Strictness for exponentially twisted regular holonomic D-modules 13

the projection and let t be the coordinate on the affine line C = P¹r{∞}. Recall that, for (M, F•M) underlying a mixed Hodge module on X × P¹, we have constructed in [ESY15, §3.1] a filtration F_•^Del(M ⊗ E^t) indexed by Q (see Definition 1.2 for the corresponding Rees construction).

Theorem 4.1. For (M, F•M) underlying a mixed Hodge module, the complex p+RF_•^Del(M ⊗ E^t) is strict and has nonzero cohomology in degree zero at most.

In the case where X is a point, this is the statement of [Sab10, Th. 6.1]. If (M, F•M) = if,+(N, F•N) for some morphism f : X → P¹ and some (N, F•N) un-derlying a mixed Hodge module on X, one can adapt the proof given in [ESY15, Prop. 1.6.9] for N = OX(∗D), where D is a normal crossing divisor, and f⁻¹(∞) ⊂ D, but this case is not enough for our purposes. The proof that we give below uses the full strength of the theory of mixed twistor D-modules of T. Mochizuki [Moc11b].

Proof of Theorem 4.1. We first note that the second assertion in the theorem (i.e., the vanishing of H^j for j 6= 0) follows from the strictness assertion together with Lemma 2.5. So let us consider the strictness assertion.

We refer to [ESY15, §§2 & 3] for the notation and results we use here. Given the filtered DX×P¹-module (M, F•M) underlying a mixed Hodge module, we associate to it the Rees module M := RFM=L

pFpM· z^p, which is a graded RFD_X×P1-module.

Its analytification M^an (with respect to the z-variable) is part of the data defining an integrable mixed twistor DX×P¹-module, according to [Moc11b, Prop. 12.5.4].

Let us consider the graded RFD_X×P1-module R_F^Del(M ⊗ E^t). Our aim is to prove the strictness (i.e., the absence of z-torsion) of the push-forward modules H^jp+RF^Del(M ⊗ E^t). Forgetting the grading, RF^Del(M ⊗ E^t) can be obtained by us-ing an explicit expression of the V -filtration as in [ESY15, Prop. 3.1.2]. It is enough to check the strictness property on the corresponding analytic object, by flatness.

Now, the analytification RF^Del(M ⊗ E^t)
^an

can be obtained by using the analytic V -filtration, by making analytic the formula of [ESY15, Prop. 3.1.2]. We then use that the V -filtration behaves well by push-forward for mixed twistor D-modules, ac-cording to results of [Moc11b]. This is the main argument for proving Theorem 4.1.

Let us denote by fM the (stupidly) localized module M (∗∞) and by^FM the (not graded) (RFD_X×P1)[τ ]hðτi-module Mf[τ ] ⊗ E^{tτ /z}. By Proposition 3.4, this is also M[τ ] ⊗ E^{tτ /z}. Similarly, (^FM)^an denotes its analytification with respect to both τ and z. We can use Proposition 3.3 together with [Moc11b, Prop. 11.3.4] to ensure that (^FM)^anunderlies an integrable mixed twistor D-module.

Let p : X × P¹× Cτ → X × Cτ denote the projection. Then p+FM^an is strict, each H^jp+FM^an is strictly specializable along τ = 0 and the V^τ-filtration satisfies V_•^τH^jp+FM^an = H^jp+(V_•^τFM^an). Indeed, these properties are satisfied according to the main results of [Moc11b].

We will now adapt the proof given in [ESY15, §3.2], which needs a supplementary argument, since we cannot argue with (3.2.2) in loc. cit.

According to [ESY15, Prop. 3.1.2], we have a long exact sequence

· · · −→ H^jp+V_α^τFM^an τ − z

−−−−−→ H^jp+V_α^τFM^an−→ H^jp+(RF^Del(M ⊗ E^t))^an−→ · · · that we can thus rewrite as

(4.2) · · · −→ V_α^τH ^jp+FM^an τ − z

−−−−−→ V_α^τH^jp+FM^an

−→ H^jp+(R_F^Del(M ⊗ E^t))^an−→ · · · Let us first check that τ − z is injective on each H^jp+FM^an. In the case considered in [ESY15, §3.2], we could use (3.2.2) of loc. cit., and when X is reduced to a point, the argument in [Sab10] uses the solution to a Birkhoff problem given by M. Saito.

We do not know how to extend the argument of [Sab10] to the case dim X > 1.

Lemma 4.3. Let Y be a complex manifold, N ^an an RY-module which underlies a mixed twistor D-module in the sense of [Moc11b]. Let h be a holomorphic function on Y . Then the action of h − z is injective on N ^an.

Proof. Since a mixed twistor D-module is in particular an object of the category MTW(Y ) (see [Moc11b, §§7.1.1 & 7.2.1]), a simple extension argument with respect to the weight filtration allows us to reduce to the case where N^an underlies a pure wild twistor D-module (as defined in [Moc11a]). Since the question is local on Y , we fix some yo∈ Y and work locally near yo.

Assume first that N^an underlies a smooth pure twistor D-module. Then it is a locally free OY-module with z-connection, and the injectivity of h − z is clear.

In general, we know that N ^an has a decomposition by the strict support (see [Sab05, §3.5], [Moc11a, §22.3.4], [Sab09, §1.4]) and we can therefore assume that, near yo, N^an has strict support a germ of an irreducible closed analytic subset Z ⊂ Y at yo. On a dense open set Z^o of the smooth part of Z, due by Kashiwara’s equivalence for pure twistor D-modules (see loc. cit.), we are reduced to the smooth case considered above and the injectivity holds. Therefore, ker[(h− z) : N^an→ N^an] is supported on a proper closed analytic subset Z^′ of Z in the neighbourhood of yo. Let F•N^an be a good filtration of N^anas an (RFD_Y)^an-module (which exists since we work locally on Y ). Then for each k, ker[(h − z) : FkN^an → FkN ^an] is a coherent OY ×Cz-submodule of N^an supported on Z^′. The (RFD_Y)^an-submodule that it generates is a coherent (RFD_Y)^an-submodule of N^an supported on Z^′. It is therefore zero since N^an has strict support equal to Z. Since this holds for any k, we conclude that ker[(h − z) : N^an→ N^an] = 0.

Since H^jp+FM^an underlies a mixed twistor D-module, we infer from Lemma 4.3 that τ − z is injective on each H^jp+FM^an. We conclude that the long exact sequence (4.2) splits into short exact sequences and therefore H^jp+(RF^DelM)^an is identified with V_α^τH^jp+FM^an/(τ − z)V_α^τH^jp+FM^an for each j. Proving that the later module is strict is a local question, near points with coordinates (τ, z) in the neighbourhood of (τo, zo) with τo= zo.

(1) If τo = zo = 0, we use that V_α^τH^jp+FM^an/τ V_α^τH^jp+FM^an is strict, due to the strict specializability of H^jp+FM^analong τ = 0 and it is enough to prove that z is injective on V_α^τH^jp+FM^an/(τ − z)V_α^τH^jp+FM^an. Due to the strictness above, if a local section m of V_α^τH^jp+FM^an satisfies zm = τ m^′ for some local section m^′ of V_α^τH^jp+FM^an, then there exists a local section m^′′ of V_α^τH^jp+FM^an such that m = τ m^′′, and since τ is injective on V_α^τH^jp+FM^anfor α ∈ [0, 1), we have m^′= zm^′′. As a consequence, if a local section m of V_α^τH^jp+FM^an satisfies zm = (τ − z)m1

for some local section m1 of V_α^τH^jp+FM^an, then there exists a local section m^′′ of V_α^τH^jp+FM^ansuch that m+ m1= τ m^′′and m1= zm^′′, hence m = (τ − z)m^′′, which gives the desired injectivity.

(2) We now assume that τo= zo6= 0. Near such a point, we have V_α^τH^jp+FM^an= H^jp+FM^an. Let us remark, however, that the V -filtration of^FM^analong τ − τo= 0 satisfies V_k^{(τ −τo)F}M^an=^FM^anfor k > 0 and V_k^{(τ −τo)F}M^an= (τ −τo)^−kFM^anfor k 6 0, according to [Sab06b, Prop. 4.1(iii)]. Applying the push-forward argument as above, we conclude that the V -filtration of H^jp+FM^analong τ − τo = 0 satisfies the same property. Therefore, setting τ^′ = τ − τo and z^′ = z − zo, we are reduced to proving

the injectivity of z^′on V_α^τ′H^jp+FM^an/(τ^′− z^′)V_α^τ′H^jp+FM^an. We can then use the same argument as we used for the case τo= 0.

5. The irregular Hodge filtration

In this section, we come back to the setup of Theorem 1.3. Let f be a meromorphic function on X with pole divisor P and let (N, F•N) be a filtered DX-module underlying a mixed Hodge module such that N = N(∗Pred). Let (M, F•M) be the mixed Hodge module on X ×P¹associated to (N, F_•N) by the construction of Remark 2.3. We know by Theorem 4.1 that the complex pX,+R_F^Del

• (M ⊗ E^t) is strict and has cohomology in degree zero at most, hence H⁰pX,+RF_•^Del(M ⊗ E^t) is equal to the Rees module of N⊗ E^f with respect to some good filtration, which we precisely define as F_•^irr(N ⊗ E^f).

Definition 5.1. The filtration F_•^irr(N ⊗ E^f) is the filtration obtained by push-forward from F_•^Del(M ⊗ E^t).

5.a. Proof of Theorem 1.3

(1) This is clear since it already holds for F_•^Del(M ⊗ E^t).

(2) Because the category of mixed Hodge modules is abelian, we have an exact sequence of filtered D-modules underlying mixed Hodge modules:

0 −→ (N0, F•N₀) −→ (N1, F•N₁) ϕ

−−→ (N2, F•N₂) −→ (N3, F•N₃) −→ 0 which gives rise to an exact sequence of filtered D-modules underlying mixed Hodge modules:

0 −→ (M0, F•M₀) −→ (M1, F•M₁) ϕ

−−→ (M2, F•M₂) −→ (M3, F•M₃) −→ 0 and therefore, according to [ESY15, Th. 3.0.1(2)], to an exact sequence of filtered D-modules:

0 → (M0⊗ E^t, F_•^Del) → (M1⊗ E^t, F_•^Del) → (M2⊗ E^t, F_•^Del) → (M3⊗ E^t, F_•^Del) → 0.

Applying H⁰p+ we keep an exact sequence, according to the second statement in Theorem 4.1.

(3) We consider the following diagram:

X × P¹ pX



π × IdP¹

//Y × P¹ pY

X π

//Y We thus have

π+RF_•^irr(N ⊗ E^f) ≃ (π ◦ pX)+RF_•^Del(M ⊗ E^t) ≃ (pY ◦ (π × Id))+RF_•^Del(M ⊗ E^t).

On the other hand, according to [ESY15, Prop. 3.2.3], (π × Id)+RF_•^Del(M ⊗ E^t) is strict and for each j,

H^j(π × Id)+RF_•^Del(M ⊗ E^t) ≃ RF_•^Del H^j(π × Id)+M

⊗ E^t. Applying now Theorem 4.1 to H^j(π × Id)+(M, F•M) we obtain the assertion.

(4) This point is similar to [ESY15, Prop. 3.2.3].

(5) The case 1.1(a) follows from [ESY15, Prop. 1.6.12]. Let us show the case 1.1(b). If i : P¹_t ֒→ P¹_t × P¹_s denotes the diagonal inclusion t 7→ (t, t) and p : P¹_t× P¹_s→ P¹_t denotes the projection (and similarly after taking the product with X), we have an isomorphism

M⊗ E^t≃ H⁰p+(i+(M ⊗ E^t)) ≃ H⁰p+ (i+M) ⊗ E^s
. We claim that, for each α ∈ [0, 1),

(5.2) F_α+^irr_•(M ⊗ E^t) = F_α+^Del_•(M ⊗ E^t).

It is enough to check

(5.3) i+ M⊗ E^t, F_α+^Del_•(M ⊗ E^t)

= (i+M) ⊗ E^s, F_α+^Del_•((i+M) ⊗ E^s)
,

and the question is non obvious in the charts t^′ = 1/t and s^′ = 1/s. Let us set δ = δ(s^′− t^′). Let us first recall that, by definition,

i+M^′ = L

k>0

M^′⊗ ∂_s^k′δ, Fp(i+M^′) = L

k>0

Fp−k−1M^′⊗ ∂^k_s′δ,

(the shift by one comes from the left-to-right transformation on RFD-modules) and, concerning the V -filtration, one checks that

V_α^s′(i+M^′) = L

k>0

V_α−k^t′ M^′⊗ ∂_s^k′δ =X

k>0

∂_s^k′s^′k(V_α^t′M^′⊗ δ).

On the other hand, we have F_α+p^Del (i+M^′) ⊗ E^1/s′

= s^′−1 Fp(i+M^′) ∩ V_α^s′(i+M^′)

⊗ e^1/s′

+ ∂s^′F_α+p−1^Del (i+M^′) ⊗ E^1/s′
, i+ F_α+^Del_•(M^′⊗ E^1/t′)

p= F_α+p−1^Del (M^′⊗ E^1/t′) ⊗ δ + ∂s^′

i+ F_α+^Del_•(M^′⊗ E^1/t′)

p−1

.

We will prove (5.3) by induction on p. Let pobe such that Fpo−2M^′= 0. We have F_α+p^Del_o (i+M^′) ⊗ E^1/s′

= s^′−1 Fpo(i+M^′) ∩ V_α^s′(i+M^′)

⊗ e^1/s′

= s^′−1 Fpo−1M^′∩ V_α^t′M^′

⊗ (δ ⊗ e^1/s′)

= t^′−1(Fpo−1M^′∩ V_α^t′M^′) ⊗ e^1/t′

⊗ δ

= i+ F_α+^Del_•(M^′⊗ E^1/t′)

po. Let us first show by induction on p that

i+ F_α+^Del_•(M^′⊗ E^1/t′)

p⊂ F_α+p^Del (i+M^′) ⊗ E^1/s′
. It is thus enough to check that

F_α+p−1^Del (M^′⊗ E^1/t′) ⊗ δ ⊂ F_α+p^Del (i+M^′) ⊗ E^1/s′
. We have

F_α+p−1^Del (M^′⊗ E^1/t′) = t^′−1(Fp−1M^′∩ V_α^t′M^′) ⊗ e^1/t′+ ∂t^′ F_α+p−2^Del (M^′⊗ E^1/t′)
.

Then on the one hand, by induction,

Let us now prove by induction on p the reverse inclusion F_α+p^Del (i+M^′) ⊗ E^1/s′

is an F -filtration, it is enough to check that

t^′−1∂_t^j−i′ (Fp−j−1M^′∩ Vα−jM^′)

⊗ e^1/t′⊗ δ ⊂ i+ F_α+^Del_•(M^′⊗ E^1/t′)

p−i. Considering the term of degree zero with respect to ∂_s^•′δ in the right-hand side, it is thus enough to check that

5.b. The irregular Hodge filtration in terms of V_α^τ. With the notation as in the beginning of this section, we consider the pull-back module N[τ ] by the projec-tion r : X × Cτ → X and the corresponding Rees object RFN[τ ] =: N [τ ], where N := RFN. Denote by N^an the analytification of N and by r⁺N ^anthat of N [τ ].

We twist r⁺N ^an by E^{τ f /z} to obtain the object ^FfN , which underlies an object of MTM^int(X×Cτ), according to Propositions 3.3 and 3.4, and to [Moc11b, Prop. 11.3.4

& 12.5.4].

Proposition 5.5. We have RF_α^irr(N ⊗ E^f)^an= V_α^τ(^FfN )/(τ − z)V_α^τ(^FfN ).

Proof. We associate to the mixed Hodge module (N, F_•N) the mixed Hodge module (M, F•M) as in Remark 2.3 (from which we keep the notation). It follows from (5.2) and [ESY15, Prop. 3.1.2] that the result holds for (M, F•M) on X × P¹ and for f equal to the projection to P¹ (note that it holds without taking “an”). Applying the same argument as in the proof of Theorem 4.1, we conclude that the operation V_α^τ/(τ − z)V_α^τ commutes with H⁰p+. On the other hand, by definition and according to (5.2), the operation RF_α^irr is compatible with H⁰p+. Therefore, the result holds for (N, F•N).

Remark 5.6. As a consequence, one can recover the graded module RF_α^irr(N ⊗ E^f) from V_α^τ(^FfN ) in the following way. As an OX[z]-module, we have an inclusion RF_α^irr(N ⊗ E^f) ⊂ N[z, z⁻¹] and RF_α^irr(N ⊗ E^f) is obtained from RF_α^irr(N ⊗ E^f)^an as the graded module with respect to the filtration on RF_α^irr(N ⊗ E^f)^an induced by the z-adic filtration of O^X ⊗O_X[z]N[z, z⁻¹]. By the proposition above, it is thus enough to identify the inclusion as OX-modules

(5.6 ∗) V_α^τ(^FfN )/(τ − z)V_α^τ(^FfN) ⊂ OX ⊗O_X[z]N[z, z⁻¹].

By using the strict specializability of ^FfN along τ = 0, one checks as in [ESY15, Proof of Prop. 3.1.2] that (τ − z)V_α^τ(^FfN ) = V_α^τ(^FfN ) ∩ (τ − z)^FfN , so that

V_α^τ(^FfN )/(τ − z)V_α^τ(^FfN ) ⊂^FfN /(τ − z)^FfN . Let us set (recall that N = N(∗Pred))