Computing regular meromorphic differential forms via Saito's logarithmic residues

Logarithmic differential forms and logarithmic residues associated to a hypersurface with an isolated singularity are considered in the context of computational complex analysis. An effective method is introduced for computing logarithmic residues. A relation between logarithmic differential forms and the Brieskorn formula on Gauss-Manin connection are discussed. Some examples are also given for illustration.


Introduction
In 1975, K. Saito introduced, with deep insight, the concept of logarithmic differential forms and that of logarithmic vector fields and studied Gauss-Manin connection associated with the versal deformations of hypersurface singularities of type A 2 and A 3 as applications. These results are published in [26]. He developed the theory of logarithmic differential forms, logarithmic vector fields and the theory of residues and published in 1980 a landmark paper [27]. One of the motivations of his study, as he himself wrote in [27], came from the study of Gauss-Manin connections ( [6,25]). Another motivation came from the importance of these concepts he realized. Notably the logarithmic residue, interpreted as a meromorphic differential form on a divisor, is regarded as a natural generalization of the classical Poincaré residue to the singular cases.
In 1990, A. G. Aleksandrov([2]) studied Saito theory and gave in particular a characterization of the image of the residue map. He showed that the image sheaf of the logarithmic residues coincides with the sheaf of regular meromorphic differential forms introduced by D. Barlet ([6]) and M. Kersken ([15,16]). We refer the reader to [5,9,10,11,13,23] for more recent results on logarithmic residues. We consider logarithmic differential forms along a hypersurface with an isolated singularity in the context of computational complex analysis. In our previous paper [33], we study torsion modules and give an effective method for computing them. In the present paper, we first consider a method for computing regular meromorphic differential forms. We show that, based on the result of A. G. Aleksandrov mentioned above, representatives of regular meromorphic differential forms can be computed by using the algorithm presented in [33] on torsion modules. Main ideas of our approach are the use of the concept of logarithmic residue and that of logarithmic vector field. Next, we show a link between logarithmic differential forms and Gauss-Manin connections, which reveals the role of the torsion module in the computation of a saturation of Brieskorn lattice of Gauss-Manin connection ( [6,28,29]).

Logarithmic differential forms and residues
In this section, we briefly recall the concept of logarithmic differential forms and that of logarithmic residues and fix notation. We refer the reader to [27] for details. Next we recall the result on A. G. Aleksandrov on regular meromorphic differential forms. Then, we recall a result of G. -M. Greuel

Logarithmic residues
Let f be a holomorphic function defined on X. Let S = {x ∈ X | f (x) = 0} denote the hypersurface defined by f .

Definition 1.
Let ω be a meromorphic differential q-form on X, which may have poles only along S. The form ω is a logarithmic differential form along S if it satisfies the following equivalent four conditions: (i) f ω and f dω are holomorphic on X.
(ii) f ω and df ∧ ω are holomorphic on X.
(iii) There exists a holomorphic function g(x) and a holomorphic (q − 1)-form ξ and a holomorphic q-form η on X, such that: (iv) There exists an (n − 2)-dimensional analytic set A ⊂ S such that the germ of ω at any where Ω q X,p denotes the module of germs of holomorphic q-forms on X at p.
For the equivalence of the condition above, see [27]. Let Ω q X (log S) denote the sheaf of logarithmic q-forms along S. Let M S be the sheaf on S of meromorphic functions, let Ω q S be the sheaf on S of holomorphic q-forms defined to be Definition 2. The residue map res : is define as follows: For ω ∈ Ω q S (log S), there exists g, ξ, η such that gω = df f ∧ ξ + η. Then the residue of ω is defined to be res(ω) = ξ g | S in M S ⊗ O X Ω q−1 S . Note that it is easy to see that the image sheaf of the residue map res of the subsheaf df f ∧ Ω q−1 X + Ω q X of Ω q X (log S) is equal to Ω q−1 X | S : See also [27] for details on logarithmic residues. The concept of residue for logarithmic differential forms can be actually regarded as a natural generalization of the classical Poincaré residue.

Barlet sheaf and torsion differential forms
In 1978, by using results of F. El Zein on fundamental classes, D. Barlet introduced in [6] the notion of the sheaf of regular meromorphic differential forms ω q S in a quite general setting. He showed that for the case q = n − 1, the sheaf ω n−1 S coincides with the Grothendieck dualizing sheaf and ω q S can also be defined in the following manner: . Then, the sheaf of regular meromorphic differential forms ω q S , q = 0, 1, . . . , n−2 on S is defined to be In 1990, A. G. Aleksandrov([2]) obtained the following result.
Let Tor(Ω q S ) denote the sheaf of torsion differential q-forms of Ω q S .
Example 1. Let X be an open neighborhood of the origin O in C 2 . Let f (x, y) = x 2 − y 3 and S = {(x, y) ∈ X | f (x, y) = 0}. Then, for stalk at the origin of the sheaves of logarithmic differential forms, we have where O X,O is the stalk at the origin of the sheaf O X of holomorphic functions and β = 2ydx − 3xdy. The differential form β, as an element of Ω 1 , is a torsion. The differential form yβ is also a torsion. Since the defining function f is quasi-homogeneous, the dimension of the vector space Tor(Ω 1 S ) is equal to the Milnor number µ = 2 of S ( [17,35]). Therefore we have Tor(Ω 1 In 1988 [1], A. G. Aleksandrov studied logarithmic differential forms and residues and proved in particular the following.
there exists an exact sequence of sheaves of O X modules, The result above yields the following observation: Tor(Ω q S ) plays a key role to study the structure of res(Ω q X (log S)).

Vanishing theorem
In 1975, in his study( [14]) on Gauss-Manin connections G. -M. Greuel proved the following results on torsion differential forms.
The dimension, as a vector space over C, of torsion module Tor(Ω n−1 S ) is equal to τ (f ), the Tjurina number of the hypersurface S at the origin defined to be Assume that the hypersurface S has an isolated singularity at the origin. We thus have, by combining the results of G. -M. Greuel above and of A. G. Aleksandrov presented in the previous subsection, the followings.
Accordingly we have the following.

Description via logarithmic residues
In this section, we recall results given in [33] to show that torsion differential forms can be described in terms of non-trivial logarithmic vector fields. We also recall basic idea for computing non-trivial logarithmic vector fields. As an application, we give a method for computing logarithmic residues.

Logarithmic vector fields
A vector field v on X with holomorphic coefficients is called logarithmic along the hypersurface S, if the holomorphic function v(f ) is in the ideal (f ) generated by f in O X . Let Der X (− log S) denote the sheaf of modules on X of logarithmic vector fields along S ( [27]).
Let ω X = dx 1 ∧ dx 2 ∧ · · · ∧ dx n . For a holomorphic vector field v, let i v (ω X ) denote the inner product of ω X by v. Proposition 2. Let S = {x ∈ X | f (x) = 0} be a hypersurface with an isolated singularity at the origin. Then, holds.
Proof . Let β = i v (ω X ), and set ω = β f . Then, f ω = β is a holomorphic differential form. Therefore, the meromorphic differential n − 1 form ω is logarithmic if and only if df ∧ β f is a holomorphic differential n-form.
is called trivial.

Lemma 1.
Let v be a germ of a logarithmic vector field. Then, the following conditions are equivalent.
Proof . The logarithmic differential form The last condition is equivalent to the triviality of the vector field v, which completes the proof.
For β ∈ Ω n−1 X,O , let [β] denote the Kähler differential form in Ω n−1 S,O defined by β, that is, [β] is the equivalence class in Ω n−1 X,O /(f Ω n−1 X,O + df ∧ Ω n−2 X,O ) of β. The lemma above amount to say that, for logarithmic vector fields v, [i v (ω X )] is a non-zero torsion differential form in Tor(Ω n−1 S,O ) if and only if v is a non-trivial logarithmic vector field. We say that germs of two logarithmic vector fields Now consider the following map It is easy to see that the map Θ is well-defined. We arrive at the following description of the torsion module. is an isomorphism.

Polar method
In [30], based on the concept of polar variety, logarithmic vector fields are studied and an effective and constructive method is considered. Here in this section, following [21,30] we recall some basics and give a description of non-trivial logarithmic vector fields. Let S = {x ∈ X | f (x) = 0} be a hypersurface with an isolated singularity. In what follows, we assume that f, ∂f ∂x 2 , ∂f ∂x 3 , . . . , ∂f ∂xn is a regular sequence and the common locus V (f, ∂f ∂x 2 , ∂f ∂x 3 , . . . , ∂f ∂xn ) ∩ X is the origin O.
We have the followings Lemma 2. Let a(x) be a germ of holomorphic function in O X,O .Then, the following are equivalent.
Since the sequence f, ∂f ∂x 2 , ∂f ∂x 3 , . . . , ∂f ∂xn is assumed to be regular, we also have the following.
Then, v ′ is trivial.
Accordingly, we have the following result.
Proposition 3. Let f, ∂f ∂x 2 , ∂f ∂x 3 , . . . , ∂f ∂xn be a regular sequence. Let v be a germ of logarithmic vector field along S of the form Then, the following conditions are equivalent.
The discussion above leads a method for computing non-trivial logarithmic vector fields: Step Step 2 For each a(x) ∈ A, compute a 2 (x), a 3 (x), ..., a n (x), b(x) ∈ O X,O , such that a(x) ∂f ∂x 1 + a 2 (x) ∂f ∂x 2 + · · · + a n−1 (x) ∂f ∂x n−1 + a n (x) give rise to a basis of Der X,O (− log S)/ ∼. Note that algorithms for computing non-trivial logarithmic vector fields is described in [33].

Regular meromorphic differential forms
Now we are ready to consider a method for computing regular meromorphic differential forms. For simplicity, we first consider three dimensional case. Assume that a non-trivial logarithmic vector field v is given. Let We introduce differential forms ξ and η as Let g(x) = ∂f ∂x 1 . Then, the following holds.
Accordingly, the logarithmic differential form ω = β f satisfies Since g(x) = ∂f ∂x 1 , we have, by definition, the following: Notice that the differential form ξ above is directly defined from the coefficients of the logarithmic vector field v.
Proposition 4. Let S = {x ∈ X | f (x) = 0} be a hypersurface with an isolated singularity at be a germ of non-trivial logarithmic vector field along S.

Examples
In this section, we give examples of computation for illustration. Data is an extraction from [33]. Let f 0 (z, x, y) = x 3 + y 3 + z 4 and let f t (z, x, y) = f 0 (z, x, y) + txyz 2 , where t is a deformation parameter. We regard z as the first variable. Then, f 0 is a weighted homogeneous polynomial with respect to a weight vector (3,4,4) and f t is a µ-constant deformation of f 0 , called U 12 singularity. The Milnor number µ(f t ) of U 12 singularity is equal to 12. In contrast, the Tjurina number τ (f t ) depends on the parameter t. In fact, if t = 0, then τ (f 0 ) = 12 and if t = 0, then τ (f t ) = 11. In the computation, we fix a term order ≻ −1 on O X,O which is compatible with the weigh vector (3,4,4). We consider these two cases separately.
These 11 elements in A are used to construct non-trivial logarithmic vector fields and regular meromorphic differential forms. We give the results of computation.
is a non-trivial logarithmic vector field, where is a non-trivial logarithmic vector field, where We omit the other nine cases.

Brieskorn formula
In 1970, B. Brieskorn studied the monodromy of Milnor fibration and developed the theory of Gauss-Manin connection ( [8]). He proved the regularity of the connection and proposed an algebraic framework for computing the monodromy via Gauss-Manin connection. He gave in particular a basic formula, now called Brieskorn formula, for computing Gauss-Manin connection. We show in this section a link between Brieskorn formula, torsion differential forms and logarithmic vector fields. We present an alternative method for computing non-trivial logarithmic vector fields. We also present some examples for illustration.

Brieskorn lattices and Gauss-Manin connection
We briefly recall some basics on Brieskorn lattice and Brieskorn formula. We refer to [7,8,29]. Let f (x) be a holomorphic function on X with an isolated singularity at the origin O ∈ X, where X is an open neighborhood of O in C n . Let where ω X = dx 1 ∧ dx 2 ∧ · · · ∧ dx n . Therefore in terms of the coordinate we have the following, known as Brieskorn formula.
is a logarithmic vector field along S.
Notice that β, yβ are non-zero torsion differential forms in Ω 1 S . The observation above can be generalized as follows.
be a germ of non-trivial logarithmic vector field along S. Let where ω X = dx 1 ∧ dx 2 ∧ · · · ∧ dx n . Then, Brieskorn formula implies the result. Now we present an alternative method for computing the module of germs of non-trivial logarithmic vector fields.
Step 1 Compute a monomial basis M of the quotient space Step 2 Compute a standard basis Sb of the ideal quotient ∂f ∂x 1 , ∂f ∂x 2 , · · · , ∂f ∂x n : (f ).
Step 3 Compute a basis B of the vector space by using Sb and M ∂f ∂x 1 , ∂f ∂x 2 , · · · , ∂f ∂x n : (f ) / ∂f ∂x 1 , ∂f ∂x 2 , · · · , ∂f ∂x n Step 4 For each b(x) ∈ B, compute a logarithmic vector field along S such that The method above computes a set of basis of non-trivial logarithmic vector fields. Note that, the number of logarithmic vector fields in the output is, as proved in [17,30], equals to the Tjurina number τ (f ).
Let v = a 1 (x) ∂ ∂x 1 + a 2 (x) ∂ ∂x 2 + · · · + a n (x) ∂ ∂x n be a germ of non-trivial logarithmic vector field along S, such that v(f ) = b(x)f (x). Then from the Proposition above, we have Therefore, the proposed method can be used as a basic procedure for computing Gauss-Manin connection. Each step can be effectively executable, as in [33], by utilizing algorithms described in [18,19,20,32]. One of the advantage of the proposed method lies in the fact that the resulting algorithm can handle parametric cases.

Examples
Let us recall that x 3 + y 7 + txy 5 is the standard normal form of semi quasi-homogeneous E 12 singularity. The weight vector of is (7,3) and the weighted degree of the quasi-homogeneous part is equal to 21 and the weighted degree of the upper monomial txy 5 is equal to 22. We examine here, by contrast, the case where the weighted degree of an upper monomial is bigger than 22. where d 1 = 4x 2 − 25x 3 y − 5y 3 , d 2 = 6xy − 25x 2 y 2 . By a direct computation, we have D(f (xω X )) = 7 10 x − 3 × 25 16 y 4 ω X mod ∂f ∂x , ∂f ∂y .
We omit the other cases.