Relation with the irregular Hodge filtration of E (v:u)f (∗H)

In this section, we set E := E^(v:u)f(∗H). We will compare the filtration Fα+•Ewith the irregular Hodge filtration F_α+^irr_•E as defined in §5, namely, we consider the case where N = O^X(∗D) (notation of §7.a) with its differential d twisted by the exponential of the rational function vf : X × P¹_v = X - - → P¹. The module ^FfN considered in

§5.b is obtained here by gluing E^{τ vf /z}[∗H] (notation of Proposition 3.3) in the v-chart with E^{τ f /uz}[∗H] in the u-chart, and we will regard these modules algebraically with respect to τ , (v:u) and z. We will use the notation introduced in §3.

Theorem 9.1. For each α ∈ [0, 1), we have

Fα+•E^vf(∗H) = F_α+^irr_•E^vf(∗H),

Fα+•G0E^{f /u}(∗H) = F_α+^irr_•E^{f /u}(∗H) ∩ G0E^{f /u}(∗H).

The proof of the theorem will be done in various steps. For the sake of simplicity, we will only treat the case where H = ∅.

• Firstly, one identifies E^{τ vf /z} as a submodule of OX(∗Pred)[v, τ, z] · e^{τ vf /z} and E^{τ f /uz} as a submodule of OX(∗Pred)[u, u⁻¹, τ, z] · e^{τ f /uz}. According to Proposition 3.4, we may have a strict inclusion only near points of Pred× {v = 0} and points of {f = 0} × {u = 0}. For the latter set, the computation is much simplified because we only consider the intersection with G0E. For the former set, we will need explicit computations of the V -filtration entering the very definition of E^{τ vf /z} in Proposi-tion 3.3.

•Secondly, one computes the terms V_α^τE^{τ vf /z} (resp. V_α^τE^{τ f /uz}) of the V -filtration relative to τ = 0, in order to apply Proposition 5.5. We will work analytically with respect to the variables of X and algebraically with respect to τ , (u : v) and z.

9.a. Computation in the v-chart. We use the algebraic version (with respect to v, τ, z) RF(DX[v, τ ]h∂v, ∂τi) of RX×Cv×Cτ. Recall that E^{τ vf /z} is a coherent RF(DX[v, τ ]h∂v, ∂τi)-submodule of OX(∗Pred)[v, τ ] · e^{τ vf /z}. We will set e = e^{τ vf /z}. Computation away from Pred. Since τ vf is holomorphic, we have E^{τ vf /z} = O_XrP

red[v, τ ] · e^{τ vf /z}. Then, from the relation ðτe = vf e, we conclude that E^{τ vf /z} is already (RFD_X[v, τ ]h∂vi)-coherent, and hence the V^τ-filtration is given by V_k^τE^{τ vf /z} = τ^max(−k,0)E^{τ vf /z}: by uniqueness of the V^τ-filtration, it is enough to check the strictness of the gr^V_k^τE^{τ vf /z}, which is clear. Therefore, only α = 0

(3)We thank É. Mann, Th. Reichelt and Ch. Sevenheck for providing us with the necessary arguments.

is relevant. In particular, V₀^τE^{τ vf /z} = E^{τ vf /z}. Hence, the quotient modulo (τ − z)E^{τ vf /z} is equal to E^vf[z].

On the other hand, we have Fα+pE^vf = E^vf for any α ∈ [0, 1) and p > 0, and Fα−1E^vf = 0, that is, RFE^vf = E^vf[z].

Computation in a neighbourhood of Pred. Near a point of Pred, let us set g = 1/f , which is holomorphic in a neighbourhood of this point. In local coordinates we have g = x^e.

First step: computation of E^{τ v/gz}. By the very definition of Proposition 3.3(1) we have, on this neighbourhood, E^{τ v/gz} = (OX(∗Pred)[τ, v, z] · e^{τ v/gz})[∗Pred]. Let Then (δ ⊗ e)1+α satisfies the following equations:

(9.3) Similarly for (δ ⊗ e)<1+α the last line of (9.3) reads

ð_xi(δ ⊗ e)<1+α= −ei

We have

We have a similar identification by using (δ ⊗ e)<1+α. Let us write the last line of (9.3) as the empty set is equal to one):

p^a,α(s, z) =Q deduce that each element of U1+α can be written as

(9.6) X

a>0

h^a,α(xI(a), v^′, τ^′, η, z)x^−ap^a,α(η, z),

and the coefficient eh^aof x^−ain its decomposition (9.5) is h^a,α(xI(a), v^′, τ^′, η, z)p^a,α(η, z).

By uniqueness, we conclude that an element written as (9.5) belongs to U1+α if and only if p^a,α(η, z) divides eh^a(xI(a), v^′, τ^′, η, z). In particular, the decomposition (9.6) is unique.

We wish to identify U1+α· (δ ⊗ e)1+αwith V_1+α^t′ (ig,+O_X(∗Pred)[v, τ, z]) · e^{τ v/t′z}and U<1+α· (δ ⊗ e)<1+α with V_<1+α^t′ (ig,+O_X(∗Pred)[v, τ, z]) · e^{τ v/t′z}. It is enough to check

(t^′ð_t′+ (1 + α)z)^mU1+α· (δ ⊗ e)1+α⊂ U<1+α· (δ ⊗ e)<1+α for m big enough and

U1+α· (δ ⊗ e)1+α/U<1+α· (δ ⊗ e)<1+α has no z-torsion (see [Sab05, Lem. 3.3.4 & §3.4.a]). For the first point, we set

Iα= {i | αei∈ Z} and, for a > 0, Iα(a) = Iα∩ I(a) and Iα(a)^c= Iα∩ I(a)^c. Then (δ ⊗ e)1+α= x^−1Iα(δ ⊗ e)<1+α, and we have the relation

Q

i∈Iα

(ðxi− eix^e−1ið_vð_τ) · (δ ⊗ e)<1+α= (−e)^1Iα(ðt^′t^′+ αz)^#Iα(δ ⊗ e)1+α

= (−e)^1Iα t^′ð_t′+ (1 + α)z
#Iα

(δ ⊗ e)1+α. For the torsion-free assertion, let us consider a section (9.6) of U1+α and let us de-compose (in a unique way) h^a,α(xI(a), v^′, τ^′, η, z) as

h^a,α(xI(a), v^′, τ^′, η, z) = X

ε∈{0,1}^Iα(a)

h^a,α,ε(xI(a+1_Iα−ε), v^′, τ^′, η, z)x^ε,

where h^a,α,ε is holomorphic in its x-variables and polynomial in v^′, τ^′, η, z. Then the decomposition (9.6) reads

X

a>0

X

ε∈{0,1}^Iα(a)

h^a,α,ε(xI(a+1_Iα−ε), v^′, τ^′, η, z)x^−(a−ε)p^a,α(η, z).

We now note that, for ε ∈ {0, 1}^Iα(a), setting b = a + 1Iα− ε, we have p^b,<α(η, z) = (η + αz)^#Iα(b)c· p^a,α(η, z).

The unique decomposition (9.6) can thus also be written uniquely as

(9.7) X

b>0

h^′b,α(xI(b), v^′, τ^′, η, z)x^−b p^b,<α(η, z)

(η + αz)^#Iα(b)c · x^1Iα, with h^′b,α= h^a,α,ε, where (a, ε) is defined by the following conditions:

ai= bi if i /∈ Iα,

ai= bi− 1 and εi= 0 if i ∈ Iαand bi>1, ai= 0 and εi= 1 if i ∈ Iα and bi= 0.

The condition that a section (9.7) · (δ ⊗ e)1+α= (9.7) · x^−1Iα(δ ⊗ e)<1+α belongs to U<1+α· (δ ⊗ e)<1+α now reads

∀ b > 0, (η + αz)^#Iα(b)c divides h^′b,α(xI(b), v^′, τ^′, η, z).

It is therefore clear that a section of U1+α· (δ ⊗ e)1+αbelongs, when multiplied by z, to U<1+α· (δ ⊗ e)<1+α if and only if it already belongs to U<1+α· (δ ⊗ e)<1+α. In other words, U1+α· (δ ⊗ e)1+α

U<1+α· (δ ⊗ e)<1+α has no z-torsion.

We conclude that

V₀^t′RF(DX[t^′, v, τ ]h∂t^′, ∂v, ∂τi) · (δ ⊗ e)1= U1· (δ ⊗ e)1

= V₁^t′(ig,+O_X(∗Pred)[v, τ, z]) · (δ ⊗ e)1, hence ig,+E^{τ v/gz}is generated by (δ ⊗ e)1. It follows that E^{τ v/gz}is generated by x⁻¹e.

We will prove the analogue of (9.2 ∗) after ig,+, from which one deduces similarly (9.2 ∗). We first notice that the equality

V₁^t′(ig,+E^{τ v/gz}) ∩ (τ − z)(ig,+O_X(∗Pred)[v, τ, z]) · e^{τ v/t′z}= (τ − z)V₁^t′(ig,+E^{τ v/gz}) immediately follows from the unique decomposition (9.6) of a local section of V₁^t′(ig,+E^{τ v/gz}). To end the proof, it therefore suffices to produce a similar unique decomposition of local sections of V_1+k^t′ (ig,+E^{τ v/gz}) := Pk

j=0ð^j_t′V₁^t′(ig,+E^{τ v/gz}) for any k > 1. This is obtained by writing

ð^k_t′(δ ⊗ e)1= x^−keð^k_t′t^′k(δ ⊗ e)1= x^−ke

k−1Q

j=0

(ðt^′t^′+ jz)(δ ⊗ e)1,

giving rise to a formula similar to (9.6) for sections of V_1+k^t′ (ig,+E^{τ v/gz}), which makes use of polynomials p^a,k (k > 1), derived from p^a,0like in [Sab97, Lem. 4.7].

Second step: computation of the V^τ-filtration of E^{τ v/gz}. For α ∈ [0, 1), let us set e_α= e^{τ v/gz}/x^[αe]+1.

Lemma 9.8. The V_•^τ-filtration of E^{τ v/gz} satisfies

V_α^τE^{τ v/gz}= V₀^τRF(DX[v, τ ]h∂v, ∂τi) · eα ∀ α ∈ [0, 1).

Proof. Since we are only interested in giving the formula for V_α^τE^{τ v/gz}, we can as well work with the localized module E^{τ v/gz}[τ⁻¹] (see [Sab05, Lem. 3.4.1]). In such a way, we can write

e_α= x^−[αe](x⁻¹e) = x^{⌈(1−α)e⌉}τ⁻¹ð_v(x⁻¹e),

showing that eα is a section of E^{τ v/gz}[τ⁻¹]. For α ∈ [0, 1) let us also set e_<α= Q

i∈Iα

xi



e_α=: x^1Iαe_α and, for p ∈ Z,

U_α+p^τ (E^{τ v/gz}[τ⁻¹]) = τ^−pV₀^τRF(DX[v, τ ]h∂v, ∂τi) · eα

U_<α+p^τ (E^{τ v/gz}[τ⁻¹]) = τ^−pV₀^τRF(DX[v, τ ]h∂v, ∂τi) · e<α, so that, clearly,

(9.9) U_<α+p^τ (E^{τ v/gz}[τ⁻¹]) ⊂ U_α+p^τ (E^{τ v/gz}[τ⁻¹]).

For p 6 0, we will set U_α+p^τ E^{τ v/gz} = U_α+p^τ (E^{τ v/gz}[τ⁻¹]) and U_<α+p^τ E^{τ v/gz} = U<α+p^τ (E^{τ v/gz}[τ⁻¹]). We will prove that U_•^τ(E^{τ v/gz}[τ⁻¹]) is the good V^τ-filtration of E^{τ v/gz}[τ⁻¹]. It is enough to prove that U_α^τE^{τ v/gz} = V_α^τE^{τ v/gz} for α ∈ [0, 1). The proof will be very similar to that of Lemma 9.2, although with the variable τ instead of the variable t^′.

By using (9.9), one first easily checks that U_α−1^τ E^{τ v/gz}⊂ U_<α^τ E^{τ v/gz} and (τ ðτ+ αz)^#IαU_α^τE^{τ v/gz} ⊂ U_<α^τ E^{τ v/gz}.

Indeed, the first point follows from the relation τ eα = x^eð_ve_α= x^e−1Iαð_ve_<α, and the second one follows from the relation

(τ ðτ+ αz)^#Iαe_α= (−1)^#Iα

Due to the uniqueness of the V^τ-filtration, the assertion of the lemma would follow from the property that gr^U_α^τE^{τ v/gz} has no z-torsion. We will argue in a way similar to that of Lemma 9.2 by finding a suitable expression for the sections of U_α^τE^{τ v/gz}.

Let us decompose OX[x⁻¹, z][v, τ ] as OX[x⁻¹, z][v, vτ ] ⊕ τ OX[x⁻¹, z][vτ, τ ]. Due We thus obtain an isomorphism of free OX[x⁻¹, z]-modules:

O_X[x⁻¹, z][v, τ ] · e−→ O^∼ X[x⁻¹, z, v][τ ðτ] ⊕ τ OX[x⁻¹, z, τ ]hτ ðτi

· e.

We can replace e with eαor e<αin the above isomorphism. We will express U_α^τE^{τ v/gz} and U_<α^τ E^{τ v/gz} as sub-OX[z]-modules of the right-hand side, and by using the gen-erator e<α in both cases, to make the computation of the quotient module easier.

We note first that U_α^τE^{τ v/gz} = OX[v, z]hðx, ðv, τ ðτi · eα, i.e., we can forget the for some nonzero constants ⋆ and with p^a,α(s, z) defined by (9.4), we conclude that, through the isomorphism (9.10), holo-morphic in its x-variables and polynomial in τ ðτ, z.

Let us check that the decomposition (9.11) is unique. The coefficient h⁽ⁿ⁾of τⁿ(n > 0) is uniquely determined by the section. If n = 0, the function h⁽⁰⁾∈ OX,0[x⁻¹, v, η, z]

decomposes uniquely asP

a>0h^(0)a (xI(a), v, η, z)x^−a. Thus h^(0)a must be divisible by pa,α(η, z) and this determines uniquely ha,α(x_I(a), v, η, z). We argue similarly for n > 0 and h⁽ⁿ⁾∈ OX,0[x⁻¹, η, z].

There is a similar decomposition for sections of U_<α^τ E^{τ v/gz}, by replacing p^a,α(τ ðτ, z) · eα with p^a,<α(τ ðτ, z) · e<α. In order to check whether a section (9.11) belongs to U_<α^τ E^{τ v/gz}, we replace eα with x^−1Iαe_<α. Let us decompose (in a unique way) ha,α(x_I(a), v, τ ðτ, z) as

h^a,α(xI(a), v, τ ðτ, z) = X

ε∈{0,1}^Iα(a)

h^a,α,ε(xI(a+1_Iα−ε), v, τ ðτ, z)x^ε,

where h^a,α,ε is holomorphic in its x-variables and polynomial in v, τ ðτ, z. We have a similar decomposition for ga,α,n(x_I(a+ne), τ ðτ, z). Then the decomposition (9.11) reads

X

a>0

X

ε∈{0,1}^Iα(a)

h^a,α,ε(xI(a+1_Iα−ε), v, τ ðτ, z)x^{−(a+1Iα−ε)}p^a,α(τ ðτ, z) · e<α

+X

a>0

X

n>0

X

ε∈{0,1}^Iα(a+ne)

τⁿg^a,α,n(xI(a+ne+1_Iα−ε), τ ðτ, z) ·

· x^{−(a+ne+1Iα−ε)}p^a,α(τ ðτ+ nz, z) · e<α. We now note that, for ε ∈ {0, 1}^Iα(a), setting b = a + 1Iα− ε, we have

p^b,<α(s, z) = (s + αz)^#Iα(b)c· p^a,α(s, z).

The unique decomposition (9.11) can thus also be written uniquely as

(9.12) X

b>0

h^′b,α(xI(b), v, τ ðτ, z)x^−b p^b,<α(τ ðτ, z)

(τ ðτ+ αz)^#Iα(b)c · e<α

+X

b>0

X

n>0

τⁿgb^′,α,n(x_I(b+ne), v, τ ðτ, z)x^−(b+ne) p^b,<α(τ ðτ+ nz, z)

(τ ðτ+ (n + α)z)^#Iα(b)c · e<α

with h^′b,α= h^a,α,ε, where (a, ε) is defined by the following conditions:

ai= bi if i /∈ Iα,

ai= bi− 1 and εi= 0 if i ∈ Iαand bi>1, ai= 0 and εi= 1 if i ∈ Iα and bi= 0,

and similarly for g^′b,α,n. The condition that a section (9.12) belongs to U_<α^τ E^{τ v/gz} now reads

∀ b > 0, (τ ðτ+ αz)^#Iα(b)c divides h^′b,α(xI(b), v, τ ðτ, z),

∀ b > 0, ∀ n > 0, (τ ðτ+ (n + α)z)^#Iα(b)c divides gb^′,α,n(xI(b+ne), τ ðτ, z).

and

It is therefore clear that a section (9.12) of U_α^τE^{τ v/gz}belongs, when multiplied by z, to U_<α^τ E^{τ v/gz}if and only if it already belongs to U_<α^τ E^{τ v/gz}. In other words, gr^U_α^τE^{τ v/gz} has no z-torsion.

Third step: End the proof of Theorem 9.1 in the v-chart. We will now use the expression of Lemma 9.8 to regard V_α^τE^{τ v/gz} as an OX[v, τ, z]-submodule of O_X[x⁻¹, v, τ, z] · eα. We can write (locally on X near P ):

V₀^τ(RFD_X[v, τ ]h∂v, ∂τi) = OX[v, τ, z]hðx, ðy^′, ðv, τ ðτi

and we notice that the action of τ ðτ on eα is equal to that of vðv, so we can for-get τ ðτ. We will also forget (y^′, ðy^′) which plays no significant role. Recall that the variables x are indexed as x1, . . . , xℓ. Working now within OX[x⁻¹, v, τ, z] · eα, we have by induction on |a|,

(xðx)^að^c_ve_α≡ z^|a|+cx^−ce

X^|a|

j=0

q^a,c,j(x)x^−jev^j



e_α mod (τ − z)OX[x⁻¹, v, τ, z], where q^a,c,j(x) is some polynomial and qa,c,|a|(x) is a nonzero constant. From this one concludes that there exist polynomials r^a,c,j(x, z) with ra,c,|a|(x, z) constant, such that

ð^a_xð^c_ve_α≡ z^|a|+cx^−(|a|+c)e

X^|a|

j=0

r^a,c,j(x, z)x^{(|a|−j)e−a}v^j



x^−[αe]−1e

mod (τ − z)OX[x⁻¹, v, τ, z], and since for 0 6 j 6 |a| we have |a − (|a| − j)e|+ 6j (see the end of the proof of Lemma 7.15), the coefficient of v^jbelongs to FjO_X(∗Pred)([(α+p)P ]) with p = |a|+c.

Using that

(τ − z)V_α^τE^{τ v/gz} = (τ − z)E^{τ v/gz}∩ V_α^τE^{τ v/gz} (see [ESY15, Proof of Prop. 3.1.2])

= (τ − z)OX[x⁻¹, v, τ, z] ∩ V_α^τE^{τ v/gz} (Lemma 9.2), we conclude that the coefficient of z^p in the gr V_α^τE^{τ v/gz}/(τ − z)V_α^τE^{τ v/gz}

(graded with respect to the z-adic filtration) is contained in Fα+pE^vf, so F_α+p^irr E^vf ⊂ Fα+pE^vf, according to Remark 5.6.

In order to obtain the reverse inclusion, we remark that, for |a| + c = p fixed, x^−av^|a|e^vf/x^[(α+p)e]+1∈ F|a|O_X(∗Pred)([(α + p)P ])v^|a|e^vf is equal, up to a nonzero constant and modulo P

j<|a|FjO_X(∗Pred)([(α + p)P ])v^je^vf, to the class of ð^a_xð^c_ve_α. We conclude by induction on |a|, the case where |a| = 0 being clear.

9.b. Computation in the u-chart

Computation away from Pred. We have Fα+pG0E^{f /u} = G0E^{f /u} = OXrP[u]e^{f /u} for p > 0. We set similarly e = e^{τ f /uz}.

Lemma 9.13. With respect to the inclusion E^{τ f /uz}⊂ OXrPred[u, u⁻¹, τ, z] · e, we have e∈ E^{τ f /uz}.

Proof. The question is local near a point of {f = 0}, since otherwise we have equality in the previous inclusion, according to Proposition 3.4, and it amounts to proving that e ∈ V₁^u(OXrPred[u, u⁻¹, τ, z] · e), so we are reduced to computing the order of e with respect to the V^u-filtration.

Let us first assume that the divisor {f = 0} has normal crossings. Let us choose local coordinates x1, . . . , xn such that f (x) = x^m with m ∈ Nⁿ (a local setting not

to be confused with that of §7.a). From the relation uðue= −(τ f /u)e we obtain ð_xi(f e) = miz

xi

x^me+mi

xi

τ f

u x^me= −mi(uðu− z)x^m−1ie, and iterating the process we find

ð^m_x (f e) = (−m)^m Qⁿ

i=1 mQi

j=1

(uðu− jz/mi) · e.

Since f e = uðτe, this gives a Bernstein relation for e showing that e ∈ V_<0^u (OXrPred[u, u⁻¹, τ, z] · e).

When {f = 0} is arbitrary, the proof proceeds exactly like in [Kas76]. We work locally near a point of {f = 0} and we choose a projective birational morphism π : X^′ → X which is an isomorphism away from {f = 0} and such that f^′ := f ◦ π defines a normal crossing divisor. Using the global section e^{τ f′/uz}of V_<0^u E^{τ f′/uz}(first part of the proof), one constructs a global section e^′ of H⁰π+V_<0^u E^{τ f′/uz} which coincides with e away from {f = 0}. This is done by using the global section 1_X←X′ of RX←X^′[u, τ ]huðui. Because E^{τ f′/uz} underlies a mixed twistor module, H⁰π+E^{τ f′/uz} is strictly specializable along u = 0 and we have V_<0^u H⁰π+E^{τ f′/uz}= H⁰π+V_<0^u E^{τ f′/uz}. Therefore, e^′ is a section of V_<0^u H⁰π+E^{τ f′/uz}, and thus satisfies a non-trivial Bernstein equation of the form

Q

β<0

(uðu+ βz)^νβ· e^′ = uP (x, u, ðx, ðτ, uðu) · e^′.

We conclude that e satisfies the same equation away from {f = 0}, hence everywhere, since OX[u, u⁻¹, τ, z] has no OX-torsion.

Due to the relation f τ ðue= −τ²ð²_τewe conclude that

V₀^τE^{τ f /uz}⊃ V₀^τRF(D(XrP )[u, τ ]h∂u, ∂τi) · e ⊃ OXrP[u, τ, z] · e.

Then, computing modulo (τ − z)E^{τ f /uz}, F_α+p^irr E^{f /u} contains OXrP[u, z] · e^{f /u} = G0E^{f /u}, hence Fα+pG0E^{f /u}= F_α+p^irr G0E^{f /u} away from Pred.

Remark 9.14. The explicit computation of F_α+p^irr E^{f /u} in the neighbourhood of f = 0 would be more complicated, and restricting to G0E^{f /u}allows us to avoid this compu-tation. Let us however note that, in the neighbourhood of the smooth locus of f⁻¹(0), an explicit formula for F_α+p^irr E^{f /u} can be obtained from Lemma 9.8 by setting there g = u and v = f . Since the order of the pole at u = 0 is one, the only interesting α is zero, and the result is:

F_p^irrE^{f /u}= 1 u^p+1

X^p

j=0

O_X[u]f^k u^k

· e^{f /u} (f smooth).

This formula extends in a natural way to F_α+p^irr E^{f /u}(∗H), provided that moreover f⁻¹(0) has no common component with H and that f⁻¹(0) ∪ H has normal crossings.

Computation near Pred. The computation is similar to, and even simpler than, the computation done in the v-chart. Indeed, due to Proposition 3.4, we have E^{τ /uxez}= O_X(∗Pred)[u, u⁻¹, τ, z] · e^{τ /uxez}, and there is no need for an analogue of Lemma 9.2.

We will consider the variable u as part of the x-variables, and the divisor u = 0 of X (see §1.b). For α ∈ [0, 1), we set eα = e^{τ /uxez}/ux^[αe]+1. The following lemma is similar to Lemma 9.8.

Lemma 9.15. The V_•^τ-filtration of E^{τ /uxez} satisfies

V_α^τE^{τ /uxez}= V₀^τRF(DX[u, τ ]h∂u, ∂τi) · eα ∀ α ∈ [0, 1).

We also obtain

ð^a_xð^b_ue_α∈ z^|a|+bF|a|+b O_X(∗Pred)[u, u⁻¹]

[(α + |a| + b)P]

· e^{τ /uxez}

mod (τ − z)E^{τ /uxez}, where F• O_X(∗Pred)[u, u⁻¹]

is the filtration by the order of the pole along Pred. Moreover, the coefficient of (ux^e)^−(|a|+b)· (ux^[αe]+1)⁻¹· e^{τ /uxez}is a nonzero constant.

It follows that

F_α+p^irr E^1/uxe = Fp O_X(∗Pred)[u, u⁻¹]

[(α + p)P]

· e^1/uxe. Intersecting with G0E^1/uxe= OX(∗Pred)[u]e^1/uxe gives

F_α+p^irr E^1/uxe∩ G0E^1/uxe= FpO_X(∗Pred) [(α + p)P ]

[u] · e^1/uxe = Fα+pG0E^1/uxe. This ends the proof of Theorem 9.1.

9.c. Another approach of Theorem 9.1 at the de Rham level. Let us assume that the zero divisor f⁻¹(0) of f : X → P¹ is smooth, that it has no component in common with D, and that f⁻¹(0) ∪ D still has normal crossings. We have a filtration Fα+•E (by using the formula given in Remark 9.14 in the u-chart). Then the proof of Theorem 9.1 gives in fact the equality Fα+•E = F_α+^irr_•E.

Let π : eX → X be a projective birational morphism such that vf extend as a morphism fvf : eX → P¹ and eD := π⁻¹(D) is a normal crossing divisor in eX. The pole divisor of vf is Pred and that of fvf , that we denote by eP_red, is contained in π⁻¹(Pred). We denote by eH the remaining components of eD. The construction of [ESY15] produces a filtration F_α+^Del_•E^fvf(∗ eH). Note that

π+E^fvf(∗ eH) = H⁰π+E^vff(∗ eH) = E^(v:u)f(∗H) =: E.

By Theorem 1.3, the push-forward π+ E^fvf(∗ eH), F_α+^Del_•E^fvf(∗ eH)

is strict, since F_α+^Del_•E^vff(∗ eH) = F_α+^irr_•E^fvf(∗ eH), and produces the filtration F_α+^irr_•E, which is nothing but Fα+•Eby Theorem 9.1 in the present setting. The strictness of the push-forward implies a quasi-isomorphism at the de Rham level:

(9.16) F_α^pDR E ≃ Rπ∗F_Del,α^p DR E^fvf(∗ eH), where, as usual, we set for a filtered D-module (M, F•M),

F^pDR M = {F−pM−→ Ω¹⊗ F−p+1M−→ · · · }.

We will show how to recover the quasi-isomorphism (9.16) for a suitable modification π : eX→ X by a direct computation. This will give, in the present setting, a proof of the degeneration at E1 of the spectral sequence attached to the hypercohomology of H^k(X, Fα+•DR E) which only relies on [ESY15] for fvf : eX → P¹, and not on the finer results of Theorem 1.3. However, the identification at the level of filtered D-modules, and not only at the level of filtered de Rham complexes, is needed for the application to Kontsevich bundles given in Theorem 1.11.

Let us set, for each p and α ∈ [0, 1), F_N,α^p DR E =n

Ω^pX(log D)([αP]) d + d(vf )

−−−−−−−−−→ Ω^p+1X (log D)([(α + 1)P]) −→ · · ·o [−p].

Such a filtration already appeared in [Yu14] in the study of the toric case, where the notation F_NP^λ (∇) was used (NP for Newton polygon).

Lemma 9.17. The natural morphism F_N,α^p DR E → F_α^pDR E is a quasi-isomorphism.

Proof. Let us prove the lemma in the v-chart for instance (the proof in the u-chart is similar), and let us assume that H = ∅ for the sake of simplicity, so that D = Pred(the general case is obtained by a Kunneth formula). Recall that n = dim X. Everything below is thus only valid on X×Cv. Consider the following complexes, with differentials induced by d + d(vf ):

Φ1(p) :n

Ω^pX(log Pred) [αP]

−→ Ω^p+1X (log Pred) [(α + 1)P]

−→ · · ·

−→ Ωⁿ⁺¹X (log Pred) [(α + n − p + 1)P]
o [−p]

and, for k such that 1 < k 6 n − p + 2, Φk(p) :n

Φ^<n−k+2₁ −→ F0O_X(∗Pred)Ω^n−k+2X [(α + n − p − k + 2)P]

−→ · · ·

−→^k−1X

j=0

FjOX(∗Pred)v^j

Ωⁿ⁺¹X [(α + n − p + 1)P]
o .

Then Φ1(p) = F_NP,α^p DR E and Φn−p+2(p) = F_α^pDR E. On each successive quotient gr^Φ_k = Φk/Φk−1, the induced differential becomes −(v/x^e)Peidxi/xi. Except at the first non-zero term, the complex gr^Φ_k decomposes into many parts of the Koszul complex associated with −(v/x^e){e1dx1/x1, . . . , eℓdxℓ/xℓ}. By a direct computation, the first non-zero chain map of gr^Φ_k is injective. In particular, gr^Φ_k is quasi-isomorphic to zero.

Let us now end the direct proof of (9.16). In the discussion of the toric case in [Yu14, §4], a specific resolution π : eX→ X of vf is constructed inductively by taking blowups along irreducible components of the intersection of the pole set Pred of vf with its zero set (f⁻¹(0) × P¹_v) ∪ (X × {v = 0}). Then it is shown in loc. cit. that (9.16) holds when we replace its left-hand side with F_N,α^p DR E. Lemma 9.17 allows us to conclude.

Appendix. Brieskorn lattices and Hodge filtration

A.a. Brieskorn lattices in dimension one. Let (M, F_•M ) be a holonomic C[t]h∂ti-module equipped with a good filtration. We denote by G the holonomic C[t]h∂ti-module C[∂t, ∂_t⁻¹] ⊗C[∂t] M . If we identify C[t]h∂ti with C[v]h∂vi by the Laplace correspondence t 7→ ∂v, ∂t 7→ −v, we also regard G as a holonomic C[v]h∂vi-module on which the multiplication by v is bijective. It is therefore also a C[v, v⁻¹]-module. We will denote by cloc the natural morphism M → G.

The Brieskorn lattice G^(F₀ ^•) of the filtration F•M is defined as the saturation of the filtration by the operator ∂_t⁻¹, that is,

(∗) G^(F₀ ^•):=X

j

∂_t^−jloc(Fc jM ) ⊂ G.

It is naturally a C[∂_t⁻¹]-module (equivalently, a C[v⁻¹]-module). We will also set G^p_(F

•)= v^−pG^(F₀ ^•)for any p ∈ Z. Let us make the link with the definition in [Sab08,

§1.d]. Let pobe an index of generation, so that Fpo+ℓM = FpoM + · · · + ∂_t^ℓFpoM for any ℓ > 0. Then the definition in loc. cit. is

(∗∗) G^(F₀ ^•)= ∂_t^−poX

j>0

∂_t^−jloc(Fc poM ).

Let us check that both definitions give the same result. Let us write (∗) as G^(F₀ ^•)= ∂_t^−poX

j

∂_t^−jloc(Fc po+jM ).

Firstly, for j 6 0, we have

∂_t^−jloc(Fc po+jM ) = cloc(∂_t^−jFpo+jM ) ⊂ cloc(FpoM ), so we can also write

G^(F₀ ^•)= ∂^−p_t ^oX

j>0

∂_t^−jloc(Fc po+jM )

= ∂^−p_t ^oX

j>0

∂_t^−j cloc(FpoM ) + · · · + ∂_t^jloc(Fc poM )

= ∂^−p_t ^oX

j>0

∂_t^−jloc(Fc poM ) = (∗∗).

We now express the Brieskorn lattice of the filtration as obtained by a push-forward operation. We consider the holonomic C[t, v]h∂t, ∂vi-module M [v, v⁻¹]e^vt. The (t, v)-Brieskorn lattice is the C[t, v⁻¹]-module defined by the following formula (see [Sab99, §1]):

G₀(M, F_•M ) =L

j

FjM · v^−je^vt⊂ M [v, v⁻¹]e^vt, G^p(M, F_•M ) = v^−pG₀(M, F_•M ).

We have ∂tG^p(M, F•M ) ⊂ G^p−1(M, F•M ) since ∂tFjM ⊂ Fj+1M and ∂te^vt= ve^vt. The relative de Rham complex

DR(M [v, v⁻¹]e^vt) :=n

M [v, v⁻¹]e^vt ∂t

−−−→ M [v, v⁻¹]e^vto

has cohomology in degree one only, and we have a natural identification as C[v]h∂v i-modules

cokerh

∂t: M [v, v⁻¹]e^vt−→ M [v, v⁻¹]e^vti

≃ G by sending P

jmjv^−je^vt to P

j(−∂t)^−jloc(mc j). This relative de Rham complex is filtered by the subcomplexes

(A.1) G^pDR(M [v, v⁻¹]e^vt) :=n

G^p(M, F_•M )−−−→ G∂t ^p−1(M, F_•M )o .

Lemma A.2 (Push-forward). The relative de Rham complex is strictly filtered by the G^•-filtration and, through the previous identification, the filtration on its H¹≃ G is equal to G^•_(F⁻¹

•).

Proof. With respect to the previous identification, G^p(M, F•M ) is sent onto G^p_(F

•)

according to the definition (∗). It remains to show that (im ∂t) ∩ G^p(M, F_•M ) =

∂tG^p+1(M, F•M ) and it is enough to check this for p = 0.

We have ∂t P

jmjv^−je^vt

=P

j(∂tmj+ mj+1)v^−je^vt and by induction on j we deduce that (∂tmj+ mj+1) ∈ FjM for all j implies mj∈ Fj−1M for all j.

Remark A.3 (Rees modules). It will also be useful to have the following interpretation in terms of Rees modules (see Proof of Theorem 4.1), for which we use the variable u instead of z here. We can twist the Rees module RFM by e^t/u by changing the action of u∂tto that of u∂t+ 1. We denote the corresponding RFC[t]h∂ti-module by RFM · e^t/u. This is nothing but G0(M, F•M ) by the change of variable u = v⁻¹.

The push-forward of an RFC[t]h∂ti-module M by the constant map q : Ct = Spec C[t] → Spec C is nothing but the de Rham complex ∂t: M → u⁻¹M, where the latter term is in degree zero. The push-forward q+(RFM · e^t/u) is thus equal to the complex (with the^• in degree zero):

RFM · e^t/u−−−→ u∂t ⁻¹RFM · e^t/u

•

. Setting u = v⁻¹ we thus have an identification

G₀(M, F•M ) ∂t

// G⁻¹(M, F•M )

RFM · e^t/u ∂t

//u⁻¹RFM · e^t/u so we can interpret G⁻¹_(F

•)as H⁰q+(RFM · e^t/u), while H⁻¹q+(RFM · e^t/u) = 0.

A.b. Brieskorn lattices in arbitrary dimension. We fix k and we will apply the previous result to (M, F•M ) = H^{k−dim X}f+(OX(∗D), (F•O_X(∗H))(∗Pred)). Here, we identify filtered C[t]h∂ti-modules and DP¹(∗∞)-modules filtered by OP¹(∗∞)-modules.

We know that the latter underlies a mixed Hodge module (up to a shift of the filtra-tion), according to [Sai90]. Working with Rees modules, the strictness property for the push-forward f+ of mixed Hodge modules can also be stated by saying that the push-forward f+

(RFO_X(∗H))(∗Pred)

is strict, and thus H^{k−dim X}f+ (RFO_X(∗H))(∗Pred)

= RFM.

On the other hand, one checks that H^{k−dim X}f+ (RFO_X(∗H))(∗Pred) · e^{f /u}

≃ H^{k−dim X}f+(RFO_X(∗H)(∗Pred)

· e^t/u, and since H^jq+(RFM · e^t/u) = 0 for j 6= 0, we conclude

H^{k−dim X}(q ◦ f )+ (RFO_X(∗H))(∗Pred) · e^{f /u}

≃ H⁰q+(RFM · e^t/u).

The left-hand term is by definition equal to

H^k X, (Ω^•_X⊗O_X(u^−•RFO_X(∗H))(∗Pred), d + u⁻¹df )
,

that is, to G0H^k_u as defined by (8.12), while the right-hand term is equal to G⁻¹ as defined above.

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在文檔中 On the irregular Hodge filtration of exponentially twisted mixed Hodge modules (頁 42-56)