The D X×C v -module E vf (∗H)

7.a. Setting. We will use the local setting and notation similar to that of [ESY15,

§1.1] that we recall now, together with the notation introduced in §1.b. In the local analytic setting, the space X^an is the product of discs ∆^ℓ× ∆^m× ∆^m′ with coor-dinates (x, y) = (x1, . . . , xℓ, y1, . . . , ym, y₁^′, . . . , y_m^′ ′) and we are given a multi-integer e= (e1, . . . , eℓ) ∈ (Z>0)^ℓ for which we set:

• f (x, y) = x^−e:=Qℓ with coefficients in O, together with its standard increasing filtration F•Dby the total order w.r.t. ∂x, ∂y, ∂y^′:

FpD= X

|α|+|β|+|γ|6p

O∂_x^α∂^β_y∂_y^γ′,

where we use the standard multi-index notation with α ∈ N^ℓ, etc. Similarly we will denote by O[t^′] the ring of polynomials in t^′ with coefficients in O and by D[t^′]h∂t^′i the corresponding ring of differential operators.

Consider the left D-modules

O(∗Pred) = O[x⁻¹], O(∗H) = O[y⁻¹], O(∗D) = O[x⁻¹, y⁻¹]

with their standard left D-module structure. They are generated respectively by 1/Qℓ

i=1xi, 1/Qm

j=1yj and 1/Qℓ

i=1xiQm

j=1yj as D-modules. We will consider on these D-modules the increasing filtration F• defined as the action of F•D on the generator: (3.12.1) p. 80], taken in an increasing way. Regarding O(∗H) as a D-submodule of O(∗D), we have FpO(∗H) = FpO(∗D) ∩ O(∗H) and similarly for O(∗Pred). On the other hand it clearly follows from the formulas above that

FpO(∗D) = X

q+q^′6p

FqO(∗H) · Fq^′O(∗Pred),

where the product is taken in O(∗D).

7.b. The V -filtration of the DX×Cv-module E= E^vf(∗H). On X × Cv we con-sider the holonomic DX×Cv-module that we denote by E^vf(∗H). It is defined by the formula

E^vf(∗H) := OX×Cv(∗(D × Cv)), d + d(vf )
.

For the sake of simplicity, we will set E = E^vf(∗H). Then E has a global section, equal to 1, that we denote by e^vf on X × Cv. Similarly, we will consider the v-algebraic version of the same object, regarded as a DX[v]h∂vi-module:

E = E^vf(∗H) := (OX(∗D)[v], d + d(vf )) = OX(∗D)[v] · e^vf.

It is standard that the DX×Cv-module E is holonomic. However, it is not of ex-ponential type as considered in [ESY15] since vf is only a rational function, but is exponentially regular according to Proposition 2.7(1), hence it enters the frame consid-ered in §2.b. It is however known to have regular singularities along v = 0 (in a sense

made precise in [Sab06b]) which has been thoroughly analyzed in [Sab97]. On the other hand, it is easy to check that F0O_X×C

v(∗H)e^vf generates E as a DX×Cv-module.

Let us recall the definition of the V -filtration (considered increasingly) along v = 0 over Cv. For each α ∈ [0, 1) and k ∈ Z, Vα+kE is a coherent DX×Cv/Cv-module (by regularity) equipped with an action of v∂v, and the minimal polynomial of v∂von Vα+kE/Vα+k−1Ehas roots contained in [−α − k, −α − k + 1). We have by definition,

There is also a notion of V -filtration for holonomic DX[v]h∂vi-modules and we have Vα+kE= OX×Cv⊗O_X[v]Vα+kE. local section of VβE has a unique decomposition

(7.2) X

Let us make explicit the action of C[v] on a section (7.2). For j > 1 we have

v^j· h^a,λ,βx^−[βe]−1x^−aP^a,λ,β(v∂v)e^vf = h^a,λ,βx^−[βe]−1x^−a+jeP^a,λ,β(v∂v− j)v^j∂_v^je^vf.

is a multiple of P^a′,0,β(s) and there is a polynomial R^a,λ,j,β(s) ∈ Q[s] such that P^a,λ,β(v∂v− j)v^j∂_v^j = R^a,λ,j,β(v∂v)P^a′,0,β(v∂v) =X

µ>0

cµP^a′,µ,β(v∂v) with cµ∈ Q. We thus obtain

(7.4) v^j· h^a,λ,βx^−[βe]−1x^−aP^a,λ,β(v∂v)e^vf

=X

µ>0

cµx^a′′h^a,λ,βx^−[βe]−1x^−a′P^a′,µ,β(v∂v)e^vf, and since I(a^′) = {i | ai− jei60} ⊃ I(a), we obtain the result in the form of (7.2).

Lemma 7.5. For any monic polynomial P (s) of degree p, there exists a monic polyno-mial Q(s) of degree p such that P (v∂v)e^vf = Q(v/x^e)e^vf in E.

Let us then denote Q^a,λ,β(s) the polynomial associated with P^a,λ,β(s) by Lemma 7.5. We thus have

(7.6) VβE =X

a>0

X

λ>0

F0O_X(∗Pred)

([βP ])(∗H)x^−aQ^a,λ,β(vf ) · e^vf,

and we note that deg Qa,λ,β = |a| + λ. Moreover, each local section has a unique decomposition

(7.7) X

a>0

X

λ>0

h^a,λ,β(xI(a), y, y⁻¹, y^′)x^−[βe]−1x^−aQ^a,λ,β(vf )e^vf.

Corollary 7.8. Let us denote by gr^[v]VβE the grading of VβE with respect to the degree in v. Then, in a neighbourhood of a point of Pred, we have

gr^[v]_p VβE ≃ FpO_X(∗Pred)

([(β + p)P ])(∗H) · v^p. 7.c. The filtrationFα+•E

Although the function vf does not extend as a map X × Cv → P¹, we can never-theless adapt in a natural way the definition given in [ESY15, (1.6.1) & (1.6.2)] for the case of the map f : X → P¹.

Definition 7.9 (The filtration). For α ∈ [0, 1) we set, over Cv, Fα+pE^vf = X

k6p

FkO_X(∗Pred)([(α + p)P ])v^k

[v] · e^vf,

Fα+pE = X

q+q^′6p

FqO_X(∗H) · Fα+q^′E^vf.

The analytification of these filtrations with respect to v are denoted Fα+pE^vf and Fα+pErespectively.

Lemma 7.10. For each α ∈ [0, 1), the filtration Fα+•E is an F•D_X[v]h∂vi-filtration which satisfies the following properties.

(1) Fα+p1E ⊂ Fβ+p2E for all p1, p2∈ Z and β ∈ [0, 1) such that α + p16β + p2. Moreover, Fα+pE = 0 for p < 0.

(2) When restricted to C^∗_v, the filtration Fα+pEis equal to F_α+p^DelE_X×C∗

v as defined in [ESY15, (1.6.2)] for the map vf : X × C^∗_v→ P¹, and for each vo∈ C^∗, we have

Fα+pE/(v − vo)Fα+pE= F_α+p^DelE^vof(∗H).

(3) The filtration Fα+•E satisfies

Fα+pE = FpD_X×C

v· FαE ; in particular, it is good with respect to F•D_X[v]h∂vi.

Proof. The exhaustivity is clear from the expression of Definition 7.9, and the first two points are straightforward. Let us check (3). It is enough to check it locally analytically on X.

Near a point of X r Pred. If H = ∅ and p > 0, we have Fα+pE = O[v]e^vf and the generation by FαE is clear.

If H = {y1· · · ym= 0} and p > 0, we have Fα+pE = X

|a|6p

y^−a−1O[v]e^vf.

Since ∂_y^a(y⁻¹O[v]e^vf) = ⋆y^−a−1O[v]e^vf mod Fα+p−1E if |a| = p, we get the genera-tion by FαE near such a point.

Near a point of Pred. From the equalities (for some nonzero constants ⋆):

∂xi x^−([αe]+1)· e^vf

= ⋆ x⁻¹_i x^−([αe]+1)· e^vf + ⋆ x⁻¹_i x−([(α+1)e]+1)v · e^vf

∂v x^−([αe]+1)e^vf

= x−([(α+1)e]+1)· e^vf, we conclude

F1D_X[v]h∂vi · FαE^vf = F0O(∗Pred) + F1O(∗Pred)v

[v] · x^−[(α+1)e]e^vf and by iterating the argument we get the generation property. The corresponding property for Fα+•E is proved similarly.

We will rely on computations made in [Sab97] and we will first express differently the filtration Fα+pE. Let us define GpE as the filtration by OX-modules (but not O_X[v]-modules), defined as

GpE = Lp k=0

(Fp−kO_X(∗H))(∗Pred)v^k· e^vf. The filtration G•E clearly satisfies

(7.11)

p < 0 =⇒ GpE = 0,

q > 0 =⇒ GpE ∩ v^qE = v^qGp−qE, p − q < 0 =⇒ GpE ∩ v^qE = 0.

For α ∈ [0, 1) and p ∈ Z, we set

(7.12) F_α+p^′ E := X

k+j6p

(GkE ∩ Vα+jE).

Then F_α+p^′ E is an OX[v]-module. Note also that F_α+^′ _•E is an F•D_X[v]h∂vi-filtration.

Indeed, GkE is stable by ∂y^′, ∂v, and we have ∂xi, ∂yjGkE ⊂ Gk+1E; moreover, Vα+jE is stable by ∂x, ∂y, ∂y^′, and we have ∂vVα+jE ⊂ Vα+j+1E.

Recall that, for j > 0, we have by definition Vα−jE = v^jVαE, so that for k > 0, (7.13) v^j : GkE ∩ VαE−→ G^∼ k+jE ∩ Vα−jE.

Therefore, we can also write

(7.14) F_α+p^′ E := C[v](GpE ∩ VαE) + Xp j=1

(Gp−jE ∩ Vα+jE).

It follows that F_α+p^′ E = 0 for p < 0 (and α ∈ [0, 1)).

Lemma 7.15. For each α ∈ [0, 1) and p ∈ Z we have F_α+p^′ E = Fα+pE.

Proof. Let us first consider an analytic neighbourhood of a point of X r Pred. Due to the relation ∂ve^vf = f e^vf, we have, near such a point, Vα+jE = v^max(−j,0)E for any j ∈ Z, and GpE =Lp

k=0FkO(∗H)v^p−ke^vf. Then, near such a point, (7.14) reads F_α+p^′ E := C[v](GpE ∩ V0E)

= C[v]GpE = FpO(∗H)[v]e^vf = Fα+pE.

We now argue locally at a point of Pred. We refine (7.2) in order to take into account the pole order along H, so a section of VβE can be written in a unique way as

(7.16) X

a>0

X

λ>0

X

c>0

h^a,c,λ,β(xI(a), yJ(c), y^′)x^−[βe]−1x^−ay^−c−1P^a,λ,β(v∂v)e^vf, with J(c) = {j | cj = 0} and ha,c,λ,β(x_I(a), y_J(c), y^′) ∈ C{x_I(a), y, y^′}. Arguing as in the proof of [Sab97, Lemma 4.11], we obtain that, for β > 0, a section (7.16) belongs to GkE∩VβE if and only if the coefficients h^a,c,λ,βare zero whenever deg P^a,λ,β+|c| =

|a| + |c| + λ is > k (note that this condition clearly defines an increasing filtration with respect to k).

We will first show that FαE = F_α^′E for α ∈ [0, 1). We have F_α^′E = C[v](G0E ∩ VαE)

FαE = F0O(∗D)([αP ])[v]e^vf. and

Here we are considering the case k = 0 and β = α. Let us consider a section (7.16) in G0E ∩ VαE. The only possible term occurs for a = 0, λ = 0 and c = 0, so G0E ∩ VαE = F0O(∗D)([αP ])e^vf. Therefore,

(7.17) FαE := F0O(∗D)([αP ])[v]e^vf = C[v](G0E ∩ VαE) = F_α^′E.

Since Fα+pE = FpD[v]h∂vi · FαE (Lemma 7.10(3)), and since F_α+p^′ E is an F•D_X[v]h∂vi-filtration, it follows that Fα+pE ⊂ F_α+p^′ E for all p.

Let us now show the reverse inclusion F_α+p^′ E ⊂ Fα+pE. Let us consider a term in GkE ∩ Vα+p−kE (0 6 k 6 p) of the form

h(xI_β(a), yJ(c), y^′)x^−[βe]−1x^−ay^−c−1Pa,λ,β(v∂v)e^vf,

with β = α + p − k, a > 0, λ + |a| + |c| 6 k. Let us rewrite P^a,λ,β(v∂v) in terms of the monomials v^j∂_v^j. For j 6 λ + |a| 6 k − |c|, the result of the action of v^j∂_v^j on h(xIβ(a), yJ(c), y^′)x^−[βe]−1x^−ay^−c−1e^vf is

h(xI_β(a), yJ(c), y^′)x−[(β+j)e]−1x^−ay^−c−1v^je^vf

= h(xIβ(a), yJ(c), y^′)x−[(α+p)e]−1x^{−a+(k−j)e}y^−c−1v^je^vf.

For a^′ ∈ Z^ℓ, let us set |a^′|+ = P

imax(0, |a^′_i|). Since |a|+ = |a| 6 k, the reverse inclusion follows from the lemma below.

Lemma 7.18. For a^′ ∈ Z^ℓ and k > 0, assume that |a^′|+ 6k. Then for j such that 0 6 j 6 k, we have |a^′− (k − j)e|+6j.

Proof. We argue by decreasing induction on j and the result is true if j = k by assumption. We are reduced to proving that, if |a^′|+>1, then |a^′− e|+6|a^′|+− 1.

There exists io such that a^′_io >1, so max(a^′_io, 0) = a^′_io >1 and max(a^′_io − e^′_io, 0) 6 a^′_io− 1. Since max(a^′_i− e^′_i, 0) 6 max(a^′_i, 0) for any i, we get |a^′− e|+6|a^′|+− 1, as wanted.

7.d. Filtration on the nearby cycles of E along v = 0. In this subsection, we analyze the filtration induced by Fα+•Eon the nearby cycles of E along v = 0. Our objective is to show that M. Saito’s criterion [Sai88, Prop. 3.3.17] applies.

Proposition 7.19.

(1) For each α ∈ [0, 1), the filtered module (E^vf(∗H), Fα+•E^vf(∗H)) is strictly specializable and regular along v = 0, in the sense of [Sai88, (3.2.1)].

(2) Let V•E^vf(∗H) be the V -filtration of E^vf(∗H) along v = 0 and, for each α ∈ [0, 1), let us set

ψv,exp(−2πiα)E^vf(∗H) := gr^V_αE^vf(∗H) = VαE^vf(∗H)/V<αE^vf(∗H).

For each α ∈ [0, 1) the jumps of the induced filtration (considered as a filtration indexed by Q)

F•ψv,exp(−2πiα)E^vf(∗H) := F•∩ VαE^vf(∗H) F•∩ V<αE^vf(∗H) occur at α + Z at most, and the filtration

FpψvE^vf(∗H) := L

α∈[0,1)

Fα+pψv,exp(−2πiα)E^vf(∗H)

is (up to a shift by dim X − 1 on ψv,6=1E^vf and by dim X on ψv,1E^vf) the Hodge filtration of a mixed Hodge module.

(3) If moreover H = ∅, this mixed Hodge module is polarized by the nilpotent part of the monodromy naturally acting on ψvE^vf, induced by the action of exp −2πiv∂v.

The latter statement means that the weight filtration of the corresponding mixed Hodge module is, up to a shift which depends on whether α = 0 or α 6= 0, the monodromy filtration of the nilpotent endomorphism induced by v∂v + α on ψv,exp(−2πiα)E^vf.

Proof of Proposition 7.19(2) and (3). It is enough to work in the algebraic setting with respect to v. Recall that we set E = E^vf(∗H) for short and that F^′ was defined by (7.12). Let β ∈ [0, 1). We claim that

(7.20) F_α+p^′ E ∩ VβE =





C[v](GpE ∩ VαE) + (Gp−1E ∩ VβE) if β > α, vC[v](GpE ∩ VαE) + (GpE ∩ VβE) if β 6 α.

This implies that and by decreasing induction one eventually finds

F_α+p^′ E ∩ Vα+1E = C[v](GpE ∩ VαE) + (Gp−1E ∩ Vα+1E).

Intersecting now with VβE gives the first line of (7.20). The second one is obtained by showing in the same way

F_α+p^′ E ∩ VαE = C[v](GpE ∩ VαE) = vC[v](GpE ∩ VαE) + (GpE ∩ VαE).

Lemma 7.15, together with (7.21) and [Sab97, Th. 4.3], proves 7.19(2) and (3).

Proof of 7.19(1). Continuing the proof of (7.20) gives, for β ∈ [0, 1) and ℓ > 1,

which is [Sai88, (3.2.1.2)] (up to changing the convention for the V -filtration), since v acts in an injective way on VβE.

We now wish to prove that Property (3.2.1.3) of [Sai88] holds, that is, for each β > 0 and for each α ∈ [0, 1) and p ∈ Z, the morphism

∂v : F_α+p^′ gr^V_β E −→ F_α+p+1^′ gr^V_β+1E

is an isomorphism. Assume first that there exists po >0 such that α + po 6 β <

α + po+ 1 (otherwise, 0 < β < α, a case which will be treated separately). Then a computation similar to that for proving (7.20) gives

F_α+p^′ E ∩ VβE =

As a consequence, we find

F_α+p^′ gr^V_β E =





Gp−po−1gr^V_βE if α + po< β < α + po+ 1, Gp−pogr^V_β E if β = α + po.

If 0 < β < α, we also get F_α+p^′ gr^V_β E = Gpgr^V_β E. So we are reduced to proving that, for any β > 0 and any k, the morphism

(7.22) ∂v : Gkgr^V_β E −→ Gkgr^V_β+1E is an isomorphism.

Away from Pred, we have gr^V_β E = 0 for each β > 0, so the assertion is empty.

Let us prove the assertion in the neighbourhood of a point of Pred. The left action of ∂v on a term of the sum (7.16) gives, since ∂ve^vf = x^−ee^vf and due to standard commutation rules,

P^a,λ,β(v∂v+ 1)h^a,c,λ,β(xI(a), yJ(c), y^′)x−[(β+1)e]−1x^−ay^−c−1e^vf.

We have P^a,λ,β(v∂v+ 1) = P^a,λ,β+1(v∂v) and we use (7.3) for β > 0 to conclude that (7.22) is an isomorphism.

It remains to be checked that F_α+^′ _•gr^V_β E is a good filtration for any β ∈ R. The previous arguments reduces us to checking this for β ∈ [0, 1], and we are reduced to proving that, for any such β, G•gr^V_β E is a good filtration. This follows from [Sab97, 4.14], since this filtration is identified (after grading by a finite filtration) to a filtration which is already known to be good (and which is the Hodge filtration of a mixed Hodge module).

7.e. Computation of Fα+•E ∩ VαE. The previous section shows that, for α, β ∈ [0, 1), the computation of Fβ+•E ∩ VαE is interesting mainly when β = α. Note that Fα+pE = 0 for p < 0 and that, away from Pred, we have Fα+pE^vf∩ VαE^vf = E^vf = O_Xe^vf.

Lemma 7.23. For α ∈ [0, 1) and p > 0, we have gr^F_p^αVαE = L

k>0

gr^G_p+kVα−kE/ gr^G_p+kVα−k−1E

≃ C[v] ⊗C

gr^G_p VαE/ gr^G_p Vα−1E . (7.23 ∗)

Note that, for p = 0, we have gr^G₀ VαE = G0E ∩ VαE.

Proof. On the one hand, the natural map GpE ∩ VαE → gr^F_p^αVαE has ker-nel equal to GpE ∩ VαE ∩ P

k>0(Gp−1+kE ∩ Vα−kE)

, according to (7.12).

The latter space is contained in GpE ∩ (Gp−1E ∩ VαE + Vα−1E), that is, in (Gp−1E ∩ VαE) + (GpE ∩ Vα−1E); but clearly the converse inclusion is also true.

We thus have an inclusion

(7.24) GpE ∩ VαE

(Gp−1E ∩ VαE) + (GpE ∩ Vα−1E)֒−→ gr^F_p^αVαE.

On the other hand,

GpE ∩ VαE ∩ X

k>1

Gp+kE ∩ Vα−kE

⊂ GpE ∩ Vα−1E ⊂ Fα+p−1E,

so (7.24) is a direct summand in gr^F_p^αVαE, and one can continue to get the first expression in (7.23 ∗). For the second expression, we use (7.13).

Lemma 7.25. For α ∈ [0, 1), p > 0, near a point of Pred, the following holds:

(1) The natural morphism, induced by the inclusion of each summand in O_X(∗Pred)[v]:

(7.25 ∗) L

a>0

|a|6p

O_I(a)x^−[αe]−1x^−aQ^a,p−|a|,α(v/x^e) −→ gr^G_p VαE^vf

is an isomorphism.

(2) For each i = 1, . . . , ℓ, the morphism ∂i: gr^G_p VαE^vf → gr^G_p+1VαE^vf induced by

−∂xi/ei is given by

∂ih^a,p−|a|,α(xI(a), y^′)x^−[αe]−1x^−aQ^a,p−|a|,α(v/x^e)

=









ha,p−|a|,α(xI(a), y^′)x^−[αe]−1x^−(a+1i)Qa+1i,p+1−|a+1i|,α(v/x^e) if i /∈ I(a), ha,p−|a|,α(0i, y^′)x^−[αe]−1x^−(a+1i)Qa+1i,p+1−|a+1i|,α(v/x^e)

+ h⁽ⁱ⁾a,p−|a|,α(xI(a), y^′)x^−[αe]−1x^−aQ^a,p+1−|a|,α(v/x^e) if i ∈ I(a), where, for i ∈ I(a), we set

h^a,p−|a|,α(xI(a), y^′) = h^a,p−|a|,α(0i, y^′) + xih⁽ⁱ⁾a,p−|a|,α(xI(a), y^′), and 0i means that xi is set to be 0 in xI(a).

Proof. The first point follows from [Sab97, Lemma 4.11]. For the second point we have, modulo GpE^vf∩ VαE^vf,

∂iha,p−|a|,α(xI(a), y^′)x^−[αe]−1x^−aQa,p−|a|,α(v/x^e)

= h^a,p−|a|,α(xI(a), y^′)x^−[αe]−1x^−(a+1i)Q^a+1i,p−|a|,α(v/x^e).

However, this is possibly not written in the form above if i ∈ I(a) (i.e., ai = 0) and we modify this expression as indicated in the statement to obtain, modulo GpE^vf∩ VαE^vf, the desired formula.

7.f. The filtered relative de Rham complex and the rescaled Yu complex We consider the relative de Rham complex DRX×Cv/CvEwhich is nothing but the complex of OC_v-modules

0 −→ E d + vdf

−−−−−−−→ Ω¹_X[v] ⊗ E −→ · · ·

and the action of ∂vby ∂/∂v +f induces a C[v]h∂vi-structure on each term compatible with the differentials.

We filter this complex as usual by subcomplexes of C[v]-modules:

Fα+pDR_X×Cv/CvE= {Fα+pE d + vdf

−−−−−−−→ Ω¹_X[v] ⊗ Fα+p+1E−→ · · · }.

As usual we set F_α^p= Fα−p. The action of ∂v on DRX×Cv/CvE induces a morphism

∂v: Fα+pDRX×Cv/CvE−→ Fα+p+1DRX×Cv/CvE.

We will use the following notation, as in [ESY15]:

Ω^k_X×Cv/Cv(log D)([(α + j)P])+=

(0 if j < 0,

Ω^k_X×Cv/Cv(log D)([(α + j)P]) if j > 0.

We define the rescaled Yu complex as being the filtered complex (α ∈ [0, 1) and p ∈ Z):

F_α+p^Yu DRX×Cv/CvE

:= OX×Cv([(α + p)P])+ d + vdf

−−−−−−−→ Ω¹_X×Cv/Cv(log D)([(α + p + 1)P])+−→ · · · which is a complex of OC_v-modules. The connection ∂/∂v + f induces a morphism

∇∂v : F_α+p^Yu DR_X×Cv/CvE→ F_α+p+1^Yu DR_X×Cv/CvE. Proposition 7.26. The natural morphism

F_α+p^Yu DRX×Cv/CvE−→ Fα+pDRX×Cv/CvE

is a quasi-isomorphism for each α ∈ [0, 1) and p ∈ Z compatible with the action of ∂v. Proof. The existence of a natural morphism follows from Lemma 7.10. The compati-bility with respect to the action of ∂v is then clear. The proof is then similar to that of [ESY15, Prop. 1.7.4]. We note that, away from Pred, we use that the morphism (7.27) (Ω^•_X×Cv/Cv(log H), d + vdf ) −→ (Ω^•_X×Cv/Cv(∗H), d + vdf ),

is a filtered quasi-isomorphism. Here the analytic version of E is needed in order to write d + vdf = e^−vf ◦ d ◦ e^vf.

7.g. Proof of Theorem 6.4. We consider the complex (Ω^•_f(α)[v], d + vdf ). We have a natural connection

∇∂v : (Ω^•_f(α)[v], d + vdf ) −→ (Ω^•_f(α + 1)[v], d + vdf ) induced by the action of f + ∂/∂v on each term of the complex.

Lemma 7.28. For α ∈ [0, 1), there is a natural filtered morphism O_X×C

v ⊗O_X[v]Ω^•_f(α)[v], d + vdf, σ^>p

−→ (DRX×Cv/CvVαE, F_α^pDRX×Cv/CvVαE) which makes the following diagram commutative:

O_X×C

v⊗O_X[v]Ω^•_f(α)[v], d + vdf

∇∂v

 //DRX×Cv/CvVαE

∇∂v



O_X×C

v⊗O_X[v]Ω^•_f(α + 1)[v], d + vdf

//DRX×Cv/CvVα+1E

Proof. Once the morphism is defined, the commutativity of the diagram is straight-forward: let d denote the differential with respect to X and dv that with respect to v;

the verification reduces to checking that e^−vf◦ d ◦ e^vf commutes with e^−vf◦ dv◦ e^vf, a statement which follow from the commutation of d with dv.

Away from Pred, the morphism is given by (7.27). At a point of Pred, we will use the algebraic version E of E for simplicity. For each k > 0, we have a natural inclusion

(F0O(∗Pred)

([αP ])(∗H)v^ke^vf ⊂ VαE.

Indeed, it is enough to prove the inclusion x^ke(F0O(∗Pred)

([αP ])(∗H) · (v/x^e)^ke^vf ⊂ VαE

and then the inclusion x^ke(F0O(∗Pred)

([αP ])(∗H)Q0,j,α(v/x^e) ⊂ VαE with P0,j,α(s) = (s + α)^j, by expressing (v/x^e)^k in terms of the Q0,j,α(v/x^e) with j 6 k. The assertion is then clear by taking the term with a = 0 in the expression of Lemma 7.1. We thus obtain the desired morphism.

In order to prove that it is filtered, we note that for each k > 0, the natural inclusion morphism Ω^k_f(α)[v] → Ω^k_X⊗O_X E factorizes through the subsheaf Ω^k_X⊗O_X FαVαE.

Indeed, according to (7.17), we have FαVαE = FαE = F0O_X(∗D)([αP ])[v] · e^vf, so we are reduced to proving the inclusion Ω^k_f(α) ⊂ Ω^k_X⊗ F0O_X(∗D)([αP ]). This is clear away from Predsince this reduces to Ω^k_X(log H) ⊂ Ω^k_X⊗ F0O_X(∗H). In the local setting of §7.a near a point of Pred, the conclusion follows from [ESY15, Formula (1.3.1)] for α = 0, and the same formula multiplied by x^−[αe] if α ∈ (0, 1).

We will show Theorem 6.4 with the filtration F_α^•introduced in Definition 7.9. That this is the filtration F_α^irr,•will be shown in Theorem 9.1. Near a point of (XrPred)×Cv

we can write d + vdf = e^−vf ◦ d ◦ e^vf to reduce the statement to the standard result proved by Deligne [Del70].

We will thus focus on Pred× Cv, and it will be enough to consider the v-algebraic version of the statement. It is also enough to prove that for each p > 0, the pth graded morphism is a quasi-isomorphism. We are thus lead to proving that for p > 0 the following vertical morphism is a quasi-isomorphism:

(7.29)

0 //Ω^p_f(α)[v]

 //0

 //· · · 0 //FαVαE ⊗ Ω^p //gr^F₁^αVαE ⊗ Ω^p+1 //· · ·

Since the question is local, we can treat separately the variables x and y, and the main problem remains the case of the x variables, so that we will assume H = ∅. We will use the computations of §7.e, from which we keep the notation.

Lemma 7.30. Near a point of Pred, for q ∈ Z and α ∈ [0, 1), the relative de Rham complex

0 −→ gr^G_q VαE^vf −→ gr^G_q+1VαE^vf⊗ Ω¹−→ · · · −→ gr^G_q+nVαE^vf⊗ Ωⁿ−→ 0 has zero cohomology in degrees > −q + 1 (recall that gr^G_q VαE^vf = 0 for q < 0).

Sketch of proof. We will forget the variables y^′ and work with the variables x ∈ C^ℓ, so we will replace n with ℓ in the de Rham complex above. We then note that this complex is the simple complex associated with the hypercubical complex built on the cube in R^ℓ with vertices ε ∈ {0, 1}^ℓ, whose vertex at ε is gr^G_q+|ε|VαE^vf and whose arrows (εi = 0) → (εi = 1) are the derivatives ∂xi. We may as well replace the arrow ∂xi with ∂i= −∂xi/ei.

The formula of Lemma 7.25(2) shows that, if εi= 0, the arrow ∂i : ε → ε + 1i is injective, with cokernel identified with

L

a′>0, a^′_i=0

|a^′|=q+1

O_I(a′)x^−[αe]−1x^−a′Qa′,0,α(v/x^e).

We use the convention that a sum indexed by the empty set is zero, a case which occurs if q + 1 < 0.

• If ℓ = 1, we only need to consider the case q > 0. The cokernel of ∂1is then equal to zero, showing that ∂1 is bijective in this case, which implies the desired assertion.

• If ℓ > 2, we replace (with a shift) the hypercubical ℓ-complex with the (ℓ − 1)-complex made of the cokernels of the injective arrows ∂1, and the formula for the induced arrows ∂2, . . . , ∂ℓ is then that of the case i /∈ I(a) in the formula of Lemma 7.25(2). Now, the maps induced by ∂2 are injective, with cokernel

L

a^′′>0, a^′′₁=a^′′₂=0

|a^′′|=q+2

O_I(a′′)x^−[αe]−1x^−a′′Q^a′′,0,α(v/x^e), etc.

From Lemma 7.30 we conclude that for α ∈ [0, 1) and each p > 0, the de Rham complex

(7.31)α 0 −→ · · · −→ 0 −→ gr^G₀ VαE^vf⊗ Ω^p−→ gr^G₁ VαE^vf ⊗ Ω^p+1 −→ · · · has zero cohomology in degrees > p + 1, while the de Rham complex

(7.31)α−1 0 −→ · · · −→ 0 = gr^G₀ Vα−1E^vf ⊗ Ω^p−→ gr^G₁ Vα−1E^vf⊗ Ω^p+1−→ · · · has zero cohomology in degrees > p + 2 since gr^G_k Vα−1E^vf ≃ gr^G_k−1VαE^vf, according to (7.13). Therefore, the quotient complex (7.31)_α/(7.31)_α−1has zero cohomology in degrees > p + 1. It follows then from Lemma 7.23 that the bottom line of (7.29) has zero cohomology in degrees > p + 1. It remains to identify the degree p cohomology of this bottom line. As noted above, we have

FαVαE^vf = FαE^vf = F0O(∗Pred)([αP ])[v]e^vf,

so the cohomology consists of sections of F0O(∗Pred)([αP ])[v] ⊗ Ω^p whose image by d + vdf belong to F0O(∗Pred)([αP ])[v] ⊗ Ω^p+1. This cohomology is then contained in Ω^p(log Pred)([αP ])[v], according to Lemma 7.32 below, and it is then easy to identify it with Ω^p_f(α)[v].

Lemma 7.32. For k > 0, a section of F0O(∗Pred)Ω^kbelongs to Ω^k(log Pred) if and only if its exterior product by Pℓ

i=1eidxi/xi belongs to F0O(∗Pred)Ω^k+1. 7.h. Some properties of the filtration F_α^•H^k

v. Recall that the DC_v-module H_v^k is defined in §1.b. For α ∈ [0, 1), we denote by VαH^k

v the free C[v]-lattice of H_v^k on which the connection ∇ induced by the DC_v-module structure has a simple pole, with residue as in Theorem 1.11(1). This is also the part of indices in [0, 1) of the Kashiwara-Malgrange V -filtration of H_v^k, which exists since it is a holonomic DC_v -module.

By a standard result on the strictness of the Kashiwara-Malgrange V -filtration with respect to proper push-forward, we have:

VαH_v^k= imh

R^kq+(DRX×Cv/CvVαE) −→ R^kq+(DRX×Cv/CvE)i

and the latter morphism is injective. We obtain, as a consequence of Proposition 7.19:

Corollary 7.33 (of [Sai88, Prop. 3.3.17]). For each k, α, p, F_α^•H_v^k satisfies the properties [Sai88, (3.2.1)].

Let us consider the restriction j^∗F_α^•H^k

Proof. The first part follows from Lemma 7.10(2) and the second part follows from the property gr^p_FYu

α H_dR^k (U, ∇) = 0 for p /∈ [0, k], which is a consequence of [ESY15, Cor. 1.5.6].

Recall that the irregular Hodge numbers h^p,q_α (f ) are defined by (1.6). As a conse-quence of Corollary 7.34 we have

h^p,q_α (f ) = rk gr^p_Fαj∗H_v^p+q. Corollary 7.35. For α ∈ [0, 1), we have F_α⁰H_v^k ⊃ VαH_v^k.

Proof. We have seen that both OC_v-modules coincide with H_v^k[v⁻¹] after tensoring with OC_v[v⁻¹] (by Corollary 7.34 for the first one, and by a standard property of the V -filtration for the second one). Hence for any m ∈ VαH^k

v there exists ℓ > 0 such that v^ℓm ∈ F_α⁰H_v^k. Let p ∈ Z be such that m ∈ F_α^pH_v^k. Corollary 7.33 implies that Property [Sai88, (3.2.1.1)] holds for the filtration F_α^•H^k

v , and thus the morphism v^ℓ: (F_α^qH^k

v ∩ VαH^k

v ) → (F_α^qH^k

v ∩ Vα−ℓH^k

v ) is an isomorphism for each q. It follows that

7.i. Nearby cycles and the monodromy filtration. We now consider the functor ψv,exp(−2πiβ) (β ∈ [0, 1)). The result of [Sai88, Prop. 3.3.17] implies then that, for each β ∈ [0, 1), the filtration naturally induced by the Q-indexed filtration F^•H^k

v on ψv,exp(−2πiβ)H^k

v is equal to

(7.36) F^•H^k(X, DR ψv,exp(−2πiβ)E) := H^k(X, F^•DR ψv,exp(−2πiβ)E)

and therefore has jumps at β + Z at most. It is then enough to consider the filtration induced by F_β^•H^k

v on ψv,exp(−2πiβ)H^k

v . Then, according to the previous results, we have

F_β^pψv,exp(−2πiβ)H_v^k =

(ψv,exp(−2πiβ)H_v^k if p 6 0,

0 if p > k.

Definition 7.37. For α ∈ [0, 1) and k > 0, the spectral multiplicity function is the function

Proof. For β ∈ [0, 1), we have an isomorphism (see Corollary 7.33):

(7.39) v : (F_α^pH^k∩ VβH^k)−→ (F^∼ _α^pH^k∩ Vβ−1H^k).

Therefore, X

β∈[0,1)

dim F_α^pψv,exp(−2πiβ)H_v^k= X

β∈[0,1)

dim F_α^pgr^V_β H_v^k = X

β∈(α−1,α]

dim F_α^pgr^V_β H_v^k

= dim F_α^pH ^k

v ∩ VαH^k

v

Fα^pH^k

v ∩ Vα−1H^k

v

= dim F_α^pH^k

v ∩ VαH^k

v

v(Fα^pH_v^k∩ VαH_v^k) (Corollary 7.33).

Since VαH^k

v is OC_v-free for α ∈ [0, 1), the OC_v-module F_α^pH^k

v ∩ VαH^k

v is O_C

v-torsionfree, hence OC_v-free, and the latter term is equal to rk(F_α^pH_v^k∩ VαH_v^k), hence to rk(F_α^pH^k

v )[v⁻¹], that is, dim F_α^Yu,pH_dR^k (U, ∇), according to Corollary 7.34.

The result follows from [ESY15, Cor. 1.4.8].

Proof of Theorem 6.1. By Lemma 7.15 and (7.21), we can apply [Sab97, Th. 5.3] to the filtration given by (7.36). It remains to identify the latter with the irregular Hodge filtration. This follows from Theorem 9.1 below.

在文檔中 On the irregular Hodge filtration of exponentially twisted mixed Hodge modules (頁 22-36)