Introduction

1.a. The irregular Hodge filtration. The category of mixed Hodge modules on complex manifolds, as constructed by M. Saito [Sai90], is endowed with the standard operations (push-forward by projective morphisms, pull-back by holomorphic maps, duality, etc.). In particular, the structure of the Hodge filtration in this category is well-behaved through these operations. For a meromorphic function f on a complex manifold X, holomorphic on the complement U of a divisor D of X, and for a mixed Hodge module with underlying filtered DX-module (N, F•N), we will define an “ir-regular Hodge filtration”, which is a filtration on the exponentially twisted holonomic D_X-module N ⊗ E^f, where E^f denote the OX-module OX(∗D) equipped with the twisted integrable connection d + df , that we regard as a left holonomic DX-module.

We note that, although N is known to have regular singularities, N ⊗ E^f has irregular singularities along the components of the divisor D where f takes the value ∞, hence cannot underlie a mixed Hodge module. Therefore, the irregular Hodge filtration we define on N ⊗ E^f, generalizing the definition of Deligne [Del07], and then [Sab10], [Yu14], [ESY15], cannot be the Hodge filtration of a mixed Hodge module in the sense of [Sai90]. There is an algebraic variant of this setting, where we assume that f is a rational function on a complex smooth variety X.

Remark 1.1. Such a filtration has been constructed in [ESY15] in the following cases:

(a) f extends as a morphism X → P¹, D is a normal crossing divisor and the filtered DX-module (N, F•N) is equal to (OX(∗D), F•O_X(∗D)), where the filtration is given by the order of the pole [Del70]. In such a setting, the filtration was denoted F_•(E^f(∗H)), where H is the union of the components of D not in f⁻¹(∞);

(b) X = Y × P¹, f is the projection to P¹ and (N, F•N) underlies an arbitrary mixed Hodge module.

Definition 1.2. By a good filtration F_• indexed by Q of a DX-module N, we mean a finite family Fα+•Nof good filtrations⁽¹⁾ indexed by Z, parametrized by α in a finite subset A of [0, 1) ∩ Q, such that Fα+pN ⊂ Fβ+qN for all α, β ∈ A and p, q ∈ Z satisfying α + p 6 β + q.

We can thus regard it as a single increasing filtration indexed by Q, such that Fα+pN/F<α+pN= 0 for any α, p, except for α in a finite set A of [0, 1) ∩ Q.

For each α, the Rees module RF_α+•Nis the graded module defined asP

pFα+pNz^p, where z is a new (Laurent) polynomial variable. Then we set RF_•N:=L

α∈ARF_α+•N. We can then regard a (usual) good filtration indexed by Z as a good filtration indexed by Q.

Theorem 1.3. Let f be a meromorphic function on X, holomorphic on U = X r D, where D is a divisor in X. For each filtered holonomic DX-module (N, F•N) un-derlying a mixed Hodge module one can define canonically and functorially a good F_•D_X-filtration F_•^irr(N ⊗ E^f) indexed by Q which satisfies the following properties:

(1) Through the canonical isomorphism (N⊗E^f)|U = N|U, we have F_•^irr(N⊗E^f)|U = F•N_|U.

(1)As usual, this is understood with respect to the filtration by the order F_•D_X, and goodness means that Fα+pN = 0 p ≪ 0locally on X, and gr^F_α+•Nis gr^FD_X-coherent.

(2) For each morphism ϕ : (N1, F•N₁) → (N2, F•N₂) underlying a morphism of mixed Hodge modules, the corresponding morphism

ϕ^f : (N1⊗ E^f, F_•^irr(N1⊗ E^f)) −→ (N2⊗ E^f, F_•^irr(N2⊗ E^f)) is strictly filtered.

(3) For each α ∈ [0, 1), the push-forward π+(N ⊗ E^f, F_α+^irr_•(N ⊗ E^f)) by any pro-jective morphism π : X → Y is strict.

(4) Let π : X → Y be a projective morphism and let h be a meromorphic func-tion on Y , holomorphic on V = Y r DY for some divisor DY in Y . Assume that DX := π⁻¹(DY) is a divisor, and set U = π⁻¹(V ) and f = h ◦ π. Then the co-homology of the filtered complex π+(N ⊗ E^f, F_•^irr(N ⊗ E^f)), which is strict by (3), satisfies

H^jπ+R_F^irr(N ⊗ E^f) = R_F^irr

(H^jπ+N) ⊗ E^h .

(5) In cases 1.1(a) and 1.1(b) above, the filtration F_•^irr coincides with the filtration F_•^Del constructed in [ESY15].

The proof of the theorem is given in §5.a and relies much on the theory of mixed twistor D-modules of T. Mochizuki [Moc11b]. This theory allows one to simplify and generalize some of the arguments given in [ESY15], by giving a general framework to treat, from the Hodge point of view, irregular D-modules like E^f. By specializing (3) to the case where Y is a point we obtain:

Corollary 1.4. For (N, F•N) underlying a mixed Hodge module on a smooth projec-tive variety X, the spectral sequence attached to the hypercohomology of the filtered de Rham complex F_α+^irr_•DR(N ⊗ E^f) degenerates at E1 for each α ∈ [0, 1).

Remark 1.5. The assumption that D := X r U is a divisor is not mandatory, but simplifies the statement. In general, higher cohomology modules supported on X r U may appear for N ⊗ E^f.

1.b. Rescaling a function. The case 1.1(a) is essentially the only case where we can give an explicit expression for F_•^irrE^f(∗H) (see [ESY15], according to 1.3(5)). Recall that we consider a smooth complex projective variety X together with a morphism f : X → P¹. We set Pred = f⁻¹(∞) and P = f^∗(∞). We also introduce a supple-mentary divisor H (which could be empty) having no common components with Pred, and we assume that D := Pred∪ H has normal crossings. We set U = X r D. We will also consider X as a complex projective manifold equipped with its analytic topology, that we will denote X^an when the context is not clear.

Our main example in this article, that we consider in Part II, is that of the rescaling of the function f : X → P¹. The rescaled function with rescaling parameter v is the function vf : U × Cv→ C, defined by (x, v) 7→ vf (x). This function does not extend as a morphism to X × P¹→ P¹.

We consider the projective line P¹_vcovered by two charts Cvand Cuwhose intersec-tion is denoted by C^∗_v, and we regard vf as a rational function vf : X × P¹_v - - → P¹. We are therefore in the situation in the beginning of the previous subsection, with underlying space X := X × P¹_v and reduced pole divisor Pred:= (Pred× P¹_v) ∪ (X × ∞), where ∞ ∈ P¹_v denotes the point u = 0. We will also set P = (P × P¹) + (X × {∞}), H= H × P¹_v and D = Pred∪ H.

We denote by E^(v:u)f(∗H) the OX×P¹_v-module OX×P¹_v(∗D) equipped with the con-nection d + d(vf ) (on the open set X × Cv) and d + d(f /u) (on the open set X × Cu).

We denote its restriction to the corresponding open subsets by E^vf(∗H) and E^{f /u}(∗H) respectively. According to Theorem 1.3, it is equipped with an irregular Hodge filtra-tion. We make it partly explicit in Theorem 9.1 (only partly, because around u = 0 we only make explicit its restriction to the Brieskorn lattice, see §9.b).

1.c. Variation of the irregular Hodge filtration and the Kontsevich bundles Regarding now v ∈ C^∗ as a parameter and considering the push-forward by q : X × P¹_v→ P¹_v of the rescaling E^(v:u)f(∗H), our aim is to describe the variation with v of the irregular Hodge filtration F_•^irrH^k U, (Ω^•_U, d + vdf )

considered in [ESY15], and its limiting behaviour when v → 0 or v → ∞.

The irregular Hodge filtration is conveniently computed with the Kontsevich com-plex. Recall that M. Kontsevich has associated to f : X → P¹as in §1.b and to k > 0 the subsheaf Ω^k_f of Ω^k_X(log D) consisting of logarithmic k-forms ω such that df ∧ ω re-mains a logarithmic (k + 1)-form, a condition which only depends on the restriction of ω to a neighbourhood of the reduced pole divisor Pred= f⁻¹(∞). For each α ∈ [0, 1), let us denote by [αP ] the divisor supported on Predwith multiplicity [αei] on the com-ponent Pi of P := f^∗(∞) with multiplicity ei. One can also define a subsheaf Ω^k_f(α) of Ω^k_X(log D)([αP ]) by the condition that df ∧ ω is a section of Ω^k+1_X (log D)([αP ]), so that the case α = 0 is that considered by Kontsevich. Clearly, only those α such that αei∈ Z for some i are relevant. If f is the constant map, then Ω^k_f = Ω^k_X(log D). One of the main results of [ESY15], suggested and proved by Kontsevich when P = Pred, is the equality, for each k,

dim H^k U, (Ω^•U, d + df )

= X

p+q=k

dim H^q(X, Ω^p_f(α)).

More precisely, for each pair (u, v) ∈ C² and each α ∈ [0, 1) one can form a com-plex (Ω^•_f(α), ud + vdf ) and it is shown that the dimension of the hypercohomology H^k X, (Ω^•_f(α), ud + vdf )

is independent of (u, v) ∈ C² and α and is equal to the above value. The irregular Hodge numbers are then defined as

(1.6) h^p,q_α (f ) = dim H^q(X, Ω^p_f(α)).

We have h^p,q_α (f ) 6= 0 only if p, q > 0 and p + q 6 2 dim X. (see Remark 8.20(3) for the mirror symmetry motivations related to the irregular Hodge filtration.) If f is the constant map, we recover the results of Deligne [Del70, Del71]:

dim H^k(U, C) = dim H^k U, (Ω^•_U, d)

= dim H^k X, (Ω^•_X(log D), d)

= X

p+q=k

dim H^q X, Ω^p_X(log D)
.

The Hodge numbers reduce here to h^p,q(X, D) = dim H^q X, Ω^p_X(log D)
.

Following the suggestion of M. Kontsevich, let us define the Kontsevich bun-dles K^k(α) on P¹_v. We set

K_v^k(α) := H^k X, (Ω^•f(α)[v], d + vdf )
, K^k

u(α) := H^k X, (Ω^•_f(α)[u], ud + df )
(1.7) .

Using the isomorphism C[u, u⁻¹]−→ C[v, v^{∼ −1}] given by u 7→ v⁻¹, we have a natural quasi-isomorphism

(1.8) u^•: (Ω^•_f(α)[v, v⁻¹], d + vdf )−→ (Ω^{∼ •}_f(α)[u, u⁻¹], ud + df )

induced by the multiplication by u^p on the pth term of the first complex. Since we know by the above mentioned results that both modules K_v^k(α), K_u^k(α) are free over their respective ring C[v] or C[u], the identification

H^k(u^•) : H^k X, (Ω^•_f(α)[v, v⁻¹], d + vdf )

≃ H^k X, (Ω^•_f(α)[u, u⁻¹], ud + df )
allows us to glue these modules as a bundle K^k(α) on P¹_v. The E1-degeneration property can be expressed by the injectivity

(1.9)

H^k X, σ^>p(Ω^•_f(α)[v], d + vdf )

֒−→ H^k X, (Ω^•_f(α)[v], d + vdf )
, H^k X, σ^>p(Ω^•_f(α)[u], ud + df )

֒−→ H^k X, (Ω^•_f(α)[u], ud + df )
, where σ^>pdenotes the stupid truncation. Since this truncation is compatible with the gluing u^•, this defines a filtration σ^>pK^k(α). When restricted to C^∗_v, this produces the family F_α^irr,pH^k U, (Ω^•_U, d + vdf )

.

We also notice that the pth graded bundle is then isomorphic to OP¹(p)^{hp,k−pα (f )}, so this filtration is the Harder-Narasimhan filtration F^•K^k(α) and the Birkhoff-Grothendieck decomposition of K^k(α) reads

(1.10) K^k(α) ≃

Lk p=0

O_P1(p)^{hp,k−pα (f )}. In particular, all slopes of K^k(α) are nonnegative and we have

deg K^k(α) = Xk p=0

p · h^p,k−p_α (f ).

We will show (see Lemma 6.2) that each K^k(α) is naturally equipped with a meromorphic connection having a simple pole at v = 0 and a double pole at most at v = ∞. It follows from a remark due to Mochizuki (see Remark 6.3) that the Harder-Narasimhan filtration satisfies the Griffiths transversality condition with respect to the connection. This is a concrete description of the variation of the irregular Hodge filtration (Corollary 6.6).

Our main result concerns the limiting behaviour of the variation of the irregular Hodge filtration when v → 0, expressed in this model.

Theorem 1.11.

(1) The meromorphic connection ∇ on K^k(α) has a logarithmic pole at v = 0 and the eigenvalues of its residue Resv=0∇ belong to [−α, −α + 1) ∩ Q.

(2) On each generalized eigenspace of Resv=0∇ the nilpotent part of the residue strictly shifts by −1 the filtration naturally induced by the Harder-Narasimhan filtra-tion.

The proof of Theorem 1.11, which is sketched in §6, does not remain however in the realm of Kontsevich bundles. It is obtained through an identification of the Kontsevich bundles with the bundles H^k(α) obtained from the push-forward D-modules H^k of E^(v:u)f(∗H) (see §1.b) by the projection q : X × P¹_v → P¹_v. Recall that H^k := R^kq∗DRX×P¹_v/P¹_vE^(v:u)f(∗H) is a holonomic DP¹_v-module for each k.

It is equipped with its irregular Hodge filtration F_•^irrH^k obtained by push-forward, according to Theorem 1.3(3). We define the bundles H^k(α) by using this filtration, and the main comparison tools with the Kontsevich bundles K (α) are provided by Theorems 6.4 and 6.5.

1.d. Motivations and open questions

We have already discussed in [ESY15, Introduction] the motivation coming from estimating p-adic eigenvalues of Frobenius (Deligne) and that coming from mirror symmetry (Kontsevich). We list below some more related questions and possible applications for further investigations.

Numerical invariants of mixed twistor D-modules. The theory of mixed twistor D-modules, as developed by T. Mochizuki [Moc11b], is the convenient framework to treat wild Hodge theory. However, this theory produces very few numerical invariants having a Hodge flavor (like Hodge numbers, degrees of Hodge bundles, etc.). The irregular Hodge filtration, when it does exist, is intended to provide such invariants. Let us emphasize that, contrary to classical Hodge theory, the irregular Hodge filtration is only a by-product of the mixed twistor structure, but is not constitutive of its definition.

Is there a suitable well-behaved category of wild Hodge D-modules with a forgetful functor to the category of mixed twistor D-modules? What about the expected functorial and degeneration properties? The exponentially twisted Hodge modules should give rise to an object in such a category. Moreover, following the definition due to Simpson of systems of Hodge bundles, we can expect that the objects in this suitable category should carry an internal symmetry (a C^∗-action in the case of tame twistor D-modules). A possible approach to this question would be to search for the desired category as the category of integrable mixed twistor D-modules endowed with supplementary structures on the object obtained by rescaling the twistor variable.

Analogies with Hodge theory. Going further in the direction of Hodge theory, one may wonder whether the irregular Hodge filtration, when it exists, shares similar properties with the usual Hodge filtration on mixed Hodge modules. For example, for a morphism f : X → P¹, the DX-module E^f underlies a pure integrable twistor D-module (see Proposition 3.3(2)) and is equipped with an irregular Hodge filtration (see Theorem 1.3 with N = (OX, d)). Let π : X → Y be a projective morphism.

According to the decomposition theorem for pure twistor D-modules [Moc11a], the push-forward π+E^f decomposes, together with its twistor structure, into a direct sum of possibly shifted simple holonomic D-modules. One can wonder whether the analogues of Kollár’s conjectures (proved by M. Saito [Sai91]) hold for the irregular Hodge filtration of E^f.

Also the question of the limiting behaviour, in the sense of Schmid, of the irregular Hodge filtration raises interesting questions. We treat the case of a tame degeneration (the case of E^vf when v → 0) in §7, but the case of a non-tame degeneration (like u → 0 in §8) remains unclear in general. We expect that the good behaviour (by definition) of the mixed twistor modules by taking irregular nearby cycles along a holomorphic function should lead to specific limiting properties for the irregular Hodge filtration, when it exists.

Extended motivic-exponential D-modules. Recall that, following [BBD82, 6.2.4], one defines the notion of a simple regular holonomic D-module of geometric origin on a smooth complex algebraic variety X if it appears as a simple subquotient in a regular holonomic DX-module obtained by using only standard geometric functors starting from the case where the variety is a point. In particular, such a simple regular holonomic DX-module is a simple summand of a regular holonomic D-module underlying a polarizable Q-Hodge module of some weight, as defined by M. Saito [Sai88, Sai90]. It therefore underlies a simple complex polarizable Hodge module.

In other words, there exists an irreducible algebraic closed subvariety Z ⊂ X, a Zariski smooth open set Z^◦⊂ Z, and an irreducible local system on Z^◦, underlying a polarizable complex variation of Hodge structure (see [Del87]), such that this regular holonomic DX-module corresponds, via the inverse Riemann-Hilbert correspondence, to the intermediate extension of this local system by the inclusion Z^◦ ֒→ X. In particular, it comes equipped with a good filtration (that induced by the polarizable Q-Hodge module) and the corresponding filtered D-module is a direct summand of the filtered D-module underlying the polarizable Q-Hodge module.

M. Kontsevich [Kon09] has defined the category of motivic-exponential D-modules by adding the twist by E^f for any rational function f to the standard permissible operations on regular holonomic D-modules of geometric origin on algebraic varieties.

By [Moc11b], any such motivic-exponential D-module underlies a pure wild twistor D-module (see [Moc11a]).

There is also the category of extended motivic-exponential D-modules, by autho-rizing extensions of such objects, but we will not consider it here.

One can expect that any motivic-exponential D-module on a complex algebraic variety is endowed with a canonical irregular Hodge filtration, and that this filtration has a good behaviour with respect to the various permissible functors (the six op-erations of Grothendieck, the nearby and vanishing cycles along a function, and the twist by some E^f). Theorem 1.3 is a step toward this expected result.

Remark 1.12 (Hodge filtration in presence of very irregular singularities)

The holonomic DY-modules one obtains as H^kπ+(N ⊗ E^f) when π is any pro-jective morphism may have irregular singularities much more complicated than an exponential twist of a regular singularity. For example, if Y is a disc, it is shown in [Rou07] that any formal meromorphic connection at 0 ∈ Y can be produced as the formalization at the origin of a connection obtained by the procedure of Th. 1.3(3) for some suitable N on X = Y × P¹.

However, these DY-modules come equipped with a good filtration F•H^kπ+(N⊗E^f) obtained by pushing-forward F_•^irr(N ⊗ E^f). If Y is projective and if for example H^kπ+(N ⊗ E^f) = 0 except for k = ko then, according to Corollary 1.4, we obtain the degeneration at E1 of the spectral sequence attached to the hypercohomology of the filtered de Rham complex F•DR H^koπ+(N ⊗ E^f). Examples of this kind can be obtained by the procedure of [Rou07] with arbitrary complicated irregular singularities.

Acknowledgments. We thank Maxim Kontsevich for suggesting us the properties stated in Theorem 1.11 and Takuro Mochizuki for explaining us some of his results on mixed twistor D-modules and his useful comments. In particular, he suggested

various improvements and simplifications to the first version of this article. We owe him the statement of Theorem 6.4. Last but not least, we thank Hélène Esnault for the many discussions we had together and for many suggestions and questions on the subject of this article.