives an explicit example in which the types of characteristic cycles are constant type A ∞ . Proposition 5.1. Let f : M → B be a n -dimensional principally polarized stable La-grangian fibration with potential function Ψ( z ) and ℓ = 1 . We denote the (univalent)period matrix by ˜ θ ( z ) = (˜ θ ji ( z )) ni,j =1 and the multi-valued period matrix θ ( z ) of f as θ ( z ) = ˜ θ ( z ) + log z n π √− (cid:18) O n − 00 1 (cid:19) . For b ∈ B for which M b is singular, define n ( b ) ( ≤ n ( b ) ≤ ∞ ) to be the order of (exp(2 π √− θ jn ( b ))) n − j =1 mod h (exp(2 π √− θ ji ( b ))) n − j =1 | ≤ i ≤ n − i in the multiplicative group ( C × ) n − / h (exp(2 π √− θ ji ( b ))) n − j =1 | ≤ i ≤ n − i . Then the characteristic cycle of M b is of type I n ( b ) . Remark 5.2. The description above is simpler when f : M → B is a 4-dimensionalprincipally polarized stable Lagrangian fibration with potential function Ψ( z ) = Ψ( z , z )and ℓ = 1, as follows. We shall use this description in Constructions (I)-(III) below. Wewrite the (univalent) period matrix ˜ θ ( z ) and the multi-valued period matrix θ ( z ) of f as˜ θ ( z ) = (cid:18) ˜ θ ( z ) ˜ θ ( z )˜ θ ( z ) ˜ θ ( z ) (cid:19) , θ ( z ) = ˜ θ ( z ) + log z π √− (cid:18) (cid:19) . For b = ( b , ∈ B for which M b is singular, the characteristic cycle of M b is then of type I n ( b ) , where n ( b ) (1 ≤ n ( b ) ≤ ∞ ) is exactly the order ofexp(2 π √− θ ( b )) mod h exp(2 π √− θ ( b )) i in the multiplicative group C × / h exp(2 π √− θ ( b )) i . Proof. In the description (III-2), M b is the quotient of ˜ M b = ∪ k ∈ Z ( C × ) n − × P k withcoordinates (( w j ) n − j =1 , y k ) ( k ∈ Z ) by the action of Γ = Z n with ordered standard basis h e , . . . , e n − , e n i . In terms of the standard basis, the action is given by: T ∗ e i : ( w j ) n − j =1 (exp(2 π √− θ ji ( b )) w j ) n − j =1 T ∗ e i : y k exp(2 π √− θ ni ( b )) y k for e = i (1 ≤ i ≤ n − e n T ∗ e n : ( w j ) n − j =1 (exp(2 π √− θ jn ( b )) w j ) n − j =1 T ∗ e n : y k exp(2 π √− θ nn ( b )) y k − . Thus ˜ M b / h e n i is ( C × ) n − × P in which (( w j ) n − j =1 , 0) and (( w ′ j ) n − j =1 , ∞ ) are identified exactlywhen the two points ( w j ) n − j =1 and ( w ′ j ) n − j =1 of ( C × ) n − are in the same orbit under the actionof the cyclic subgroup G ( b ) := h (exp(2 π √− θ jn ( b ))) n − j =1 i of ( C × ) n − . On the other hand, (( C × ) n − × P ) / h e i i n − i =1 is the normalization of M b . Thus, M b is obtained from (( C × ) n − × P ) / h e i i n − i =1 by identifying the two ( n − − dimensionalcomplex tori (( C × ) n − × {∞} ) / h e i i n − i =1 and (( C × ) n − × { } ) / h e i i n − i =1 ,by the action of G ( b )above. Here, as a subgroup of ( C × ) n − , the group h e i i n − i =1 is the multiplicative subgroupgenerated by the n − π √− θ ji ( b ))) n − j =1 , ≤ i ≤ n − . This implies the result. (cid:3) Construction I. Under the notation of Section 5, we set ℓ = 1 andΨ( z , z ) := ( z + 5 √− + ( z + 5 √− + 3 z z + 3 z z , Then˜ θ ( z ) = (cid:18) z + z + 5 √− z + z z + z z + z + 5 √− (cid:19) , θ ( z ) = ˜ θ ( z ) + log z π √− (cid:18) (cid:19) and Im ˜ θ ( z ) = (cid:18) y + y + 5 y + y y + y y + y + 5 (cid:19) . OCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 19 Here and hereafter x i and y i are the real and imaginary part of z i respectively. Since t + 5 > t + 5) − t > − < t < 2, it follows that Im θ ( z ) is positive definiteon the polydisk { ( z , z ) | | z i | < } . Taking a smaller 2-dimensional polydisk B with multi-radius ǫ , we then obtain a 4-dimensionalLagrangian fibration f : M → B , associated with the potential function Ψ( z ) and ℓ = 1.The discriminant set is z = 0. Define N = N ( z ) to be the order of e π √− z mod h e π √− z +5 √− i in the multiplicative group C × / h e π √− z +5 √− i . By abuse of language, we include N = ∞ when the order is not finite. Then, the characteristic cycle of M ( z , is of type I N . Proposition 5.3. In Construction 1, the characteristic cycle on M ( z , is of Type I k with k < ∞ if and only if z ∈ Q ( √− . So, the singular fibers of finite characteristic cycle I k ( k < ∞ ) and the singular fibersof infinite characteristic cycle I ∞ are both dense over the disciminant set. Moreover, thecharacteristic cycle of M ( z , is precisely of type I k ( k < ∞ ) for z = 1 /k . So, the singularfibers with characteristic cycles of type I k with any sufficiently large k appear in this family.Proof. By the definition of N = N ( z ), it follows that N < ∞ for M ( z , if and only if thereare integers k > m such that( e π √− z ) k = ( e π √− z +5 √− ) k . The last condition is equivalent to kz − m ( z + 5 √− ∈ Z which is also equivalent to( k − m ) x ∈ Z and ( k − m ) y − m = 0 . Note that k − m = 0 in the last equivalent condition, as otherwise k = m = 0. It isimmediate to see that two integers k > m satisfying last equivalent condition exist ifand only if z ∈ Q ( √− | e π √− z +5 √− | > | z | < 1, whereas | e π √− /k | = 1for k ∈ Z , it follows that the order N ( z ) for z = 1 /k is precisely the order of e π √− /k inthe multiplicative group C × . This implies the last statement. (cid:3) Construction II. Under the notation of Section 5, we set ℓ = 1 andΨ( z , z ) := √− z + z )2 + z z k , where n is a positive integer. Then˜ θ ( z ) = (cid:18) √− /k /k √− (cid:19) , θ ( z ) = ˜ θ ( z ) + log z π √− (cid:18) (cid:19) and Im ˜ θ ( z ) = (cid:18) (cid:19) . The matrix Im ˜ θ ( z ) is positive definite. So, taking a smaller 2-dimensional polydisk B ,we obtain 4-dimensional Lagrangian fibration f : M → B , associated with the potentialfunction Ψ( z ) and ℓ = 1 above. The discriminant set is z = 0. The order of e π √− /k mod h e − π i is exactly k in the multiplicative group C × / h e − π i , where − π = 2 π √− · √− 1. Then, thecharacteristic cycle of M ( z , is of type I k , and in particular, the type is constant. Construction III. Under the notation of Section 5, we set ℓ = 1 andΨ( z , z ) := √− z + z )2 + αz z , where α is any irrational, real number, say √ 2. Then˜ θ ( z ) = (cid:18) √− αα √− (cid:19) , θ ( z ) = ˜ θ ( z ) + log z π √− (cid:18) (cid:19) and Im ˜ θ ( z ) = (cid:18) (cid:19) . The matrix Im θ ( z ) is positive definite. So, taking a smaller 2-dimensional polydisk B ,we obtain 4-dimensional Lagrangian fibration f : M → B , associated with the potentialfunction Ψ( z ) and ℓ = 1 above. The discriminant set is z = 0. Since α is irrational realnumber, the element e π √− · α mod h e − π i is of infinite order in the multiplicative group C × / h e − π i , where − π = 2 π √− ·√− 1. Then,the characteristic cycle of M ( z , is of type A ∞ , and in particular the type is constant. References [BHPV] Barth, W., Hulek, K., Peters, C., Van de Ven, : Compact complex surfaces . Second enlarged edition.Springer Verlag, Berlin-Heidelberg, 2004[DM] Donagi, R., Markmann, E.: Cubics, integrable systems, and Calabi-Yau threefolds , Proceedings ofthe Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), 199-221, Israel Math.Conf. Proc. Bar-Ilan Univ., Ramat Gan, 1996[Gr] Griffiths, P. A.: Periods of integrals on algebraic manifolds. III. Some global differential-geometricproperties of the period mapping , Inst. Hautes ´Etudes Sci. Publ. Math. (1970) 125–180.[GH] Griffiths, P. A., Harris, J.: Principles of algebraic geometry . Wiley-Interscience, 1978[GS] Guillemin, V., Sternberg, S.: Symplectic techniques in physics . Second edition. Cambridge Univer-sity Press, Cambridge, 1990[Hw] Hwang, J.-M.: Base manifolds for fibrations of projective irreducible symplectic manifolds , Invent.math. (2008) 625-644[HO1] Hwang, J.-M., Oguiso, K.: Characteristic foliation on the discriminant hypersurface of a holomor-phic Lagrangian fibration , Amer. J. Math. (2009) 981-1007 OCAL STRUCTURE OF PRINCIPALLY POLARIZED STABLE LAGRANGIAN FIBRATIONS 21 [HO2] Hwang, J.-M., Oguiso, K.: Multiple fibers of holomorphic Lagrangian fibrations , to appear inCommunications Contemporary Math.[Kd] Kodaira, K.: On compact complex analytic surfaces II, III , Ann. of Math. (1963) 563–626, ibid (1963) 1–40.[Na] Nakamura, I. : Relative compactification of the N´eron model and its application , Complex analysisand algebraic geometry, pp. 207–225. Iwanami Shoten, Tokyo, 1977.[Vo] Voisin, C.: Hodge theory and complex algebraic geometry II , Cambridge University Press, 2003 Jun-Muk Hwang, Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Korea E-mail address : [email protected] Keiji Oguiso, Department of Mathematics, Osaka University, Toyonaka 560-0043 Osaka,Japan and Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Korea E-mail address ::