manuscripta math.

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Takuro Mochizuki

A twistor approach to the Kontsevich complexes Received: 26 January 2017 / Accepted: 4 November 2017 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 quasiisomorphic to the family of Kontsevich complexes. It reveals a generalized Hodge theoretic meaning of Kontsevich complexes. On the basis of the quasi-isomorphism, 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.

1. Introduction 1.1. Mixed twistor D-modules The theory of mixed twistor D-modules was developed by C. Sabbah and the author [4,5,7–9] on the basis of the fundamental ideas for mixed Hodge modules due to M. Saito [11,12]. 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 + d f ) 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 λpλ∗ ΘY . Here, pλ : Cλ × Y −→ Y denotes the projection, and T. Mochizuki (B): Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. e-mail: [email protected] Mathematics Subject Classification: 14F10 · 32C38 · 32S35

https://doi.org/10.1007/s00229-017-0989-5

T. Mochizuki

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), with 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 -module of T . Note that ΞDR (M2 ) := ι−1 1 M2 /(λ − 1)M2 is naturally a DY -module, where ι1 : {1} × Y −→ 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 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 D X (1) -module M := O X (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) ).  := L∗ (τ f, D (1) )(∗τ ). We obtain the R X (∗τ )-module M We consider the sheaf of subalgebras τV0 R X (1) in R X (1) generated by OCλ ×X (1) and λpλ∗ Θ X (1) (log τ ), where Θ X (1) (log τ ) denote the sheaf of vector fields which are  is logarithmic along τ = 0. By a general theory of mixed twistor D-modules, M τ uniquely equipped with an increasing sequence of coherent V0 R X (1) -submodules  (α ∈ R) with the following property: Uα M   = M.  – Uα M  = Uα+ M.  – For any α ∈ R, we have > 0 such that Uα M  = Uα−1 M  and ðτ Uα M  ⊂ Uα+1 M  for any α ∈ R. – We have τ Uα M  := Uα M/U  <α M  are M – The induced endomorphisms τ ðτ + λα on GrU α  :=  M. nilpotent for any α ∈ R. Here, we set U<α M U β β<α  – GrU α M are strict, i.e., flat over OCλ . Note that the V -filtrations of R-modules underlying mixed twistor D-modules are  is naturally equipped with characterized by a more involved condition. Because M the action of λ2 ∂λ , the condition is simplified as above.  explicitly. Then, we shall show that their de Rham We shall describe Uα M complex relative to X (1) −→ Cτ is quasi-isomorphic to an explicitly given family of complexes, called Kontsevich complexes. Kontsevich complexes Let us recall the concept of Kontsevich complexes. Put P := f ∗ (∞) as an effective divisor. The reduced divisor is denoted by Pred . The multiplication of d f induces a morphism Ω Xk (log D) −→ Ω Xk+1 (log D) ⊗ O X (P).

A twistor approach to the Kontsevich complexes

The inverse image of Ω Xk+1 (log D) ⊂ Ω Xk+1 (log D) ⊗ O X (P) by d f is denoted by Ω kf . The multiplication of d f induces a morphism d f : Ω kf −→ Ω k+1 f . The exterior derivative induces d : Ω kf −→ Ω k+1 f . k For any 0 ≤ α < 1, we set Ω f (α) := Ω kf ⊗ O X ([α P]). Here, for real numHi (i = 1, . . . , N ), we set [ai ] := max{n ∈ bers ai and reduced   hypersurfaces  N N := a H Z | n ≤ ai } and i i i=1 i=1 [ai ] Hi . We have the induced operators k+1 k k d : Ω f (α) −→ Ω f (α) and d f : Ω f (α) −→ Ω k+1 f (α). For any (λ, τ ) ∈ C2 , we have the derivative λd + τ d f : Ω kf (α) −→ Ω k+1 f (α). They satisfy the integrability condition (λd + τ d f ) ◦ (λd + τ d f ) = 0. Thus, we   obtain complexes Ω •f (α), λd + τ d f . They are called Kontsevich complexes. We clearly have the family version of the complexes. Let q X : Cλ × X (1) −→ X j := λ− j q ∗ Ω j . Then, we have the derivative denote the projection. We set Ω X f f j −→ Ω j+1 satisfying (d + λ−1 τ d f ) ◦ (d + λ−1 τ d f ) = 0. d + λ−1 τ d f : Ω f

f

  • (α), d +λ−1 τ d f is naturally quasiTheorem 1. (Theorem 5) For 0 ≤ α < 1, Ω f  relative to the projection X (1) −→ Cτ . isomorphic to the de Rham complex of Uα M This theorem reveals a generalized Hodge theoretic property of the Kontsevich complexes, and provides a bridge between the theory of mixed twistor D-modules and the study of Kontsevich complexes. We shall show that it is useful by giving an alternative proof of the interesting results of H. Esnault, M. Kontsevich, C. Sabbah, M. Saito and J.-D. Yu, which we will explain in the next subsection.

1.3. An application The following theorem was conjectured by M. Kontsevich. It was proved by H. Esnault, C. Sabbah and J.-D. Yu [1]. Some interesting cases were proved by Kontsevich and M. Saito. (See [3] and the appendix of [1].)  Theorem 2. The dimension of the hypercohomology groups Hi X, Ω •f (α), λd +  τ d f are independent of (λ, τ ) ∈ C2 and any 0 ≤ α < 1. We set X := P1τ × X . Let pi denote the projection of X onto the i-th components. We identify OP1 (1) with OP1 ({∞}). The constant function 1 and the coordinate function τ are naturally regarded as sections of OP1 (1). We consider the bundles Kkf (α) := p1∗ OP1 (k) ⊗ p2∗ Ω kf (α). We have the relative differential operators p2∗ (d) + τ p2∗ (d f ) : Kkf (α) −→ Kk+1 f (α). Thus, we obtain a complex of sheaves K•f (α) on X. We have the decreasing filtration F • of the complex K•f (α) defined as follows:

F j Kif (α) :=

0 (i < j) i K f (α) (i ≥ j)

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T. Mochizuki

  By Theorem 2, we obtain a vector bundle Kif (α) := Ri p1∗ K•f (α) on P1 . Let   F j Kif (α) denote the image of Ri p1∗ F j K•f (α) . We have   j j Gr F Kif (α) H i− j X, Ω f (α) ⊗ OP1 ( j). Hence, F • is the Harder-Narasimhan filtration of Kif (α). Moreover, Sabbah and Yu proved the following theorem in [10]. Theorem 3. Let 0 ≤ α < 1. (i) The bundle Kif (α) is equipped with a naturally induced meromorphic connection ∇ such that   ∇ · Kif (α) ⊂ Kif (α) ⊗ ΩP11 {0} + 2{∞} . (2)  Note that the condition (2) implies that ∇ F j Kif (α) ⊂ F j−1 Kif (α)⊗ΩP11 {0}+  2{∞} . (ii) Let Res0 (∇) denote the endomorphism of Kif (α)|0 obtained as the residue. Then, any eigenvalue β of Res0 (∇) is a rational number such that −α ≤ β < −α + 1. Let Eβ Kif (α)|0 denote the generalized eigen space of Res0 (∇) corresponding to  β. By setting Uγ Kif (α)|0 := −β≤γ Eβ Kif (α)|0 , we obtain an increasing filtration i U• Kif (α)|0 indexed by α − 1 < γ ≤ α. Note that GrU γ K f (α)|0 is naturally isomorphic to E−γ Kif (α)|0 . The filtration F on Kif (α)|0 induces filtrations on i GrU γ K f (α)|0 , which are also denoted by F:

  i j i U i F j GrU γ K f (α)|0 := Im F ∩ Uγ K f (α)|0 −→ Gr γ K f (α)|0 . Because Res0 (∇) preserves the filtration U , we have the induced endomorphisms U i U GrU γ Res0 (∇) of Gr γ K f (α)|0 . Let Nγ denote the nilpotent part of Gr γ Res0 (∇). Then, Sabbah and Yu also proved the following in [10]. Theorem 4. For any α − 1 < γ ≤ α, Nγ gives a strict morphism i U i (GrU γ K f (α)|0 , F) −→ (Gr γ K f (α)|0 , F[−1]),

where denotes the filtration defined by F[−1]a := F a−1 . Namely, we have  jF[−1] i U Nγ F Gr γ Kif (α)|0 = Im(Nγ ) ∩ F j−1 GrU γ K f (α)|0 . In Sect. 3.3.2, we shall explain how to deduce Theorems 2, 3 and 4 from Theorem 1 and general results for mixed twistor D-modules. We remark that our argument is based on a generalized Hodge theory and some concrete computations of V -filtrations. Hence, eventually, it is not completely different from those in [1,10]. But, the argument in this paper looks more direct. The author thinks that it would be desirable to have many ways to explain the theorems because the complexes (Ω •f (α), λd + τ d f ) and the bundles Kif seem quite basic. He also hopes that this paper might also be useful to explain how the general theory of twistor D-modules could be applied to the study of specific objects.

A twistor approach to the Kontsevich complexes

Outline of the paper In Sect. 2, we study the D-module associated to the meromorphic function as in Sect. 1.2. In particular, we explicitly describe the relative de Rham complexes of their V -filtration along τ (Propositions 2 and 3), which is essentially a simpler version of Theorem 1. Then, we explain how to deduce a part of Theorem 3. We could include the computations for D-modules in Sect. 2 to the computations for R-modules in Sect. 3 after minor modifications. But, the author expects that it would be useful to give an explanation in this simpler situation. In Sect. 3, we study the R-module in Sect. 1.2, and we revisit Theorems 2, 3 and 4 from the viewpoint of mixed twistor D-modules. After the preliminaries in Sects. 3.1–3.2, we explain in Sect. 3.3 the main theorem (Theorem 5) and how we deduce Theorems 2, 3 and 4 from general results of mixed twistor D-modules.  In Sect. 3.4, we give a description of the V -filtration of the R X (1) (∗τ )-module M (Theorem 6). Besides the argument in Sect. 2 for D-modules, we need an additional task to check the strictness. Then, we establish Theorem 5 in Sect. 3.5.

2. The case of D-modules 2.1. Meromorphic flat bundles We continue to use the setting in Sect. 1. We set X (1) := Cτ × X . Here, Cτ is just an affine line with a coordinate τ . We use the notation D (1) , P (1) , etc., with a similar meaning. For any complex manifold Y with a hypersurface Y1 , let OY (∗Y1 ) denote the sheaf of meromorphic functions on Y whose poles are contained in Y1 . For any OY -module M, let M(∗Y1 ) := M ⊗OY OY (∗Y1 ). If Y1 is given as { f = 0} for a holomorphic function f , we also use the notation OY (∗ f ) and M(∗ f ). We shall consider the meromorphic flat bundle M := O X (1) (∗D (1) ) v with ∇v = v d(τ f ) of rank one, where v denotes a global frame. The corresponding D X (1) -module is also denoted by M. By the isomorphism O X (1) (∗D (1) ) M; 1 ←→ v, the connection ∇ is identified with the connection d + d(τ f ) on O X (1) (∗D (1) ). 2.1.1. Local coordinate systems When we study M locally around a point (1) (τ, Q) ∈ Dred , we shall use a holomorphic local coordinate neighbourhood (U, x1 , . . . , xn ) of Q in X with the following property:  1   1 −ki – Pred ∩ U = i=1 {xi = 0}, H ∩ U = i= {xi = 0}, f |U = i=1 xi for 1 +1 ki > 0 (i = 1, . . . , 1 ).

We set ki := 0 (i = 1 + 1, . . . , ), and the tuple i | i = 1, . . . , ) ∈ Z is  (km i

m denoted by k. For any m ∈ Z , we set x := i=1 xi . In particular, we have f = x −k on U . We set U (1) := Cτ × U which is equipped with a coordinate system (τ, x1 , . . . , xn ).

T. Mochizuki

We have ∂i v = −ki τ f xi−1 v for i = 1, . . . , , and ∂i v = 0 for i = +1, . . . , n. We also have ∂τ v = f v. We set δ := (1, . . . , 1) ∈ Z . We have τ ∂τ (x −δ+m v) = τ f x −δ+m v, ∂i xi (x −δ+m v) = (m i − ki τ f )x −δ+m v. Hence, for i = 1, . . . , 1 , we have (τ ∂τ + ki−1 ∂i xi )(x −δ+m v) = (m i /ki )x −δ+m v. (3)    When we are given α, we set [αki ] := max n ∈ Z  n ≤ αki ,  a real number   and [αk] := [αki ]  i = 1, . . . , ∈ Z . 2.2. V -filtration along τ Let π : X (1) −→ X denote the projection. We naturally regard π ∗ D X as the subalgebra of D X (1) . Let τV0 D X (1) ⊂ D X (1) denote the sheaf of subalgebras generated by τ ∂τ over π ∗ D X . We shall construct τV0 D X (1) -submodules Uα M (α ∈ R) of M, and we shall prove that the filtration U• M is a V -filtration  of M.  For 0 ≤ α ≤ 1, we have an O X (1) -submodule O X (1) D (1) + [α P (1) ] v ⊂ M. We set   Uα M := π ∗ D X · O X (1) (D (1) + [α P (1) ]) v ⊂ M. Lemma 1. Outside {τ = 0} ∩ P (1) , we have Uα M = M for 0 ≤ α ≤ 1. Proof. The claim is clear outside D (1) . Let (τ0 , Q) be any point of D (1) . We use a coordinate system as in Sect. 2.1.1. Suppose Q ∈ / Pred . For any m ∈ Z ≥0 , we have −1 −δ−m −δ−m ∂i (x v) = −(m i + 1)x xi v. Hence, we easily obtain   π ∗ D X · O X (1) (D (1) ) v = O X (1) (∗D (1) )v on a neighbourhood of (τ0 , Q). Suppose Q ∈ Pred and τ0 = 0. For any m ∈ Z ≥0 , we have ∂i (x −δ−m v) = −(1 + m i + ki τ x −k )x −δ−m xi−1 v. Then, we easily obtain O(∗D (1) )v ⊂ Uα M.   Lemma 2. We have ∂τ U0 M ⊂ U1 M and τ U1 M ⊂ U0 M. Proof. We have ∂τ (gx −δ v) = (∂τ g)x −δ v + gx −δ−k v ∈ U1 M. Hence, we have ∂τ U0 M ⊂ U1 M. Let us prove τ U1 M ⊂ U0 M. We have only to check it locally (1) around any point of Pred by using a coordinate system as in Sect. 2.1.1. We have   τ gx −δ−k v = gx −δ (τ f )v = −ki−1 ∂i (xi gx −δ v) + ki−1 (∂i g)x −δ xi v ∈ U0 M. Thus, we are done.

 

If α ≤ 0, we take the integer n such that 0 ≤ α + n < 1, and we set Uα M := τ n Uα+n M. For α ≥ 1, we define Uα M := β+n≤α ∂τn Uβ M. If α ≤ α  , we have Uα M ⊂ Uα  M. By the construction, forany α ∈ R, we have > 0 such that Uα M = Uα+ M. We define U<α M := β<α Uβ M.

A twistor approach to the Kontsevich complexes

Proposition 1. U• M is a V -filtration of M along τ indexed by the rational numbers with the standard order (up to shift of the degree by 1). More precisely, the following holds:  – Uα M are coherent τV0 D X (1) -modules such that α Uα M = M. – We have τ Uα M ⊂ Uα−1 M and ∂τ Uα M ⊂ Uα+1 M.  dim X +1 =0 – τ ∂τ + α is nilpotent on Uα M/U<α M. Indeed, we have τ ∂τ + α on Uα M/U<α M.  

Proof. We divide the claim into several lemmas.

Lemma 3. We have a natural action of τV0 D X (1) on Uα M.   Proof. It is enough to prove τ ∂τ Uα M ⊂ Uα M. We have only to consider the (1) case 0 ≤ α ≤ 1. We have only to check it locally around any point of Pred . We use a local coordinate system as in Sect. 2.1.1. We set p := [αk]. We have O X (1) (D (1) + [α P (1) ])v = O X (1) x −δ− p v on U (1) . We have τ ∂τ (x −δ− p v) + ( pi /ki )x −δ− p v = −ki−1 ∂i (xi x −δ− p v).  

Then, the claim is clear. Lemma 4. τV0 D X (1) -modules Uα M are coherent.

Proof. They are pseudo-coherent over O X (1) and locally finitely generated over τV D τ 0 X (1) . Hence, they are coherent over V0 D X (1) (See [2]). Let [τ ∂τ ] denote the endomorphism of Uα M/U<α M induced by τ ∂τ .  dim X Lemma 5. For any 0 < α ≤ 1, [τ ∂τ ]+α is nilpotent. Indeed, [τ ∂τ ]+α =0 on Uα M/U<α M. (1)

Proof. Because the support of the sheaf Uα M/U<α M is contained in Pred , we (1) have only to check the claim locally around any point of Pred by using a coordinate ) := [αk]. If α = pi /ki , we have system as in Sect. 2.1.1. We set p = ( p i   τ ∂τ + α (x −δ− p v) = −ki−1 ∂i (xi x −δ− p v). Let S := {i | α = pi /ki }. Then, we have

   |S| τ ∂τ + α (x −δ− p v) = (−ki−1 ∂i ) x −δ− p xi · v ∈ U<α M. i∈S

i∈S

We also have the following equality for any holomorphic function g: (τ ∂τ + α)(gx −δ− p v) = g · (τ ∂τ + α)(x −δ− p v) + (τ ∂τ g)(x −δ− p v). We have the following for any i with ki = 0:

ki τ (∂τ g)x −δ− p v = −∂i xi (∂τ g)x −δ− p+k v + (∂i ∂τ g)x −δ− p+k xi v +(∂τ g)(ki − pi )x −δ− p+k v ∈ U<α M. Then, we can easily deduce the claim of the lemma.

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T. Mochizuki

Lemma 6. If N ≥ dim X + 1, we have (τ ∂τ ) N U0 M ⊂ τ U<1 M. (1)

Proof. We have only to check the claim locally around any point of Pred by using

1

   a coordinate system as in Sect. 2.1.1. Set δ 1 := (1, . . . , 1, 0, . . . , 0) ∈ Z . We have  1  −1  −(δ−δ ) 1 v. Hence, we have (τ ∂τ ) 1 (x −δ v) = i=1 −ki ∂i · x (τ ∂τ ) 1 +1 (x −δ v) =

1 

(−ki−1 ∂i )(τ f x −(δ−δ 1 ) v)

i=1

=τ

1  (−ki−1 ∂i )(x −δ−(k−δ 1 ) v) ∈ τ U<1 M.

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i=1

For any section s of U0 M and any holomorphic function g, we have τ ∂τ (gs) = τ (∂τ g) s + g τ ∂τ s, and τ (∂τ g)s ∈ τ U<1 (M). Then, we can deduce the claim of the lemma.    j Lemma 7. We have ∞ j=0 ∂τ U1 M = M. Proof. Let M denote the left hand side. We clearly have M ⊂ M. Let m ∈ Z ≥0 . For i = 1 + 1, . . . , , we have ∂i (x −δ−m v) = −(m i + 1)x −δ−m xi−1 v. We also  have ∂τ (x −δ−m v) = x −δ−m−k v. Then, we obtain M = O X (1) (∗D (1) ) ⊂ M .  We obtain Proposition 1 from Lemmas 3–7.   Remark 1. We set Vα M := Uα+1 M, then −∂τ τ − α is nilpotent on Vα /V<α . 2.3. Primitive expression 2.3.1. Primitive expression for sections of O X (1) (∗D (1) ) Let (U, x1 , . . . , xn ) be a holomorphic coordinate neighbourhood of X as in Sect. 2.1.1. We set U (1) := Cτ × U . We set Y := {τ = x1 = · · · = x = 0} ⊂ U (1) . We shall consider local sections O X (1) (∗D (1) ) on a small neighbourhood of Y . Such a section s has the unique Laurent expansion:   s= h m, j x m τ j . m∈Z j∈Z≥0

Here, h m, j are holomorphic functions on Y . Note that if N > 0 is sufficiently large, depending on s, we have h m, j = 0 unless m i ≥ −N (i = 1, . . . , ). We also have the following unique expression:   (1) s= h m, j x m · (τ f ) j . m∈Z j∈Z≥0 (1) Here, h (1) m, j are holomorphic functions on Y . Indeed, we have h m, j = h m− j k, j .

A twistor approach to the Kontsevich complexes

Definition 1. A section s of O X (1) (∗D (1) ) on a neighbourhood of Y is called (m, j)primitive if we have s = gx m (τ f ) j for a section g of O X (1) with g|Y = 0. Definition 2. A primitive expression of a section s of O X (1) (∗D (1) ) on a neighbourhood of Y is a decomposition  s= sm, j (m, j)∈S

where S ⊂ Z × Z≥0 is a finite subset, and each sm, j is (m, j)-primitive. Lemma 8. Any section s of O X (1) (∗D (1) ) on a neighbourhood of Y has a primitive expression.  Proof. We have an expression s = (m, j)∈T gm, j x m τ j for a finite subset T ⊂ Z × Z≥0 and holomorphic functions gm, j with gm, j|Y = 0. Then, we have s =  m+ j k (τ f ) j . Each g m+ j k (τ f ) j is (m + j k, j)-primitive.   m, j x (m, j)∈T gm, j x def

We consider the partial order on Z defined by (ai ) ≤ (bi ) ⇐⇒ ai ≤ bi (∀i). For any T ⊂ Z , let min(T ) denote the set of the minimal elements in T with respect to the partial order. We also use a similar partial order on Z × Z≥0 . Let π : Z × Z≥0 −→ Z denote the projection.  Lemma 9. Let T ⊂ Z ×Z≥0 be a finite subset. Suppose (m, j)∈T gm, j x m (τ f ) j = 0 on a neighbourhood of Y . – For any m ∈ min π(T ) and any j ∈ Z≥0 , we have gm, j|Y = 0. – For any (m, j) ∈ min(T ), we have gm, j|Y = 0. Proof. For any (m, j), we have the Laurent expansion   gm, j = gm, j;n,q x n τ q . n∈Z ≥0 q∈Z≥0

Here, gm, j;n,q are holomorphic functions on Y . We have    gm, j;n,q x m+n+q k (τ f ) j+q . 0= (m, j)∈T n∈Z q≥0 ≥0

If m ∈ min π(T ), the coefficient of x m (τ f ) j is gm, j;0,0 = gm, j|Y . Hence, we obtain gm, j|Y = 0. If (m, j) ∈ min(T ), the coefficient of x m (τ f ) j is gm, j;0,0 = gm|Y .   Hence, we obtain gm, j|Y = 0. (1) ) on a neighbourhood of Corollary 1. Let s be a non-zero section  of O X (1) (∗D m Y with a primitive expression s = (m, j)∈S gm, j x (τ f ) j . Then, the following holds:

– The set min π(S) is well defined for s. For any m ∈ min π(S) and any j ∈ Z≥0 , gm, j|Y is well defined for s. – The set min(S) is well defined for s. For any (m, j) ∈ min(S), gm, j|Y is well defined for s.

T. Mochizuki

2.3.2. Subsheaf Mα0

For 0 ≤ α ≤ 1 and for N ∈ Z≥0 , we define

G N Mα0 :=

N 

  O X (1) D (1) + [α P (1) ] (τ f ) j v ⊂ M.

j=0



We set Mα0 := N ≥0 G N Mα0 . Let V0 D X denote the subalgebra of D X generated by O X and the logarithmic tangent sheaf Θ X (log D) of X with respect to D. We naturally regard π ∗ V0 D X as a subsheaf of D X (1) .   Lemma 10. We have Mα0 = π ∗ V0 D X · O X (1) D (1) + [α P (1) ] v . (1)

Proof. We have only to check the claim locally around any point of Pred . We use a local  coordinate system as in Sect. 2.1.1. We set p = ( pi ) := [αk]. Because ∂i xi vx −δ− p (τ f ) j = −( pi + jki )v x −δ− p (τ f ) j − ki vx −δ− p (τ f ) j+1 , we easily obtain the claim of the lemma. By the construction, we have Uα M = π ∗ D X · Mα0 for any 0 ≤ α ≤ 1. 2.3.3. Primitive expression for sections of Uα M (0 ≤ α ≤ 1) We use the notation in Sect. 2.3.1. Let 0 ≤ α ≤ 1. Set p := [αk] ∈ Z . For any n ∈ Z ≥0 , we set ∂ n :=   ni n i . For any m ∈ Z , we have the unique decomposition i=1 ∂i and |n| = m = m+ − m− such that m± ∈ Z ≥0 and {i | m +,i = 0} ∩ {i | m −,i = 0} = ∅. Any section s of Uα M on a neighbourhood of Y has an expression  ∂ n sn s= n∈Z ≥0

as an essentially finite sum, where sn are sections of Mα0 . Here, “essentially finite” means there exists a finite subset T ⊂ Z ≥0 such that sn = 0 unless n ∈ T . Definition 3. Let (m, j) ∈ Z × Z≥0 . A section s of Uα M on a neighbourhood of Y is called (m, j)-primitive if s = ∂ m− (gx − p−δ+m+ (τ f ) j v) for a holomorphic function g with g|Y = 0, i.e., gx m+ (τ f ) j is (m+ , j)-primitive as a section of   O X (1) (∗D (1) ). Definition 4. Let s be a non-zero section of Uα M on a neighbourhood of Y . A primitive expression of s is a decomposition  sm, j s= (m, j)∈S

where S ⊂ Z × Z≥0 is a finite set, and sm, j are (m, j)-primitive sections of Mα0 . This kind of expressions have been used in the study of pure twistor D-modules [4], for example. Lemma 11. Any non-zero section s of Uα M on a neighbourhood of Y has a primitive expression.

A twistor approach to the Kontsevich complexes

Proof. We have the following expression as an essentially finite sum:

   s= ∂ n gn,q, j x −δ− p+q (τ f ) j v . n∈Z ≥0 q∈Z ≥0 j∈Z≥0

Here, “essentially finite” means that there exists a finite subset T ⊂ Z ≥0 × Z ≥0 × Z≥0 such that gn,q, j = 0 unless (n, q, j) ∈ T . We consider the following claim.     (Pa ): If s has an expression s = |n|≤a q, j ∂ n gn,q, j x −δ− p+q (τ f ) j v as an essentially finite sum, then s has a primitive expression s = (m, j)∈S sm, j such that |m− | ≤ a for any m ∈ π(S). If a = 0, the claim is given by Lemma 8. We prove (Pa ) by assuming (Pa−1 ). Note that if qi > 0, we have

  ∂i xi gx −δ− p+q xi−1 (τ f ) j v = xi ∂i g + (− pi + qi − 1 − ki j)g x −δ− p+q xi−1 (τ f ) j v − ki gx −δ− p+q xi−1 (τ f ) j+1 v.

(6)

By using (6), s has an expression   (1)  ∂ n gn,q, j x −δ− p+q (τ f ) j v s= n,q, j

with the following property: (1)

– gn,q, j = 0 unless |n| ≤ a. (1)

(1)

– If |n| = a and gn,q, j = 0, we have {i | n i = 0, qi = 0} = ∅ and gn,q, j|Y = 0.  (1)    By applying (Pa−1 ) to the lower term |n|
 ∂ m− gm, j x −δ− p+m+ (τ f ) j v = 0. (7) (m, j)∈S

Lemma 12. We have gm, j|Y = 0 for any m ∈ min π(S) and j ∈ Z. We also have gm, j|Y = 0 for any (m, j) ∈ min S. Proof. In general, we have the following equality for any section g of O X (1) on a neighbourhood of Y :

  ∂i gx −δ− p+n (τ f ) j v = xi ∂i g − (1 + pi − n i + jki )g x −δ− p+n xi−1 (τ f ) j v − ki gx −δ− p+n xi−1 (τ f ) j+1 v.

(8)

T. Mochizuki

Hence, we have the following expression:   ∂ m− gm, j x −δ− p x m+ (τ f ) j v =



h m, j,k x −δ− p+m (τ f ) j+k v,

0≤k≤|m− |

where h m, j,k are sections of O X (1) on a neighbourhood of Y such that h m, j,k|Y = C m, j,k · gm, j|Y for some C m, j,k ∈ Q. Because {i | m +,i = 0, m −,i = 0} = ∅, we have C m, j,0 = 0. By (7), we have the following in O X (1) (∗D (1) ): 



h m, j,k x −δ− p+m (τ f ) j+k = 0.

(m, j)∈S 0≤k≤|m− |

Take m ∈ min π(S). According to Lemma 9, for any p ≥ 0, we have 

C m, j,k gm, j|Y = 0.

j+k= p

We obtain gm, j|Y = 0 by an ascending induction on j. Take (m, j) ∈ min S. By Lemma 9, we obtain h m, j,0|Y = C m, j,0 gm, j|Y = 0. Hence, we obtain gm, j|Y = 0.   Corollary 2. Let s be a section of Uα M on a neighbourhood of Y with a primitive expression 

s=

  ∂ m− gm, j x −δ− p+m+ (τ f ) j v .

(m, j)∈S

– The set min π(S) is well defined for s. For any m ∈ min π(S) and j ∈ Z≥0 , gm, j|Y is well defined for s. – The set min S is well defined for s. For any (m, j) ∈ min(S), gm, j|Y is well defined for s.  

2.4. Quasi-isomorphism of complexes 2.4.1. Statements We set Mα0 (−D (1) ) := Mα0 ⊗ O X (1) (−D (1) ). The action of π ∗ V0 D X on Mα0 (−D (1) ) naturally induces a complex Mα0 (−D (1) ) ⊗ Ω X• (1) /C (log D (1) ). We have a natural inclusion of complexes: τ

Mα0 (−D (1) ) ⊗ Ω X• (1) /C (log D (1) ) −→ Uα M ⊗ Ω X• (1) /C . τ

τ

We shall prove the following proposition in Sect. 2.4.2. Proposition 2. For 0 ≤ α ≤ 1, the morphism (9) is a quasi-isomorphism.

(9)

A twistor approach to the Kontsevich complexes

We set Ω kf (α) := Ω kf ⊗ O([α P]). We set Ω kf,τ (α) := π ∗ Ω kf (α). In the case α = 0, we also use the symbols Ω kf and Ω kf,τ . We obtain a complex Ω •f,τ (α) with the differential given by d + τ d f . We have a natural morphism of complexes induced by the correspondence 1 −→ v: Ω •f,τ (α) −→ Mα0 (−D (1) ) ⊗ Ω X• (1) /C (log D (1) ). τ

(10)

We shall prove the following proposition in Sect. 2.4.3. Proposition 3. For 0 ≤ α ≤ 1, the morphism (10) is a quasi-isomorphism. Let p2 : X (1) −→ Cτ denote the projection.   Corollary 3. Ri p2∗ Ω •f (α) (0 ≤ α < 1) is a locally free OCτ -module equipped with a naturally induced logarithmic connection ∇. Any eigenvalue β of Res0 (∇) is a rational number such that −α ≤ β < −α + 1.   Proof. By the construction, it is easy to see that Ri p2∗ Ω •f (α) is a coherent OCτ   module. By the above propositions, Ri p2∗ Ω •f (α) is equipped with the action of   the differential operator τ ∂τ . The restriction Ri p2∗ Ω •f (α) |C∗ is a DC∗τ -module. τ Because it is coherent as an OC∗τ -module, it is a locally free OC∗τ -module. Although we may finish the proof by applying the functoriality of the V filtration, let us recall the proof in this easy situation with the argument in §3.2 of [11]. The multiplication of τ n (n ≥ 0) induces an isomorphism of π ∗ D X -modules Uα M Uα−n M because α < 1. We have the exact sequence of π ∗ D X -modules: 0 −→ Uα−1 M −→ Uα M −→ Uα M/Uα−1 M −→ 0. It induces the following exact sequence of coherent OCτ -modules:     Ri p2∗ Uα−1 M ⊗ Ω X• (1) /C −→ Ri p2∗ Uα M ⊗ Ω X• (1) /C τ τ   −→ Ri p2∗ (Uα M/Uα−1 M) ⊗ Ω X• (1) /C τ   ϕ i+1 • −→ R p2∗ Uα−1 M ⊗ Ω X (1) /C τ

(11)

Suppose that ϕ = 0, and we will deduce a contradiction. For N ≥ 1, the image of the morphism     Ri+1 p2∗ Uα−N M ⊗ Ω X• (1) /C −→ Ri+1 p2∗ Uα−1 M ⊗ Ω X• (1) /C τ τ  is equal to the image of the multiplication of τ N −1 on Ri+1 p2∗ Uα−1 M ⊗   Ω X• (1) /C . Hence, by applying Nakayama’s lemma to Ri+1 p2∗ Uα−1 M ⊗ τ Ω X• (1) /C , there exists N0 such that the image of the composite of the following τ morphisms is non-zero:   ϕ   • • −→ Ri p2∗ (Uα M/Uα−1 M) ⊗ Ω X −→ Ri+1 p2∗ Uα−1 M ⊗ Ω X (1) /C (1) /C τ τ     • • −→ Ri+1 p2∗ Uα−1 M ⊗ Ω X C := Cok Ri+1 p2∗ Uα−N0 M ⊗ Ω X (1) /C (1) /C τ

τ

(12)

T. Mochizuki

  • For any β ∈ R, τ ∂τ + β on Ri p2∗ GrU β (M) ⊗ Ω X (1) /Cτ is nilpotent. Hence,   the eigenvalues of the endomorphism τ ∂τ of Ri p2∗ Uβ M/Uγ M ⊗ Ω X• (1) /C τ   are contained in [−β, −γ [= −β ≤ y < −γ . In particular, the eigenvalues   of the endomorphism τ ∂τ of Ri p2∗ Uα M/Uα−1 M ⊗ Ω X• (1) /C are contained in τ   [−α, −α+1[. Because C is contained in Ri+1 p2∗ Uα−1 M/Uα−N0 M⊗Ω X• (1) /C , τ the eigenvalues of the endomorphism τ ∂τ on C are contained in [−α +1, −α + N0 [. Hence, the image of (12) is 0, and we arrived  at a contradiction,  i.e., ϕ = 0. It implies that the multiplication of τ on Ri p2∗ Uα M ⊗ Ω X• (1) /C is injective. Thus, τ   we obtain that Ri p2∗ Uα M ⊗ Ω X• (1) /C is locally free at τ = 0. Moreover, we τ      have Ri p2∗ Uα M ⊗ Ω X• (1) /C |τ =0 Ri p2∗ (Uα M Uα−1 M) ⊗ Ω X• (1) /C , and τ τ the eigenvalues of τ ∂τ are contained in [−α, −α + 1[.   2.4.2. Proof of Proposition 2 We have only to check the claim around any point (τ0 , Q) of D (1) . We use a coordinate system (U, x1 , . . . , xn ) around Q as in Sect. 2.1.1. Set p := [αk]. (1) )v Let us consider the case τ0 = 0. We have Mα0 (−D (1) ) = O X (1) (∗Pred and Uα M = M = O X (1) (∗D (1) ) v around (τ0 , Q). For 1 ≤ p ≤ , we set p− S( p) := {(0, . . . , 0)}×Z≥0 1 ×{(0, . . . , 0)} ⊂ Z 1 ×Z p− 1 ×Z − p = Z . We put   (1) (1) H≤ p := 1 +1≤i≤ p {xi = 0} and H> p := p




 (1) (1) M≤ p := ∂ n Mα0 = O X (1) H>(1)p ∗Pred ∗H≤ p v. n∈S( p)

We have M≤ 1 = Mα0 . ∂p

∂p

Lemma 13. If p ≥ 1 + 1, the complexes x p M≤ p−1 −→ M≤ p−1 and M≤ p −→ M≤ p are quasi-isomorphic with respect to the inclusion. Proof. For j ∈ Z≥0 , we consider the following:





 − j−1 (1) p ∗H≤(1)p−1 x p F j M≤ p := ∂ n Mα0 = O X (1) H>(1)p ∗Pred v. n∈S( p) n p≤ j

We have pF0 M≤ p = M≤ p−1 . We have the morphisms of sheaves ∂ p : pF M≤ p −→ pF ≤p j j+1 M . We can easily check the following by direct computations.  – If j ≥1, the induced morphisms of O X (1) /x p O X (1) -modules pF j pF j−1 −→ pF p j+1 F j are isomorphisms. – The morphism pF0 M≤ p = M≤ p−1 −→ pF1 / pF0 is a surjection, and the kernel is x p M≤ p−1 . Then, the claim of the lemma follows.

 

A twistor approach to the Kontsevich complexes

By Lemma 13, the following inclusion of the complexes of sheaves are quasiisomorphisms for p ≥ 1 + 1:

(1)

(1)

a1

M≤ p−1 (−H> p−1 ) −→ M≤ p−1 (−H> p−1 ) · d x p /x p

(1) a2 (1) −→ M≤ p (−H> p ) −→ M≤ p (−H> p ) · d x p .

(13)

Here, ai (i = 1, 2) are induced by the exterior derivative in the x p -direction. Hence, the following inclusions of the complexes of sheaves are quasi-isomorphisms:



(1) (1) M≤ p−1 −H> p−1 ⊗ Ω X• (1) /C log H> p−1 τ



−→ M≤ p −H>(1)p ⊗ Ω X• (1) /C log H>(1)p .

(14)

τ





(1) (1) We have M≤ −H> ⊗ Ω X• (1) /C log H> = M ⊗ Ω X• (1) /C , and τ

τ







(1) (1) M≤ 1 −H> 1 ⊗ Ω X• (1) /C log H> 1 = Mα0 −H (1) ⊗ Ω X• (1) /C τ τ

α (1) • = M0 −D ⊗ Ω X (1) /C

τ



log H (1)

log D (1) . (15)

Hence, we are done in the case τ0 = 0. Let us consider the case τ0 = 0. For 0 ≤ p ≤ , we regard Z p = Z p ×

− p

   {(0, . . . , 0)} ⊂ Z . We set Uα M≤ p :=



∂ n Mα0 ,

F j Uα M≤ p =

p

p



∂ n Mα0 .

p

n∈Z≥0

n∈Z≥0 n p≤ j

We have Uα M≤ = Uα M and pF0 Uα M≤ p = Uα M≤ p−1 . We consider the following maps ∂ p : pF j Uα M≤ p −→ pF j+1 Uα M≤ p . The following lemma is easy to see by Corollary 2. Lemma 14. Let s be a section of Uα M on a neighbourhood of Y with a primitive expression s=



sm, j .

(m, j)∈S

Then, s is a section of pF j Uα M≤ p if and only if we have m i ≥ 0 (i > p) and   m p ≥ − j for any m ∈ min π(S).

T. Mochizuki

Lemma 15. If j ≥ 1, the following induced morphism of sheaves is an isomorphism: ≤p j Uα M pF ≤p j−1 Uα M pF

∂p

−→

≤p j+1 Uα M pF U M≤ p j α

pF

p ≤p Proof. It is surjective by construction. Let s be a non-zero section of  F j Uα M on a neighbourhood of Y with a primitive decomposition s = (m, j)∈S sm, j  p ≤ p  such that ∂ p s is also a section of F j Uα M . We set s := m p =− j sm, j and  s  := m p >− j sm, j . Because ∂ p s  ∈ pF j Uα M≤ p , we obtain ∂ p s  ∈ pF j Uα M≤ p .  If s  is non-zero, ∂ p s  = m p =− j ∂ p sm, j is a primitive expression of ∂ p s  . We / pF j Uα M≤ p , and thus we arrive at a contradiction. Hence, s  = 0, obtain ∂ p s  ∈ p   i.e., s ∈ F j−1 Uα M≤ p .

Lemma 16. The kernel of the following induced surjection is x p Uα M≤ p−1 : ∂p

Uα M≤ p−1 −→

pF U M≤ p 1 α Uα M≤ p−1

≤ p−1 . We take a primitive expression s = Proof. Let s be a section of U αM     (m, j)∈S sm, j . We set s := m p =0 sm, j and s := m p >0 sm, j . We have s  ∈ x p Uα M≤ p−1 . Because

    ∂ p · x p · g · x −δ− p (τ f ) j v = x p ∂ p (g) − ( p p + k p j)g x −δ− p (τ f ) j v − k p g · x −δ− p (τ f ) j+1 v, we have∂ p s  ∈ Uα M≤ p−1 . Hence, we have ∂ p s  ∈ Uα M≤ p−1 . If s  = 0, / Uα M≤ p−1 , ∂ p s  = m p =0 ∂ p sm, j is a primitive expression of s  . We obtain ∂ p s  ∈  and we arrive at a contradiction. Hence, we have s  = 0, i.e., s ∈ Uα M≤ p−1 x p .  By Lemma 15 and Lemma 16, for 1 ≤ p ≤ , the inclusions of the complexes 

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

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

(1)

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

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

(16)

τ

(1)

(1)

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

τ

(1)

τ

have Uα M≤0 (−D>0 ) ⊗ Ω X• (1) /C (log D>0 ) = Mα0 (−D (1) ) ⊗ Ω X• (1) /C (log D (1) ). τ τ Hence, Proposition 2 is proved.  

A twistor approach to the Kontsevich complexes

2.4.3. Proof of Proposition 3 We have only to check the claim around any point of P (1) . We use the coordinate system as in Sect.   2.1.1.  Set p := [αk]. For any non-negative integer N , we set G N Mα0 (−D (1) ) := Nj=0 O X (1) x − p (τ f ) j v. We set  k (1) ) := G Mα (−D (1) ) ⊗ Ω k G N Mα0 (−D (1) ) ⊗ Ω X (log D (1) ). N (1) /C (log D 0 X (1) /Cτ τ

The derivative d of the complex Mα0 (−D (1) ) ⊗ Ω X• (1) /C (log D (1) ) induces τ

  d : G N Mα0 (−D (1) ) ⊗ Ω Xk (1) /C (log D (1) ) τ   (1) −→ G N +1 Mα0 (−D (1) ) ⊗ Ω Xk+1 ) . (1) /C (log D

(17)

τ

  We also set G −1 Mα0 (−D (1) ) ⊗ Ω Xk (1) /C (log D (1) ) := Ω kf,τ (α)v = Ω kf,τ x − p v. τ Let N ≥ 0. Take a section ω=

N 

  ω j x − p (τ f ) j · v ∈ G N Mα0 (−D (1) ) ⊗ Ω Xk (1) /C (log D (1) ) , τ

j=0

where ω j ∈ Ω Xk (1) /C (log D (1) ). Suppose that τ

  (1) ) . dω ∈ G N Mα0 (−D (1) ) ⊗ Ω Xk+1 (1) /C (log D τ

  (1) ). Then, τ d f ∧ω N (τ f ) N x − p v is a section of G N Mα0 (−D (1) ) ⊗Ω Xk+1 (1) /C (log D τ

(1) ), i.e., ω is a section of Ω k . Lemma 17. We have d f ∧ ω N ∈ Ω Xk+1 N (1) /C (log D f,τ τ

 Proof. Let s be a local section of O X (1) such that (τ f ) N +1 s ∈ Nj=0 O X (1) (τ f ) j . We obtain that τ N +1 s ∈ O X (1) f −1 , and s ∈ O X (1) f −1 . There exists a holomorphic function t such that s = t f −1 . We obtain (τ f ) N +1 s = (τ f ) N τ t ∈ O X (1) (τ f ) N .  (1) )·(τ f ) j . Note that (d f / f )∧ω N ·(τ f ) N +1 is a section of Nj=0 Ω Xk+1 (1) /C (log D τ

By the above argument, we obtain that (d f / f ) ∧ ω N · (τ f ) N +1 is a section of (1) )(τ f ) N . Hence, τ d f ∧ ω is a section of Ω k+1 (log D (1) ). Ω Xk+1 N (1) /C (log D X (1) /C τ

(1) ). Then, we obtain that d f ∧ ω N is a section of Ω Xk+1 (1) /C (log D τ

τ

 

Lemma 18. We have an expression ω N = (d f / f ) ∧ κ1 + f −1 κ2 , where κ1 and κ2 (1) ) and Ω k are local sections of Ω Xk−1 (log D (1) ), respectively. (1) /C (log D X (1) /C τ

τ

Proof. Let  Q be any point of Pred . Let U be a neighbourhood of Q in X . The complex Ω X• (log D), d f / f is acyclic on U because d f / f is a nowhere vanishing section of Ω X1 (log D) on U . If U is sufficiently small, we can take decompositions Ω Xk (log D) = B k ⊕C k such that the multiplication of d f / f induces an isomorphism   C k B k+1 . Then, the claim of the lemma follows.

T. Mochizuki

If N ≥ 1, we have f −1 κ2 (τ f ) N = τ κ2 (τ f ) N −1 . Hence, we have     ω − d κ1 (τ f ) N −1 x − p v ∈ G N −1 Mα0 (−D (1) ) ⊗ Ω Xk (1) /C (log D (1) ) . τ

We also have κ1 (τ f ) N −1 x − p v ∈ G N −1 .  Let ω be a local section of G N Mα0 (−D (1) )⊗Ω Xk (1) /C (log D (1) ) such that dω τ   is a local section of G −1 Mα0 (−D (1) )⊗Ω Xk (1) /C (log D (1) ) . By applying the previτ  ous argument successively, we can find a local section τ of G N −1 Mα0 (−D (1) ) ⊗   α (1) ) such that ω − dτ is a local section of G (1) Ω Xk−1 −1 M0 (−D ) ⊗ (1) /C (log D τ  Ω Xk (1) /C (log D (1) ) . τ As a consequence, we have the following.   – If a local section ω of G N Mα0 (−D (1) )⊗Ω Xk (1) /C (log D (1) ) satisfies dω = 0, τ   (1) ) such we can find a local section τ of G N −1 Mα0 (−D (1) )⊗Ω Xk−1 (1) /C (log D τ   that ω − dτ is a local section of G −1 Mα0 (−D (1) ) ⊗ Ω Xk (1) /C (log D (1) ) . τ   – Let ω be a local section of G −1 Mα0 (−D (1) ) ⊗ Ω Xk (1) /C (log D (1) ) such that τ   (1) ) dω = 0. If we have a local section τ of G N Mα0 (−D (1) )⊗Ω Xk−1 (1) /C (log D  τ such that ω = dτ , then we can find a local section σ of G N −1 Mα0 (−D (1) ) ⊗  α  (1) ) such that τ − dσ is a local section of G (1) Ω Xk−2 −1 M0 (−D ) ⊗ (1) /C (log D τ  (1) ) . We have ω = d(τ − dσ ). Ω Xk−1 (1) /C (log D τ

 

Then, we obtain the claim of Proposition 3. 3. The case of mixed twistor D-modules 3.1. Preliminary

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 Y denote the sheaf of subalgebras of DY ΘY denote the tangent sheaf of Y . Let R generated by λ2 ∂λ over RY . A left RY -module is equivalent to an OY -module M with a relative flat mero Y -module is equivmorphic connection ∇ rel : M −→ M ⊗ λ−1 ΩY1 /Cλ . A left R alent to an OY -module M with a flat meromorphic connection ∇ : M −→ M ⊗ λ−1 ΩY1 (log λ). 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 j projection. For any RY -module N , we have the RW -modules π† N (− dim Z ≤ j := j ≤ dim Z ) as follows. Let q Z : Y −→ Z denote the projection. We set Ω Z

A twistor approach to the Kontsevich complexes

• ⊗ N on Y induced by λ− j q Z∗ Ω Z . Then, we have a naturally defined complex Ω Z • the exterior derivative of Ω Z and the relative flat meromorphic connection ∇ rel of N . We have

j Z• ⊗ N . π† N R j+dim Z (idCλ ×π )∗ Ω j

Y -module, then π j N are naturally R 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    V (λ0 ) N (∗t) |B(λ0 )×Y , where we set u∈K(a,λ0 ) (−ðt t + e(λ, u)) is nilpotent on Gr a e(λ, (b, β)) = β − λb − βλ2 for (b, β) ∈ R × C. Y -module. As in [8, Proposition Suppose moreover that N is enhanced to an R 7.3.1], we have K(a, λ0 ) = {(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.    – We have a∈R Va N (∗t) = N (∗t).     – For any a ∈ R,  we have  > 0such that  Va N (∗t)  = Va+ N (∗t)  .  – We have t Va 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     GraV (N (∗t)) := Va N (∗t) V
T. Mochizuki

field θ = d(τ f ), and the metric h given by h(e, e) = 1. Then, (E, ∂ E , θ, h) is a wild harmonic bundle. It is homogeneous with respect to the S 1 -action on X (1) given by t (τ, Q) = (tτ, Q) and t ∗ e = e, in the sense of [6]. (1) (1) (1) We set X (1) := Cλ × X (1) and Pred := Cλ × Pred . Let p : X (1) \ Pred −→ (1) X (1) \ Pred denote the projection. Let us recall that a family of λ-flat bundles is associated to a harmonic bundle in this situation. The C ∞ -bundle p −1E is equipped with the holomorphic structure ∂ 1 given by ∂ 1 p −1 (e) = p −1 (e) · λd τ f . It is also equipped with D given by D p −1 (e) = p −1 (e) ·  family of flat λ-connections   the −1 d(τ f ) + λd τ f . We set υ := p (e) exp(−λτ f ). It is a holomorphic frame of ( p −1 E, ∂ 1 ). We have Dυ = υ d(τ f ). Let E denote the sheaf of holomorphic sections of ( p −1 E, ∂ 1 ). Multiplying λ−1 to the (1, 0)-part of D, we obtain the family of flat connections

. D f : E −→ E ⊗ λ−1 Ω 1 (1) (1) (X

\Pred )/Cλ

In terms of the frame υ, it is given by D f υ = υ · λ−1 d(τ f ). The bundle p −1 E is naturally S 1 -equivariant with respect to the action given by t (λ, τ, Q) = (tλ, tτ, Q), for which we have t ∗ υ = υ. We have t ∗ D f = D f . Hence, D f is extended to a meromorphic flat connection

   (1)  . ∇ : E −→ E ⊗ λ−1 Ω 1 (1) (1) log {0} × X (1) \ Pred X

\Pred

Namely, the derivative in the λ-direction is induced. (See [6], for example.) In terms of the frame υ, it is given by ∇υ = υd(λ−1 τ f ). The bundle E with D f gives an R X (1) \P (1) -module, which is also denoted by E. red

Because D f is extended to the meromorphic connection ∇ as mentioned above, E (1) (1) -module. is naturally enhanced to an R X \P red

(1)

(1)

We set S := {|λ| = 1}. Let σ : S × (X (1) \ Pred ) −→ S × (X (1) \ Pred ) be given by σ (λ, τ, Q) = (−λ, τ, Q). We have the sesqui-linear pairing induced by the metric h: C h : E|S×(X (1) \D (1) ) × σ ∗ E|S×(X (1) \D (1) ) −→ Db S×(X (1) \D (1) )/S . (See [8] for the concept of sesqui-linear pairings.)    √ In this case,  it is given by C h (υ, σ ∗ υ) = exp −λ(τ f ) + λ(τ f ) = exp 2 −1 Im(λτ f ) . Thus, we obtain (1) an R-triple (E, E, C h ) on X (1) \ Pred . It is naturally enhanced to an R-triple. 3.2.2. The associated pure twistor D-modules According to [5], (E, E, C h ) is uniquely extended to a pure twistor D-module T = (M, M, C) on X (1) of weight 0 with the polarization (id, id) such that (i) the strict support of T is X (1) , (ii) T|X (1) \P (1) is identified with (E, E, C h ). Let us describe the R X (1) -module M. red

(1)

The holomorphic bundle E is naturally extended to an OX (1) (∗Pred )-module (1) QE := OX (1) (∗Pred )υ with a meromorphic connection    ∇ : QE −→ QE ⊗ λ−1 ΩX (1) log {0} × X (1)

A twistor approach to the Kontsevich complexes

X (1) -module denoted by given by ∇υ = υ · d(λ−1 τ f ). It naturally induces an R QE. Lemma 19. We have a natural isomorphism of R X (1) -modules M QE. Proof. We take a projective birational morphism F : X (2) −→ X (1) such that (i) (2) (1) Pred := F −1 (Pred ) is simply normal crossing, (ii) X (2) \ P (2) X (1) \ P (1) , (iii) the zeroes and the poles of F ∗ (τ f ) are separated. Let (E  , ∂ E  , θ  , h  ) be the pull back of the harmonic bundle (E, ∂ E , θ, h) by F. Then, it is a good wild harmonic bundle on (X (2) , D (2) ). We have the associated pure twistor D-module T = (M , M , C ) on X (2) . By the construction of M in [5], we have a natural isomorphism M (∗P (2) )   (1) F ∗ QE. Then, we have a natural isomorphism F† M (∗Pred ) QE. Note that M is a direct summand of F† (M ) such that M|X (1) \P (1) = F† (M )|X (1) \P (1) , and red

red

that M is strictly S-decomposable. Because the strict support of M is X (1) , a local section of M is 0 if its support is contained in P (1) . Hence, we obtain M ⊂ (1) (1) (1) M(∗Pred ) F† (M )(∗Pred ) QE. Moreover, we have OX (1) (−N Pred )υ ⊂ M for some positive integer N . By using ðτ υ = f υ, we obtain that υ is a section of M, and hence M ⊃ QE.   X (1) -module. In particular, M is naturally an R

3.2.3. The associated mixed twistor D-modules Recall that H is a hypersurface of X such that D = H ∪ Pred and codim(H ∩ Pred ) ≥ 2. We set H (1) := Cτ × H . We set D(1) := Cλ × D (1) and H(1) := Cλ × H (1) . We have the localization of T along H (1) in the category of mixed twistor Dby T[∗H (1) ]. It consists of an R X (1) -triple modules on X (1) . (See [7].) It is denoted   (1) (1) (1) M[!H ], M[∗H ], C[∗H ] with a weight filtration W. We have a natural morphism of R X (1) -triples T −→ T[∗H (1) ]. Let us describe the R X (∗τ )-module M[∗H (1) ](∗τ ). We have the R X (1) (∗τ )-module QE(∗H(1) )(∗τ ) and the OX (1) (∗τ )-submodule  denote the R X (1) (∗τ )-submodule of QE(∗H(1) )(∗τ ) genQE(H(1) )(∗τ ). Let M (1) erated by QE(H )(∗τ ) over R X (1) :

 := R X (1) · QE(H(1) )(∗τ ) ⊂ QE(∗H(1) )(∗τ ). M X (1) (∗τ )-submodule of QE(∗H(1) )(∗τ ). It is naturally an R Lemma 20. We have a natural isomorphism of R X (1) (∗τ )-modules:  M[∗H (1) ](∗τ ) M.

(18)

to X (1) \ Proof. We have a natural injection M −→ M[∗H (1)] whose restriction  (1) (1) (1) (1) H is an isomorphism. It induces M(∗H )(∗τ ) M[∗H ] (∗H )(∗τ ). We obtain a morphism M[∗H (1) ](∗τ ) −→ M(∗H(1) )(∗τ ) = QE(∗H(1) )(∗τ ). Take a locally defined holomorphic function g on U ⊂ X (1) such that −1 g (0) = H (1) . Let  ιg : U −→ U × Ct denote the graph of g. We have ιg† (M[∗H (1) ]) ιg† M [∗t] ⊂ ιg† (M(∗H(1) )). (See §3.1.2, §3.3 and §5.4.1

T. Mochizuki

of [7].) It implies that a local section of M[∗H (1) ] is 0 if its support is contained in H(1) . Hence, we may regard M[∗H (1) ](∗τ ) as an R X (1) -submodule of QE(∗H(1) )(∗τ ). By the construction of R-module M[∗H (1) ]|X (1) \{τ =0} in §5.3.3 of [7], outside {τ = 0}, M[∗H (1) ] is generated by QE(H(1) ) over R X (1) . Hence, |X (1) \{τ =0} . Because the actions of τ we have M[∗H (1) ](∗τ )|X (1) \{τ =0} = M (1)  are invertible, we obtain M[∗H (1) ](∗τ ) = M  in on M[∗H ](∗τ ) and M (1)   QE(∗H )(∗τ ). X (1) \D (1) Because the restriction of (18) to X (1) \ D(1) is an isomorphism of R modules, we can easily deduce that (18) is an isomorphism of R X (1) -modules. We  := Vα−1 M  for any α ∈ R, which is a  We put Uα M have the V -filtration of M. unique filtration satisfying the condition in Sect. 1.2. The filtration Uα (α ∈ R) is also called the V -filtration in this paper. We shall explicitly describe the filtration in Sect. 3.4. 3.2.4. Push-forward Let p1 : X (1) −→ Cτ denote the projection. As the cohomology of the push-forward, we obtain RCτ -triples for j ∈ Z:

j −j j j p1† T[∗H (1) ] = p1† M[!H (1) ], p1† M[∗H (1) ], p1† C[∗H (1) ] . They are mixed twistor D-modules with the induced filtrations W.   i T[∗H (1) ] (∗τ ) are admissible variations of mixed Lemma 21. For any i, p1† i M[H (1) ](∗τ ) twistor structure on (Cτ , 0) in the sense of [7, §9]. In particular, p1† are locally free OC2 (∗τ )-modules. λ,τ

 Proof. We set M1 [H (1) ] := M[H (1) ] (λ − 1)M[H (1) ]. We have    1    i i i p1† M[H (1) ] (λ − 1) p1† M[H (1) ] p1† M [H (1) ] .   i M1 [∗H (1) ] Because M1 [∗H (1) ] M, p1† is a locally free OC∗τ -module |C∗τ according to Corollary 3. (We the claim directly more easily than  can check  i T[∗H (1) ] is a smooth RC∗τ -triple. Because Corollary 3.) Hence, p1† ∗ |Cτ   i W i (1) (1) Gr W w p1† (T[∗H ]) are polarizable pure twistor D-modules, Gr w p1† T[∗H ] (∗τ ) are obtained as the canonical prolongation of good wild polarizable variation of pure twistor structure of weight w.Hence, the tuple of the identity id : Cτ −→ Cτ , i T[∗H (1) ] (∗τ ) with W gives a cell of the mixed the open subset C∗τ ⊂ Cτ , and p1†   i (1) twistor D-module p1† T[∗H ] in the sense of §11.1 of [5]. Then, according to   i T[∗H (1) ] (∗τ ) is an admissible variation of mixed Proposition 11.1 of [5], p1† twistor structure.   i M[∗H (1) ](∗τ ) are strictly specializable along τ . As The RCτ (∗τ )-modules p1†    we have a unique filtration Va pi M[∗H (1) ](∗τ ) (a ∈ R) by in the case of M, 1†   τV R -coherent submodules of p i M[∗H (1) ](∗τ ) satisfying the conditions in 0 Cτ 1†     i M[∗H (1) ](∗τ ) := V i (1) Sect. 3.1. We set Uα p1† α−1 p1† M[∗H ](∗τ ) .

A twistor approach to the Kontsevich complexes

 i  Corollary 4. Uα p1† M[∗H (1) ](∗τ ) are locally free OC2 -modules for any α ∈ λ,τ R.   i T[∗H (1) ] (∗τ ) are admissible variations of mixed twistor Proof. Because p1† i M[∗H (1) ](∗τ ) are good-KMS in the sense of structure, the RCτ (∗τ )-modules p1† i M[∗H (1) ] ∗ [7, §5.11]. By using the C -equivariance, we can easily observe that p1†  i  (∗τ ) are regular-KMS in the sense of [7, §5.11]. Hence, Uα p1† M[∗H (1) ](∗τ )   are locally free OC2 -modules. λ,τ

3.2.5. Hodge structure on the nearby cycle sheaf We consider the C∗ -action on Cλ × Cτ given by t (λ, τ ) = (tλ, tτ ). It induces a C∗ -action on X (1) . The R X (1) modules M[H (1) ] ( = ∗, !) are C∗ -equivariant, and the sesqui-linear pairing of M[!H (1) ] and M[∗H (1) ] is equivariant with respect to the S 1 -action. i M[H (1) ]. The induced We have the induced natural C∗ -actions on p1† −i S 1 -action is compatible with the sesqui-linear pairing of p1† M[!H (1) ] and i (1) p1† M[∗H ]. For i ∈ Z and a ∈ R, we have the nearby R{0} -triple:   i i i a p1† a p −i M[!H (1) ], ψ a p1† a p1† ψ T[∗H (1) ] = ψ M[∗H (1) ], ψ C[∗H (1) ] . 1† a pi C[∗H (1) ] denotes the induced sesqui-linear pairing. With the relative Here, ψ 1† monodromy filtration W , it is a mixed twistor structure. We have the induced C∗ a pi M[∗H (1) ], which are compatible with the a p −i M[!H (1) ] and ψ actions on ψ 1† 1† a pi C[∗H (1) ] is compatible with the S 1 filtration W . The sesqui-linear pairing ψ 1† action. Hence, the mixed twistor structure comes from a complex mixed Hodge structure. a pi M[∗H (1) ] Namely, let Hia denote the vector space obtained as the fiber of ψ 1† over 1 ∈ Cλ . Then, we have the decreasing filtrations F and G on Hia such that the a pi M[∗H (1) ] and ψ a p −i M[!H (1) ] with the C∗ -action are isomorR-modules ψ 1† 1† phic to the analytification of the Rees modules of the filtered vector spaces (Hia , F) and (Hia , G)∨ . Moreover, we have the increasing filtration W on Hia induced by the a pi M[∗H (1) ] so that (Hia , F, G, W ) is a complex mixed Hodge filtration W of ψ 1† structure. a p k M[H ] a p k M[H ]−→λ−1 ψ For any k ∈ Z, let Nak denote the morphisms ψ 1† 1† ( = ∗, !) induced as the nilpotent part of τ ∂τ . Let Nia denote the endomorphism of Hia induced by Nai . Recall that (−Na−i , −Nai ) gives a morphism of mixed twistor a pi (T[∗H ])⊗T a pi (T[∗H ]) −→ ψ T (−1). Hence, Nia induces a morstructures ψ 1† 1† i phism of mixed complex Hodge structure (Ha , F, G, W ) −→ (Hia , F, G, W )(−1), where (−1) indicates the Tate twist. In particular, we obtain the following. Lemma 22. Nia (F j ) = Im(Nia ) ∩ F j−1 for any i ∈ Z, j ∈ Z and a ∈ R.

 

T. Mochizuki

3.2.6. Functoriality of V -filtrations Let p1 : X (1) −→ Cλ × Cτ denote 1 1 p1∗ ΩC1 2 . We the projection. Let ΩX (1) /C2 denote the kernel of ΩX (1) −→ λ,τ λ,τ k 1 (1) 2 . Let π : 1 (1) 2 := λ−1 Ω 1 (1) 2 and Ω k (1) 2 := Ω set Ω X /C X /C X /C X /C λ,τ

λ,τ

λ,τ

λ,τ

X (1) −→ X denote the projection. Put π := idCλ ×π . We set π ∗ R X := −1 ∗ π R X . For any π R X -module B, we have a naturally defined OX (1) ⊗ π −1 OX • complex B ⊗ ΩX (1) /C2 . λ,τ

By the functoriality of mixed twistor D-modules with respect to the pushj (1) forward via projective morphisms, p1† GrU α M[∗H ] are strict for any α ∈ R. Hence, we obtain the following isomorphism (see the proof of Theorem 3.18 of [8]): 

 j   • (1) 2 [dim X ] . p1∗ Uα M[∗H (1) ](∗τ ) ⊗ Ω Uα p1† M[∗H (1) ](∗τ ) R j X /C λ,τ

  ⊗ Ω • (1) 2 [dim X ] . p1∗ Uα M It is isomorphic to R j X /C

(19)

λ,τ

3.3. Main theorem 3.3.1. Quasi-isomorphism Let q X : X (1) −→ X denote the projection. We set k (α) := λ−k q ∗ Ω k (α). We have the following morphism of sheaves: Ω X f f,λ,τ kf,λ,τ (α) −→ Ω k+1 (α). d + λ−1 τ d f : Ω f,λ,τ 2  We have d + λ−1 τ d f = 0. We shall prove the following theorem later in Sect. 3.5. Theorem 5. For 0 ≤ α < 1, we have a natural quasi-isomorphism   •   f,λ,τ (α), d + λ−1 τ d f Uα M[∗H (1) ](∗τ ) ⊗ Ω • (1) 2 . Ω X /C λ,τ

It is equivariant with respect to the C∗ -action. 3.3.2. Consequences

We shall explain some consequences of Theorem 5.

j (α) := R j • (α). By Theorem 5 and the Proof of Theorem 2. We set K p1∗ Ω f,λ,τ f isomorphism (19), we have   j (α) Uα p j+dim X M[∗H (1) ](∗τ ) K 1† f  j  for any j. Because Uα p1† M[∗H (1) ](∗τ ) are locally free OC2 -modules for any λ,τ

j (α) at (λ, τ ) is quasi-isomorphic to the j-th hyperj, we obtain that the fiber of K f • cohomology group of (Ω f (α), λd + τ d f ) by using a general result on any perfect complexes (Lemma 23 below). In particular, the dimension of the hypercohomology groups are independent of (λ, τ ), i.e., Theorem 2 holds.  

A twistor approach to the Kontsevich complexes

Lemma 23. Let Y be any complex manifold. Let C • be a bounded complex of locally free OY -modules. Let F be any OY -module. If the cohomology sheaves H j (C • ) ( j ∈ Z) are locally free OY -modules, we naturally have H j (C • ) ⊗ F   H j (C • ⊗ F) for any j. j Cτ -module, Proof of Theorem 3. The RCτ -module p1† M[∗H (1) ] is naturally an R 2 2 i.e., it is equipped with the action of λ ∂λ . The action of λ ∂λ preserves   j+dim X (α) = Uα p j M[∗H (1) ](∗τ ) . It means that K i (α) is equipped with K 1† f f i (α) ⊂ K i (α) ⊂ i (α) and ∇λ2 ∂ K a meromorphic connection ∇ such that λ∇τ ∂τ K f f f λ i (α) is C∗ -equivariant with respect to the i (α). The holomorphic bundle K K f f action t (λ, τ ) = (tλ, tτ ). The meromorphic connection ∇ is equivariant by the construction, and the C∗ -action on the bundle is equal to the parallel transport by the connection. The induced vector bundle on P1 is isomorphic to Kif (α). It is equipped with the induced meromorphic connection ∇ such that ∇Kif (α) ⊂ Kif (α) ⊗ ΩP11 ({0} + 2{∞}). Here, 0 corresponds to [λ : τ ] = [1 : 0] and ∞ corresponds to [λ : τ ] = [0 : 1]. By the construction, it is equal to the connection in Corollary 3 around 0. Thus, we obtain Theorem 3.    i  (α)|τ =0 denote the image of Proof of Theorem 4. For α − 1 < b ≤ α, let Ub K f     i−dim X i−dim X (1) i (α)|τ =0 . Ub p1† M[∗H ] |τ =0 −→ Uα p1† M[∗H (1) ] |τ =0 = K f     i−dim X U i (α)|τ =0 . We also have the filtration b−1 p We have ψ M[∗H (1) ] Gr b K 1† f i U on K f (α)|0 as in Sect. 1.3. By the constructions, we have the following natural identifications:  i  (α)|τ =0 Hi−dim X GrU K GrU Ki (α)|0 . b−1

b

f

|λ=1

b

f

Hak .)

(See Sect. 3.2.5 for i (α) induced by the truncations By Theorem 2, we have the filtration F (1) on K f i (α)|Cλ ×{0} . We also as in Sect. 1.3 (see (1)). In particular, it induces a filtration of K f i (α)|Cλ ×{0} corresponding to the C∗ -action, according have a filtration F (2) on K f to the Rees construction. It is easy to observe that F (1) = F (2) . Hence, the Hodge X is equal to the filtration induced by F (1) . Then, the claim filtration F on Hi−dim b−1 of Theorem 4 follows from Lemma 22.   3.4. A description of the V -filtration  We consider the R X (1) (∗τ )-module M  given 3.4.1. The R X (1) (∗τ )-module M in Sect. 3.2.3. It has the global section υ. For a local coordinate system of X around a point of D as in Sect. 2.1.1, we have ði υ = −ki (τ f xi−1 )υ and ðτ υ = f υ. We have τ ðτ (x −δ+m υ) = x −δ+m τ f υ, ði xi x −δ+m υ = x −δ+m (λm i − ki τ f )υ. Hence, for any i with ki = 0, we have (τ ðτ + ki−1 ði xi )(x −δ+m υ) = x −δ+m (m i /ki )λυ.

(20)

T. Mochizuki

 along τ = 0 Let π : X (1) −→ X denote the 3.4.2. The V -filtration of M (1) −→ X denote the induced morphism. Set π ∗ R X := projection. Let π : X −1 π R X . Let τV0 R X (1) denote the sheaf of subalgebras in R X (1) OX (1) ⊗ π −1 OX  generated by π ∗ R X and τ ðτ . We shall define τV0 R X (1) -coherent submodules Uα M  for any α ∈ R. For 0 ≤ α ≤ 1, we set of M

 := π ∗ R X OX (1) (D(1) + [αP (1) ])υ ⊂ M.  Uα M For any real number α, we take the integer n such that 0 ≤ α + n < 1, and we set   := τ n Uα+n  M. Uα M  We use it in the  for the construction of Uα M. Remark 2. We do not need U1 (M) proof of Theorem 6. We shall prove the following theorem in Sects. 3.4.3–3.4.7.  is the V -filtration of M.  Theorem 6. U• M Remark 3. Theorem 6 is not precisely the twistor version of Proposition 1 because |{λ=1} = M(∗τ ), where M is the D-modules studied in Sect. 2. M  We may prove the following lemma by the argu3.4.3. Easy property of U• M ment in the proof of Lemma 1.  = M.  Lemma 24. Outside {τ = 0}, we have Uα M

 

 Lemma 25. We have a natural action of τV0 R X (1) on Uα M. (1)

Proof. We have only to check it around any point of Pred by using a coordinate system as in Sect. 2.1.1, and the relation (20). The argument is the same as that in Lemma 3.    ⊂ Uβ M.  As a result, we have Lemma 26. For α − 1 < β ≤ α, we have τ Uα M  ⊂ Uα  M  for any α ≤ α  . Uα M (1)

Proof. We have only to prove the claim around any point of Pred by using a coordinate system in Sect. 2.1.1. Set p := [αk]. It is enough to consider the case   0 ≤ α < 1. If α − 1 < β < 0, we have τ Uα M  ⊂ Uβ M by the construction. By −δ− p k−δ− p using the relation τ x υ = τ ðτ x υ , we can prove the claim in the case 0 ≤ β < α.   The following lemma is proved as in the case of the D-modules (Lemma 4).  are coherent.   Lemma 27. The τV0 R X (1) -modules Uα M   :=  We set X 0 := {0} × X ⊂ X (1) . We set U<α M β<α Uβ M. We obtain an   U<α M.  It is equipped with an endomorphism [τ ðτ ] induced R X 0 -module Uα M by τ ðτ . The following lemma is a generalization of Lemma 5.

A twistor approach to the Kontsevich complexes

  U<α M  is nilpotent. Lemma 28. For 0 < α < 1, the action of [τ ðτ ]+αλ on Uα M dim X  is 0. Indeed, the action of [τ ðτ ] + αλ (1)

Proof. We have only to check the claim around any point of Pred by using a coor  dinate system as in Sect. 2.1.1. Set p := [αk]. By using τ ðτ + λα x −δ− p υ = −ki−1 ði (xi x −δ− p υ), we obtain  (τ ðτ + λα) N (x −δ− p υ) ∈ U<α M for any N ≥  1 as in the proof of Lemma 5. We have τ ðτ + λα (gx −δ− p υ) = g · (τ ðτ + λα)(x −δ− p υ) + (τ ðτ g)x −δ− p υ.  ⊂ U<α M.  Then, the claim of the lemma We have (τ ðτ g)x −δ− p υ ∈ τ Uα M follows.   The following lemma can be also checked as in the case of the D-modules (Lemma 6).  ⊂ τ U<1 M.  Lemma 29. If N ≥ dim X + 1, we have (τ ðτ ) N U0 M (1) Proof. We have only to check the claim around any point of Pred by using a coordinate system as in Sect. 2.1.1. Let δ 1 be as in the proof of Lemma 6. We have  1  −1  −(δ−δ ) 1 υ. Hence, we have −ki ði · x (τ ðτ ) 1 (x −δ υ) = i=1

1





−ki−1 ði τ f x −(δ−δ 1 ) υ (τ ðτ ) 1 +1 x −δ υ = i=1

=τ

1 

−ki−1 ði



 x −δ−(k−δ 1 ) υ ∈ τ U<1 M.

(21)

i=1

 and any holomorphic function g, we have τ ðτ (gs) = For any section s of U0 M  Then, we obtain the claim of the τ (ðτ g) s + g τ ðτ s, and τ (ðτ g)s ∈ τ U<1 (M). lemma.   We obtain the following from Lemmas 28 and 29.  <α M  for any α ∈ R. Lemma 30. [τ ðτ ] + αλ is nilpotent on Uα M/U   = M.  Lemma 31. We have α∈R Uα M

 

(1) Proof. We have only to  prove the claim locally around any point of D . By the    construction, M := α∈R Uα M is an R X (1) (∗τ )-module. By the construction, ⊂ M  for any α, and hence M  ⊂ M.  We have ðτN x −δ− p υ = we have Uα M −δ− p−N k (1)   ⊂ M  . υ. Hence, we have QE(H )(∗τ ) ⊂ M . We obtain M   x   U<α M  are strict for any To prove Theorem 6, it remains to prove that Uα M   U<α M  is α, i.e., we need to establish that the multiplication of λ − λ1 on Uα M injective for any λ1 ∈ C. We shall prove it in Sect. 3.4.7 after some preparations.

T. Mochizuki

3.4.4. Preliminary Let (U, x1 , . . . , xn ) and Y be as in Sect. 2.3.1. We set Y := Cλ × Y and Y λ0 := {λ0 } × Y . In the following, Y (λ0 ) denotes a neighbourhood   of Y λ0 in Y, and N (λ0 ) denote the product of Y (λ0 ) and (x1 , . . . , x )  |xi | < for some > 0. For any section s of OX (1) (∗D(1) ) on N (λ0 ) , we have the Laurent expansion   s= h m, j x m τ j , m∈Z j∈Z≥0

where h m, j are holomorphic functions on Y (λ0 ) . As before, we have the unique expansion   (1) h m, j x m (τ f ) j , s= m∈Z j∈Z≥0 (1)

(1)

where h m, j are holomorphic functions on Y (λ0 ) . Indeed, we have h m, j = h m− j k, j . Definition 5. A section s of OX (1) (∗D(1) ) on N (λ0 ) is called (m, j)-primitive if it is expressed as s = gx m (τ f ) j for a holomorphic function g on N (λ0 ) with   g|Y (λ0 ) = 0. Definition 6. A primitive expression of a section s of OX (1) (∗D(1) ) on N (λ0 ) is an expression  gm, j x m (τ f ) j , s= (m, j)∈S

where S is a finite subset in Z × Z≥0 and gm, j are holomorphic functions on N (λ0 ) with gm, j|Y (λ0 ) = 0.   We use the partial orders on Z and Z × Z≥0 as in Sect. 2.3.1. We reword Lemma 8 and Corollary 1 in this context. (1) (λ0 ) has a primitive expression Lemma  32. Any section s of OX (1) (∗D ) on N s = (m, j)∈S gm, j x m (τ f ) j . Moreover the following claims hold.

– The set min π(S) is well defined for s. For any m ∈ min π(S), gm, j|Y (λ0 ) is well defined for s. – The set min(S) is well defined for s. For any (m, j) ∈ min(S), gm, j|Y (λ0 ) is well defined for s.   We reword Lemma 9 in this context. we are given holomorphic Lemma 33. Let T ⊂ Z ×Z≥0 be a finite subset. Suppose  functions gm, j ((m, j) ∈ T ) on N (λ0 ) such that (m, j)∈T gm, j x m (τ f ) j = 0. – For any m ∈ min π(T ) and any j ∈ Z≥0 , we have gm, j|Y (λ0 ) = 0. – For any (m, j) ∈ min(T ), we have gm, j|Y (λ0 ) = 0.

 

A twistor approach to the Kontsevich complexes

be any effective divisor of X such that P red ⊂ 3.4.5. Primitive expressions Let P (1) := Cλ × P (1) := Cτ × P and P (1) . For any non-negative integer Pred . We set P N , we set  := GNM P

N 

  (1) (τ f ) j υ ⊂ M.  OX (1) D(1) + P

j=0

  . We set U M  := π ∗ R X · M  be the sheafification of  in Let M N G N MP P P P  M. Let V0 R X ⊂ R X be generated by λΘX /C (log D) over OX . Set π ∗ V0 R X := π −1 V0 R X . (See Sect. 3.2.6 for π .) We naturally regard π ∗ V0 R X ⊂ OX (1) ⊗ π −1 OX R X (1) . We can check the following lemma as in the case of the D-modules (Lemma 10).   (1) υ , and hence  = π ∗ V0 R X · OX (1) D(1) + P Lemma 34. We have M P    = π ∗ R X · OX (1) D(1) + P (1) υ . U P M    for Uα M

= By the lemma, we have U[α P] M any 0 ≤ α ≤ 1. We use the notation in Sect. 3.4.4. Take any λ0 ∈ C.  We set Di = {z i = 0}

on U . pi Di = P on the neighbourhood U . Let p ∈ Z ≥0 be determined by i=1  > 1 . Let s be a non-zero section of U P M on N (λ0 ) . Note we have pi = 0 for i  We have an expression s = n∈Z ðn sn as an essentially finite sum, where sn are ≥0  on N (λ0 ) . sections of M P

 on N (λ0 ) is called (m, j)-primitive if it is Definition 7. A section s of U P M expressed as   s = ðm− gx −δ− p+m+ (τ f ) j υ for a holomorphic function g on N (λ0 ) with g|Y (λ0 ) = 0.

 

 on N (λ0 ) is a decomDefinition 8.  A primitive expression of a section s of U P M position s = (m, j)∈S s(m, j) , where S is a finite subset of Z × Z≥0 , and s(m, j)  are (m, j)-primitive sections of U P M.  on N (λ0 ) has a primitive expression Lemma 35. Any section s of U P M    s= ðm− gm, j x −δ− p+m+ (τ f ) j υ . (m, j)∈S

Proof. We give an algorithm to obtain a primitive expression. Let s be any section  on N (λ0 ) . We have the following expression as an essentially finite sum: of U P M

   s= ðn gn,q, j x −δ− p+q (τ f ) j υ . (22) n∈Z ≥0 q∈Z ≥0 j∈Z≥0

For a ∈ Z≥0 , we say that an expression (22) has the property (Ra ) if the following holds for any (n, q, j) with |n| ≥ a:

T. Mochizuki

– We have gn,q, j = 0 unless {i | n i = 0, qi = 0} = ∅.  – If gn,q, j = 0, then ðn gn,q, j x −δ− p+q (τ f ) j υ are (q − n, j)-primitive.    n −δ− p+q (τ f ) j υ . If a is sufficiently Take any expression s = n,q, j ð gn,q, j x large, the expression has the property (Ra ). In general, if qi > 0, we have

  ði xi gx −δ− p+q xi−1 (τ f ) j υ = xi ði g − λ( pi − qi + 1 + jki )g x −δ− p+q xi−1 (τ f ) j υ − ki gx −δ− p+q xi−1 (τ f ) j+1 υ.



 n

(23)



Let s = n,q, j ð gn,q, j x −δ− p+q (τ f ) j υ be an expression with the property (Ra )   such that a ≥ 1. Applying (23) and Lemma 32 to each ðn gn,q, j x −δ− p+q (τ f ) j υ with |n| = a − 1, we can obtain an expression with the property (Ra−1 ). We can arrive at an expression with the property (R0 ), which is a primitive expression of s.   Suppose that we are given a finite set S ⊂ Z ×Z≥0 and sections of gm, j of OX (1)

 on N (λ0 ) for (m, j) ∈ S, such that 0 = (m, j)∈S ðm− gm, j x −δ− p+m+ (τ f ) j υ , where m = m+ − m− is the decomposition as in Sect. 2.3.3. Lemma 36. For any m ∈ min π(S) and j ∈ Z≥0 , we have gm, j|Y (λ0 ) = 0. For any (m, j) ∈ min S, we have gm, j|Y (λ0 ) = 0. Proof. In general, we have the following for any section g of OX (1) on N (λ0 ) :

  ði gx −δ− p+n (τ f ) j υ = xi ði g − (1 + pi − n i + jki )λg x −δ− p+n xi−1 (τ f ) j υ − ki gx −δ− p+n xi−1 (τ f ) j+1 υ. Hence, we have the following expression:    ðm− gm, j x −δ− p x m+ (τ f ) j υ =

(24)

h m, j,k x −δ− p+m (τ f ) j+k υ,

0≤k≤|m− |

where h m, j,k are sections of OX (1) on N (λ0 ) such that h m, j,k|Y (λ0 ) = C m, j,k · gm, j|Y (λ0 ) for some C m, j,k ∈ Q. By {i | m +,i = 0, m −,i = 0} = ∅, we have C m, j,0 = 0. We have the following in OX (1) (∗D(1) ): 0=





h m, j,k x −δ− p+m (τ f ) j+k .

(m, j)∈S 0≤k≤|m− |

 For m ∈ min π(S) and p ≥ 0, we have j+k= p C m, j,k gm, j|Y (λ0 ) = 0. We obtain gm, j|Y (λ0 ) = 0 by an ascending induction on j. For (m, j) ∈ min S, we have   C m, j,0 gm, j|Y (λ0 ) = 0. Thus, the proof of Lemma 36 is finished.

A twistor approach to the Kontsevich complexes

 on N (λ0 ) with a primitive expression Corollary 5. Let s be a section of U P M    ðm− gn, j x −δ− p+m+ (τ f ) j υ . s= (m, j)∈S

– The set min π(S) is well defined for s. For any m ∈ min π(S) and any j ∈ Z≥0 , gm, j|Y (λ0 ) is well defined for s. – The set min S is well defined for s. For any (m, j) ∈ min(S), gm, j|Y (λ0 ) is well defined for s.   3.4.6. Variant of primitive expression Let P1 be an irreducible component of  ⊂ U M.  We assume that P1 = {x1 = 0} Pred . We have the inclusion U P M P+P1 on (U, x1 , . . . , xn ).  on N (λ0 ) has an expression as an essentially Lemma 37. Any section s of U P+P 1M finite sum    s = s + ðn− gn (λ, τ, x)x −δ− p+n+ x1−1 υ , n∈Z

   (ii) each ðn− gn (λ, τ, x)x −δ− p+n+ x −1 υ is such that (i) s  is a section of U P M, 1  (n, 0)-primitive as a section of U P+P 1 M. Proof. We prepare two procedures which are used to obtain an expression with the desired property. (λ ) (A) Suppose that a section s of U P+P 1 M on N 0 has an expression as an essentially finite sum     ðn gn, j (λ, τ, x)x −δ− p x1−1 (τ f ) j υ . (25) s= n∈Z ≥0 j≥0

Then, by using the relations (24), we obtain an expression of s as an essentially finite sum:    ðn gn(1) (λ, τ, x)x −δ− p x1−1 υ . (26) s = s + n∈Z ≥0

  (1)       Here,  (i) s is a section of U P M, (ii) max |n| gn = 0 ≤ max |n| ∃ j gn, j = 0 .   (B) Suppose that we are given a section ðn g(λ, τ, x)x −δ− p x1−1 υ . By applying   Lemmas 32 and (23), we obtain an expression of ðn g(λ, τ, x)x −δ− p x1−1 υ as an essentially finite sum:    ðn (gq(2) (λ, τ, x)x −δ− p+q x1−1 υ) ðn g(λ, τ, x)x −δ− p x1−1 υ = s  + q

+

  m∈Z ≥0 j≥0

 (3)  ðm gm, j x −δ− p x1−1 (τ f ) j υ . (27)

T. Mochizuki (2)

Here, (i) s  is a section of U P M, (ii) if gq = 0, {i | n i = 0, qi = 0} is empty and  (2)  (3) ðn gq (λ, τ, x  )x −δ− p+q x1−1 υ is (q − n, 0)-primitive, (iii) if gm, j = 0, we have |m| < |n|. Suppose that we have an expression (25) of s such that |n| ≤ a for any (n, j) such that gn, j = 0. By applying (A), we obtain an expression (26). We have  (1)  (1) |n| ≤ a if gn = 0. By applying (B) to ðn gn (λ, τ, x)x −δ− p x1−1 υ , we obtain an expression of s as follows:     ðn gn(4) (λ, τ, x)x −δ− p+q x1−1 υ s = s  + |n|=a q

 

+

|n|
 (5)  ðn gn, j x −δ− p x1−1 (τ f ) j υ .

(28)

j

Here, (i) s  is a section of U P M, (ii) if gn(4) = 0, {i | n i = 0, qi = 0} is empty  (4)  and ðn gn (λ, τ, x)x −δ− p+q x1−1 υ is (q − n, 0)-primitive. Then, we can obtain an expression of s with the desired property by an inductive argument.   Lemma 38. Let T1 be a finite subset in Z such that m 1 ≤ 0 for any m ∈ T1 . Suppose that we are given a section s  of U P M and holomorphic functions gm (λ, τ, x) (m ∈ T1 ) on N (λ0 ) such that the following holds:    s = ðm− gm (λ, τ, x)x −δ− p+m+ x1−1 υ . (29) m∈T1

Then, gm|Y (λ0 ) = 0 for any m ∈ min T1 .  Proof. Take a primitive expression of s  in U P M: s =

0 (n)  j

  ðn− h n, j x −δ− p+n+ (τ f ) j υ .

n∈T2 j=0

   

 Here,  T2 bea finite subset of Z . We set T2+ := n ∈ T2 n1 ≥ 0 and T2− := n ∈ T2  n 1 < 0 . For n, we set n− := n− −(1, 0, . . . , 0) and n+ := n+ +(1, 0, . . . , 0). Then, by using (23), we obtain an expression as follows: s =

0 (n)  j

   ðn− h n, j x −δ− p+n+ xi−1 (τ f ) j υ

n∈T2+ j=0

+



j0 (n)+1

n∈T2−

j=0

   (1) ðn− h n, j x −δ− p+n+ x1−1 (τ f ) j υ . (30)

   (1) Here, we can observe that ðn− h n, j0 (n)+1 x −δ− p+n+ x1−1 (τ f ) j0 (n)+1 υ is primitive.  := {n + (1, 0, . . . , 0) | n ∈ We set T2 := {n + (1, 0, . . . , 0) | n ∈ T2 } and T2±    , i.e., m = n + (1, 0, . . . , 0) for T2± }. Take any m ∈ min T1 ∪ T2 . If m ∈ T2−

A twistor approach to the Kontsevich complexes (1)

n ∈ T2− , we obtain h n, j|Y (λ0 ) = 0 for any j ∈ Z>0 by using (29), (30) and Lemma

 . Then, 36. But, it contradicts with h (1) = 0. Hence, we obtain m ∈ / T2− Y (λ0 )   n, j0 (n)+1| we obtain min(T1 ) ⊂ min T1 ∪ T2 . Then, we obtain the claim of the lemma by using (29), (30) and Lemma 36 again.  

 with an expression as in Lemma 37: Corollary 6. Let s be a section of U P+P 1M s = s +



  ðn− gn (λ, τ, x)x −δ− p+n+ x1−1 υ .

(31)

n∈T

Then, the set min(T ) and the functions gn|Y (λ0 ) (n ∈ min(T )) are well defined for s.   We obtain the following corollary.   U M  is strict. Corollary 7. U P+P 1M P (1)

Proof. We have only to consider the issue locally around any point of Pred by using  on N (λ0 ) . Take an expression the notation above. Let s be a section of U P+P 1M  if and only if (31) of s as in Lemma 37. By Corollary 6, s is a section of U P M T = ∅. Then, for any λ1 ∈ C, the conditions for s and (λ − λ1 )s are equivalent. ⊂ U P . Note that we have τ U P+P (λ0 ) with a primitive expression  Lemma  39. Let s be a section of U P M on N  M if and only if j = 0 for any s = (m, j)∈S sm, j . It is a section of τ U P+P (m, j) ∈ S.

Proof. Note that the condition is independent of the choice of a primitive expression, by Corollary 5. Then, the claim is clear.     τU M  is strict. Corollary 8. U P M P+P

 

3.4.7. The end of the proof of Theorem 6 Let us finish the proof of Theorem 6.  <α M  is strict for any α ∈ R. We have only to It remains to prove that Uα M/U prove the claim locally around any point of D(1) by using a coordinate system as in Sect. 2.1.1. It is enough to consider the case 0 ≤ α < 1. Take any λ0 ∈ Cλ .  = Uα M  for 0 ≤ α ≤ 1. We have an effective divisor P(< α) Note that U[α P] M  M,  = U<α  (ii) [α P] − P(< α) is effective. for 0 < α ≤ 1 such that (i) U P(<α) M    U<α  is strict M Hence, by using Corollary 7 successively, we obtain that Uα M for 0 < α < 1.  = τ U<1 M  ⊂ τUM  ⊂ U0 M.  We obtain that We have U<0 M 1     is strict as above. By using Corollary 8, U0 M  (τ U  M)  is strict. (τ U1 M)/U <0 M 1 Thus, we are done.  

T. Mochizuki

3.5. Proof of Theorem 5 3.5.1. Quasi-isomorphisms We generalize the results in Sect. 2.4.1 in the con1 (1) ) denote the kernel of text of mixed twistor D-modules. Let ΩX (1) /C2 (log D λ,τ

1 (log D (1) )−→ 1 (1) 2 (log D(1) ):=λ−1 Ω 1 (1) 2 (log D(1) ) ΩX p1∗ ΩC1 2 . We set Ω (1) X /Cλ,τ X /Cλ,τ λ,τ  1 (1) 2 (log D(1) ). k (1) 2 (log D(1) ) := k Ω and Ω

X

/Cλ,τ

X

/Cλ,τ

[α P] ⊗ O(−D(1) ). We have an [α P] (−D(1) ) := M Let 0 ≤ α < 1. We set M inclusion of complexes: ⊗ Ω [α P] (−D(1) ) ⊗ Ω • (1) 2 • (1) 2 (log D(1) ) −→ Uα M M X /C X /C λ,τ

(32)

λ,τ

The following proposition is an analogue of Proposition 2, which we shall prove in Sect. 3.5.2. Proposition 4. The morphism (32) is a quasi-isomorphism. The correspondence 1 −→ υ induces a natural morphism of complexes: [α P] (−D(1) ) ⊗ Ω •f,λ,τ (α) −→ M • (1) 2 (log D(1) ). Ω X /C

(33)

λ,τ

We obtain the following proposition as in the case of Proposition 3, which we shall prove in Sect. 3.5.3. Proposition 5. The morphism (33) is a quasi-isomorphism. We immediately obtain Theorem 5 from Proposition 4 and Proposition 5. 3.5.2. Proof of Proposition 4 We essentially repeat the argument in Sect. 2.4.2. We have only to check the claim around any point (λ0 , τ0 , Q) of D(1) . We use a (1) coordinate system as in Sect. 2.1.1. We set Di := {xi = 0} on the coordinate neighbourhood. Set p := [αk].

1

   Let us consider the case τ0 = 0. For 1 ≤ p ≤ , we set S( p) := {(0, . . . , 0)}×

− p

p− Z≥0 1

   × {(0, . . . , 0)} ⊂ Z 1 × Z p− 1 × Z − p = Z . We consider ≤ p := M



[α P] . ðn M

n∈S( p)

For any 0 ≤ α < 1, it is equal to the following in OX (1) (∗D(1) ) around (λ0 , τ0 , Q):

    (1)  (1) (1) ðn OX (1) (D(1) )(∗Pred )υ = λ|n| OX (1) D(1) + n i Di (∗Pred )υ. n∈S( p)

≤ 1 = M [α P] . We have M

n∈S( p)

A twistor approach to the Kontsevich complexes ð

p ≤ p−1 −→ ≤ p−1 and Lemma 40. For 1 + 1 ≤ p ≤ , the complexes x p M M

ð

p ≤ p −→ ≤ p are quasi-isomorphic with respect to the inclusion. M M  p−1 p (1) (1) Proof. We set D< p := i=1 {xi = 0} and D≤ p := i=1 {xi = 0} on the neighbourhood. For j ∈ Z, we consider the following:   p (1) F j L≤ p := OX (1) D(1) + jD(1) p (∗D< p ).

 (1)   p (1) ≤p = O We set L≤ p := (∗D≤ p ). We have pF j L≤ p −→ X (1) D j Fj L pF ≤ p induced by ð p . If j ≥ 0, the induced morphisms j+1 L   p F j L≤ p pF j−1 L≤ p −→ pF j+1 L≤ p pF j L≤ p  are injective. Indeed, the induced morphism x p ð p on pF j L≤ p pF j−1 L≤ p is the multiplication of ( j + 1)λ. By using the case j = 0, we obtain that the kernel of pF −→ pF pF is x pF . 0 1 0 p 0 For j ∈ Z≥0 , we consider the following:

   (1) p ≤ p := [α P] = Fj M ðn M λ|n| OX (1) D(1) + n i Di(1) (∗Pred )υ n∈S( p) n p≤ j

n∈S( p) n p≤ j

≤ p ⊂ pF j L≤ p and x p pF0 M ≤ p ⊂ We have pF0 M≤ p = M≤ p−1 . We have pF j M pF L≤ p . −1 Let us consider the following commutative diagram:   a1 pF M ≤ p x pF M ≤p − ≤ p pF−1 L ≤ p −−−→ pF0 L 0 p 0 ⏐ ⏐ ⏐ ⏐ b1  b2    a2 pF M ≤ p pF ≤ p −−− ≤ p pF j−1 L ≤ p . −→ pF j L j  j−1 M j

Here, ai are induced by the inclusions, and bi are induced by ð p . By using an explicit description, we can check that a1 is a monomorphism. As mentioned, b2 is a monomorphism. By the construction, b1 is epimorphism. Hence, we obtain that b1 is an isomorphism, and that a2 is a monomorphism. We can deduce the following claims on the morphisms induced by ð p :   ðp – If j ≥ 1, the induced morphisms pF j pF j−1 −→ pF j+1 pF j are isomorphisms.  ≤ p = M ≤ p−1 . ≤ p−1 −→ pF1 pF0 is x p M – The kernel of pF0 M Then, the claim of Lemma 40 follows.    (1)

For 1 ≤ p ≤ , we set H> p := i= p+1 {x i = 0}. By using Lemma 40, the following inclusions of the complexes of sheaves are quasi-isomorphisms (see the argument after Lemma 13):







(1) (1) (1) • • ≤ p −H> ≤ p−1 −H(1) log H> p−1 −→ M log H> M p ⊗ ΩX (1) /C2 p . > p−1 ⊗ ΩX (1) /C2 λ,τ

λ,τ

T. Mochizuki (1)

(1)

≤ (−H ) ⊗ Ω ⊗ Ω • (1) 2 (log H ) = M • (1) 2 , and We have M > > X /C X /C

λ,τ





λ,τ



(1) [α P] −H(1) ⊗ Ω • (1) 2 log H(1) = M • (1) 2 log H(1) M≤ 1 −H> 1 ⊗ Ω > 1 X /Cλ,τ X /Cλ,τ



(1) • (1) [α P] −D (1) 2 log D ⊗Ω . (34) =M X /C λ,τ

Hence, we are done in the case τ0 = 0. Let us consider the case τ0 = 0. For 1 ≤ p ≤ , we regard Z p = Z p ×

− p

   {(0, . . . , 0)} ⊂ Z . We set ≤ p := Uα M



[α P] , ðn M

≤ p = F j Uα M

p

p n∈Z≥0



[α P] . ðn M

p n∈Z≥0 n p≤ j

≤ = Uα M  and pF0 Uα M ≤ p = Uα M ≤ p−1 . We consider the We have Uα M following maps: ≤ p −→ pF j+1 Uα M ≤ p . ð p : pF j Uα M The following lemma is easy to see.  on N (λ0 ) with a primitive expression s = Lemma 41. Let s be a section of Uα M  p ≤ p if and only if we have m i ≥ 0 (m, j)∈S sm, j . Then, s is a section of F j Uα M   (i > p) and m p ≥ − j for any m ∈ min π(S). Lemma 42. If j ≥ 1, the following induced morphism of OX (1) /x p OX (1) -modules is an isomorphism: ≤ p j Uα M ≤ p j−1 Uα M

pF pF

ðp

−→

≤ p j+1 Uα M pF U M ≤p j α

pF

p ≤ p Proof. It is surjective by construction. Let s be a non-zero section of F j Uα M on (λ ) N 0 with a primitive decomposition s = (m, j)∈S sm, j such that ð p s is also a    ≤ p . We set s  := section of pF j Uα M m p =− j sm, j and s := m p >− j sm, j .  p ≤ p  p ≤ p   Because ð p s  ∈ F j Uα M , we obtain ð p s ∈ F j Uα M . If s  is non  zero, ð p s = m p =− j ð p sm, j is a primitive expression of ð p s . We obtain  p ≤ p  , and thus we arrive at a contradiction. Hence, s  = 0, i.e., ðps ∈ / F j Uα M p ≤ p .   s ∈ F j−1 Uα M

≤ p−1 : Lemma 43. The kernel of the following induced morphism is x p Uα M ðp

≤ p−1 −→ Uα M

pF U M ≤p 1 α ≤ p−1  Uα M

A twistor approach to the Kontsevich complexes

≤ p−1 , and we take a primitive expression s = Proof. Let s be a section of Uα M    p−1 . We set s  := (m, j)∈S sm, j such that ∂ p s ∈ Uα M m p =0 sm, j and s :=   ≤ p−1 , we have ð p s  ∈ Uα M ≤ p−1 as in m p >0 sm, j . Because s ∈ x p Uα M ≤ p−1 . If s  = 0, ð p s  = the proof of Lemma 16. Hence, we have ð p s  ∈ Uα M    / U M ≤ p−1 , and α m p =0 ð p sm, j is a primitive expression of s . We obtain ð p s ∈  ≤ p−1 .   we arrive at a contradiction. Hence, we have s = 0, i.e., s ∈ x p Uα M By Lemmas 42 and 43, the inclusions of the complexes    ðp ðp ≤ p−1 −→ ≤ p−1 −→ Uα M ≤ p −→ ≤ p Uα M Uα M x p Uα M  (1) are quasi-isomorphisms. For 0 ≤ p ≤ , we set D> p := i= p+1 {x i = 0} on the neighbourhood. We obtain that the following inclusions of the complexes of sheaves are quasi-isomorphisms:



(1) ≤ p−1 −D(1) • log D> p−1 Uα M > p−1 ⊗ ΩX (1) /C2 λ,τ



≤ p (1) • (1)  (1) 2 log D> −D> p ⊗ Ω (35) −→ Uα M p X /C 

λ,τ

≤ (−D(1) ) ⊗ Ω ⊗Ω • (1) 2 . We • (1) 2 (log D(1) ) = Uα M We have Uα M > > X /Cλ,τ X /Cλ,τ (1) (1) • ≤0 (1)   • (1) 2 also have Uα M (−D>0 ) ⊗ ΩX (1) /C2 (log D>0 ) = M[α P] (−D ) ⊗ Ω X /Cλ,τ λ,τ (log D(1) ). Hence, Proposition 4 is proved.   3.5.3. Proof of Proposition 5 We essentially repeat the argument in Sect. 2.4.3. (1) . We use the coordinate We have only to check the claim around any point of Pred system as in Sect. 2.1.1. We set p := [αk]. For any non-negative integer N , we set N     (1)  OX (1) x − p (τ f ) j υ. := G N M[α P] − D j=0

 [α P] (−D(1) ) ⊗ Ω k We define G N M



X (1) /C2λ,τ

log D(1)



as

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

  [α P] (−D(1) )⊗Ω k (1) 2 (log D(1) ) := Ω k (α)υ = Ω k x − p υ, We set G −1 M f,λ,τ f,λ,τ X /C λ,τ

k k where Ω f,λ,τ := Ω f,λ,τ (0). Let N ≥ 0. Take a section ω=

N  j=0

 j      [α P] − D(1) ⊗ Ω k (1) 2 log D(1) , ω j x− p τ f · υ ∈ G N M X /C λ,τ

  [α P] (−D(1) )⊗Ω k (1) 2 (log D(1) ). If dω∈G N M k+1 (log D(1) ) , where ω j ∈Ω X /C X (1) /C2 then we have

λ,τ

λ,τ

 N      [α P] − D(1) ⊗ Ω k+1 log D(1) . λ−1 τ d f ∧ ω N τ f x − p υ ∈ G N M X (1) /C2 λ,τ

T. Mochizuki

k+1 As in Lemma 17, we obtain d f ∧ ω N ∈ Ω (log D(1) ), i.e., ω N is a section X (1) /C2 λ,τ

k . of Ω f,λ,τ

Lemma 44. We have an expression ω N = (d f / f ) ∧ κ1 + f −1 κ2 , where κ1 and κ2 k−1 k (1) 2 (log D(1) ), respectively. are sections of Ω (log D(1) ) and Ω   X (1) /C2 X /C λ,τ

f −1 κ

λ,τ

(τ f ) N

(τ f ) N −1 .

= τ κ2 Hence, we have If N ≥ 1, we have 2     [α P] (−D(1) ) ⊗ Ω k (1) 2 (log D(1) ) . ω − d κ1 (τ f ) N −1 x − p υ ∈ G N −1 M X /C λ,τ

We also have f −1 κ1 (τ f ) N x − p υ ∈ G N −1 .  [α P] (−D(1) ) ⊗ Ω k (1) 2 (log D(1) ) such Let ω be a local section of G N M X /Cλ,τ   [α P] (−D(1) ) ⊗ Ω k (1) 2 (log D(1) ) . Then, that dω is a local section of G −1 M X /C λ,τ

by applying the previous argument successively,   we can find a local section τ of (1) ) such that ω − dτ is a local section [α P] (−D(1) ) ⊗ Ω k−1 (log D G N −1 M 2 X (1) /Cλ,τ   k (1) (1) ) .  of G −1 M[α P] (−D ) ⊗ ΩX (1) /C2 (log D λ,τ   [α P] (−D(1) ) ⊗ Ω k (1) 2 (log D(1) ) satis– If a local section ω of G N M X /Cλ,τ  [α P] (−D(1) ) ⊗ fies dω = 0, we can find a local section τ of G N −1 M   [α P] (−D(1) )⊗ k−1 Ω (log D(1) ) such that ω−dτ is a local section of G −1 M X (1) /C2λ,τ  k (1) 2 (log D(1) ) . Ω X /Cλ,τ   [α P] (−D(1) ) ⊗ Ω k (1) 2 (log D(1) ) such – Let ω be a local section of G −1 M X /Cλ,τ  [α P] (−D(1) ) ⊗ that dω = 0. Suppose that we have a local section τ of G N M  k−1 (log D(1) ) such that ω = dτ . Then, we can find a local section σ Ω X (1) /C2λ,τ   [α P] (−D(1) ) ⊗ Ω k−2 of G N −1 M (log D(1) ) such that τ − dσ is a local X (1) /C2λ,τ   [α P] (−D(1) ) ⊗ Ω k−1 section of G −1 M (log D(1) ) . We have ω = d(τ − X (1) /C2 λ,τ

dσ ). Then, we obtain the claim of Proposition 5. The proof of Theorem 5 is also completed.  

Acknowledgements The author thanks the referee for valuable suggestions to improve this paper. This note is written to understand the intriguing work of H. Esnault, C. Sabbah, M. Saito and J.-D. Yu on the Kontsevich complexes [1,10,13]. I thank Sabbah for sending an earlier version of [10] and for discussions on many occasions. I also thank him for clarifying the statement of Theorem 4. I am grateful to Esnault for some discussions and for her kindness. I thank M.-H. Saito for his kindness and support. I thank A. Ishii and Y. Tsuchimoto for their constant encouragement. This study was partially supported by the Grant-in-Aid for Scientific Research (C) (No. 22540078), the Grant-in-Aid for Scientific Research (C) (No. 15K04843), the Grant-in-Aid for Scientific Research (A) (No. 22244003), the Grant-in-Aid for Scientific Research (S) (No. 24224001), the Grant-in-Aid for Scientific Research (S) (No. 17H06127) and the Grant-in-Aid for Scientific Research (S) (No. 16H06335), Japan Society for the Promotion of Science.

A twistor approach to the Kontsevich complexes

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