arXiv:1501.04145v2 [math.AG] 18 Aug 2017 [PDF]

A twistor approach to the Kontsevich complexes

arXiv:1501.04145v2 [math.AG] 18 Aug 2017

Takuro Mochizuki Abstract We study the V -filtration of the mixed twistor D-modules associated to algebraic meromorphic functions. We prove that their relative de Rham complexes are quasi-isomorphic to the family of Kontsevich complexes. It reveals a generalized Hodge theoretic meaning of Kontsevich complexes. On the basis of the quasiisomorphism, we revisit the results on the Kontsevich complexes due to H. Esnault, M. Kontsevich, C. Sabbah, M. Saito and J.-D. Yu from a viewpoint of mixed twistor D-modules. Keywords: Mixed twistor D-module, Kontsevich complex, V -filtration. 14F10, 32C38, 32S35.

1 1.1

Introduction Mixed twistor D-modules

The theory of mixed twistor D-modules was developed by C. Sabbah and the author ([4], [5], [7], [8], [9]). Very roughly, mixed twistor D-modules are holonomic D-modules equipped with mixed twistor structure. We have the standard 6-operations on the derived category of algebraic mixed twistor D-modules on complex algebraic manifolds, which are compatible with the standard 6-operations for algebraic holonomic D-modules. For any algebraic function f on a complex algebraic manifold Y , the associated algebraic flat bundle (OY , d + df ) is naturally enhanced to an algebraic pure twistor D-module on Y . As a result, we can say that many holonomic D-modules are naturally enhanced to mixed twistor D-modules. One of general issues is to describe such mixed twistor D-modules as explicitly as possible. Once we have an explicit description of a mixed twistor D-module, we might have a chance to relate it with a more concrete object, and to apply a general theory of mixed twistor D-modules for the study of the object. R-modules Let us recall the concept of R-modules which is one of the ingredients to formulate mixed twistor D-modules. Let Cλ denote just a complex line with the coordinate λ. For any complex manifold Y , let RY denote the sheaf of algebras on Cλ × Y obtained as the subalgebra of the sheaf of holomorphic differential operators DCλ ×Y generated by λp∗λ ΘY . Here, pλ : Cλ × Y −→ Y denotes the projection, and ΘY denotes the tangent sheaf of Y . When we are given a local coordinate system (x1 , . . . , xn ), then λ∂xi are denoted by ðxi or ði . Mixed twistor D-modules T on Y are formulated as a pair of RY -modules Mi (i = 1, 2), a sesqui-linear pairing C of M1 and M2 , and a weight filtration W , satisfying some conditions. (See [8] and [7] for more details on sesqui-linear pairings, weight filtrations, and the conditions.) In this paper, M2 is called the underlying RY
M2 /(λ − 1)M2 is naturally a DY -module, where ι1 : {1} × Y −→ module of T . Note that ΞDR (M2 ) := ι−1 1 C × Y is the inclusion. We call ΞDR (M2 ) the D-module underlying T . Then, we reword the general issue as follows: We would like to describe the R-modules underlying mixed twistor D-modules as explicitly as possible.

1.2

Main result

In this paper, we study the R-modules underlying the mixed twistor D-module associated to algebraic meromorphic functions. More precisely, let X be a smooth complex projective manifold with a morphism f : X −→ P1 . Let D be a hypersurface of X such that f −1 (∞) ⊂ D. We assume that D is normal crossing. We put 1

X (1) := Cτ × X and D(1) := Cτ × D. We obtain the meromorphic function τ f on (X (1) , D(1) ). We have the holonomic DX (1) -module M := OX (1) (∗D(1) ) v with the flat connection ∇ given by ∇v = v d(τ f ). It is naturally enhanced to a mixed twistor D-module T∗ (τ f, D(1) ). We have the underlying R-module L∗ (τ f, D(1) ). f := L∗ (τ f, D(1) )(∗τ ). We obtain the RX (∗τ )-module M We consider the sheaf of subalgebras τV0 RX (1) in RX (1) generated by OCλ ×X (1) and λp∗λ ΘX (1) (log τ ), where ΘX (1) (log τ ) denote the sheaf of vector fields which are logarithmic along τ = 0. By a general theory of mixed f f is uniquely equipped with an increasing sequence of coherent τV0 RX (1) -submodules Uα M twistor D-modules, M (α ∈ R) with the following property: S f = M. f • Uα M f f = Uα+ǫ M. • For any α ∈ R, we have ǫ > 0 such that Uα M

f for any α ∈ R. f ⊂ Uα+1 M f and ðτ Uα M f = Uα−1 M • We have τ Uα M

f f f • The induced endomorphisms τ ðτ + λα on GrU α M := Uα M/Uℓ1 ) ⊗ Ω•X (1) /Cτ (log H>ℓ1 ) = Mα ) ⊗ Ω•X (1) /Cτ (log H (1) ) 0 (−H

(1) ) ⊗ Ω•X (1) /Cτ (log D(1) ). (15) = Mα 0 (−D

Hence, we are done in the case τ0 6= 0. ℓ−p

z }| { Let us consider the case τ0 = 0. For 0 ≤ p ≤ ℓ, we regard Z = Z × {(0, . . . , 0)} ⊂ Zℓ . We set X X p Uα M≤p := ∂ n Mα Fj Uα M≤p = ∂ n Mα 0, 0. p

p

n∈Zp ≥0

n∈Zp ≥0 np ≤j

We have Uα M≤ℓ = Uα M and pF0 Uα M≤p = Uα M≤p−1 . We consider the following maps ∂p : pFj Uα M≤p −→ pFj+1 Uα M≤p . The following lemma is easy to see by Corollary 2.20. Lemma 2.25 Let s be a section of Uα M on a neighbourhood of Y with a primitive expression X sm,j . s= (m,j)∈S

Then, s is a section of pFj Uα M≤p if and only if we have mi ≥ 0 (i > p) and mp ≥ −j for any m ∈ min π(S). Lemma 2.26 If j ≥ 1, the following induced morphism of sheaves is an isomorphism: Fj Uα M≤p ∂p pFj+1 Uα M≤p −→ p pF ≤p Fj Uα M≤p j−1 Uα M p

Proof It is surjective by construction.PLet s be a non-zero section of pFj Uα M≤p on a neighbourhood of Y with a primitive decomposition s = (m,j)∈S sm,j such that ∂p s is also a section of pFj Uα M≤p . We set P P s′ := mp =−j sm,j and s′′ := mp >−j sm,j . Because ∂p s′′ ∈ pFj Uα M≤p , we obtain ∂p s′ ∈ pFj Uα M≤p . If s′ is P non-zero, ∂p s′ = mp =−j ∂p sm,j is a primitive expression of ∂p s′ . We obtain ∂p s′ 6∈ pFj Uα M≤p , and thus we arrive at a contradiction. Hence, s′ = 0, i.e., s ∈ pFj−1 Uα M≤p . Lemma 2.27 The kernel of the following induced surjection is xp Uα M≤p−1 : ∂p

Uα M≤p−1 −→ 12

p

F1 Uα M≤p Uα M≤p−1

Proof Let s be a section of Uα M≤p−1 . We take a primitive expression s = P P ′′ ′′ ≤p−1 . Because mp =0 sm,j and s := mp >0 sm,j . We have s ∈ xp Uα M

P

(m,j)∈S

sm,j . We set s′ :=



∂p · xp · g · x−δ−p (τ f )j v = xp ∂p (g) − (pp + kp j)g x−δ−p (τ f )j v − kp g · x−δ−p (τ f )j+1 v, P we have ∂p s′′ ∈ Uα M≤p−1 . Hence, we have ∂p s′ ∈ Uα M≤p−1 . If s′ 6= 0, ∂p s′ = mp =0 ∂p sm,j is a primitive expression of s′ . We obtain ∂p s′ 6∈ Uα M≤p−1 , and we arrive at a contradiction. Hence, we have s′ = 0, i.e., s ∈ Uα M≤p−1 xp . By Lemma 2.26 and Lemma 2.27, for 1 ≤ p ≤ ℓ, the inclusions of the complexes

∂p ∂p xp Uα M≤p−1 −→ Uα M≤p−1 −→ Uα M≤p −→ Uα M≤p

Sℓ (1) are quasi-isomorphisms. For 0 ≤ p ≤ ℓ, we set D>p := i=p+1 {xi = 0} on the neighbourhood of (0, Q). We obtain that the following inclusions of the complexes of sheaves are quasi-isomorphisms: (1)

(1)

(1)

(1)

Uα M≤p−1 (−D>p−1 ) ⊗ Ω•X (1) /Cτ (log D>p−1 ) −→ Uα M≤p (−D>p ) ⊗ Ω•X (1) /Cτ (log D>p ) (1)

(16)

(1)

We have Uα M≤ℓ (−D>ℓ ) ⊗ Ω•X (1) /Cτ (log D>ℓ ) = Uα M ⊗ Ω•X (1) /Cτ . We also have (1)

(1)

(1) Uα M≤0 (−D>0 ) ⊗ Ω•X (1) /Cτ (log D>0 ) = Mα ) ⊗ Ω•X (1) /Cτ (log D(1) ). 0 (−D

Hence, Proposition 2.21 is proved. 2.4.3

Proof of Proposition 2.22

We have only to check the claim around any point of P (1) . We use coordinate system as in §2.1.1. Set
the PN (1) −p (−D ) := (τ f )j v. We set p := [αk]. For any non-negative integer N , we set GN Mα 0 j=0 OX (1) x

(1) (1) ) ⊗ ΩkX (1) /Cτ (log D(1) ). ) ⊗ ΩkX (1) /Cτ (log D(1) ) := GN Mα GN Mα 0 (−D 0 (−D

(1) The derivative d of the complex Mα ) ⊗ Ω•X (1) /Cτ (log D(1) ) induces 0 (−D


(1) ) ⊗ ΩkX (1) /Cτ (log D(1) ) −→ d : GN Mα 0 (−D


(1) ) ⊗ Ωk+1 (log D(1) ) . (17) GN +1 Mα 0 (−D X (1) /Cτ


(1) ) ⊗ ΩkX (1) /Cτ (log D(1) ) := Ωkf,τ (α)v = Ωkf,τ x−p v. We also set G−1 Mα 0 (−D Let N ≥ 0. Take a section ω=

N X j=0


(1) ) ⊗ ΩkX (1) /Cτ (log D(1) ) , ωj x−p (τ f )j · v ∈ GN Mα 0 (−D

where ωj ∈ ΩkX (1) /Cτ (log D(1) ). Suppose that
(1) ) ⊗ Ωk+1 (log D(1) ) . dω ∈ GN Mα 0 (−D X (1) /Cτ


(1) ) ⊗ Ωk+1 (log D(1) ). Then, τ df ∧ ωN (τ f )N x−p v is a section of GN Mα 0 (−D X (1) /Cτ

Lemma 2.28 We have df ∧ ωN ∈ Ωk+1 (log D(1) ), i.e., ωN is a section of Ωkf,τ . X (1) /Cτ

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PN Proof Let s be a local section of OX (1) such that (τ f )N +1 s ∈ j=0 OX (1) (τ f )j . We obtain that τ N +1 s ∈ OX (1) f −1 , and s ∈ OX (1) f −1 . There exists a holomorphic function t such that s = tf −1 . We obtain (τ f )N +1 s = (τ f )N τ t ∈ OX (1) (τ f )N . PN (log D(1) ) · (τ f )j . By the above argument, Note that (df /f ) ∧ ωN · (τ f )N +1 is a section of j=0 Ωk+1 X (1) /Cτ we obtain that (df /f ) ∧ ωN · (τ f )N +1 is a section of Ωk+1 (log D(1) )(τ f )N . Hence, τ df ∧ ωN is a section of X (1) /Cτ

Ωk+1 (log D(1) ). Then, we obtain that df ∧ ωN is a section of Ωk+1 (log D(1) ). X (1) /Cτ X (1) /Cτ

Lemma 2.29 We have an expression ωN = (df /f ) ∧ κ1 + f −1 κ2 , where κ1 and κ2 are local sections of k−1 (1) ΩX ) and ΩkX (1) /Cτ (log D(1) ), respectively. (1) /C (log D τ
Proof Let Q be any point of Pred . Let U be a neighbourhood of Q in X. The complex Ω•X (log D), df /f is acyclic on U because df /f is a nowhere vanishing section of Ω1X (log D) on U . If U is sufficiently small, we can take decompositions ΩkX (log D) = B k ⊕ C k such that the multiplication of df /f induces an isomorphism C k ≃ B k+1 . Then, the claim of the lemma follows. If N ≥ 1, we have f −1 κ2 (τ f )N = τ κ2 (τ f )N −1 . Hence, we have

(1) ) ⊗ ΩkX (1) /Cτ (log D(1) ) . ω − d κ1 (τ f )N −1 x−p v ∈ GN −1 Mα 0 (−D

We also have κ1 (τ f )N −1 x−p v ∈ GN −1 .
(1) ) ⊗ ΩkX (1) /Cτ (log D(1) ) such that dω is a local section of Let ω be a local section of GN Mα 0 (−D
(1) ) ⊗ ΩkX (1) /Cτ (log D(1) ) . G−1 Mα 0 (−D

By applying the previous argument successively, we can find a local section τ of
k−1 (1) (1) ) ⊗ ΩX ) GN −1 Mα (1) /C (log D 0 (−D τ


(1) ) ⊗ ΩkX (1) /Cτ (log D(1) ) . such that ω − dτ is a local section of G−1 Mα 0 (−D As a consequence, we have the following.
(1) ) ⊗ ΩkX (1) /Cτ (log D(1) ) satisfies dω = 0, we can find a local section • If a local section ω of GN Mα 0 (−D
k−1 (1) (1) (1) )⊗ ) ⊗ ΩX ) such that ω − dτ is a local section of G−1 Mα τ of GN −1 Mα (1) /C (log D 0 (−D 0 (−D τ
(1) k ΩX (1) /Cτ (log D ) .
(1) ) ⊗ ΩkX (1) /Cτ (log D(1) ) such that dω = 0. If we have a local • Let ω be a local section of G−1 Mα 0 (−D
k−1 (1) (1) ) ⊗ ΩX ) such that ω = dτ , then we can find a local section section τ of GN Mα (1) /C (log D 0 (−D τ
k−2 (1) (1) (1) )⊗ ) ⊗ ΩX ) such that τ − dσ is a local section of G−1 Mα σ of GN −1 Mα (1) /C (log D 0 (−D 0 (−D τ
k−1 (1) ΩX (1) /Cτ (log D ) . We have ω = d(τ − dσ). Then, we obtain the claim of Proposition 2.22.

3 3.1

The case of mixed twistor D-modules Preliminary

e R-modules and R-modules Here, let us recall the concept of R-modules [8]. Let Y be any complex manifold. We set Y := Cλ × Y . Let p : Y −→ Y denote the projection. Let DY denote the sheaf of holomorphic differential operators on Y. Let RY denote the sheaf of subalgebras of DY generated by λp∗ ΘY over OY , where ΘY denote e Y denote the sheaf of subalgebras of DY generated by λ2 ∂λ over RY . the tangent sheaf of Y . Let R A left RY -module is equivalent to an OY -module M with a relative flat meromorphic connection ∇rel : e Y -module is equivalent to an OY -module M with a flat meromorphic connection M −→ M⊗λ−1 Ω1Y/Cλ . A left R −1 1 ∇ : M −→ M ⊗ λ ΩY (log λ). 14

Push-forward by projection We recall the functoriality of R-modules with respect to the push-forward by a projection [8]. Suppose that Y = Z × W for complex manifolds Z and W , and that Z is projective. Let π : Y −→ W denote the projection. For any RY -module N , we have the RW -modules π†j N (− dim Z ≤ j ≤ dim Z) e j := λ−j q ∗ Ωj . Then, we have a naturally defined as follows. Let qZ : Y −→ Z denote the projection. We set Ω Z Z Z • e ⊗ N on Y induced by the exterior derivative of Ω• and the relative flat meromorphic connection complex Ω Z Z ∇rel of N . We have   e •Z ⊗ N . π†j N ≃ Rj+dim Z (idCλ ×π)∗ Ω e Y -module, then π j N are naturally R e W -modules. If N is an R †

V -filtrations Suppose that Y = Y0 × Ct . For simplicity, we assume that Y0 is relatively compact and open in a larger complex manifold Y0′ . Let V0 RY be the sheaf of subalgebras in RY generated by λp∗ ΘY (log t) over OY . Let N be an RY -module underlying a mixed twistor D-module on Y , which can be extended to a mixed twistor D-module on Y0′ × Ct . Then, by the definition of mixed twistor D-module, N (∗t) is strictly specializable along t as an RY (∗t)-module. Namely, for any λ0 ∈ Cλ , we have a neighbourhood B(λ0 ) ⊂ Cλ of λ0 and a unique filtration V (λ0 ) N (∗t)|B(λ0 )×Y by coherent V0 RY -submodules of N (∗t)|B(λ0 )×Y satisfying the conditions as in [5, Definition 22.4.1]. Note that, for each a ∈ R, we have the finite subset K(a, λ0 ) ⊂ R × C such that
Q V (λ0 ) N (∗t)|B(λ0 )×Y , where we set e(λ, (b, β)) = β − λb − βλ2 for u∈K(a,λ0 ) (−ðt t + e(λ, u)) is nilpotent on Gra (b, β) ∈ R × C. e Y -module. As in [8, Proposition 7.3.1], we have K(a, λ0 ) = Suppose moreover that N is enhanced to an R {(a, 0)}, and

λ2 ∂λ Va(λ0 ) N (∗t)|B(λ0 )×Y ⊂ Va(λ0 ) N (∗t)|B(λ0 )×Y
for any a ∈ R. We obtain a global filtration Va N (∗t) (a ∈ R) of V0 RY -coherent submodules of N (∗t) by gluing V (λ0 ) (λ0 ∈ C), which is uniquely characterized by the following conditions.
S • We have a∈R Va N (∗t) = N (∗t).

• For any a ∈ R, we have ǫ > 0 such that Va N (∗t) = Va+ǫ N (∗t) .



• We have tVa N (∗t) = Va−1 N (∗t) and ðt Va N (∗t) ⊂ Va+1 N (∗t) for any a ∈ R. • The induced endomorphisms ðt t + λa are nilpotent on

for any a ∈ R. Here, we set V0 X /C X /C λ,τ

λ,τ

Hence, Proposition 3.35 is proved. 3.5.3

Proof of Proposition 3.36 (1)

We essentially repeat the argument in §2.4.3. We have only to check the claim around any point of Pred . We use the coordinate system as in §2.1.1. We set p := [αk]. For any non-negative integer N , we set N X
f[αP ] (−D(1) ) := OX (1) x−p (τ f )j υ. GN M j=0

f[αP ] (−D(1) ) ⊗ Ω e k (1) 2 We define GN M X /C

λ,τ


(log D(1) ) as


f[αP ] (−D(1) ) ⊗ Ω e k (1) 2 (log D(1) ). GN M X /C λ,τ

f[αP ] (−D(1) ) ⊗ Ω e k (1) 2 We set G−1 M X /C

λ,τ

N ≥ 0. Take a section

ω=

N X j=0


e k (α)υ = Ω e k x−p υ, where Ω ek ek (log D(1) ) := Ω f,λ,τ f,λ,τ f,λ,τ := Ωf,λ,τ (0). Let


f[αP ] (−D(1) ) ⊗ Ω e k (1) 2 (log D(1) ) , ωj x−p (τ f )j · υ ∈ GN M X /C λ,τ


f[αP ] (−D(1) ) ⊗ Ω e k+1 e k (1) 2 (log D(1) ). If dω ∈ GN M where ωj ∈ Ω (log D(1) ) , then we have X /C X (1) /C2 λ,τ

λ,τ


f[αP ] (−D(1) ) ⊗ Ω e k+1 λ−1 τ df ∧ ωN (τ f )N x−p υ ∈ GN M (log D(1) ) . X (1) /C2 λ,τ

e k+1 ek . As in Lemma 2.28, we obtain df ∧ ωN ∈ Ω (log D(1) ), i.e., ωN is a section of Ω f,λ,τ X (1) /C2 λ,τ

Lemma 3.41 We have an expression

ωN = (df /f ) ∧ κ1 + f −1 κ2 , e k−1 e k (1) 2 (log D(1) ), respectively. where κ1 and κ2 are sections of Ω (log D(1) ) and Ω X /C X (1) /C2 λ,τ

λ,τ

If N ≥ 1, we have f

−1

N

κ2 (τ f )

N −1

= τ κ2 (τ f ) . Hence, we have

f[αP ] (−D(1) ) ⊗ Ω e k (1) 2 (log D(1) ) . ω − d κ1 (τ f )N −1 x−p υ ∈ GN −1 M X /C λ,τ

We also have f −1 κ1 (τ f )N x−p υ ∈ GN −1 .
f[αP ] (−D(1) ) ⊗ Ω e k (1) 2 (log D(1) ) such that dω is a local section of Let ω be a local section of GN M X /Cλ,τ
f[αP ] (−D(1) ) ⊗ Ω e k (1) 2 (log D(1) ) . Then, by applying the previous argument successively, we can G−1 M X /Cλ,τ
f[αP ] (−D(1) ) ⊗ Ω e k−1 (log D(1) ) such that ω − dτ is a local section of find a local section τ of GN −1 M X (1) /C2λ,τ
f[αP ] (−D(1) ) ⊗ Ω e k (1) 2 (log D(1) ) . G−1 M X /C λ,τ

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f[αP ] (−D(1) )⊗ Ω e k (1) 2 (log D(1) ) satisfies dω = 0, we can find a local section • If a local section ω of GN M X /Cλ,τ
f[αP ] (−D(1) )⊗ Ω e k−1 f[αP ] (−D(1) )⊗ τ of GN −1 M (log D(1) ) such that ω−dτ is a local section of G−1 M X (1) /C2λ,τ
e k (1) 2 (log D(1) ) . Ω X /C λ,τ


f[αP ] (−D(1) ) ⊗ Ω e k (1) 2 (log D(1) ) such that dω = 0. Suppose that • Let ω be a local section of G−1 M X /Cλ,τ
f[αP ] (−D(1) ) ⊗ Ω e k−1 (log D(1) ) such that ω = dτ . Then, we can we have a local section τ of GN M X (1) /C2λ,τ
f[αP ] (−D(1) ) ⊗ Ω e k−2 find a local section σ of GN −1 M (log D(1) ) such that τ − dσ is a local section of X (1) /C2λ,τ
f[αP ] (−D(1) ) ⊗ Ω e k−1 G−1 M (log D(1) ) . We have ω = d(τ − dσ). X (1) /C2 λ,τ

Then, we obtain the claim of Proposition 3.36. The proof of Theorem 3.6 is also completed.

References [1] H. Esnault, C. Sabbah, J.-D. Yu, (with an appendix by M. Saito), E1 -degeneration of the irregular Hodge filtration, J. reine angew. Math. (2015), doi:10.1515/crelle-2014-0118, [2] M. Kashiwara, D-modules and microlocal calculus, Translations of Mathematical Monographs, 217. American Mathematical Society, Providence, (2003). [3] L. Katzarkov, M. Kontsevich, T. Pantev, Bogomolov-Tian-Todorov theorems for Landau-Ginzburg models, J. Differential Geom. 105 (2017), no. 1, 55–117. [4] T. Mochizuki, Asymptotic behaviour of tame harmonic bundles and an application to pure twistor Dmodules I, II, Mem. AMS. 185, (2007). [5] T. Mochizuki, Wild harmonic bundles and wild pure twistor D-modules, Ast´erisque 340, Soci´et´e Math´ematique de France, Paris, 2011. [6] T. Mochizuki, Harmonic bundles and Toda lattices with opposite sign II, Comm. Math. Phys. 328, 1159– 1198, DOI:10.1007/s00220-014-1994-0. [7] T. Mochizuki, Mixed twistor D-modules, Lecture Notes in Mathematics, 2125. Springer, 2015. [8] C. Sabbah, Polarizable twistor D-modules Ast´erisque, 300, (2005) [9] C. Sabbah, Wild twistor D-modules, in Algebraic analysis and around, Adv. Stud. Pure Math., 54, Math. Soc. Japan, Tokyo, (2009), 293–353. [10] C. Sabbah, J.-D. Yu. On the irregular Hodge filtration of exponentially twisted mixed Hodge modules, Forum Math. Sigma 3 (2015), 71 pp. [11] M. Saito, Modules de Hodge polarisables, Publ. RIMS., 24, (1988), 849–995. [12] M. Saito, Mixed Hodge modules, Publ. RIMS., 26, (1990), 221–333. [13] J.-D. Yu, Irregular Hodge filtration on twisted de Rham cohomology, Manuscripta Math. 144(12) (2014), 99–133. Address: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan Email: [email protected]

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