Hodge ideals of free divisors

We consider the Hodge filtration on the sheaf of meromorphic functions along free divisors for which the logarithmic comparison theorem holds. We describe the Hodge filtration steps as submodules of the order filtration on a cyclic presentation in terms of a special factor of the Bernstein-Sato polynomial of the divisor and we conjecture a bound for the generating level of the Hodge filtration. Finally, we develop an algorithm to compute Hodge ideals of such divisors and we apply it to some examples.

If D is free, then we have the equality Hence the terms of the so-called logarithmic de Rham complex of (X, D), i.e., the complex (Ω • X (log D), d), are locally free O X -modules. Notice that the most basic example of a free divisor is a divisor with simple normal crossings, in which case the logarithmic de Rham complex is a well studied object. It is particularly useful for the construction (due to Deligne, see, e.g., [PS08,II.4]) of a mixed Hodge structure on the cohomology of the complement U = X\D, in case it is quasi-projective. If D has simple normal crossings, then it is classical that the logarithmic de Rham complex computes the cohomology of U , in other words, we have a quasi-isomorphism Rj * C U ∼ = (Ω • X (log D), d). This is not always true for any free divisor, but if it is, we say that the logarithmic comparison theorem holds for D (see [CJNMM96]). This is in particular the case under a condition called strongly Koszul (see [NM15,Corollary 4.5]), which we recall in the next section (see Definition 2.3 below). Strongly Koszul free divisors are the objects of study of this paper. Many interesting free divisors, such as free hyperplane arrangements, or more generally locally quasi-homogeneous free divisors satisfy the strong Koszul hypothesis. A nice feature of divisors in this class is that we have a natural isomorphism where V D X is the sheaf of logarithmic differential operators with respect to D, from which we obtain an explicit representation of O X ( * D) ∼ = D X /I, where I is a left ideal in D X . As a consequence, we can consider another filtration (besides the Hodge filtration) F ord • O X ( * D), called the order filtration, which is simply given by the image of F • D X (the filtration by order of differential operators) under the isomorphism → O X ( * D) (or, equivalently, is induced from F • D X when writing O X ( * D) ∼ = D X /I for the ideal I mentioned above). The main tool to describe the Hodge filtration F H • O X ( * D) is to look at the graph embedding i h : X ֒→ C t × X, where h is a local defining equation of D, and to consider i h,+ O X ( * D). It also underlies a mixed Hodge module (on C t × X), and it is well-known that the Hodge filtration on O X ( * D) can be deduced from the one on Hence we are reduced to determine F H • i h,+ O X ( * D). In order to do so, we use a key property of mixed Hodge modules, which is known as strict specializability. It can be rephrased as a formula (see [Sai93,Proposition 4.2]) describing the Hodge filtration on a module which is the extension of its restriction outside a smooth divisor (which is the hyperplane {t = 0} ⊂ C t × X in our case). In order to use it, we have to compute some steps of the canonical V -filtration along {t = 0} of the module i h,+ O X ( * D), denoted by V • can i h,+ O X ( * D). Here we rely crucially on a previous result of the second named author ( [NM15,Theorem 4.1]), namely, that the roots of the Bernstein-Sato polynomial b h (s) of h are contained in (−2, 0) and that they are symmetric around −1. As we will see below, the set of roots of b h (s) bigger than −1 plays a particular role, and we put for later reference. For an element α i ∈ B ′ h , we write l i for the multiplicity of the root α i − 1 in b h . The polynomial αi∈B ′ h (s − α i + 1) li ∈ C[s] is the special factor of the Bernstein-Sato polynomial b h (s) mentioned in the abstract. We can consider another V -filtration on the module i h,+ O X ( * D), induced from V • D Ct×X for a cyclic presentation D Ct×X /I ′ of i h,+ O X ( * D) obtained from the cyclic presentation D X /I of O X ( * D) mentioned above. We will denote it by V Theorem. (see Proposition 3.3 and Theorem 4.4 below) Let D ⊂ X be a strongly Koszul free divisor, and suppose that it is globally given by a reduced equation h ∈ O X . Then: 1. We have the following inclusions of coherent O X -modules: 2. The zeroth step of the canonical V -filtration on i h,+ O X ( * D) can be described as follows: 3. For all k ∈ Z, we have the following recursive formula for the Hodge filtration on O X ( * D) (recall the shift convention between F H • O X ( * D) and F H • i h,+ O X ( * D)): Notice that part 2 actually holds under weaker assumptions on D, see Proposition 3.3 below for more details. Combining these three results and taking into account equation (1) allow us to determine the Hodge ideals of D. Theorem 5.2 below (and specifically Formula (48)) gives a concrete and explicit way to calculate the Hodge filtration steps on i h,+ O X ( * D) resp. on O X ( * D) and the Hodge ideals of D. In the second part of section 5, we apply our method to certain interesting examples. In applications, it is sometimes useful to study the behaviour of a Hodge module under the duality functor. It is defined for objects of the category MHM, but it restricts to the usual holonomic dual of the underlying D-module. We give (see Theorem 4.10) some statements estimating the Hodge filtration on the dual Hodge module with underlying D X -module DO X ( * D), following standard convention, this D X -module is denoted by O X (!D). Interestingly, our results on the dual Hodge filtration also involve the cardinality of the set B ′ h , i.e. the number of roots (counted with multiplicity) of b h lying strictly in the interval (−1, 0). Finally, we state a conjecture (see Conjecture 4.12) estimating the generating level of the Hodge filtration F H • O X ( * D), namely, we expect that it is always generated at level |B ′ h |. Computation of examples in section 5 supports this conjecture. The paper is organized as follows: In section 2, we first introduce rigorously the class of divisors we are interested in, and define the order filtration on O X ( * D) globally. We prove an important fact for the filtered module (O X ( * D), F ord • ), which is known as the Cohen-Macaulay property (see Proposition 2.9). It is known from Saito's theory that the same property holds for (O X ( * D), F H • ), i.e., the filtered module underlying the mixed Hodge module j * Q H U [n]. We are using the Cohen-Macaulay property of both filtrations to give the estimation of the Hodge filtration on the dual module of O X ( * D) (see Theorem 4.10) referred to above.
In section 3, we recall some general facts on V -filtrations and then describe (see Proposition 3.3) the canonical V -filtration on the graph embedding module of O X ( * D). In the subsequent section 4, we use these results to prove formula (3) (see Theorem 4.4). Finally, in section 5, we develop techniques for the computation of Hodge ideals, and perform them for some significant examples. Acknowledgements: We would like to thank the anonymous referee for valuable suggestions to improve the readability, as well as for pointing out a mistake in the proof of Corollary 3.8 in a former version of this paper.
2 Filtration by order on O X ( * D) The purpose of this section is to introduce the class of divisors we are going to study in this paper, these are free divisors satisfying the strong Koszul hypothesis. In that case, we can define a particular good filtration F ord • on the sheaf O X ( * D). We call it order filtration, because if locally we choose a reduced equation h for D, then the strong Koszul hypothesis implies that there is a canonical representation of O X ( * D) as a cyclic left D X -module (generated by h −1 ) D X /I(h), and then our order filtration is the filtration on D X /I(h) induced by the filtration on D X by the order of differential operators. Nevertheless, as we will see below, F ord • O X ( * D) is globally well defined. As mentioned in the introduction, one of our main results is that for each k ∈ Z the Hodge filtration F H k O X ( * D) is a coherent O X -submodule of F ord k O X ( * D) as well as a precise description of the inclusion F H k O X ( * D) ⊂ F ord k O X ( * D) (see Theorem 4.4 below). We will also show in this section that the order filtration F ord • O X ( * D) shares a key feature with F H • O X ( * D), known as the Cohen-Macaulay property. This is used in section 4 (see the proof of Theorem 4.10) to give some statement about the dual Hodge filtration. For the remainder of this paper (except in subsection 5.4, where we explicitly relax these assumptions), we will assume X to be an n-dimensional complex manifold, and D ⊂ X a free divisor. We will be specifically working with those free divisors satisfying an additional hypothesis called strongly Koszul, that we define now. Denote as before by Θ X the locally free O X -module of rank n of vector fields and by D X the sheaf of linear differential operators with holomorphic coefficients, endowed with the filtration F • D X by the order of differential operators. For each divisor D ⊂ X we write O X ( * D) for the sheaf of meromorphic functions with poles along D and Ω • X ( * D) the meromorphic de Rham complex. If D is a free divisor, we denote by Θ X (− log D) ⊂ Θ X the locally free O X -module of rank n of logarithmic vector fields and by Ω • X (log D) the logarithmic de Rham complex. Moreover, we let V D X := P ∈ D X | P (I(D) k ) ⊂ I(D) k ∀k ∈ Z ⊂ D X be the sheaf of rings of logarithmic differential operators with respect to D. It is filtered by the order of differential operators as well, namely F k V D X := V D X ∩ F k D X for all k ∈ Z, and there is a canonical isomorphism of graded O X -algebras (see [CM99,Remark 2 The sheaf O X (D) ⊂ O X ( * D) of meromorphic functions with poles of order ≤ 1 is a left V D X -module by the definition of V D X and we have a canonical D X -linear map ( [CMNM05,§4]) We quote the following result (see [CMNM05,Corollaire 4.2]): Theorem 2.1. Let D ⊂ X be a free divisor. The following properties are equivalent: (i) The logarithmic comparison theorem (LCT) holds for D (i.e. the inclusion Ω • X (log D) ֒→ Ω • X ( * D) is a quasi-isomorphism of complexes of sheaves of C-vector spaces).
(ii) The map (4) is a quasi-isomorphism of complexes of left D X -modules.
Let us notice that property (ii) in the above theorem means that the complex D X is concentrated in degree 0 and the canonical D X -linear map Definition 2.2. Assume that D ⊂ X is free and that the logarithmic comparison theorem holds for D. We define the order filtration F ord which by definition is a filtration by O X -coherent submodules.
Let us give a more explicit local description. Let p ∈ D, and take a basis δ 1 , . . . , δ n of Θ X (− log D) p as well as a local equation h ∈ O X,p of D around p, such that δ i (h) = α i h. Assume that the LCT holds for D, then we have the following explicit local presentation where the class [1] is send to h −1 (see the explanation after Corollaire 4.2 in [CMNM05]). Under this isomorphism, the filtration F ord • O X ( * D) p is the induced filtration on D X,p /D X,p (δ 1 + α 1 , . . . , δ n + α n ) by F • D X,p , which explains its name.
We now introduce the strong Koszul hypothesis. It implies that the LCT holds for D ⊂ X, but it is stronger. Its additional assumptions will be needed in the next section.
Definition 2.3. Let D ⊂ X be a free divisor and let p ∈ D. Let h ∈ O X,p be a local reduced equation of (D, p) and let δ 1 , . . . , δ n be any O X,p -basis of Θ X,p (− log D) with δ i (h) = α i h.

We say that
(Here and at later occasions, for an element P in a filtered ring, we denote by σ(P ) its symbol, i.e. its class in the associated graded ring.) 2. We say that D is strongly Koszul at p ∈ D if the sequence is the filtration on D X,p [s] for which vector fields on X as well as the variable s have order 1, we will call it the total order filtration on D X,p [s]).
We say that D is Koszul resp. strongly Koszul if it is so at any p ∈ D. Since the strong Koszul assumption will be our main hypothesis later, we will sometimes abbreviate it by saying that D is an SK-free divisor.
Let us notice that, in the above definition, the ordering of the sequences is not relevant because their elements are homogeneous. Let us also notice that if D has an Euler local equation h ∈ O X,p , i.e. h belongs to its gradient ideal (with respect to some local coordinates), then there is a basis δ 1 , . . . , δ n−1 , χ of Θ X (− log D) p with δ i (h) = 0 and χ(h) = h. In such a case, D is strongly Koszul at p if and only if h, σ(δ 1 ), . . . , σ(δ n−1 ) is a regular sequence in Gr F • D X,p (compare with Definition 7.1 and Proposition 7.2 of [GS10] in the case of linear free divisors). For D to be strongly Koszul is equivalent to be of "linear Jacobian type", i.e. the ideal (h, h ′ x1 , . . . , h ′ x1 ) is of linear type, and any strongly Koszul free divisor is Koszul (see Propositions (1.11) and (1.14) of [NM15]). Any plane curve is a Koszul free divisor. Any locally quasi-homogeneous free divisor is strongly Koszul [CMNM05, Theorem 5.6], and so are free hyperplane arrangements or discriminants of stable maps in Mather's "nice dimensions". In particular, any normal crossing divisor is strongly Koszul free. Let us recall the following properties of SK-free divisors.
Proposition 2.4. If D is a strongly Koszul free divisor, then we have: 1. D is locally strongly Euler homogeneous, that is, for each p ∈ D there is a vector field χ ∈ Θ X,p which vanishes at p and such that χ(h) = h for some reduced local equation h of D at p.
2. The LCT holds for D, in particular, we have the local cyclic presentation of O X ( * D) from formula (5) above. More specifically, if for each p ∈ D, we take h and χ as in point 1 and if we let δ 1 , . . . , δ n−1 to be a basis of germs at p of vector fields vanishing on h, in such a way that δ 1 , . . . , δ n−1 , χ is a basis of Θ X (−log D) p , then there is an isomorphism of left D X,p -modules sending the class of 1 ∈ D X,p to h −1 ∈ O X,p ( * D).
3. Let p ∈ D be a point, h ∈ O X,p be a reduced local equation of D at p and b h (s) the Bernstein-Sato polynomial of h (i.e., the monic generator of the ideal of polynomials Proof. Part 1 is a consequence of Propositions (1.9) and (1.11) of [NM15]. Part 2 is a consequence of Corollary (4.5) of [NM15]. Part 3 is a consequence of Corollary (4.2) of [NM15].
Remark 2.5. Definition 2.3 is purely algebraic and makes sense in rings other than analytic local rings O X,p , for instance in polynomial rings R := C[x 1 , . . . , x n ], algebraic local rings of polynomial rings at maximal ideals or formal power series rings. Under this scope, if h ∈ C[x 1 , . . . , x n ] is a non-constant reduced polynomial, D = V(h) is the affine algebraic hypersurface with equation h = 0 and D an ⊂ C n is the corresponding analytic hypersurface, we know that D is a (algebraic) free divisor if and only if D an is a (analytic) free divisor, and if so, the following properties are equivalent: (a) The (affine algebraic) divisor D is strongly Koszul, i.e. for some (and hence for any) basis δ 1 , . . . , δ n of , s] is the graded ring of W m [s] with respect to the total order filtration.
(c) The (analytic) divisor D an is strongly Koszul (in the sense of Definition 2.3 above).
The next step is to discuss some deeper properties of the order filtration F ord • O X ( * D). The main result is Proposition 2.9 below, which is concerned with the dual filtered module D(O X ( * D), F ord • ). In order to do this, we need to recall a few facts about the duality theory of filtered modules. The original reference is [Sai88, Section 2.4], but for what follows below, [Sch14, § 29] provides enough background information. For any complex manifold X, we consider the (sheaf of) Rees ring(s) . In local coordinates x 1 , . . . , x n on U ⊂ X, we have The ring R F D X will be denoted by D X . For what follows, we will need an interpretation of this sheaf in terms of Lie algebroids, more precisely, we will use the fact that D X is the enveloping algebra of the Lie algebroid Lie algebroids are the sheaf version of Lie-Rinehart algebras (see [Rin63]). They were originally studied in the setting of Differential Geometry (the book [Mac05] is a complete reference here), and for instance in [Che99] the complex analytic case is considered. For the ease of the reader, let us recall the notions of Lie-Rinehart algebras and Lie algebroids. Let us take a commutative base ring k and a commutative k-algebra A (resp. a sheaf of commutative k-algebras A over a topological space X). A Lie-Rinehart algebra over (k, A) is a (left) A-module L which is also a k-Lie algebra, endowed with an anchor map ̺ : L → Der k (A): a (left) A-linear morphism of k-Lie algebras such that for all λ, λ ′ ∈ L and a ∈ A. Respectively, a Lie algebroid over (k, A) is a (left) A-module L which is also a sheaf of k-Lie algebras, endowed with an anchor map ̺ : L → Der k (A): a (left) A-linear morphism of sheaves of k-Lie algebras such that [λ, for all local sections λ, λ ′ of L and a of A.
We usually assume that L is a projective A-module (resp. L is a locally free A-module) of finite rank. It is clear that for each point p ∈ X, by using the natural morphism Der k (A) p → Der k (A p ), any Lie algebroid L over (k, A) gives rise to a Lie-Rinehart algebra L p over (k, A p ). For any Lie-Rinehart algebra L over (k, A) (resp. for any Lie algebroid L over (k, A)), there is a universal k-algebra U(L) (resp. sheaf of k-algebras U(L)), endowed with canonical maps A → U(L) and L → U(L) (resp. A → U(L) and L → U(L)), called the enveloping algebra of L (resp. of L). One easily proves that for each p ∈ X, there is a canonical isomorphism U(L p ) ∼ = U(L) p . Enveloping algebras are naturally (positively) filtered. Over a complex analytic manifold X, our basic example is k = C, A = O X and L = Θ X = Der C (O X ), where the anchor is the identity. In this case, the corresponding enveloping algebra is the sheaf of differential operators D X (cf. [Che99,§ 2] On the other hand, the natural filtration on D X as enveloping algebra of zΘ X [z] can be derived from its graded structure in the following way (notice that it is not the usual filtration associated to the grading): We have a canonical commutative diagram of graded O X [z]-algebras and where the horizontal arrows are isomorphisms by the Poincaré-Birkhoff-Witt theorem for Lie algebroids resp. Lie-Rinehart algebras (see, e.g., [Rin63, Theorem 3.1]). Let us notice that all objects in the above diagram are bigraded (where the additional grading is given by the z-degree), and all maps in the diagram are morphisms of bigraded algebras.
Moreover, for a sequence of filtered D X -modules the following properties are equivalent: The left D X -module O X will be always endowed with the trivial filtration is a graded left D X -module with the usual z-grading, that will be denoted by O X . The canonical right D X -module (see [SS19, Example 8.1.9]) can also be seen as the Rees module R F ω X associated with the canonical right D X -module ω X endowed with the filtration The main advantage to work with (graded) D X -modules rather than with filtered D X -modules is that the former category is abelian whereas the latter is not.
Let k be a commutative graded ring (e.g., k = C[z]) and B a sheaf of graded k-algebras over a topological space X (e.g., B = D X or B = O X for X a complex manifold). We recall the sheaf version of some well-known definitions for graded modules over graded rings (cf. [FF74,§1]). where the N (i) is the shifted graded module defined as N (i) j = N (i + j) for all i, j ∈ Z. The above inclusion is an equality whenever M is locally of finite presentation. For any graded right B-module Q and any left B-module M, the graded tensor product Q * ⊗ B M is the sheaf of k-modules Q ⊗ B M endowed with the grading If M is a left resp. right B-module, then * Hom B ( M, B) is a graded right resp. left B-module. The complex R * Hom DX ( O X , D X ) can be computed by means of the Spencer resolution of O X The complex R * Hom DX ( O X , D X ) is concentrated in homological degree n and we have a canonical isomorphism of graded right D X -modules As for the non-graded case, we have natural equivalences of categories between graded left D X -modules and graded right D X -modules: 1. For any graded left D X -module M, we define 2. For any graded right D X -module M, we define For a complex of graded left (resp. right) D X -modules M (which may not come from a complex of filtered D X -modules (M, F • )), we define its dual complex to be * We have canonical isomorphisms of graded left (resp. right) D X -modules (actually, complexes concentrated in degree 0) * We also have canonical isomorphisms * for all i ∈ Z. Given a filtered holonomic module, one can consider the dual of its Rees module. However, this is not in general the Rees module of a filtered module. The next lemma gives a criterion to know when this is the case. From (13), the canonical left (resp. right) D X -module O X (resp. ω X ) is self-dual as filtered module with the trivial filtration (resp. with the filtration (10)). It is clear that if (M, F • ) is a filtered holonomic D X -module having the Cohen-Macaulay property, then (M, F (i) • ) also has the Cohen-Macaulay property and (see (14) above) for any integer i.
Let us come back to the situation of a free divisor D ⊂ X. The sheaf of logarithmic differential operators V D X ⊂ D X is the enveloping algebra of the Lie algebroid Θ X (− log D) ⊂ Θ X over the C-algebra O X (this is basically [ The Lie algebroids zΘ X [z] and zΘ X (− log D)[z] with their z-grading are graded Lie algebroids in the sense that they are graded as O X [z]-modules and as sheaves of C[z]-Lie algebras, and that their anchor maps are graded too. Moreover, the inclusion zΘ X (− log D)[z] ⊂ zΘ X [z] is graded and so is a map of graded Lie algebroids in the sense we leave the reader to write down. We know that the relative dualizing module (see Definition (A.28) of [NM15]) for the inclusion of Lie algebroids In a completely similar way, the relative dualizing module of the inclusion , and so we have a canonical isomorphism ω L/ L0 = O X (D) [z], but in this case, due to the graded structures, this relative dualizing module is also naturally graded since it can be defined by using * Hom OX instead of Hom OX , and of course, this grading coincides with the z-grading of O X (D)[z].
We have seen before (see Theorem 2.1) that if D ⊂ X is such that the LCT holds (e.g. if D is strongly Koszul), then there is an isomorphism of left D X -modules X -module that is locally free (actually, of rank one) over O X . This situation can be generalized by replacing O X (D) by an integrable logarithmic connection (ILC) E with respect to D, which by definition is a locally free O X -module of finite rank endowed with a left V D X -module structure. This notion is useful when one aims at studying the cohomology of X\D with respect to some local coefficient system (i.e., the local system of horizontal sections of E). The next two results will be stated and proved in this greater generality. Although we will use them in this paper only for the case E = O X (D), we believe that they may be useful for future applications. Moreover, for the final proof of Proposition 2.9 below (even if we are only interested in the case E = O X (D)) some intermediate steps need to be carried out for an arbitrary ILC. Given an ILC E with respect to D, we define (in complete analogy to Definition 2.2) the order filtration on D X ⊗ V D X E to be: is the filtered tensor product of D X with its order filtration and E with its trivial filtration given by F k E = E for k ≥ 0 and F k E = 0 for k < 0. We have a natural D X -linear graded map Lemma 2.7. Let D be a Koszul free divisor, and let E be an ILC with respect to D. Then: E is concentrated in degree 0 and so we have an isomorphism in the derived category of complexes of left graded D X -modules Proof. Both properties can be proved by forgetting the graded structures. It is easy to see that under the Koszul hypothesis on D, the inclusion of Lie algebroids (over the is a Koszul pair in the sense of [CMNM09, Definition 1.16], i.e., some (or any ) (see diagram (8) above) and the fact that the Koszul property means exactly that the inclusion of Lie algebroids They have augmentations to E resp. to E and are a locally free V D X -resp. V D X -resolution of E resp. E. We can also consider the complex Sp respectively. According to [CMNM09, Proposition 1.18], since both inclusions of Lie algebroids are Koszul pairs, the cohomology of both complexes is concentrated in degree 0, and equal to D ⊗ V D X E and D ⊗ V D X E, respectively. This proves in particular the second statement. But actually the proof of this result in loc. cit. gives us an additional strictness property for Sp • V D X ,DX (E). Namely, if we filter this complex as we obtain that and this complex is concentrated in degree 0 by the Koszul hypothesis (and so is Sp • V D X ,DX (E)), but this result also implies that the differentials . . , n are strict for the above filtrations. In particular, the right exact sequence is strict, or equivalently, the sequence is exact. Notice also that we clearly have To finish, consider the commutative diagram of graded left D X -modules the second row is so as we have just explained (sequence (17) above). By (18), the first and second vertical arrows are isomorphisms and so is the third one.
To prove our main result in this section, we will use a graded (and Lie-algebroid) version of Theorem (A.32) of [NM15]. However, instead of stating it in full generality (for general graded Lie algebroids or Lie-Rinehart algebras), we will only state it for the case we need, namely, the inclusion of graded Lie and the corresponding map of graded enveloping algebras V D X ⊂ D X .
Proposition 2.8. Let F be a graded locally free O X -module of finite rank endowed with a graded left module structure over V D X . We have a canonical isomorphism in the derived category of graded D X -modules . This proof is a straightforward translation of the proof of Theorem (A.32) of [NM15] to the graded case. It essentially consists of replacing functors Hom and ⊗ by * Hom and * ⊗ in all the constructions in the Appendix of loc. cit., as well as the fact that the relative dualizing module ω L/ L0 is naturally graded. We sketch it for the convenience of the reader, and, actually, we propose a shorter and slightly different approach. Let us recall that where i : V D X → D X is the inclusion, turns out to be left V D X -linear. It induces a natural graded D X -linear map and so a natural map in the derived category of left graded D X -modules which, by standard reasons, is an isomorphism whenever N is coherent. To finish, notice that if N is a graded locally free O X -module, then we have a canonical isomorphism * We are now ready to prove the announced result on the dual of the filtered module ( Proposition 2.9. Assume that D ⊂ X is a Koszul free divisor and that E is an ILC with respect to D. In particular, the filtered module (D X ⊗ V D X E, F ord • ) satisfies the Cohen-Macaulay property and its dual filtered Proof. By Lemma 2.7, we have and so we have * In conclusion, the dual D X -module of M is strict, and the dual filtration F D • DM is given as the order filtration F ord • D X of the operators δ 1 , . . . , δ n−1 , χ + 1 form a regular sequence. However, later (see the proof of Theorem 4.10 below) we need to know that the dual filtration , which is provided by the proof above.
The following is an easy variant of Proposition 2.9 which we will need later in section 4.
Corollary 2.11. Let, as above, D ⊂ X be a Koszul free divisor and E be an ILC. Then for any k ∈ Z, the filtered holonomic module (D ⊗ V D X E, F ord •+k ) has the Cohen-Macaulay property, and its dual filtered module is given by The statement of the corollary follows simply by combining the proof of Proposition 2.9 with formula (15) and the remark that surrounds it.

Canonical and induced V -filtration
In this section we are discussing in detail the V -filtration on the module i h,+ O X ( * D), where i h is the graph embedding for some local reduced equation h of D ⊂ X. This will be used in the next section in order to obtain information on the Hodge filtration on i h,+ O X ( * D), and on O X ( * D) itself. Recall that for any complex manifold M , and for a divisor H ⊂ M with V 0 D M is a sheaf of rings, notice that it equals the sheaf of logarithmic differential operators (with respect to H), which was denoted by V H X in the previous chapter. Moreover, all V k D M are sheaves of V 0 D M -modules. We will usually suppose that H is smooth, and moreover that it is given by a globally defined equation We start by recalling the canonical V -filtration on a holonomic D M -module. In general, it is indexed by the complex numbers, and in order to define it, one needs to choose an ordering on C such that for all α, β ∈ C, we have α < α + 1, α < β ⇐⇒ α + 1 < β + 1 and α < β + m for some m ∈ Z. Notice however that for Hodge modules, only rational indices can occur. Moreover, we will later only use the integer parts of this filtration.
and we call the filtration V • ind M the induced V -filtration on M. In particular, if M is holonomic, then V • ind M has a (minimal and monic) Bernstein We will mainly use the above definitions for the case where H is the divisor {t = 0} ⊂ M = C t × X and where M = i h,+ O X ( * D), i h being the graph embedding of a defining equation for a divisor D ⊂ X. Using a construction that goes back to Malgrange (see [Mal75]), one can find a cyclic generator for this module, so that it has an induced V -filtration. It is essentially well-known that the Bernstein polynomial of this filtration is given by the Bernstein polynomial b h (s) of the equation h defining D, up to a change of variables. However, we recall the proof for the convenience of the reader. Notice also that this result holds quite generally for any divisor, and does not depend on freeness or any Koszul assumption. We therefore let D ⊂ X be a divisor defined locally at a point p ∈ X by a reduced equation h = 0, and we denote by i h : X → C t × X its graph embedding. We put as O Ct×X,(0,p) -modules. It becomes an isomorphism of D Ct×X,(0,p) -modules when equipping the right hand side with the D Ct×X,(0,p) -structure given by for any m ∈ O X,p ( * D) and any a ∈ O X,p .
Lemma 3.2. In the above situation, write j ∈ Z >0 for the negative of the smallest integer root of b h (s).
The fact that −j is the smallest integer root of b h (s) is equivalent to say that O X,p ( * D) is generated as a D X,p -module by h −j . Therefore, N (h) is generated by h −j · h s , that we will write h s−j from now on. Hence we can consider the filtration where we have used that s = −∂ t t and t = h in our alternative representation of N (h). As a consequence, we have isomorphisms for every k ≥ 0. On the other hand, for k ≤ 0, we have that so we obtain surjections .
As a consequence, we obtain a surjection for any k ≤ 0. We finally obtain surjections for any k ≤ 0, where the first isomorphism holds because for any l ≥ 0 and any l ′ ∈ Z, s is a non-zero divisor on the modules (i h, annihilates the left-hand side of equation (20) and b h (−∂ t t − j + k − 1) annihilates the left-hand side of the surjection (21), so they annihilate the respective right-hand sides as well.
Proof. Since the roots of b h (s) are in (−2, 0), we know thanks to Lemma 3.2 that the roots of b where the two inclusions marked as * ⊂ are due to the definition of the filtration V • N (h), i.e., due to formula holds for all k ∈ Z, and this shows Formula (22).
Remark 3.4. Without the assumption that the roots of b h (s) are in (−2, 0) (i.e., for general h ∈ O X,p ) the previous result is no longer true in general. Namely, since in general the roots of b h (s) are negative rational numbers, we know by Lemma 3.2 that the roots of b has roots bigger than 1, and this prevents the inclusion In the remaining part of this section, we restrict our attention to SK free divisors. We study more in detail the induced V -filtration on the module N (h), since as we have seen, it helps to understand (at least the integer steps of) the canonical V -filtration on that module. In particular, we will prove (see Proposition 3.10 below) a compatibility property between the induced V -filtration and the order filtration on that module (which is similar to the order filtration F ord • O X ( * D) studied in section 2). Its statement is similar to [RS20, Proposition 4.9]), although the presentation of the proof is slightly different. We first give a concrete cyclic presentation of the graph embedding module for equations defining an SK free divisor, starting from the presentation for O X ( * D) discussed in the previous section (see Equation (5) above).
Lemma 3.5. Let D ⊂ X be a free divisor and let p ∈ D be such that D is strongly Koszul at p. Let h ∈ O X,p be a reduced equation for D near p. Let i h : X ֒→ C t × X, x → (h(x), x) be the graph embedding of h, then we have an isomorphism of D Ct×X,(0,p) -modules .
We still denote this module (resp. this cyclic presentation) by N (h).
We will be later interested in calculating the Hodge filtration on a mixed Hodge module which has i h,+ O X.x ( * D) as underlying D Ct×X,(0,p) -module. For that purpose, we consider the filtration F ord • N (h) which is induced on N (h) by the filtration F • D Ct×X,(0,p) by the order of differential operators. Then we have G = Gr F • D (where, as before, F • D is the filtration on D by the order). Obviously, G is graded by the degree of the variables T, X 1 , . . . , X n , and we write G l for the degree l part.
Lemma 3.6. Let D ⊂ X be a free divisor and let p ∈ D such that D is strongly Koszul at p and locally defined by some h ∈ O X,p . Write so that N (h) = D/I(h). Then the set {t − h, δ 1 , . . . , δ n−1 , χ + ∂ t t + 1} is an involutive basis of the ideal I(h), that is, we have the equality of ideals in G. Recall that we denote for an element P ∈ D its symbol in G by σ(P ) and by σ(I(h)) the ideal {σ(P ) | P ∈ I(h)} in G.
Proof. The proof basically follows the argument in [CM99, Proposition 4.1.2] once we know that symbols of the generators of I(h), i.e., the elements t − h, σ(δ 1 ), . . . , σ(δ n−1 ), σ(χ) + T · t form a regular sequence in the ring G. Notice also that Lemma 3.7 below is a similar statement (in a commutative ring though), and we will give some indications of the proof there.
We denote by V • G the filtration induced by V • D (the V -filtration on D with respect to the divisor {t = 0} ⊂ C t × X) on the ring G. It can be described as For any f ∈ G, we write ord V for the maximal b ∈ Z such that f ∈ V b G. The graded ring of G with respect to the filtration V looks quite similar to G itself, namely, we have since V • induces the t-adic filtration on the ring C{t, x 1 , . . . , x n }.
As usual, for f ∈ G we denote by σ V (f ) ∈ Gr • V G the symbol with respect to V , i.e., the class of f in Then we have the following fact (which is analogous to the statement of Lemma 3.6 above, and in fact holds in a more general setting for certain filtered rings, although we do not consider such a generality here).
Lemma 3.7. Suppose that σ V (f 1 ), . . . , σ V (f k ) is a regular sequence in Gr • V G. Then we have the equality of ideals in Gr • V G. Proof. We follow the argument given (in a more algebraic situation though) in [SST00, Proposition 4.3.2]. We obviously have σ V (I) ⊃ σ V (f 1 ), . . . , σ V (f k ) and we need to show the converse inclusion. Assume that it does not hold, that is, take Now by assumption the symbols σ V (f 1 ), . . . , σ V (f k ) form a regular sequence in Gr • V G, and this implies that the module of syzygies between these elements of G is generated by the so-called Koszul relations, i.e., In other words, we have an equality of elements of G k (where e i is the i-th canonical generator of G k ). Reordering the right hand side of the above equation yields . Hence if we replace those g i in equation (24) by which contradicts the above choice of a relation g with maximal o(g). Hence we must have that σ V (f ) ∈ σ V (f 1 ), . . . , σ V (f k ) and so σ V (I) = σ V (f 1 ), . . . , σ V (f k ) , as required.
We apply the previous lemma in the situation where I is given as the ideal σ(I(h)) ⊂ G (recall that σ denotes the usual symbol with respect to the order filtration F • D). For notational convenience, put G 0 := t − h, G 1 := δ 1 , . . . , G n−1 := δ n−1 and G n := χ + ∂ t t + 1, so that I(h) = (G 0 , . . . , G n ) ⊂ D. Then we obtain the following consequence.
Corollary 3.8. Let D ⊂ X be a free divisor and let x ∈ D such that D is strongly Koszul at x and locally defined by some h ∈ O X,p . Then the set is an involutive basis of the ideal σ(I(h)) ⊂ G with respect to the filtration V • G induced from the filtration V • D, that is, we have As a consequence, any f ∈ σ(I(h)) ⊂ G has a standard representation with respect to V • , that is, there are For the last statement, we remind the reader that the filtration V • G is descending, and that consequently, for g ∈ G, ord V (g) denotes the maximum of all l such that g ∈ V l G.
Corollary 3.9. Let D ⊂ X be a free divisor, let p ∈ D such that D is strongly Koszul at p and locally defined by some h ∈ O X,p . Then Proof. Since the generators of I(h) belong all to V 0 D, which is a ring, the inclusion I(h) ∩ V 0 D ⊃ V 0 D(t − h, δ 1 , . . . , δ n−1 , χ + ∂ t t + 1) is trivial. Let us show the reverse one. Let P be an operator in I(h) ∩ V 0 D and let us prove that it lies within V 0 D(t − h, δ 1 , . . . , δ n−1 , χ + ∂ t t + 1) by induction on the order of P with respect to F • D, written ord F (P ). If ord F (P ) = −1, the claim is trivial, for then P = 0. Let us now assume that the statement is true for all Q ∈ F d−1 D for a given non-negative integer d, and let us prove it for any P ∈ F d D. Since P ∈ I(h), we have the expression Thanks to Lemma 3.6, we know that A ∈ F d D and the B i and C are in F d−1 D and also that σ(P ), which belongs to V 0 G, can be written as On the other hand, we know after Corollary 3.8 that we can chooseÃ Then we have that is a syzygy of the tuple (t − h, σ(δ 1 ), . . . , σ(δ n−1 ), σ(χ) + T t). By Lemma 3.6 again, since this tuple is a regular sequence in G, ς will be a sum of Koszul syzygies (i.e., those of the form a i e j − a j e i ∈ G n+1 for some elements a i , a j of the regular sequence, e i and e j being the corresponding unit vectors). Now notice that for all k, l, the canonical map can be written as a sum of Spencer syzygies of (t − h, δ 1 , . . . , δ n−1 , χ + ∂ t t + 1) (that is, the respective lifts of the Koszul ones in G that come from the expression of the [G i , G j ], for any i, j, as a linear combination of the form k α k G k with coefficients in O) plus another tuple (A ′′ , B ′′ 1 , . . . , B ′′ n−1 , C ′′ ), where now A ′′ ∈ F d−1 D and B ′′ i , C ′′ ∈ F d−2 D. Let us call now Consequently, Summing up, we obtain that P ′′ ∈ F d−1 D ∩ V 0 D ∩ I(h), so by the induction hypothesis, it can be written as a linear combination of t − h, δ 1 , . . . , δ n−1 , χ + ∂ t t + 1 with coefficients in V 0 D and thus P too.
Now we can show the following result expressing a certain compatibility between the induced V -filtration and the order filtration on N (h).
Proposition 3.10. For all k, l ∈ Z, the canonical morphism is surjective.
Proof. We reformulate the surjectivity we want to prove as a compatibility property between three filtrations on D, the first two being F • D and V • D. The third one is defined (somewhat artificially) as It is clear that the statement of the proposition is equivalent to the surjectivity of the map for r = 0, where we denote by ′ F • resp. by ′ V • the filtrations induced by F • D resp. V • D on Gr J r D (notice that for r = 0, these are nothing but F ord • N (h) resp. V • ind N (h)). Since both the J-and the F -filtration are exhaustive, we can now apply [Sai88, Corollaire 1.2.14] from which we conclude that surjectivity of the map (25) holds (for all l, k, r) if and only if the map is surjective for all l, k, r. Again, J • G l is the filtration induced by J • D on G l , i.e., J r G l = G l if r ≥ 0 and J r G l = σ l (I(h) ∩ F l D) if r < 0, where, as usual, σ l : F l D ։ G l denotes the map sending an operator of degree l to its symbol in G l . However, the surjectivity of (26) is a nontrivial condition only for r < 0 (and it is the same condition for all negative r): For r ≥ 0, it means that the map F l D ∩ V k D −→ V k G l is surjective, which is true, as we have already noticed in the proof of Corollary 3.9. If r < 0, the map (26) is nothing but We will see that the surjectivity of the latter map follows from the involutivity properties proved above. Namely, let σ(P ) = σ l (P ) ∈ V k G l be given, where P ∈ I(h) ∩ F l D. Then by Lemma 3.6 we have an expression recall that we had put G 0 := t − h, G 1 := δ 1 , . . . , G n−1 := δ n−1 and G n := χ + ∂ t t + 1 for the involutive basis of I(h). Since obviously the symbols σ(G i ) are homogeneous elements in the ring G = Gr F • D, we have that deg(k i ) = l − deg(σ(G i )) = l − ord(G i ). On the other hand, we know by Corollary 3.8 that ord V (k i ) ≥ ord V (σ(P )) − ord V (σ(G i )), that is, Moreover, it follows from the concrete form of the operators G 0 , . . . , G n that ord V (σ(G i )) = ord V (G i ) = 0, hence k i ∈ V ord V (σ(P )) G. Now choose any lift of k i to an operator K i ∈ F l−ord(Gi) D ∩ V ord V (σ(P )) D. Such a lift exists since, as we have already noticed above, the map is surjective for all l, k. Then the element P ′ := n i=0 K i · G i ∈ I(h) is the preimage we are looking for, i.e., σ(P ′ ) = σ(P ) and we have recall that we had chosen σ(P ) ∈ V k G l ∩ σ l (I(h) ∩ F l D), meaning that k ≤ ord V (σ(P )). Hence we have shown the surjectivity of , and as we said at the beginning of the proof, using [Sai88, Corollaire 1.2.14], the surjectivity of as required.

Description of the Hodge filtration
This section is the central piece of the article. We apply the results on the canonical V -filtration from the last section to compute the Hodge filtration on the mixed Hodge module which has O X ( * D) as underlying D Xmodule. The main result is Theorem 4.4, which gives a precise description of F H • O X ( * D). We also complement it with some statements about the Hodge filtration on the dual module O X (!D), see Theorem 4.10, which we expect to be useful for future applications. We conjecture (see Conjecture 4.12) some bound for the so-called generating level of the Hodge filtration (see Definition 4.11 for this notion), which is supported by computation of examples in section 5 below. Notice that some of the results in this section (first part of Theorem 4.4 or Theorem 4.12) hold globally on X, however, for most of the proofs we will need to require that D ⊂ X is defined by an equation h. As already indicated, we will determine the Hodge filtration using the graph embedding i h : X ֒→ C t × X and by considering extensions of Hodge modules from C * t × X to C t × X. Hence, let again X be an n-dimensional complex manifold and D ⊂ X a reduced free divisor, which is strongly Koszul at each point p ∈ D. We assume for now (until and including Corollary 4.3) that D is given by a reduced equation h ∈ Γ(X, O X ). Notice also that contrary to the last section, we will consider all the objects as sheaves since this is more convenient when applying the functorial constructions from the theory of mixed Hodge modules. Put U := X\D, and consider the following basic diagram: Here j and j ′ are open embeddings, whereas i h and i ′ h (both are given by x → (x, h(x)) and may be called graph embeddings) are closed. We are considering the pure Hodge module Q H U [n] on U . By the functorial properties of the category of mixed Hodge modules, we know that there is an object j * Q H U [n] ∈ MHM(X), whose underlying D X -module is the module O X ( * D) of meromorphic functions. We are interested in describing the Hodge filtration F H • on O X ( * D).
recall from Proposition 2.4 that the basis δ 1 , . . . , δ n−1 , χ of Der X (− log D) was chosen such that δ i (h) = 0 for i = 1, . . . , n − 1 and χ(h) = h. Hence we have the following presentation identifying the class of 1 ∈ D U on the right-hand side with the function h −1 ∈ O U . Under this isomorphism, . . . , δ n−1 , χ + 1), here F ord • denotes the filtration induced on a cyclic D-module by the filtration on D by the order of differential operators. From this presentation, we deduce the following statement.
Next we will deduce from this result a description of the Hodge filtration on the Hodge module j ′ . This is the crucial step towards our first main result (Theorem 4.4 below). We will use the results from the last section concerning the canonical V -filtration of the graph embedding module i h,+ O X ( * D) in an essential way. The result we are after can be stated as follows.
from Lemma 3.5 (which, as indicated at the beginning of this section, we write here as an isomorphism of sheaves of D Ct×X -modules rather than of germs) we have the following inclusion of coherent O Ct×X -modules for all k ∈ Z: Proof. First put for notational convenience , δ 1 , . . . , δ n−1 , χ + 1) Since for all r ∈ N. We have seen in Proposition 3.3 that V k can N (h) ⊂ V k ind N (h) holds for all k ∈ Z (recall that since we assume in this section that D is SK free, we know that the roots of b h (s) are included in (−2, 0) by [NM15,Theorem 4.1], so that the assumptions of Proposition 3.3 are satisfied). Thus it suffices to show that we have we are done by putting p = 0. Otherwise, since m ∈ j ′ * F ord r N ′ (h) (which means by definition that m |C * t ×X ∈ F ord r N ′ (h)) it follows that the class of m in the quotient F ord hence there is some p such that the class of t p · m is zero in this quotient, that is t p · m ∈ F ord r N (h), and obviously we have . Now by Proposition 3.10 we know that there exists an operator . By definition, P ′ can be written as P ′ = t p · P , where P ∈ V 0 D Ct×X ∩ F r D Ct×X . Then the class [P ] of P in N (h) satisfies [P ] ∈ F ord r N (h) and obviously we have Using the description of the canonical V -filtration on the module N (h) along the divisor {t = 0} from Corollary 3.3, we can give a more precise description of the Hodge filtration on that module. We also recall from such corollary that B ′ h := {α ∈ Q ∩ (0, 1) | b h (α − 1) = 0}, and for α ∈ B ′ h , we write l α for the multiplicity of α − 1 in b h (s). For the sake of brevity, we will sometimes write in the following F H N (h), translating the Hodge filtration on Proof. From the proof of the last lemma we know already that so that in particular F H k i h,+ O X ( * D) = 0 for all k < 1 since F ord l N ′ (h) = 0 for negative l. The same holds for F ord l N (h), hence, the above formulas are proved when k < 1. We rewrite formula (35) in a recursive way, namely where the last equality uses again formula (35). Now we conclude using Proposition 4.2: We know from equation (30) by the very definition of the functor (j ′ ) * and where the inclusion * ⊂ follows from equation (36) above. As a consequence, the first inclusion V 0 is in fact an equality, and therefore equation (36) becomes so that the recursive Formula (33) is shown. Formula (34) follows by replacing the term V 0 can N (h) with the expression from Proposition 3.3. Moreover, it is clear that the non-recursive Formula (32) follows from the recursive one (33) by induction.
Our main purpose is to describe the Hodge filtration on O X ( * D). This description will be obtained as a consequence of Proposition 4.    Proof. It is sufficient to prove the second statement, since locally where D is defined by an equation h = 0, formula (37) exhibits F H k O X ( * D) as a submodule of F ord k M(h). However, the statement is as in Definition 2.2) holds globally on X, once it is shown at each p ∈ D. First notice that we have the equality in other words, we obtain for all k > 1. As we have noticed in Corollary 4.3, we have the inclusion

we obtain by plugging in formula (34) that
Shifting the indices by one, the inclusion is then clear. But we also have the inclusion in the other direction: as required.
Remark 4.5. (see also subsection 5.1 below) As shown in [Sai93, Proposition 0.9] we have is the pole order filtration. In our situation of a strongly Koszul free divisor, we obviously have We have seen so far that F H , with equality iff −1 is the only root of b h (s) for any local reduced equation h of D. However, we can actually give an inclusion in the reverse direction, but with a specific shift, which we conjecture to be the generating level of the Hodge filtration on O X ( * D) (see Conjecture 4.12 below). As a preparation, we need the following result.
We obtain the following two consequences that we will use below to discuss the dual Hodge filtration.
Corollary 4.7. With the aforementioned notations, we have 1. Recall Formula (39), which says that Now obviously the element [1] ∈ N (h) belongs to the submodule M(h) ⊗ 1 ⊂ N (h), hence we obtain from Lemma 4.6 that the element [1] ∈ M(h) lies in F H r M(h). 2. The statement is trivial for k < r. On the other hand, we have by definition that for any k ≥ r any class [P ] ∈ F ord k−r M(h) can be represented by an operator P ∈ D X of order k − r. Moreover, Corollary 4.8. Under the above conditions, we have: We will finish this section by showing some results about the Hodge filtration on the dual Hodge module Dj * Q H U [n]. Its underlying D X -module is DO X ( * D), which we denote by O X (!D). From the logarithmic comparison theorem we know that D X ⊗ V D X O X (D) ∼ = O X ( * D), and so we can apply Proposition 2.9 (for the case E = O X (D)) to obtain that the holonomic filtered module (O X ( * D), F ord • ) has the Cohen-Macaulay property and moreover that the dual filtered module of (O X ( * D), F ord . We recall that a similar property holds for the Hodge filtration, more precisely, we have the following.
With these preparations, we obtain the following result concerning the Hodge filtration on the dual module O X (!D).
Theorem 4.10. Let, as above, D ⊂ X be a strongly Koszul free divisor, then we have the following inclusions of coherent O X -submodules of O X (!D): In particular, since F ord −1 O(!D) = 0, we obtain the vanishing Proof. Equation (40) from Corollary 4.8 says Applying the Rees functor R(−) defined on page 7 to the first inclusion yields a short exact sequence of graded recall that for a graded D X -module M, we write M(l) for the same module with grading M(l) k = M k+l . Notice that the cokernel C is a z-torsion module, since, e.g., for any section s ∈ F H k O X ( * D) we have s ∈ F ord k O X ( * D) (due to the second inclusion from Formula (4.8)). Hence [s] · z k ∈ (R F ord O X ( * D)) k = (R F ord O X ( * D)(−r)) k+r and therefore [s] · z k maps to 0 in C. Now apply the duality functor for graded left D X -modules (defined by formula (11)) to the short exact sequence (42). From Corollary 2.11 we deduce that * On the other hand, by Theorem 4.9 we know that the filtered module (O X ( * D), F H • ) satisfies the Cohen-Macaulay property as well, and we write as before (O(!D), F H,D • ) for its dual filtered module. Hence, the triangle resulting from applying * D to the sequence (42) has the following long exact cohomology sequence: The D X -module C is a z-torsion module, which implies that H 0 * DC is so (since z is a central element in D X ).
The D X -module R F H,D O(!D) (and similarly (R F ord O X (!D))(r)) has no z-torsion, since it is the Rees module of a filtered D X -module, hence H 0 * DC = 0. Notice however that H 1 * DC does not necessarily vanish, since the homological dimension of the rings R U defined in Formula (7) is bounded from above by 2n + 1 only (which is the global homological dimension of Gr F for all k ∈ Z, which completes the proof.
Finally, we will state a conjecture about the so-called generating level of the Hodge filtration. Let us recall the following definition from [Sai09].
Definition 4.11. Let X be a complex manifold and let a well filtered module (M, F • ) be given. Then we say that the filtration F • M is generated at level k if holds for all l ≥ 0, or, equivalently, if for all k ′ ≥ k, we have The smallest integer k with this property is called the generating level of F • M.
Based on our calculations in section 5, we conjecture the following bound for the generating level of F H • on O X ( * D). Remark 4.13.
1. As we will see in section 5, this bound for the generating level is not always better than the known general bound, which is n − 2 (see [MP19a,Theorem B]). For the specific class of linear free divisors, it is known however that deg(b h (s)) = n, and in this case the bound r would be a drastic improvement, see subsection 5.3 below. Notice also that the general bound n − 1 − ⌈ α D ⌉ from [MP20a, Theorem A] is not helpful in our situation, since the minimal exponent α D , which is the negative of the biggest root of the reduced Bernstein polynomial b h (s)/(s + 1), lies in (0, 1), so that this bound is again n − 2.
2. There is a criterion in [Sai94, Lemma 2.5] that guarantees generating level smaller or equal to k for a filtered module (M, F • ) satisfying the Cohen-Macaulay property in terms of vanishing of the dual filtration. It cannot, however, directly be applied in our situation since for this we would need the vanishing which is provided by Theorem 4.10. One may hope though that some refinement of this argument (see, e.g., [MP20a,Proposition 3.3] for a statement in a related situation) can give the desired result.
3. As an immediate consequence of the conjecture, we would obtain a local vanishing result for a log resolution of an SK-free divisor, as provided by [MP19a,Theorem 17.1]. Namely, for a log resolution π : X −→ X which is an isomorphism over X\D and where E := (π −1 (D)) red (a reduced normal crossing divisor on X) we would have (provided that Conjecture 4.12 holds true) R k π * Ω n−k X (log E) = 0 for all k > r.

Computations of Hodge ideals and examples
The purpose of this section is to develop some techniques to calculate Hodge ideals of a strongly Koszul free divisor using our main result (Theorem 4.4). We will illustrate these methods by concrete computations of Hodge ideals and of the generating level of the Hodge filtration for some interesting examples. We start with the following preliminary result.
Lemma 5.1. Let D ⊂ X be a free divisor, let p ∈ D such that D is strongly Koszul at p and take a reduced local defining equation h ∈ O X,p of (D, p) ⊂ (X, p). Consider the vector fields δ 1 , . . . , δ n−1 ,χ as in Lemma 4.1. Let β(s) be any polynomial in C[s]. Then V 0 D Ct×X,(0,p) (t, β(∂ t t), t − h, δ 1 , . . . , δ n−1 ,χ + 1) ∩ D X,p [∂ t t] = D X,p [∂ t t](h, β(∂ t t), δ 1 , . . . , δ n−1 ,χ + 1). (45) Proof. For the sake of simplicity, let us denote D Ct×X,(0,p) just by D. Before starting the proof, let us explain something we will take for granted throughout it: we know that V 0 D = C{x, t}[∂ x , ∂ t t] and that any P ∈ V 0 D can be expressed in a unique way as a series with constant coefficients, and for each (β, m), the series α,ℓ a β,m α,ℓ x α t ℓ is convergent. Now, by using the identity t ℓ (∂ t t) m = (∂ t t − ℓ) m t ℓ we find another unique formal representation and one easily sees that for fixed β, i, ℓ the series α c β,i α,ℓ x α is convergent. So we have a unique formal expression and it makes sense to consider P ∈ D X,p [∂ t t][[t]], where the last ring is the completion of D X,p [∂ t t][t] with respect to the (t)-adic topology. Here one has to use that the left ideal generated by t coincides with the right ideal generated by t, and so it is a bilateral ideal, and also that the monomials t ℓ , ℓ ≥ 0, form a basis of D X,p [∂ t t][t] as a left and as a right D X,p [∂ t t]-module.
Let us now begin with the actual proof and denote by I the ideal (t, β(∂ t t), t − h, δ 1 , . . . , δ n−1 ,χ + 1) of V 0 D. The inclusion ⊃ in equation (45) is trivial, let us show the reverse one. We will use a more suitable set of generators of I, namely {t,β, h, δ 1 , . . . , δ n−1 , χ + ∂ t t + 1}, whereβ = β(−χ − 1). Let then P ∈ I ∩ D X,p [∂ t t]. Then there exist operators Q, R, A, B 1 , . . . , B n−1 , C ∈ V 0 D such that Our goal is to show that Q must vanish and that the other operators belong actually to D X,p [∂ t t]. Let us write , and analogously for R, A, the B i and C. We can express the right-hand side of equation (46) as a new operator S = k S k t k , where again S k ∈ D X,p [∂ t t]. Moreover, since t,β, h, δ 1 , . . . , δ n−1 and χ + 1, when seen as elements of , are homogeneous in t, we have for every k ≥ 0 that Comparing the S k ∈ D X,p [∂ t t] with the terms of degree k in t at the left-hand side (recall that P ∈ D X,p [∂ t t]), we find that S k = 0 for every k > 0. Let us write nowR = R − R 0 and similarly with the other operators. Each of them can be written as a series in t and we have checked that Therefore, since k>0 C k ( χ − k + 1)t k =C( χ + 1), Summing up, we have been able to write P as a linear combination of h,β, the δ i and χ + 1 with coefficients in D X,p [∂ t t], as we wanted. 2. For any k ≥ 0, we have the following equality of O Ct×X,(0,p) -modules: and so Proof. Following the same convention as in Lemma 5.1 above, we will write O and D for O Ct×X,(0,p) and D Ct×X,(0,p) , respectively. We will define on D X,p [s] an action of O by putting t · P (s) := P (s + 1)h and extending it by linearity. Let us check that such an action is well-defined. Indeed, let a(x, t) = α,k a αk x α t k be a series in O and let us show that we can multiply by a in D X,p [s]. As a first step, we will restrict ourselves to consider an element P ∈ D X,p . In order to show that a · P lies within D X,p [s], it is equivalent by linearity to show it for P = ∂ β , with β ∈ N n . Then we will have where we use the common component wise partial ordering and the standard multi-index notation for the factorial and the binomial numbers. Therefore, a · ∂ β is a finite sum of monomials in the ∂ i times certain convergent series, so it belongs to D X,p ⊂ D X,p [s]. Consider now P s j ∈ D X,p [s]. Then, which is another finite sum of elements in D X,p [s]. Thus a · P s j belongs clearly to D X,p [s], as we wanted to show. Now the action of O on D X,p [s] can be extended to an action of V 0 D. Indeed, we have [t, s] · P (s) = P (s + 1)h = t · P (s) for any P (s) ∈ D X,p [s], so we can take the action of ∂ t t as that of −s. Now using that V 0 D ∼ = D X,p [s] ⊗ OX,p O as O X,p -modules, we can extend both actions by linearity to get the desired one of V 0 D. Let us check now that π p becomes V 0 D-linear with the new structure on D X,p [s]. Indeed, consider an operator P (s) ∈ D X,p [s] and a series a = α,k a αk x α t k ∈ O as before. Then, Note that we are replacing the powers of t by those of h in the third equality. To justify this step, let us rewrite the p β,m being convergent functions in O. Now dividing all such functions by t − h we can write them as p βm (x, t) = q βm (x, t)(t − h) + r βm (x). (This is a very particular instance of the Weierstraß division theorem for convergent power series whose proof is elementary in this case.) Consequently, For every β and m we know that r βm (x) = p βm (x, h), hence, we have an expression , we know that there is a representative P ∈ D of ξ and that there are operators Q, R ∈ V 0 D and A, B 1 , . . . , B n−1 , C ∈ D such that It is clear that t Q can be rewritten as Q ′ t for a suitable Q ′ ∈ V 0 D. Regarding β(∂ t t) R, let us expand R as k R k t k with R k ∈ V 0 D in an analogous fashion as in the proof of Lemma 5.1. Then, where R ′ ∈ V 0 D. Therefore, renaming Q ′ + R ′ as Q ∈ V 0 D, we have a new expression Notice that since , it follows from Proposition 3.10 that we can pick a new representative P ∈ F k D ∩ V 0 D of ξ, that is, we have P − P ∈ I(h). Moreover, when writing P as an element in D X,p [∂ t t][[t]], we can replace any positive power of t by h, and this will not change the order. Hence, there exists an operator P ∈ F k D∩D X,p [∂ t t] with P − P = A(t−h). It follows that P − P ∈ I(h), or, said otherwise, there are coefficients A ′ , B ′ 1 , . . . , B ′ n−1 , C ′ ∈ D such that Hence (by replacing P with the expression from equation (52)) we obtain that the class ξ ∈ V 0 can N (h)∩F ord k N (h) can be represented by an operator P ∈ F k D ∩ D X,p [∂ t t] which has an expression where A := A + A ′ , B i := B i + B ′ i and C := C + C ′ . By construction, the operators P , Q and R are elements in V 0 D. Then we apply Corollary 3.9 and conclude that A, B 1 , . . . , B n−1 and C belong to V 0 D as well. Summing up, we can choose P inside Applying Lemma 5.1 above, we obtain that P ∈ D X,p [∂ t t](h, β(∂ t t), δ 1 , . . . , δ n−1 ,χ + 1) ∩ T k D X,p [∂ t t], and replacing ∂ t t by −s provides the desired claim. For the last part, Formula (48), we combine Formula (33) (or its non-recursive version, Formula (32)) with Formula (47).
As a first application of Theorem 5.2 and Theorem 4.4, we can give a formula to calculate the zeroth Hodge ideal I 0 (D) p .
Corollary 5.3. Under the assumptions of the previous Proposition, let (J 0 ) p be the ideal of O X,p defined as Proof. Point 1 is a consequence of Corollary 4.3 and Theorem 5.2 above. For the second point we use formula (39). We know that is seen as a O X,p -submodule of N (h). However, since (J 0 ) p ⊂ O X,p , there is actually no proper intersection and the claim follows. In order to obtain the formula for the zeroth Hodge ideal we just need to recall its definition and the isomorphism between O X,p ( * D) and M (h) (see point 2 of Proposition 2.4). Since F H 0 O X,p ( * D) = I 0 (D) p · O X,p (D), we just need to multiply by h the elements of (J 0 ) p · h −1 ⊂ O X,p ( * D), but again, (J 0 ) p ⊂ O X,p , so I 0 (D) p = (J 0 ) p and we are done.
Remark 5.4. Notice that the ideal J p from Theorem 5.2 equals, up to changing s by s + 1 the ideal where β(s) = α∈B h (s − α) lα , with B h and l α being, respectively, the set of roots of b h in the interval (−1, 0) and the multiplicity of each root α. Then we have Our next aim is to describe an algorithm to calculate effectively the filtration steps F H k O X ( * D) resp. the Hodge ideals I k (D) for an SK-free divisor. The starting point is Formula (48). Recall (see the explanation before Lemma 3.2) that there is a left D Ct×X,(0,p) -linear isomorphism which we need to make explicit in order to describe our algorithmic procedure to obtain the modules F H k O X ( * D). Let us consider the C-algebra automorphism ϕ : D Ct×X,(0,p) → D Ct×X,(0,p) defined as It is clear that ϕ(δ) = δ − δ(h)∂ t and ϕ −1 (δ) = δ + δ(h)∂ t for each δ ∈ Θ X,p , and ϕ −1 (t) = t − h. Now given a class [Q] ∈ N (h), by division by t we find unique operators A ∈ D Ct×X,(0,p) and Q ′ ∈ O X,p ∂ x1 , . . . , ∂ xn , ∂ t such that ϕ(Q) = A t + Q ′ (ord(Q ′ ) ≤ ord(Q)).
Then we write and taking classes in M (h) we obtain the element which is readily verified to be well-defined. Notice that the inverse map Ψ −1 sends an element . Formula (48) becomes: where we put Ψ = Ψ • π p . We denote Φ : M (h) ∼ − → O X,p ( * D) the isomorphism of left D X,p -modules given by Φ([Q]) = Q(h −1 ), for each Q ∈ D X,p (see formula (5)), and by Φ : We recall from Formulas (38) With these preliminary remarks, we can now explain an effective procedure to compute a system of generators of the O X,p -modules F H k O X,p ( * D), k ≥ 0: Step 1: We compute an involutive basis B = {P 1 , . . . , P N } of J p with respect to the total order filtration T on D X,p [s], i.e. P 1 , . . . , P N are such that where for P ∈ D X,p [s], σ T (P ) denotes the symbol of P in Gr T • (D X,p [s]). Let us write ord T (P j ) = d j . Consequently, for each k ≥ 0, a system of generators of the O X,p -module J p ∩ T k D X,p [s] is given by Step 2: After equation (54), a system of generators of the O X,p -module is Step 3: By means of the isomorphism Φ, we obtain a system of generators of the O X,p -module F H k+1 (O X,p ( * D)[∂ t ]), that we call ξ e , e = 1, . . . , R.
is a submodule of the rank (k + 1) free O X,pmodule P k (O X,p ( * D)[∂ t ]), we have Finally, a syzygy computation allows us to compute a system of generators of the O X,p -module In the following subsections we apply this algorithm to calculate the Hodge ideals and the generating level for some interesting classes of examples. Since it is known ([MP19a, Proposition 10.1]) that I 0 (D) = J ((1 − ε)D), and since an algorithm for the calculation of multiplier ideals exists already (see [BL10]), our approach first of all provides an alternative way to compute these multiplier ideals.

Divisors with normal crossings
If X is any complex manifold and D ⊂ X is a reduced divisor with only normal crossings, then it is elementary to check that D is free and strongly Koszul at each of its points. As we have already pointed out in Remark 4.5, it easily follows from the fact that −1 is the only root of b D (s) that we have Consequently, since obviously the order filtration is generated at level 0, so is the Hodge filtration (a fact that is of course well known from Saito's theory). For the Hodge ideals themselves, we obtain that I k (D) · F ord k O X ( * D) = P k O X ( * D), and an easy local calculation shows that this coincides with [MP19a, Proposition 8.2]. Summarizing, if the divisor D has only normal crossings, then our results easily imply the known facts about the Hodge filtration resp. the Hodge ideals for such divisors.

Surfaces
We consider two kinds of free surfaces in C 3 , namely (two-dimensional hyper-)plane arrangements and Sekiguchi's free divisors ( [Sek09]). Let D be a central hyperplane arrangement k i=1 H i ⊂ X = C n . It is a longstanding question to detect when an arrangement is a free divisor; for a recent account, see [Dim17,Chapter 8]. However, if D is free, then it is strongly Koszul since it is locally quasi-homogeneous (see, e.g., [CMNM02] for this implication), so that the methods from this paper do apply. Below are some results for low-dimensional free arrangements:

Name
Equation (x 2 − y 2 )(x 2 − z 2 )(y 2 − z 2 ) (y 2 z − z 3 , x 2 z − z 3 , y 3 − yz 2 , xy 2 − xz 2 , x 2 y − yz 2 , x 3 − xz 2 ) In both cases the Hodge filtration on O X ( * D) is generated at level zero. Of course, these results do not depend on the fact that dim(D) = 2, similar calculations are possible for other arrangements, but the ideal I 0 (D) becomes difficult to print.
Let us consider now the examples of divisors in C 3 that are discussed in the paper [Sek09]. We refer to loc. cit. for details on their construction and only produce the results of the calculation of I 0 (D) here. All these examples are given by weighted homogeneous equations in three variables, and we write (d; d x , d y , d z ) for a weight vector, where d x , d y and d z are the weights of x, y and z and d is the degree of the defining equation for the divisor D ⊂ C 3 .

Linear free divisors
The paper [BM06] introduced a large class of examples of free divisors which appear as discriminants in prehomogenous vector spaces. Since the module Θ(− log D) has a basis of linear (in the global coordinates) vector fields in these examples, the divisor D is called linear free. A rich source of linear free divisors comes from representation of quivers. It can be shown that the strong Koszul assumption is satisfied in these cases if the underlying graph of the quiver is of ADE-type. Quivers of type A yield free divisors with normal crossings, so the first non-trivial example is the discriminant in the representation space of the D 4 -quiver. Here we have D ⊂ Mat(2 × 3, C) = a 11 a 12 a 13 a 21 a 22 a 23 | a ij ∈ C , where D = V (h) and h = ∆ 1 · ∆ 2 · ∆ 3 , with ∆ 1 , ∆ 2 and ∆ 3 being the three maximal minors of an element of Mat(2 × 3, C) (hence, h is a homogenous equation of degree 6). In this case, we have b h (s) = (s + 1) 4 (s + 2/3)(s + 4/3) and hence we consider J = (h, s − 1/3, δ 1 , . . . , δ 5 , χ − s + 1) ⊂ D C 6 [s].
Notice that for this example, the computation of the multiplier ideal J ((1 − ε)D) with the methods from [BL10] does not terminate. The generating level of the Hodge filtration on O X ( * D) is two here. The first Hodge ideal I 1 (D) has a minimal generating set consisting of 24 polynomials of degree 13 at most, whereas I 2 (D) is minimally generated by 124 polynomials, their maximal degree being 22. In fact, Conjecture 4.12 would yield the following estimate for the generating level of the full D n -series.
Conjecture 5.5. Let D ⊂ X = C 4n−10 be the linear free divisor D n . Then, the Hodge filtration on O X ( * D) is generated at level n − 3.
Proof assuming Conjecture 4.12. The dimension of the divisors D n can be found in [GMS09, p. 1347]. The result then is just a consequence of applying the statement of Conjecture 4.12, once we know a general formula for the Bernstein-Sato polynomials of the divisors D n . It can be found at [Sev11, .
Note that this conjecture, if true, would give a bound for the generating level which is exactly a quarter of the general one given in [MP19a,Theorem B]. We also believe that such bounds are attained for all of the D n (i.e. that the generating level is exactly n − 3), as we have checked for n = 4, 5.
Notice that similar computations are possible up to some extent for other linear free divisors of SK-type, such as those given by the E 6 -and E 7 -quivers. Under the assumption of Conjecture 4.12, one could give generating level bounds, using the calculation of Bernstein polynomials from [Bar18].
We finish this subsection with some examples of non-reductive linear free divisors in low dimension, as found in [GMNS09, Example 5.1, Table 6.1], that we summarize in the table below.

The Whitney umbrella and the cross caps
We finish this section by mentioning two examples that slightly fall out of the general setup of this paper. Namely, consider the Whitney umbrella D = V(h) = V(x 2 − y 2 z) ⊂ C 3 . This is not a free divisor, indeed, we have Θ(− log D) = 4 i=1 O C 3 δ i where δ 1 = y∂ y − 2z∂ z , δ 2 = −yz∂ x − x∂ y , δ 3 = −y 2 ∂ x − 2x∂ z , δ 4 = χ = (1/2)x∂ x + (1/3)y∂ y + (1/3)z∂ z , so that Θ(− log D) is not O C 3 -locally free. However, the isomorphism of left D C 3 -modules still holds true, and we find that b h (s) = (s + 1) 2 (s + 3/2). According to Lemma 3.2 from above, we have b = s 2 (s − 1/2). One checks that the main results of this paper can also be applied to this example. In particular, it follows that the integer r appearing in Lemma 4.6 is zero, and hence we deduce from Corollary 4.7 that F H • O C 3 ( * D) ∼ = F ord • O C 3 ( * D). In particular, in this example the generating level of the Hodge filtration is zero since F ord • O C 3 ( * D) is generated at level 0. A similar reasoning applies to the higher dimensional example D = V(h) := V(x 1 x 2 x 2 3 x 4 + x 3 3 x 2 4 − x 2 1 x 2 3 x 5 + x 3 2 x 4 − x 1 x 2 2 x 5 + 3x 2 x 3 x 4 x 5 − 2x 1 x 3 x 2 5 − x 3 5 ) ⊂ C 5 , called cross-cap, which again is not free (the module of logarithmic vector fields has 9 generators), here we have b h (s) = (s + 1) 3 (s + 3/2)(s + 4/3)(s + 5/3).
As a conclusion, we obtain that for both the Whitney umbrella and the cross-cap we have I 0 (D) = O X , but I k (D) O X for all k > 1. Actually, these two examples are the first two of a whole series, which is discussed in [HL09]. However, it is unclear at this point whether we always have the equality F H • O C n ( * D) ∼ = F ord • O C n ( * D) since this needs the fact that the roots of b h (s) are contained in (−2, −1], and a general formula for the Bernstein polynomial for the elements in this series is not known (actually, it seems computational impossible to obtain b h (s) even for the next example, which is a divisor in C 7 ).