The case (IV-2) Let us consider the case?=∗. By considering 0−→∂Rn≥0, we obtain Extj CH∗,CH×∂Rn≥0∗
P '
0 (j≤n−2) C (j=n−1)
Let us consider the case?=!. We have an exact sequence 0−→CH!−→CH∗−→C∂H∗−→0. Let us look at Extj C∂H∗,CH×∂Rn≥0
. For 0−→[0,1[×Rn−1, we haveExtj C0,C[0,1[×Rn−1
= 0 for anyj. Hence, we obtain Extj CH!,CH×∂Rn≥0
=
0 (j≤n−2) C (j=n−1) The case (IV-3) It is easy to showExtj CH!,CK×∂Rn≥0
= 0 for anyj. By using the argument in (IV-2), we can showExtj CH∗,CK×∂Rn
= 0 for anyj. Thus, the proof of Theorem 5.15 is finished.
5.5.6 Some uniqueness of K-structure
We use the notation in Subsection 5.5.1. Let V be a good meromorphic flat bundle on (X, D). Let g be a holomorphic function on X such that g−1(0) = D, and let ig be the graph X −→ X ×C. We regard DRnil
X×eC(ig†V) as a cohomologically constructible sheaf onXe. LetKbe a subfield ofC. AK-structure of DRnil
X×eC ig†V
is defined to be aK-cohomologically constructible complexF onXe with an isomorphismα:F ⊗C'DRnil
X×eC(ig†V) in the derived category. Two K-structures (Fi, αi) (i= 1,2) are called equivalent, if there exists an isomorphism β :F1 −→ F2 for which the following diagram is commutative:
F1⊗C −−−−→β⊗C F2⊗C
α1
y α2
y DRnilX×e
C(ig†V) −−−−→= DRnilX×e
C(ig†V) Lemma 5.19 Let (Fi, αi) (i = 1,2) be K-structures of DRnil
X×eC ig†V
. If their restriction to π1−1(X−D[2]) are equivalent, then they are equivalent onX.e
Proof We putFiC:=Fi⊗C. We have the following commutative diagram:
Hom(F1,F2)⊗C −−−−→ Hom F1|π−1
1 (X−D[2]),F2|π−1
1 (X−D[2])
⊗C
y'
y' Hom F1C,F2C
−−−−→ Hom FC
1|π−11 (X−D[2]),FC
2|π−11 (X−D[2])
According to Theorem 5.15, the horizontal arrows are injective. Hence, Hom(F1,F2) is the intersection of Hom F1|π−1
1 (X−D[2]),F2|π−1
1 (X−D[2])
and Hom F1C,F2C
in Hom FC
1|π−11 (X−D[2]),FC
2|π1−1(X−D[2])
. Then, the el-ement of Hom(F1C,F2C) corresponding to the identity of DRnil
X×eC(ig†V) comes from Hom(F1,F2).
6 Good pre-K-holonomic D-modules
LetD =D1∪D2 be a decomposition. Recall that the complex DR<D1≤D2
X(D)e M
is quasi-isomorphic to its 0-th cohomology sheafL<D1≤D2. (See Subsection 5.1.2.) It is naturally equipped with aK-structureL<DK 1≤D2, ifV has a goodK-structure. By Corollary 3.16 and (45), we obtain a pre-K-Betti structure
FV<D1≤D2 :=Rπ∗L<DK 1≤D2 of the holonomicD-moduleV !D1
. These pre-K-Betti structures are called canonical. LetD10 ∪D02=D be another decomposition such thatD1⊂D01. The natural morphismV !D10
−→V !D1
is compatible with the pre-K-Betti structures. We also use the symbols FV∗ andFV! to denoteFV≤D andFV<D, respectively. We also use the symbolFV to denoteFV∗ for simplicity.
More generally, letι :Z ⊂X be a complex submanifold with a normal crossing hypersurfaceDZ. LetVZ
be a meromorphic flat bundle on (Z, DZ). We say ι†VZ has a goodK-structure if VZ has a good K-structure in the above sense. The canonical pre-K-Betti structures forι†VZ(!DZ,1) are also defined in a similar way for a decompositionDZ =DZ,1∪DZ,2.
6.1.2 Induced pre K-Betti structures on the nearby cycle functor and the maximal functor We setX := ∆n and D :=S`
i=1{zi = 0}. LetV be a good meromorphic flat bundle on (X, D) with a good K-structure. For eachI⊂`, we setI!i:=I∪ {i} andI∗i:=I\ {i}. TheD-module
Πa,bi? V(!D(I))
=
V ⊗Ia,bz
i
(!D(I?i)) has the canonical pre-K-Betti structure, where ? = ∗,!. Hence, ψi V(!D(I))
and Ξi V(!D(I))
have the induced pre-K-Betti structures.
Lemma 6.2 The induced pre-K-Betti structure of ψi(V)|Di\∂Di is good, i.e., it is compatible with the Stokes filtrations. Moreover, the induced pre-K-Betti structure of ψi V(!D(I))
is canonical for eachI⊂`.
Proof We have only to consider the casei= 1. We give a preparation. By Lemma 3.17, we have the following commutative diagram:
DRX
Π−∞,01! V(!D(I))
−−−−→ DRX
Π−∞,01∗ V(!D(I))
'
x
'
x
DR<D(IX ∗1) Π−∞,01! V
−−−−→ DR<D(IX ∗1) Π−∞,01∗ V
'
x
'
x
DR<D(IX !1) V ⊗I−∞,0z1
−−−−→ DR<D(IX ∗1) V ⊗I−∞,0z1
(70)
By the upper square, the inducedK-structure of DRXψ1 V(!D(I))
can be identified with theK-structure of the following:
DR<D(IX ∗1)ψ1(V)'Cone
DR<D(IX ∗1) Π−∞,01! V
−→DR<D(IX ∗1) Π−∞,01∗ V
(71) We setD0 :=S`
i=2Di. Letπ1:X(De 0)−→X be the real blow up. We obtain (71) as the push-forward of the following onX(De 0):
DR<D(I∗1)≤D(`−I!1)
X(De 0) ψ1(V)'Cone
DR<D(I∗1)≤D(`−I!1)
X(De 0) Π−∞,01! V
−→DR<D(I∗1)≤D(`−I!1)
X(De 0) Π−∞,01∗ V (72) We prepare some commutative diagram in a general setting. For a holonomicDX-moduleM, we put
DR<D(I!1)≤D(`−I!1)
X(De 0) M:= Ω<D(I!1)≤D(`−I!1)
X(De 0) ⊗π−1
1 OXπ1−1M[dimX]
DR<D(I∗1)≤D(`−I∗1)
X(De 0) M:= Ω<D(I∗1)≤D(`−I!1)
X(De 0) (∗D1)⊗π−1
1 OXπ−11 M[dimX]
We have the following commutative diagram obtained from a commutative diagram similar to (14):
DR<D(I∗1)≤D(`−I!1)
X(De 0) M(!D1) −−−−→ DR<D(I∗1)≤D(`−I!1)
X(De 0) M(∗D1) x
=
x
DR<D(I!1)≤D(`−I!1)
X(De 0) M −−−−→ DR<D(I∗1)≤D(`−I∗1)
X(De 0) M
IfMis a good meromorphic flat bundle, the left vertical arrow is also quasi-isomorphism, which can be shown by the argument in the proof of Proposition 3.15.
Letρ:X(D)e −→X(De 0) be the induced map. We have the following natural commutative diagram, where the vertical arrows are quasi-isomorphisms by Theorem 4.9:
DR<D(I!1)≤D(`−I!1)
X(De 0) M −−−−→ DR<D(I∗1)≤D(`−I∗1)
X(De 0) M
'
y '
y ρ∗DR<D(I!1)≤D(`−I!1)
X(D)e M −−−−→ ρ∗DR<D(I∗1)≤D(`−I∗1)
X(D)e M
Thus, we obtain the following commutative diagram, in which the vertical arrows are quasi-isomorphisms:
DR<D(I∗1)≤D(`−I!1)
X(De 0) Π−∞,01! V
−−−−→ DR<D(I∗1)≤D(`−I!1)
X(De 0) Π−∞,01∗ V
'
x
'
x
ρ∗DR<D(I!1)≤D(`−I!1)
X(D)e V ⊗I−∞,0z1
−−−−→ ρ∗DR<D(I∗1)≤D(`−I∗1)
X(D)e V ⊗I−∞,0z1
(73)
Because DR<D(I!1)≤D(`−I!1)
X(D)e V ⊗I−∞,0z1
and DR<D(I∗1)≤D(`−I∗1)
X(D)e V ⊗I−∞,0z1
are equipped withK-structures compatible with the morphism, we obtain a K-structure of DR<D(I∗1)≤D(`−I!1)
X(De 0) ψ1(V) from (72) and (73).
Moreover, the lower square in (70) is obtained as the push-forward of (73). Hence, the K-structure of DRXψ1 V(!D(I))
is obtained as the push-forward of theK-structure of DR<D(I∗1)≤D(`−I!1)
X(De 0) ψ1(V).
Let us consider the case I =∅. By the above consideration, we obtain that F≤0P is compatible with the K-structure, whereFP denotes the Stokes filtration of ψ1(V) at each point P ∈ π−11 (∂D1). By considering the tensor product with meromorphic flat bundles with rank one, we can deduce that FP is defined over K.
Since the pre-K-Betti structure of ψ1 V(!D(I))
comes from theK-structure of DR<D(I∗1)≤D(`−I!1)
X(De 0) ψ1(V), it is canonical.
6.1.3 Good holonomic D-module with good K-structure LetMbe a good holonomicD-module on (X, D).
Definition 6.3 We say that M has a good K-structure, if (i) φI(M) ∗D(Ic)
has a good K-structure, (put φ∅(M) :=M), (ii) the induced morphisms
ψiφI(M)(∗D(Ic))−→φiφI(M)(∗D(Ic))−→ψiφI(M)(∗D(Ic)) (i6∈I) are compatible with theK-structures.
A morphism of good holonomicD-module with goodK-structuresf :M1−→ M2is defined to be a morphism of D-modules such that φI(f) are compatible with K-structures for anyI ⊂`. Let Holgood(X, D, K) denote the category of good holonomicD-modules with good K-structures on (X, D). It is an abelian category.
Let V be a good meromorphic flat bundle on X with a good K-structure. Then, we naturally have φJ V !D(I)
∗D(Jc)
' ψJ(V) for any I, J ⊂ `, which are equipped with good K-structures. Via these K-structures, we regardV(!D(I))∈Holgood(X, D, K).
Lemma 6.4 LetM ∈Holgood(X, D, K)such thatM ∗D(J)
=M. LetI⊂`such that|I|=n−dim SuppM andVI =φI(M) ∗D(Ic)
6= 0. Then, the morphismsi†VI !D(Ic)∗D(J)
−→ M −→i†VI are compatible with goodK-structures.
Proof We set M1 := i†VI !D(Ic)∗D(J)
and M2 := i†VI. Let us show that φL(M1) ∗D(Lc)
−→
φL(M) ∗D(Lc)
−→ φL(M2) ∗D(Lc)
are compatible with K-structures for any L ⊂ `. We have only to consider the caseL⊃I. In the caseL=I, it is clear. Assume that we have already known the compatibility forL. We setLj:=Lt {j} ⊂`. We obtain the following morphisms compatible withK-structures:
ψj
φL M1
(∗D(Lc))
−→ψj
φL(M)(∗D(Lc))
−→φjφL(M) ∗D(Lcj)
−→ψj
φL(M) ∗D(Lc)
−→ψj
φL(M2) ∗D(Lc) (74) It implies the compatibility forLj.
6.1.4 Pre-K-Betti structures for ΞIψJ(ι†VI)
LetKtJtI=L⊂`. LetVI be a good meromorphic flat bundle on (DI, ∂DI). Letι:DI −→X. For a map f :KtJ −→ {0,1}, we set K0(f) :=f−1(0)∩K. We put
Cf(J, K, ι†VI) :=
ι†VI⊗ O
k∈K0(f)
I−∞,1zk ⊗ O
k6∈K0(f)
I−∞,0zk
!D(f−1(0)) .
Let0denote the constant map valued in{0}. Letδi denote the map such thatδi(j) = 0 (j 6=i) andδi(i) = 1.
We can identify ΞKψJ ι†VI
as the kernel of the following morphism:
C0 J, K, ι†VI
−→ M
i∈KtJ
Cδi J, K, ι†VI
(75)
IfVI has a goodK-structure, we obtain a pre-K-Betti structure of ΞKψJ(ι†VI) by (75).
Lemma 6.5 Fori6∈L, we setKi:=K∪ {i} andJ i:=J∪ {i}. The following morphisms are compatible with the pre-K-Betti structures:
ΞKψJ i ι†VI
−→ΞKiψJ ι†VI
−→ΞKψJ i ι†VI Proof It is clear by construction.
Recall that we have the naturally induced goodK-structure onψi ι†VI
fori6∈I(Lemma 6.2).
Lemma 6.6 For i 6∈ L, the natural isomorphism ΞKψJ i ι†VI
' ΞKψJ
ψi ι†VI
is compatible with the induced K-structures.
Proof Both theK-structures are obtained as the kernel of the morphism (75) for (J i, K).
6.1.5 Functoriality
LetD1∪D2 be a decomposition ofD. LetD3 be a hypersurface ofX. Letϕ:X0 −→X be a proper birational morphism such that (i) D0 := ϕ−1(D3∪D) is normal crossing, (ii) X0−D0 ' X −(D3∪D). Let V be a good meromorphic flat bundle on (X, D) with a goodK-structure. We putV0 := ϕ∗V ⊗ OX0(∗D0). We set D10 :=ϕ−1(D1). We takeD02⊂D0 such thatD01∪D20 is a decomposition.
Proposition 6.7 V0 is equipped with an inducedK-good structure. Moreover, the natural morphisms V(!D1)−→ϕ†V0(!D01), ϕ†V0(!D20)−→V(!D2)
are compatible with the canonical pre-K-Betti structures.
Proof The first claim is easy to see. Let us show the second claim. We use the notation in Subsection 5.3. Let ϕe : Xe0(D0) −→ X(D) be the induced map. By construction, it is easy to see that the morphismse DR<D1≤D2
X(D)e (V)−→ Rϕe∗DR<D
0 1≤D02
Xe0(D0) (V0) and Rϕe∗DR<D
0 2≤D10
Xe0(D0) (V0)−→DR<D2≤D1
X(D)e (V) are compatible with the inducedK-structures. Then, the second claim follows from Theorem 5.12.