Behaviour of some Hodge invariants by middle convolution

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Behaviour of some Hodge invariants by middle convolution

Nicolas Martin

To cite this version:

Nicolas Martin. Behaviour of some Hodge invariants by middle convolution. 2018. �hal-01861831v2�

MIDDLE CONVOLUTION

by Nicolas Martin

Abstract. — Following a paper of Dettweiler and Sabbah, this article studies the behaviour of various Hodge invariants by middle additive convolution with a Kummer module. The main result gives the behaviour of Hodge numerical data at infinity. We also give expressions for Hodge numbers and degrees of some Hodge bundles without making the hypothesis of scalar monodromy at infinity, which generalizes the results of Dettweiler and Sabbah.

R´esum´e(Comportement d’invariants de Hodge par convolution interm´ediaire)

— Suivant les travaux de Dettweiler et Sabbah, cet article s’int´eresse au comporte- ment d’invariants de Hodge par convolution interm´ediaire additive par un module de Kummer. Le r´esultat principal pr´ecise le comportement de donn´ees num´eriques de Hodge `a l’infini. Nous explicitons ´egalement le comportement des nombres de Hodge et des degr´es de certains fibr´es de Hodge sans faire l’hypoth`ese de monodromie scalaire

`

a l’infini, g´en´eralisant ainsi les r´esultats de Dettweiler et Sabbah.

The initial motivation to study the behaviour of various Hodge invariants by middle additive convolution is Katz’s algorithm [Kat96], which makes it possible to reduce a rigid irreducible local systemL on a punctured projective line to a rank-one local system. This algorithm is a successive application of tensor products with a rank-one local system and middle additive convolutions with a Kummer local system, and terminates with a rank-one local system.

We assume that the monodromy at infinity of L is scalar, so this property is preserved throughout the algorithm.

Nicolas Martin, Centre de math´ematiques Laurent Schwartz, ´Ecole polytechnique, Uni- versit´e Paris-Saclay, F-91128 Palaiseau cedex, France

E-mail :nicolas.martin@polytechnique.edu

2000 Mathematics Subject Classification. — 14D07, 32G20, 32S40.

Key words and phrases. — D-modules, middle convolution, rigid local system, Katz algo- rithm, Hodge theory.

If we assume that the eigenvalues of the local monodromies ofL have abso- lute value one, such a local system underlies a variation of polarized complex Hodge structure unique up to a shift of the Hodge filtration [Sim90, Del87], and this property is preserved at each step of Katz’s algorithm. The work of Dettweiler and Sabbah [DS13] is devoted to computing the behaviour of Hodge invariants at each step of the algorithm.

Our purpose in this article is to complement the previous work of Dettweiler and Sabbah without assuming that the monodromy at infinity is scalar, and to do that, we take up the notation introduced in [DS13,§2.2] and recalled in§1.1.

More precisely, our main result consists in making explicit the behaviour of the nearby cycle local Hodge numerical data at infinity by middle additive convo- lution with the Kummer module Kλ₀. Considering a regular holonomic D_A¹- moduleM verifying various assumptions, whose singularities at finite distance belong tox={x1, . . . , xr}, we denote by MCλ₀(M) =M∗midKλ₀ this convo- lution and show the following theorem (see§1.1 for the notation and assump- tions). In the following, we set γ0∈(0,1) such that exp(−2iπγ0) =λ06= 1.

Theorem 1. — Let M^min be the D_P¹-module minimal extension of M at in- finity. Given γ∈[0,1) andλ= exp(−2iπγ), we have:

ν_∞,λ,`^p (MC_λ0(M)) =



















ν_∞,λλ^p−1

0,`(M) ifγ∈(0,1−γ0) ν_∞,λλ^p

0,`(M) ifγ∈(1−γ0,1) ν_∞,λ^p

0,`+1(M) ifλ= 1

ν<sub>∞,1,`−1</sub><sup>p−1</sup> (M) ifλ=λ<sub>0</sub>, `≥1 h^pH¹(P¹,DRM^min) ifλ=λ₀, `= 0.

This result has applications beyond Katz’s algorithm since it enables us to give another proof of a theorem of Fedorov [Fed18], which completely deter- mines the Hodge numbers of the variations of Hodge structures corresponding to hypergeometric differential equations; this work is developed in [Mar18b].

In addition, we get general expressions for Hodge numbersh^pof the variation and degreesδ^pof some Hodge bundles (recalled in§1.1) which generalize those of Dettweiler and Sabbah. The results are the following.

Theorem 2. — The local invariantsh^p(MC_λ0(M))are given by:

h^p(MCλ₀(M)) = X

γ∈[0,γ₀)

ν_∞,λ^p (M) + X

γ∈[γ₀,1)

ν_∞,λ^p−1(M) +h^pH¹(A¹,DRM)−ν_∞,λ^p−1

0,prim(M).

Theorem 3. — The global invariants δ^p(MCλ₀(M))are given by:

δ^p(MCλ₀(M)) = δ^p(M) + X

γ∈[γ0,1)

ν_∞,λ^p (M)−

r

X

i=1

µ^p_xi,1(M) + X

γ∈(0,1−γ0)

µ^p−1_x

i,λ(M)

.

1. Hodge numerical data and modules of normal crossing type 1.1. Hodge invariants. — In this section, we recall the definition of local and global invariants introduced in [DS13, §2.2] (all references to [DS13] are made to the published paper). Let ∆ be a disc centered at 0 with coordinate t and let (V, F^•V,∇) be a variation of polarizable Hodge structure on ∆^∗ =

∆\ {0} of weight 0. We denote byM the correspondingD∆-module minimal extension at 0.

Nearby cycles. Fora∈(−1,0] andλ=e^−2iπa, the nearby cycle space at the originψλ(M) is equipped with the nilpotent endomorphism N =−2iπ(t∂t−a) and the Hodge filtration is such that NF^pψλ(M)⊂F^p−1ψλ(M). The mono- dromy filtration induced by N enables us to define the spaces P`ψλ(M) of prim- itive vectors, equipped with a polarizable Hodge structure (see [SS19, §3.1.a]

for more details). The nearby cycle local Hodge numerical data are defined by ν<sub>λ,`</sub><sup>p</sup> (M) :=h<sup>p</sup>(P`ψλ(M)) = dim gr^p_FP`ψλ(M),

with the relationν_λ^p(M) :=h^pψλ(M) = P

`≥0

`

P

k=0

ν_λ,`^p+k(M). We set ν_λ,prim^p (M) :=X

`≥0

ν_λ,`^p (M) and ν_λ,coprim^p (M) :=X

`≥0

ν<sub>λ,`</sub><sup>p+`</sup>(M).

Vanishing cycles. Forλ6= 1, the vanishing cycle space at the origin is given byφλ(M) =ψλ(M) and comes with N andF^pas before. Forλ= 1, the Hodge filtration onφ₁(M) is such thatF^pP_{`</sub>φ<sub>1</sub>(M) = N(F<sup>p</sup>P<sub>`+1}ψ₁(M)). Similarly to nearby cycles, the vanishing cycle local Hodge numerical data is defined by

µ^p_{λ,`</sub>(M) :=h<sup>p</sup>(P<sub>`}φ_λ(M)) = dim gr^p_FP_`φ_λ(M).

Degrees δ^p. For a variation of polarizable Hodge structure (V, F^•V,∇) on A¹\x, we denote byM the underlyingDA¹-module minimal extension at each point of x. The Deligne extensionV⁰ of (V,∇) onP¹ is contained inM, and we set

δ^p(M) = deg gr^p_FV⁰.

In this paper, we are mostly interested in the behaviour of the nearby cycle local Hodge numerical data at infinity by middle convolution with the Kummer moduleKλ₀ =D_A¹/D_A¹·(t∂t−γ0), withγ0∈(0,1) such that exp(−2iπγ0) =λ0. This operation is denoted by MCλ₀. Note thatKλ₀ is equipped with the trivial Hodge filtration with jump at zero : F^pKλ₀ =Kλ₀ forp≤0 and F^pKλ₀ = 0 forp≥1.

Assumptions. As in [DS13, Assumption 1.2.2(1)], we assume in what follows that M is an irreducible regular holonomic D_A¹-module, not isomorphic to (C[t],d) and not supported on a point.

1.2. Modules of normal crossing type. — Let us consider X a poly- disc in Cⁿ with analytic coordinates x1, ..., xn, D the divisor {x1· · ·xn = 0}

and M a coherent DX-module of normal crossing type (notion defined in [Sai90, §3.2]). For every α = (α1, ..., αn) ∈ Rⁿ, we define the sub-object Mα=Tn

i=1

S

k_i≥0ker(xi∂xi−αi)^ki ofM. There exists a finite setA⊂[−1,0)ⁿ such that Mα= 0 forα6∈A+Zⁿ. If we set M^alg:=⊕αMα, the natural mor- phismM^alg⊗_{C[x1,...,xn]h∂x}

1,...,∂_xniDX→M is an isomorphism.

To be precise, only the casen= 2 will occur in this paper. In the situations that we will consider, it will be possible to make the above decomposition explicit and then apply the general theory of Hodge modules of M. Saito. For a complete review of the 6 operations formalism forD-modules, see [Meb89].

2. Proof of Theorem 1

Steps of the proof. Let us begin to list the different steps of the proof:

1. We write the middle convolution MCλ₀(M) as an intermediate direct image by the sum map. By changing coordinates and projectivizing, we can consider the case of a proper projection.

2. We use a property of commutation between nearby cycles and projective direct image in the theory of Hodge modules, in order to carry out the local study of a nearby cycle sheaf.

3. To be in a normal crossing situation and use the results of the theory of Hodge modules, we perform a blow-up and make completely explicit the nearby cycle sheaf previously introduced (Lemma 2.3).

4. We take the monodromy and the Hodge filtrations into account, using the degeneration atE₁ of the Hodge to de Rham spectral sequence and the Riemann-Roch theorem (following [DS13]) to get the expected theorem.

Geometric situation. Let s : A¹x×A¹y → A¹t be the sum map. We can change the coordinates so thatsbecomes the projection onto the second factor and projectivize to get es : P¹x×P¹t → P¹t. We set x⁰ = 1/x and t⁰ = 1/t as coordinates on a neighbourhood of (∞,∞) ∈P¹x×P¹t, Mλ₀ =M Kλ₀ and Mλ₀= (Mλ₀)min(x⁰=0)the minimal extension ofMλ₀along the divisor{x⁰ = 0}.

A reasoning similar to that of [DS13, Prop 1.1.10] gives MCλ₀(M) =es+Mλ₀. With some abuse of notation, we still denote byMthe push-forward in the sense ofD-modules by the inclusionA¹x,→P¹xand we denote byM⁰its restriction to the affine chart centered at∞. A similar abuse of notation is made forM_λ0.

Let us specify the geometric situation that we will consider in the following, in which we blow up the point (∞,∞) inP¹x×P¹t. We setX = Bl_(∞,∞)(P¹x×P¹t), e: X →P¹x×P¹t and j :A¹x×A¹t ,→X the natural inclusion. There are two charts of the blow up : one given by coordinates (u1, v1)7→(t⁰ =u1v1, x⁰=v1) and the other one by (u2, v2)7→(t⁰ =v2, x⁰ =u2v2). The strict transform of the line {t⁰ = 0} is called P¹x, and the exceptional divisor is calledP¹exc. We denote by 0 ∈ P¹exc the point given by u2 = 0, 1 ∈ P¹exc the point given by u2= 1 and∞ ∈P¹x∩P¹exc. We have the following picture:

OnA¹x×A¹t, we have

Mλ₀ =MKλ₀ =M[t]⊗

C[x, t,(t−x)⁻¹],d_(x,t)+γ0

d(t−x) t−x

=

M[t,(t−x)⁻¹],∇(x,t)+γ₀d(t−x) t−x

.

On the affine chart centered at (∞,∞), we have M_λ0 =

M⁰[t⁰, t⁰⁻¹,(x⁰−t⁰)⁻¹],∇_(x0,t⁰)+γ₀

−dx⁰ x⁰ −dt⁰

t⁰ +d(x⁰−t⁰) x⁰−t⁰

.

Notation 2.1. — Let us set Nλ₀ = e⁺Mλ₀, Nλ₀ = (Nλ₀)_min(x⁰_◦e=0) and T^λ=ψ_t⁰_◦e,λNλ0 equipped with a nilpotent endomorphim denoted by N.

Lemma 2.2. — Mλ0[t⁰⁻¹] =e₊Nλ0[t⁰⁻¹].

Proof. — By definition of the minimal extension,Nλ0 is the image of the map j_†j⁺N_λ0 −→j₊j⁺N_λ0. For i6= 0,Hⁱe₊Nλ0 is supported on (∞,∞), thereby Hⁱe₊Nλ0[t⁰⁻¹] = 0. As the kernel ofH⁰e₊Nλ0−→M_λ0 is similarly supported on (∞,∞), we deduce that e₊Nλ0[t⁰⁻¹] is a submodule ofM_λ0.

We have e₊j₊j⁺N_λ0 = (e◦j)₊(e◦j)⁺M_λ0 and, as e is proper, we can write e₊j_†j⁺N_λ0=e_†j_†j⁺N_λ0= (e◦j)_†(e◦j)⁺M_λ0. ThenMλ0[t⁰⁻¹] is the image of the map e₊j_†j⁺N_λ0[t⁰⁻¹]−→e₊j₊j⁺N_λ0[t⁰⁻¹]. Outside of (∞,∞), Mλ0[t⁰⁻¹] and e₊Nλ0[t⁰⁻¹] are submodules of M_λ0 which are isomorphic. Now, we can consider the intersection of these two submodules ofM_λ0, with two morphisms from the intersection to each of them. The kernel and the cokernel of these two morphisms are a priori supported on (∞,∞), but ast⁰ is invertible, they are zero. Therefore, Mλ₀[t⁰⁻¹] and e+Nλ₀[t⁰⁻¹] are isomorphic.

Let us fixγ∈[0,1) andλ= exp(−2iπγ). Asesandeare proper, the nearby cycles functor is compatible with (se◦e)₊ [Sai88, Prop 3.3.17], so we get

ψ_∞,λ(MC_λ0(M)) =ψ_t⁰_,λ(es₊Mλ0) =ψ_t⁰_,λ(es₊e₊Nλ0) =se₊e₊T^λ. Lemma 2.3. — We set (M⁰)γ = ker(x⁰∂x⁰ −γ)^r acting on ψx⁰M⁰ for r 0 and(M⁰)^λ= (M⁰)_γ[x⁰, x⁰⁻¹]. Let us set

T₀^λ=

(M⁰)^λλ0[(x⁰−1)⁻¹],∇+γ0

−dx⁰

x⁰ + dx⁰ x⁰−1

in the chartP¹exc\ {∞}(with the coordinatex⁰ instead ofu2) and denote simi- larly its meromorphic extension at infinity, with the action of the nilpotent endomorphism x⁰∂x⁰ −(γ+γ0). For the extension by zero (instead of mero- morphic), we use the notation (T₀^λ)⁰. Then:

(Case 1) For λ6∈ {1, λ0},T^λ is supported on P¹exc andT^λ=T₀^λ.

(Case 2) Forλ= 1,T^λ is supported onP¹exc and is isomorphic to the minimal extension of T₀¹ atx⁰ = 0.

(Case 3) For λ = λ0, T^λ is supported on P¹x∪P¹exc and comes in an exact sequence

0→(T₀^λ0)⁰→T^λ→T₁→0

compatible with the nilpotent endomorphism, where T₁ is supported onP¹x and is isomorphic to the meromorphic extension ofM at infinity with the action by 0 of the nilpotent endomorphism.

Proof. — We make a local study of the problem, reasoning in the three follo- wing charts:

(i) in the chart (u2, v2), calledChart 1;

(ii) in the neighbourhood of P¹x\ {∞}, called Chart 2; (iii) in the neighbourhood of ∞, calledChart 3.

The cases of Charts 1 and 2 do not contain any significant problem and are treated in [Mar18a, 4.2.4]. In Chart 1, we find

ψv₂,λNλ₀ =

( T₀^λ ifλ6= 1 (T₀^λ)_min({0}) ifλ= 1.

In Chart 2, in which one can use the coordinates (x, t⁰), we find ψt⁰,λMλ₀=

( 0 ifλ6=λ0

M ifλ=λ0.

Let us make the case of Chart 3 precise. Forα= (α1, α2)∈R², we set (N_λ0)_α= [

r₁≥0

ker(u₁∂_u1−α₁)^r1∩ [

r₂≥0

ker(v₁∂_v1−α₂)^r2.

By writing the expression of the connection in coordinates (u1, v1), we get that (Nλ₀)_{(−γ0,α−γ0)}can be identified with (M⁰)α with actions ofu1∂u₁ andv1∂v₁

respectively expressed as −γ0Id andx⁰∂x⁰−γ0Id. Hereψt⁰◦e,λNλ₀ =ψg,λNλ₀

whereg=u1v1, and we are in the situation of a calculation of nearby cycles of a coherentC[u1, v1]h∂u₁, ∂v₁i-module of normal crossing type alongD={g= 0}

where g=u1v1 is a monomial function. The general question is developed in [Sai90, §3.a] (see also [SS19, §13.3]); let us make it precise in the particular case at hand. We consider the commutative diagram

X  ^ig //

g ##

X×Cz

p2

C^z where i_g is the graph embedding.

Then we can see that (i_g)₊Nλ0 = Nλ0[∂_z] = L

k≥0 (Nλ0 ⊗∂_z^k) is a left C[u1, v1, z]h∂u₁, ∂v₁, ∂zi-module equipped with the following actions:

(i) action ofC[u1, v1] : f(u1, v1)·(m⊗∂_z^k) = (f(u1, v1)m)⊗∂_z^k. (ii) action of∂_z : ∂_z(m⊗∂^k_z) =m⊗∂_z^k+1.

(iii) action of∂_u1 : ∂_u1(m⊗∂_z^k) = (∂_u1m)⊗∂_z^k−v₁m⊗∂_z^k+1. (iv) action of∂_v1 : ∂_v1(m⊗∂_z^k) = (∂_v1m)⊗∂_z^k−u₁m⊗∂_z^k+1. (v) action of z: z·(m⊗∂^k_z) =u₁v₁m⊗∂_z^k−km⊗∂_z^k−1.

Let us denote bySu(resp. Sv) the operator defined bySu(m⊗∂_z^k) = (u1∂u₁m)⊗

∂_z^k (resp. Sv(m⊗∂_z^k) = (v1∂v₁m)⊗∂_z^k). WithE=z∂z, we get the relations u₁∂_u1(m⊗∂^k_z) = (S_u−E−(k+ 1))(m⊗∂_z^k)

and

v1∂v1(m⊗∂_z^k) = (Sv−E−(k+ 1))(m⊗∂^k_z).

Letting V^•(Nλ0[∂_z]) denote the V-filtration with respect to z, we have T = ψg,λNλ₀ = gr^γ_V(Nλ₀[∂z]). We have the decompositionsN_λ^alg₀ =L

α∈R²(Nλ₀)α

and (T^λ)^alg = L

β∈R²Tβ, and by arguing in a way similar to [SS19, Lemma 13.2.26] (with left modules), we show that the only indicesβ that appear are those such thatα=β+(γ+k+1)(1,1) forαin the decomposition ofN_λ^alg₀ and k∈Z. In particular, we cannot haveα2∈Zand having a minimal extension along{v1= 0}does not play any role here. In other words, we can identify in this partNλ₀ andNλ₀.

More precisely, forβ₁, β₂≥ −1, we deduce from [Sai90, Th. 3.3] (or [SS19, Cor.

13.2.32]) the following expressions for T_β:

Tβ=











0 ifβ₁6=−1, β₂6=−1 coker(S_u−E ∈End((N_λ0)_γ,β2+γ+1[E])) ifβ₁=−1, β26=−1 coker(Sv−E∈End((Nλ₀)β₁+γ+1,γ[E])) ifβ16=−1, β2=−1 coker((Su−E)(Sv−E)∈End((Nλ₀)γ,γ[E])) ifβ= (−1,−1).

In what follows, we will work with the point of view of quivers. A good reference can be found in Chapters 3.1.b and 9.4 of [SS19]. LetA⊂(−1,0] be the (finite) set of those α∈ (−1,0] such that (M⁰)α 6= 0 and let us look at the different cases for β∈[−1,0]² :

(i) Forβ26=−1, we haveT_(−1,β2)6= 0 if and only if γ=−γ0 andβ2∈A.

(ii) For β1 6= −1, we have T_(β1,−1) 6= 0 if and only if γ = α−γ0 with α ∈ A modZandβ1=−αmodZ.

(iii)T_(−1,−1)6= 0 if and only ifγ=−γ0 and 0∈A.

We deduce from these relations that:

(Case 1+2) If γ6=−γ0 thenT^λ is supported on P¹exc and, according to (ii), is determined by the only data of coker(Sv−E∈End((Nλ₀)_−γ0,γ[E])) equipped with an action ofE−γ, that we can identify with (Nλ₀)_−γ0,γ where the action of E−γ can be identified with S_v−γ. Consequently, T^λ is determined by (M⁰)_γ+γ0 with an action ofx⁰∂_x⁰−(γ+γ₀).

(Case 3) Ifγ=−γ0 thenT^λis determined by two data:

•firstly by the data, according to (i), of coker(Su−E∈End((Nλ0)−γ0,α−γ0[E])) for α ∈ A modZ, α 6∈ Z, supported on P¹x and equipped with an action of E+γ₀. We can identify it with (N_λ0)_{−γ0,α−γ0} where the action ofE+γ₀ is identified with S_u+γ₀, that we identify with (M⁰)_α with an action by 0.

• secondly by the data of the biquiver T_(−1,−1) ^v1 //

u1

T_(−1,0)

∂_v1

oo

T_(0,−1)

∂u1

OO

As u⁻¹₁ acts on (N_λ0)_{−γ0+1,−γ0}[E] and v⁻¹₁ on (N_λ0)_{−γ0,−γ0+1}[E], it can be assumed that T_(−1,−1), T_(0,−1) and T_(−1,0) are all three cokernels of maps of End((N_λ0)_{−γ0,−γ0}[E]). Now we set C_uv = coker(S_u−E)(S_v−E), C_u = coker(S_u −E) and C_v = coker(S_v −E), and the previous biquiver can be identified with the following

Cuv ϕv ////

ϕu

Cu Sv−E

oo

Cv S_u−E

OO

where ϕu : Cuv → Cv is induced by the inclusion im(Su−E)(Sv −E) ⊂ im(Sv −E), and the same for ϕv. As Su −E ∈ End((Nλ₀)_{−γ0,−γ0}[E]) is injective (because Su is identified on (M⁰)0with−γ0Id) and

im(Su−E:Cv→Cuv) = im(Su−E)

im(S_u−E)(S_v−E) = kerϕv, we deduce the following exact sequence:

0→Cv →Cuv →Cu→0.

Therefore, we have an exact sequence of biquivers:

0 −→ C_v //

Su−E

0 −→

oo C_uv

ϕ_v ////

ϕ_u

C_u −→

S_v−E

oo C_u ^Id //

C_u −→ 0

S_v−E

oo

Cv Id

OO

Cv Su−E

OO

0

OO

• The left biquiver is a quiver representing the extension by zero supported on P¹exc and identified with (Nλ₀)_{−γ0,−γ0} where the action of E+γ0 can be identified with Sv+γ0, in other words (M⁰)0 with the action ofx⁰∂x⁰. This is the following biquiver:

(M⁰)₀ //

−x⁰∂_x0

oo 0

(M⁰)0 Id

OO

• The right biquiver is a quiver representing the meromorphic extension sup- ported on P¹x and identified with (Nλ0)−γ0,−γ0 where the action of E+γ0 is equal to 0, in other words (M⁰)₀ with the action by 0. This is the following biquiver:

(M⁰)0

Id //

(M⁰)0 x⁰∂_x0

oo

0

OO

• It is possible to make the central biquiver explicit in terms of (M⁰)₀, insofar as we can identifyC_uv with ((M⁰)₀)²with an action of

E+γ0=

γ0Id γ0(x⁰∂x⁰−γ0Id) Id x⁰∂x⁰−γ0Id

,

and we get the following biquiver:

((M⁰)0)²

pv ////

pu

(M⁰)0 (x⁰∂_x0−γ0Id,−Id)

oo

(M⁰)0 (−γ0Id,−Id)

OO

wherepu=p1+ (x⁰∂x⁰−γ0)p2andpv=p1−γ0p2, withp1 the projection onto the first factor and p2 onto the second factor.

Finally, forλ=λ0 we have an exact sequence 0→(T₀^λ0)⁰→T^λ→T1→0

where (T₀^λ0)⁰ is the extension by zero of T₀^λ0 at infinity equipped with the nilpotent endomorphism x⁰∂x⁰, and T1 is supported on P¹x and given by the meromorphic extension ofM at infinity equipped with the nilpotent endomor- phism 0.

By gluing the expressions obtained for the different values of λin each of the three charts, we get the announced result of the lemma.

By construction, the complexK^•=se+e+T^λ has cohomology in degree zero only. More precisely, asse◦e:P¹exc∪P¹x→ {pt}, that amounts to saying that RΓ(P¹exc∪P¹x,DR^anT^λ) is a two-term complex with a kernel reduced to zero.

If we take the monodromy filtration M• into account, we have the following more precise result:

Lemma 2.4. — H^j(gr^M<sub>`</sub> K<sup>•</sup>) = 0forj6= 0 and`∈Z.

Proof. — (Case 1+2) Ifλ6=λ₀, thenT^λis supported onP¹exc and localized at infinity. Consequently, we can see T^λ as a C[x⁰]h∂x⁰i-module, that is a C[x⁰]- module with a connection∇∂_x0. The question is to show that the morphism

gr^M_{`</sub> T<sup>λ∇</sup>−→<sup>∂x0</sup> gr<sup>M</sup><sub>`} T^λ

has a kernel reduced to zero. With N the nilpotent endomorphism, let us set T^λ=

(M⁰)^λλ0,∇∂_x0 = ∂

∂x⁰ + γ x⁰Id + N

x⁰

where T^λ is minimally extended at 0 ifλ= 1, and 1_γ0 =

C[x⁰,(x⁰−1)⁻¹], ∂

∂x⁰ + γ₀ x⁰−1Id

so thatT^λ=T^λ⊗1γ₀. Form∈T^λand m⁰∈C[x⁰,(x⁰−1)⁻¹], we have

∇_∂x0(m⊗m⁰) =∇_∂x0(m)⊗m⁰+m⊗ ∂

∂x⁰ + γ₀ x⁰−1Id

m⁰.

Now, we have gr^M_{`</sub> T<sup>λ</sup>= gr<sup>M</sup><sub>`} T^λ⊗1γ₀ and we want to show that ker

gr^M_{`</sub> T<sup>λ</sup>⊗1<sub>γ0</sub><sup>∇</sup>−→<sup>∂x0</sup> gr<sup>M</sup><sub>`} T^λ⊗1_γ0

= 0.

Form∈gr^M_` T^λ, let us notice the equality

∇∂_x0

m⊗ 1 (x⁰−1)^k

=∇∂_x0(m)⊗ 1

(x⁰−1)^k − km

(x⁰−1)^k+1 + γ0m (x⁰−1)^k+1, from which we deduce, asγ₀6∈Z, thatm⊗(x⁰−1)^−k has a pole at 1 of order k+ 1 ifm6= 0. Therefore, if an elementm⊗m⁰is such that∇∂_x0(m⊗m⁰) = 0, thenm= 0.

(Case 3) If λ=λ₀, we have the exact sequence 0→T₀→T^λ→T₁→0 where T0= (T₀^λ0)⁰ is supported onP¹exc andT1 is supported onP¹x. Equivalently, we can think in terms of primitive parts instead of graded parts, as we shall do.

The reasoning of the previous case applies in the same way toK₀^• =es₊e₊T₀, yielding that the complexes P<sub>`</sub>K<sub>0</sub><sup>•</sup> are concentrated in degree 0 for all `∈ N.

As N is zero on T1, we have N(T^λ)⊂T0. Let us show we have equality and, for that, let us go back to the description ofT^λat the neighbourhood of∞in terms of biquivers:

((M⁰)0)²

pv ////

p_u

(M⁰)0 (x⁰∂_x0−γ0Id,−Id)

oo

(M⁰)0 (−γ0Id,−Id)

OO

The action ofE+γ₀on ((M⁰)₀)²is given by E+γ₀=

γ0Id γ0(x⁰∂x⁰−γ0Id) Id x⁰∂_x⁰−γ₀Id

, whose rank is equal to the dimension of (M⁰)0, and thus

(E+γ0)((M⁰)0)²={(γ0m, m)∈((M⁰)0)² |m∈(M⁰)0} '(M⁰)0. Now, a calculation shows that the following diagram

((M⁰)₀)^{2 p2◦(E+γ0)}////

p_u

(M⁰)₀

−x⁰∂_x0

(M⁰)₀

(−γ0Id,−Id)

OO

−x⁰∂_x0 //// (M⁰)₀

Id

OO

is commutative, so the image by N of the biquiver representing T^λ gives the biquiver representing T0, in other words N(T^λ) =T0. Consequently, we are in a situation of a minimal extension quiver:

T^λ

N ////

T0.

? _

oo

We deduce from [KK87, Prop. 2.1.1(iii)] that P`T<sup>λ</sup> 'P`−1T0 for ` ≥1, and thus the complexes P`K^•are concentrated in degree 0 for all`≥1. Moreover, as the total complex K^• is concentrated in degree 0, the complexP0K^• is also concentrated in degree 0. We have the same property for graded parts instead of primitive parts.

Now, with these two lemmas, we are able to show the main theorem. We can equip T^λ with a Hodge filtration by properties of nearby cycles, and (M⁰)^λλ0 as well, in such a way that the formula of Lemma 2.3 is compatible with the Hodge filtrations. Note that these two objects are not variations of polarized Hodge structures, but variations of mixed Hodge structures. However, one can adapt the arguments of [DS13] to the generalized context below.

Remark 2.5. — Let us begin making the Hodge filtration more explicit. In the chart (u2, v2), let us consider theD-modulee⁺(M⁰C[t⁰, t⁰⁻¹]), on which u2 andv2 act in an invertible way. This is

e^∗(M⁰C[t⁰, t⁰⁻¹]) =C[u2, u⁻¹₂ , v2, v₂⁻¹]⊗_C[x0,x⁰⁻¹]M⁰, on which the action of the connection is induced by

u₂∂_u2(1⊗m) =v₂∂_v2(1⊗m) =x⁰∂_x⁰(1⊗m).

We consider the localized Hodge filtration Fe^pM⁰ = C[x⁰, x⁰⁻¹]⊗_C[x⁰] F^pM⁰, Fe^p(M⁰ C[t⁰, t⁰⁻¹]) = (Fe^pM⁰) C[t⁰, t⁰⁻¹] and its pull-back Fe^pe^∗(M⁰ C[t⁰, t⁰⁻¹]) =C[u2, u⁻¹₂ , v2, v⁻¹₂ ]⊗_C[x⁰_,x⁰⁻¹_]Fe^pM⁰. In order to compute nearby cycles alongv2, we make use of theV-filtrationV^•(e⁺(M⁰C[t⁰, t⁰⁻¹])) along v2, that we will compare with the V-filtration of M⁰ along x⁰ denoted by V^•(M⁰). The above relations link the Bernstein polynomial with respect to v2

with that with respect tox⁰ and lead to the identification

C[u2, u⁻¹₂ ]⊗_C[u2]V^a(e⁺(M⁰C[t⁰, t⁰⁻¹])) =C[u2, u⁻¹₂ , v2]⊗_C[x⁰_]V^aM⁰. For λ = e^−2iπa, let us define the functor ψev₂,λ as the functor ψv₂,λ followed by localization C[u2, u⁻¹₂ ]⊗_C[u₂] •. We thus find an isomorphism of filtered C[u₂, u⁻¹₂ ]hu2∂_u2i-modules

(ψe_v2,λe⁺(M⁰C[t⁰, t⁰⁻¹]),Fe^•ψe_v2,λe⁺(M⁰C[t⁰, t⁰⁻¹]))

'C[u2, u⁻¹₂ ]⊗_C(ψx⁰,λM⁰, F^•ψx⁰,λM⁰), since, for a ∈ (−1,0], the filtration induced by Fe^•M⁰ on gr^a_VM⁰ is equal to that induced by F^•M⁰, and where the action ofu2∂u₂ on the right-hand side is induced by u2∂u₂(1⊗m) = 0.

Considering instead the pull-back of M_λ0 amounts to twisting e⁺(M⁰ C[t⁰, t⁰⁻¹]) by the pull-back connection e⁺(C[x⁰, x⁰⁻¹]Kλ0). This leads to also invert the action ofu₂−1 and to localize theFe-filtration along the divisor u2 = 1. On the other hand, it twists the monodromies around u2 = 0 and v2 = 0 by λ0. The localization of the Hodge filtrationF^•T^λ is by definition the localization of the filtration induced by Fe^•Nλ₀ onψv₂,λNλ₀. As a conse- quence, if we setTe^λ=C[u₂, u⁻¹₂ ,(u₂−1)⁻¹]⊗_C[u₂]T^λthat we endow with the localized filtration Fe^•Te^λ = C[u2, u⁻¹₂ ,(u2−1)⁻¹]⊗_C[u2]F^•T^λ, we obtain an isomorphism (notation of Lemma 2.3)

(2.6) (Te^λ,Fe^•Te^λ)'(T₀^λ,Fe^•T₀^λ),

where Fe^•T₀^λ is defined byFe^pT₀^λ=C[u₂, u⁻¹₂ ,(u₂−1)⁻¹]⊗_C(F^pψ_x⁰_,λλ0M⁰).

Proof of Theorem 1. — (Case 1) Let us begin with the caseλ6∈ {1, λ0} and, as a first step, without taking the monodromy filtration into account. As se◦e:P¹exc→ {pt}, we haveψ_∞,λ(MCλ₀(M)) =es+e+T^λ =RΓ(P¹exc,DRT^λ).

Setting x ={0,1,∞} and DRT^λ =i_∗V, wherei : P¹exc\x,→ P¹exc denotes the open inclusion and V is a non-constant local system onP¹exc\x, we have H^m(P¹exc,DRT^λ) =H^m(P¹exc, i_∗V). This quantity is zero form = 0 because there is no section supported on a point, andV has no global section since one of its local monodromies has not 1 as an eigenvalue. Using the Poincar´e duality theorem, we have alsoH²(P¹exc, i_∗V) = 0 if we argue with the dualV^∨. Then

ν_∞,λ^p (MCλ₀(M)) = dim gr^p_Fψ_∞,λ(MCλ₀(M)) = dim gr^p_FH¹(P¹exc,DRT^λ).

Adapting [DS13, Prop. 2.3.3] (we use the formula given at the end of their proof, without usingν_∞,1,prim^p−1 (T^λ)), we have

ν_∞,λ^p (MC_λ0(M)) =δ^p−1(T^λ)−δ^p(T^λ)−h^p(T^λ)−h^p−1(T^λ) (2.7)

+X

x∈x

X

µ6=1

ν_x,µ^p−1(T^λ)

| {z }

=h^p−1(T^λ)

+µ^p_x,1(T^λ)

| {z }

=0

! .

=δ^p−1(T^λ)−δ^p(T^λ)−h^p(T^λ) + 2h^p−1(T^λ)

since we remark that for each of the three singular points of T^λ, we have a single eigenvalue for the local monodromy (different from 1) and only one term ν_x,µ^p−1(T^λ) is non zero, hence equal toh^p−1(T^λ).

Now, asλλ06= 1, we haveH¹(P¹x,DR(M⁰)^λλ0) = 0. The same reasoning with (M⁰)^λλ0 instead ofT^λ, which has two singularities, gives

(2.8) 0 =δ^p−1((M⁰)^λλ0)−δ^p((M⁰)^λλ0)−h^p((M⁰)^λλ0) +h^p−1((M⁰)^λλ0).

In accordance with Lemma 2.3, we have

T^λ= (M⁰)^λλ0⊗

C[x⁰, x⁰⁻¹,(x⁰−1)⁻¹],d +γ0

−dx⁰

x⁰ + dx⁰ x⁰−1

,

and then we can infer, according to [DS13, 2.2.12], thath^p(T^λ) =h^p((M⁰)^λλ0).

Let us apply the formula for the twist by a unitary rank-one connection given in [DS13, Prop. 2.3.2]:

(2.9) δ^p(T^λ) =δ^p((M⁰)^λλ0)−h^p((M⁰)^λλ0) + X

α∈[γ0,1)

ν_∞,e^p −2iπα((M⁰)^λλ0)

+ X

α∈[1−γ0,1)

ν^p_1,e−2iπα((M⁰)^λλ0)

| {z }

=0 becauseα6=0

.

The remaining sum can be expressed as X

α∈[γ0,1)

ν_∞,e^p −2iπα((M⁰)^λλ0) =

(h^p((M⁰)^λλ0) ifγ∈(0,1−γ0) 0 ifγ∈(1−γ0,1), so we deduce that

(2.10) δ^p(T^λ) =

( δ^p((M⁰)^λλ0) ifγ∈(0,1−γ0) δ^p((M⁰)^λλ0)−h^p((M⁰)^λλ0) ifγ∈(1−γ0,1).

We are now able to apply the formula (2.7):

(i) Forγ∈(0,1−γ₀), we have:

ν^p_∞,λ(MCλ₀(M)) =δ^p−1(T^λ)−δ^p(T^λ)−h^p(T^λ) + 2h^p−1(T^λ)

=δ^p−1((M⁰)^λλ0)−δ^p((M⁰)^λλ0)−h^p((M⁰)^λλ0) + 2h^p−1((M⁰)^λλ0)

=h^p−1((M⁰)^λλ0) (according to 2.8).

(ii) Forγ∈(1−γ0,1), we have:

ν_∞,λ^p (MC_λ0(M)) =

δ^p−1((M⁰)^λλ0)−h^p−1((M⁰)^λλ0)

−

δ^p((M⁰)^λλ0)−h^p((M⁰)^λλ0)

−h^p((M⁰)^λλ0) + 2h^p−1((M⁰)^λλ0)

=δ^p−1((M⁰)^λλ0)−δ^p((M⁰)^λλ0) +h^p−1((M⁰)^λλ0)

=h^p((M⁰)^λλ0) (according to 2.8).

We deduce from the isomorphism given in 2.6 thath^p(T^λ) =ν_∞,λλ^p

0(M), then h^p((M⁰)^λλ0) =ν_∞,λλ^p

0(M).

To sum up, we have

(2.11) ν_∞,λ^p (MCλ₀(M)) =

(ν_∞,λλ^p−1

0(M) ifγ∈(0,1−γ₀) ν_∞,λλ^p

0(M) ifγ∈(1−γ0,1).