Preliminary

Proof It follows from Lemma 7.36.

Lemma 7.40 LetΦ_K andΦ^†_σ(K) satisfy the above conditions. We set H :=S0− S Φ_K⊗σ^∗Φ^†_σ(K)

:Q0E_0|X ⊗σ^∗ Q<δE_0|σ(X)^†

−→ O_X Then,H =O Λ(M, )

with respect toh0 for some >0andM >0.

Proof It can be shown by the argument in the proof of Lemma 7.32.

Local C^∞-isomorphisms Let Φ^p_K (p= 0, . . . , m) be as in Lemma 7.37, and let ap (p= 0, . . . , m) be non-negativeC^∞-functions on K such thatP

ap = 1. We set Φ_K:=Pm

p=0ap·Φ^p_K. We also setG:= (Φ⁰_K)⁻¹◦Φ_K andG^p:= (Φ⁰_K)⁻¹◦Φ^p_K. By Lemma 7.39, |G^p−id|h₀ =O Λ0()

, and hence|G−id|h₀ =O Λ0()

for some >0 andM >0.

We can show the following estimate by using an argument in the proof of Lemma 7.33 with Lemma 7.38:

Φ⁻¹_K ◦ ∇_λ(λ²∂_λ)◦Φ_K− ∇_λ,0(λ²∂_λ) =O Λ(M, )

(81) Let Φ^†_σ(K)^q (q= 0,1, . . . , m⁰) be as in Lemma 7.31, and letb_q be non-negative C^∞-functions on σ(K) such that Pbq = 1. We set Φ^†_σ(K):=Pbq·Φ^†_σ(S)^q . We setH :=S Φ_K⊗σ^∗(Φ^†_σ(K))

− S0. Then, we can show the following estimate with respect toh0for some >0 andM >0, by using Lemma 7.40:

H =O Λ(M, ) , ∂_E4

0 ,P¹H =O Λ(M, )

(82) Construction We take 0< R₁< R₂<1. We setK₁:=

λ

|λ| ≤R₂ andK₂:=

λ

R₁≤ |λ| ≤R⁻¹₁ . We take a partition of unity χK₁, χK₂, χ_σ(K1)

onP¹which subordinates to{K1,K2, σ(K1)}.

We take a holomorphic isomorphism ΦK₁ : QE_0|K×X −→ QE_|K×X as in Lemma 7.37. Similarly, we take a holomorphic isomorphism Φ^†_σ(K

1):Q<δE_0|σ(K)×X^† † −→ Q<δE_|σ(K)×X^† † as in Lemma 7.37.

We can take a flat isomorphism Φ_K2 : E0,De^f0

|K2×(X−D)−→ E,De^f

|K2×(X−D). We set Φ :=χK1·ΦK1+χK2·ΦK2+χ_σ(K1)·Φ^†_σ(K

1).

By using (81) and (82), we can check that Φ satisfies the estimates in (63). Thus, the proof of Proposition 7.23 is finished.

8 An application to HS-orbit

(See also Subsection 5.1.3.) IfX is a point, it means thatH requires no ramification in the sense of [8].

By a classical theory (see also Subsection 5.1.3), we have the Stokes filtrationF^S indexed by Irr(∇),≤S

for each small sectorS of X − X⁰. We say that the real structure and the Stokes structure are compatible, if the Stokes filtrations on any small sectorsS come from a flat filtration ofH_R|S⁰ . (See [14].)

By taking Gr of (H,∇) with respect to the Stokes filtrations, we obtain a TE-structure Gra(H,∇) fora ∈ Irr(∇). As observed in [8], if the real structure and the Stokes structure are compatible, Gra(H,∇) is enriched to a TER-structure denoted by Gr_a(H, H_R⁰ ,∇). If (H, H_R⁰ ,∇) is enriched to a TERP-structure (H, H_R⁰ ,∇, P, w), Gr_a(H, H_R⁰ ,∇) is also naturally enriched to a TERP-structure denoted by Gr_a(H, H_R⁰ ,∇, P, w).

Another formulation In [8], a compatibility of real structure and Stokes structure is formulated in a slightly different way. Let us check that it is equivalent to the above. For simplicity, we consider the case in whichX is a point.

Let H be a vector bundle on Cλ with a meromorphic flat connection∇ : H −→H ⊗Ω¹_C

λ(∗0) such that H requires no ramification with the good set of irregular values Irr(∇) ⊂ λ⁻¹·C. Take θ0 ∈ R such that Re(a−b)(r·e

√−1θ0)6= 0 for any distincta,b∈Irr(∇). Take a sufficiently small >0, and let us consider the sector

S:=

r·e

√−1θ

θ₀−≤θ≤θ₀+π+

LetS denote the closure of S in the real blow upCeλ(0)−→Cλ along 0. LetZ:=S ∩π⁻¹(0). As a version of Hukuhara-Turrittin theorem, it is well known that we have auniqueflat decomposition

(H,∇)_|S= M

a∈Irr(∇)

Ha,S,∇a,S

(83)

such that the restriction of (83) toZbis the same as the pull back of the irregular decomposition ofH_|b0. Assume that the flat bundle (H,∇)_|C^∗

λ is equipped with a real structure, i.e., aC-anti-linear flat involution κ:H −→H. In other words, (H,∇, κ) is a TER-structure. In Section 8 of [8], the real structure and the Stokes structure are defined to be compatible, ifκ(H_a,S) =H_a,S for any a∈Irr(∇) and anyS as above.

If a small sectorS is contained in S, the restriction of (83) toS gives a splitting ofF^S. Hence, ifH_a,S are preserved byκ for anya, the filtration F^S is also preserved by κ. Let S1 and S2 be small sectors containing the rays{r·e

√−1θ0|r >0}and{−r·e

√−1θ0|r >0}, respectively. Then,a≤S₁bif and only ifa≥S₂b. By the parallel transform onS, the flat bundleH_|S is trivialized, and we can observe thatH_a,S =F_a^S1∩ F_a^S2. Hence, if F_a^Si (i= 1,2) are preserved byκ,H_a,S is also preserved byκ. The equivalence of two notions of compatibilities follows from these considerations.

8.1.2 Two Stokes filtrations of integrable twistor structures

Let (V,De⁴) be a variation of integrable twistor structures over P¹×X. It is obtained as the gluing of T E-structure (V0,De^f0) onX :=Cλ×X andT E-structure (Ve _∞,De^†∞^f) onX^† :=Cµ×X^†. We setX⁰:={0} ×X ⊂ X andX^†0:={0} ×X^† ⊂Cµ×X^†.

Definition 8.1 We say that(V,De⁴)is unramifiedly pseudo-good, if both(V₀,De^f0)and(V_∞,De^†∞^f)is unramifiedly pseudo-good. In that case, letIrr(eD^f0)andIrr(eD^†∞^f)denote the sets of irregular values ofDe^f0 andDe^†∞^f, respectively.

If X is a point, it is also said that (V,De⁴)requires no ramification.

Definition 8.2 Assume (V,De⁴) is unramifiedly pseudo-good.

• We say that the sets of the irregular values of (V,De⁴) are compatible, ifIrr De^f0

andIrr De^f∞

bijectively correspond bya←→γ^∗a.

• We say that(V,De⁴)has compatible Stokes structures, if the following holds:

– The sets of irregular values of(V,De⁴) are compatible.

– For a small sector S of X − X⁰, we have the Stokes filtration F^S of (V0,De^f0). We also have the Stokes filtrationF^γ(S)of(V∞,De^f∞), where we regardγ(S)as a small sector ofX^†− X^†0. Then,F^S andF^γ(S)are the same under the parallel transform along any rays connecting S andγ(S).

Remark 8.3 In the above definition, a ray means a line (t·e

√−1ϕ, P)

0< t <∞ in C^∗_λ× {P} ⊂C^∗_λ×X. We say that it connects S and γ(S), if (i) (t·e

√−1ϕ, P) is contained in S for any sufficiently small t, (ii) (t·e

√−1ϕ, P)is contained inγ(S) for any sufficiently larget.

Lemma 8.4 If (V,De⁴) is equipped with either a real structure κor a perfect pairing S of weight w, then the irregular values of De^f0 andDe^f∞ are compatible.

Proof We have Irr γ^∗De^f∞

= γ^∗a

a∈Irr(eD^f∞) . If (V,De⁴) is equipped with a real structure,γ^∗(V_∞,De^f∞)' (V₀,De^f0). Hence, the irregular values ofDe^f0 andDe^f∞are compatible.

We have Irr σ^∗De^f∞

= σ^∗a

a ∈ Irr(eD^f∞) . Note that a ∈ Irr(eD^f∞) are of the form µ⁻¹a⁰, where a⁰ are holomorphic functions on X^†. Hence, σ^∗a =−γ^∗a. If (V,De⁴_f) is equipped with a perfect pairing, (V0,De^f0) is isomorphic to the dual ofσ^∗(V∞,De^f∞). Therefore, the irregular values ofDe^f0 andDe^f∞are compatible.

If (V,De⁴) is unramifiedly pseudo-good, we obtainT E-structure Gra(V0,De^f0) onX fora∈Irr(eD^f0), andT E-e structure Grb(V∞,De^f∞) onX^† forb∈Irr(eD^f∞), by taking Gr with respect to the Stokes filtrations. If (V,De⁴) has compatible Stokes structures, we have the natural isomorphism

Gr_a(V₀,De^f0)_{|X −X}0 'Gr_γ∗a V_∞,De^f∞

|X^†−X^†0.

Hence, we obtain a variation of integrable twistor structures Gr_a(V,De⁴) for eacha ∈Irr(eD^f0) as the gluing of them. We have the following functoriality (Lemma 5.17).

Lemma 8.5 Let(V^(a),De^(a)4)be unramifiedly pseudo-good. Assume (i)(V^(a),De^(a)4) (a= 1,2)have compatible Stokes filtrations, (ii) the union I := Irr(eD⁽¹⁾0 ^f)∪Irr(eD⁽²⁾0 ^f) is good. Then, a morphism (V⁽¹⁾,De⁽¹⁾⁴) −→

(V⁽²⁾,De⁽²⁾⁴)inducesGra V⁽¹⁾,De⁽¹⁾⁴

−→Gra V⁽²⁾,De⁽²⁾⁴

for eacha∈ I.

We have the natural isomorphisms

γ^∗Gra(V,De⁴)'Gra γ^∗(V,De⁴)

, σ^∗Gra(V,De⁴)'Gr−a σ^∗(V,De⁴) . The following lemma follows from functoriality.

Lemma 8.6 Assume (V,De⁴) has compatible Stokes structures. If (V,De⁴) is equipped with a real structure, (resp. a perfect pairing of weightw), each Gr_a(V,De⁴)is also equipped with an induced real structure (resp. an induced perfect pairing of weightw).

Lemma 8.7 Let (H, H_R⁰ ,∇, P⁰,−w) be a variation of TERP-structures, and let (V,De⁴,S, κ,−w) be the cor-responding variation of twistor-TERP structures. (See Subsection2.1.8 for the correspondence.) Assume that (H, H_R⁰ ,∇, P⁰,−w)is unramifiedly pseudo-good, or equivalently, (V,De⁴,S, κ,−w)is unramifiedly pseudo-good.

• The real structure and the Stokes structures of(H,∇)are compatible, if and only if(V,De⁴)has compatible Stokes structures.

• If the real structures and the Stokes structures are compatible, Gra(V,De⁴,S, κ,−w) is the variation of twistor-TERP structures corresponding to Gra(H, H_R⁰ ,∇, P⁰,−w).

Proof Note that the Stokes filtrations of γ^∗(H,∇) onγ^∗(S) is given by the composite of the conjugate with respect toH_R⁰ and the parallel transport along the rays connectingS andγ(S), with the change of the index sets from Irr(∇) to

γ^∗a

a∈Irr(∇) . Then, the first claim follows.

Let us consider the second claim. We have only to consider the case w = 0. We may assume that (H, H_R⁰ ,∇, P⁰) is obtained from (V,∇,S, κ) by the procedure explained in Subsection 2.1.8. By construc-tion, we have Gr_a(H,∇) = Gra(V₀,∇0). For comparison of induced real structures and pairings, we have only to consider the case in whichX is a point.

Let us compare the induced real structures. The flat real structure ofH⁰ is obtained as the composite:

H_|λ parallel transform

−−−−−−−−−−−→ H

|λ⁻¹ κ|λ

−−−−→ H_|λ

Hence, we have the following factorization of the real structure on Gr_a(H)_|λ obtained as Gr of the Stokes filtration:

Gr_a(H)_|λ parallel transform

−−−−−−−−−−−→ Gr_a(H)

|λ⁻¹

Gr_a(κ)_|λ

−−−−−−→ Gr_a(H)_|λ It is the same as the real structure induced by Gra(κ) on Gra(V,∇).

LetP :H⊗j^∗H −→ OC_λ be the pairing induced by κandS as in (6), whose restriction to H⁰ isP⁰. Let S be a small sector inC^∗_λ. We have the following factorization ofP_|S:

F_a^S(H)⊗j^∗F_b^j(S)(H) =F_a^S(V₀)⊗σ^∗γ^∗F_b^j(S)(V₀) ^1⊗σ

∗κ

−−−−→ F_a^S(V₀)⊗σ^∗F^σ(S)

γ^∗(b)(V_∞) −−−−→ O^S _S

The restriction toF_a^S(H)⊗j^∗F_b^j(S)(H) is 0 unlessa−b≥_S 0. The induced pairingP_a for Gr_a(V₀) is factorized as follows:

Gr_a(V₀)_|S⊗j^∗Gr_a(V₀)_|j(S) ^1⊗σ

∗Graκ

−−−−−−−→ Gr_a(V₀)_|S⊗σ^∗Gr_γ∗(a)(V_∞)_|σ(S) −−−−→ O^GraS S

Hence, it is the same as the pairing induced by Gr_a(V,∇,S, κ). Thus, the proof of Lemma 8.7 is finished.

8.1.3 Preliminary for pull back We setX :=C_z, D=

0 ,X :=C_λ×X,D:=C_λ×D andW :=D ∪({0} ×X). Letπ:Xe(W)−→ X be a real blow up ofX along W. Letπ1 :Ceλ(0)−→Cλ be the real blow up ofCλ along{0}. Letφ0:X −→Cλ

be given byφ0(λ, z) =λ·z. It induces the mapφe0:Xe(W)−→Ceλ(0).

Let H be a vector bundle on Cλ with a meromorphic flat connection∇ : H −→H ⊗Ω¹_C

λ(∗0) such that (H,∇) requires no ramification with the good set of irregular valuesI ⊂C·λ⁻¹. LetVdenote the flat bundle on Ce_λ(0) associated toH_|C^∗

λ. For eachQ∈π⁻¹₁ (0), we have the Stokes filtrationF^Q ofV_|Q for the meromorphic prolongmentH. (See Subsection 5.1.5) We can naturally regardφe^∗₀Vas the flat bundle onXe(W) associated to (φ^∗₀H)_{|X −W}.

Lemma 8.8 The following holds:

• φ^∗₀(H,∇) is unramifiedly pseudo-good in the level m = (−1,−1). (See Subsection 5.1.3.) The set of irregular values is given by φ^∗₀I:=

φ^∗₀a

a∈ I .

• For eachP ∈π⁻¹(W), the Stokes filtrationF^P ofφe^∗₀(V)_|P forφ^∗₀H is the pull back of the Stokes filtration of V_|eφ

0(P).

• We have the natural isomorphism φ^∗₀Gra(H)'Grφ^∗₀a φ^∗₀H . Proof We have the decomposition (H,∇)_|b0 = L

a∈I(H_a,∇b_a), where ∇b_a−da are regular. It induces the decomposition ofφ^∗₀(H,∇)_|cW. Hence, the first claim is clear.

We set Q:=φe₀(P). Note that the orders ≤Q and ≤P are the same under the identificationI 'φ^∗₀I. Let H₁ ⊃H be an unramifiedly good lattice. Then,φ^∗₀H₁ is an unramifiedly good lattice. We take a small sector S_Q ∈ MS(Q,C^∗_λ,I) such that there exists the Stokes filtration F^SQ ofH_1|S

Q. We take a small multi-sector SP ∈ MS(P,X −W, φ^∗₀I) such thatφ0(SP)⊂SQ. Then, we obtain the filtrationφe^∗₀F^SQ ofφ^∗₀(H1)_|S

P indexed by φ^∗₀I,≤_P

. It gives the Stokes filtration ofφ^∗₀(H₁)_|S

P, which follows from the characterization in Proposition

5.5. Since the filtration of φe^∗₀(V)_|P induced byφe^∗₀F^SQ is the same as the pull back ofF^Q on V_|Q, we obtain the second claim. Note that we also obtain that the Stokes filtration ofφ^∗₀(H)_|S

P is given by the pull back of the Stokes filtration ofH_|S

Q.

LetS_P be a small multi-sector as above. By the above compatibility of the Stokes filtrations and Lemma 5.21, we obtain the natural isomorphisms

φ^∗₀ Gra(H)

|SP 'Grφ^∗₀a(φ^∗₀H)_|S

P. (84)

By varyingSP and gluing them, we obtainφ^∗₀ Gra(H)

|eU(W)'Grφ^∗₀a(φ^∗₀H)_|eU(W), whereU is a neighbourhood of W, andUe(W) denote the real blow up ofUalongW. By using the flatness, it is extended toφ^∗₀ Gr_a(H)

|X(W)e ' Gr_φ^∗

0a(φ^∗₀H)_|

X(We ). Hence, we obtain an isomorphism onX. 8.1.4 Rescaling and HS-orbit

We recall a rescaling construction in [7] and [8]. See also [22]. We setX :=Cz, D=

0 and X^∗ :=X−D.

ForR >0, we set X(R) :=

z∈X

|z|< R and X^∗(R) :=X(R)∩X^∗. We set X :=C_λ×X. We use the symbolsX^∗, D, X(R) andX^∗(R) in similar meanings. Letφ₀ : X −→C_λ be given by φ₀(λ, z) =λ·z. The restriction toX^∗ is denoted by ψ₀.

TERP-structure We consider only TERP-structures of weight 0. Hence, we omit to specify weights.

Let (H, H_R⁰ ,∇, P) be a TERP-structure. Hertling and Sevenheck studied the variation of TERP-structures ψ₀^∗(H, H_R⁰ ,∇, P) onX^∗. If there exists an R >0 such thatψ^∗₀(H, H_R⁰ ,∇, P)_|X^∗_(R) is pure and polarized, the variation is called an HS-orbit (Hertling-Sevenheck orbit), and we say in this paper that (H, H_R⁰ ,∇, P) induces an HS-orbit.

Remark 8.9 An HS-orbit is called a “nilpotent orbit” in [8]. We use “HS-orbit” for distinction from twistor nilpotent orbit. It matches their terminology “Sabbah-orbit”.

Lemma 8.10 We assume (i) (H,∇) requires no ramification, (ii) the Stokes structure and the real structure of(H, H_R⁰ ,∇)are compatible. Then, the following holds:

• ψ^∗₀(H,∇)is unramifiedly pseudo-good. The set of irregular values is given by ψ₀^∗a

a∈Irr(∇) .

• The real structure and the Stokes structure of ψ₀^∗(H,∇)are compatible.

• We have the natural isomorphism ψ^∗₀Gra H, H_R⁰ ,∇, P

'Grψ₀^∗aψ₀^∗ H, H_R⁰ ,∇, P .

Proof The first two claims follow from Lemma 8.8. To show the third claim, we have only to compare the induced flat pairings. It can be done directly, or by considering the restriction toCλ× {1}.

Integrable twistor structure We set X^† := Cµ ×X^†, D^† := Cµ×D^†, X^{∗ †} := X^† − D^† and W^† :=

D^†∪ {0} ×X^†

. Letφ_∞:X^† −→C_µ be given byφ_∞(µ, z) =µ·z. The restriction toX^{∗ †}is denoted byψ_∞. Let (V,∇) be an integrable twistor structure on P¹ which requires no ramification. It is obtained as the gluing of (V₀,∇₀) and (V_∞,∇_∞). The gluing is denoted byg:V_0|C^∗

λ 'V_∞|C^∗

µ, which is flat with respect to∇.

We set HS(V)₀ := ψ₀^∗(V₀) and HS(V)_∞ := ψ_∞^∗ (V_∞). They are naturally equipped with T E-structure HS(∇)0 and T E-structure HS(∇)e _∞. Note that HS(V,∇)0 and HS(V,∇)_∞ are unramifiedly pseudo good. Let us construct a flat isomorphism Φ between HS(V,∇)_0|C^∗

λ×X^∗ and HS(V,∇)_∞|C^∗

µ×X^†∗. The fibers HS(V)_0|(λ,z) and HS(V)_∞|(µ,z) are naturally identified withV_0|λ·z andV_∞|µ·z, respectively. Ifλ=µ⁻¹, we have (λ·z)⁻¹= µ·z· |z|⁻². Hence, we have an isomorphism Φ_(λ,z):H(V)_0|(λ,z)'H(V)_∞|(λ−1,z)induced by the gluingg with the parallel transform along the segments connectingλ⁻¹·zandλ⁻¹·z· |z|⁻². Thus, we obtain the isomorphism Φ as desired.

Let HS(V,∇) denote the variation of integrable twistor structures obtained as the gluing of HS(V,∇)0and HS(V,∇)∞. The following lemma is clear from the construction and the functoriality (Lemma 5.17).

Lemma 8.11

• Let F : (V⁽¹⁾,∇⁽¹⁾)−→(V⁽²⁾,∇⁽²⁾)be a morphism of integrable pure twistor structures. Then, we have the induced morphisms HS(F) : HS(V⁽¹⁾,∇⁽¹⁾)−→HS(V⁽²⁾,∇⁽²⁾).

• Let f beγ or σ. Then,HS◦f^∗(V,∇) is naturally isomorphic tof^∗HS(V,∇).

By the above lemma, a real structureκof (V,∇) induces a real structure HS(κ) of HS(V,∇). Since we have the natural isomorphism HS T(0)

'T(0)X^∗, a paringS of (V,∇) with weight 0 induces a pairing HS(S) of HS(V,∇) with weight 0. Hence, an integrable twistor structure with a pairing (V,∇,S) induces HS(V,∇,S) onP¹×X^∗, and if (V,∇,S) is equipped with a real structure, HS(V,∇,S) is also equipped with a naturally induced real structure.

Lemma 8.12 Assume that(V,∇)has compatible Stokes structures. Then,HS(V,∇)also has compatible Stokes structures, and we have the natural isomorphism

HS Gra(V,∇)'Grψ₀^∗aHS(V,∇) (85)

If(V,∇)is equipped with a pairing of weight0(resp. a real structure),(85)preserves the induced pairings (resp.

real structures).

Proof It follows from Lemma 8.8.

Lemma 8.13 Let(H, H_R⁰ ,∇, P⁰)be a TERP-structure, and let(V,∇,S, κ)be the corresponding twistor-TERP structure. Then,HS(V,∇,S, κ) is the variation of twistor-TERP structure corresponding toψ₀^∗(H, H_R⁰ ,∇, P⁰).

Proof By construction, we have the natural isomorphism HS(V,∇)₀'(H,∇). We have only to compare the induced real structures and pairings on them. Since they are flat, we have only to compare them on the fiber overz= 1. Then, the claim is clear.

If there exists anR >0 such that HS(V,∇,S)_|P1×X^∗(R)is pure and polarized, it is called a twistor HS-orbit, and we say that (V,∇,S) induces a twistor HS-orbit.

ドキュメント内 R ESEARCH I NSTITUTEFOR M ATHEMATICAL S CIENCESKYOTOUNIVERSITY,Kyoto,Japan ByTakuroMOCHIZUKINovember2008 AsymptoticbehaviourofvariationofpurepolarizedTERPstructures RIMS-1647 (ページ 72-77)