# Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues

Logarithmic differential forms and logarithmic vector fields associated to a hypersurface with an isolated singularity are considered in the context of computational complex analysis. As applications, based on the concept of torsion differential forms due to A.G. Aleksandrov, regular meromorphic differential forms introduced by D. Barlet and M. Kersken, and Brieskorn formulae on Gauss-Manin connections are investigated. (i) A method is given to describe singular parts of regular meromorphic differential forms in terms of non-trivial logarithmic vector fields via Saito's logarithmic residues. The resulting algorithm is illustrated by using examples. (ii) A new link between Brieskorn formulae and logarithmic vector fields is discovered and an expression that rewrites Brieskorn formulae in terms of non-trivial logarithmic vector fields is presented. A new effective method is described to compute non trivial logarithmic vector fields which are suitable for the computation of Gauss-Manin connections. Some examples are given for illustration.

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## 1 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 and as applications. These results were published in [33]. He developed the theory of logarithmic differential forms, logarithmic vector fields and the theory of residues and published in 1980 a landmark paper [34]. One of the motivations of his study, as he himself wrote in [34], came from the study of Gauss–Manin connections [5, 32]. 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  [5] and M. Kersken [15, 16]. We refer the reader to [4, 8, 9, 10, 12, 29, 30] 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 [40], 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 adapting the method presented in [40] on torsion modules. Main ideas of our approach are the use of the concept of logarithmic residues and that of logarithmic vector fields. Next, we discuss a relation between logarithmic differential forms and Brieskorn formulae [5, 35, 37] and we show that Brieskorn formulae can be rewritten in terms of logarithmic vector fields. Applications to the computation of Gauss–Manin connections are illustrated by using examples.

In Section 2, we briefly recall some basics on logarithmic differential forms, logarithmic residues, Barlet sheaf and torsion differential forms. In Section 3, we first recall the notion of logarithmic vector fields and a result gave in [40] to show that torsion differential forms can be described in terms of non trivial logarithmic vector fields. Next, we recall our previous results to show that non-trivial logarithmic vector fields can be computed by using a polar method and local cohomology. Lastly in Section 3, we present Theorem 3.11 which say that regular meromorphic differential forms can be explicitly computed by modifying our previous algorithm on torsion differential forms. In Section 4, we give some examples to illustrate the proposed method of computing non-trivial logarithmic vector fields and regular meromorphic differential forms. In Section 5, we consider Brieskorn formulae on Gauss–Manin connections. We show that Brieskorn formulae described in terms of logarithmic differential forms can be rewritten in terms of non-trivial logarithmic vector fields. We give a new method for computing non-trivial logarithmic vector fields which is suitable in use to compute a connection matrix of Gauss–Manin connections. Finally, we show that the use of integral dependence relations provides a new effective tool for computing saturations of Gauss–Manin connection.

## 2 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 [34] for details. Next we recall the result of A.G. Aleksandrov on regular meromorphic differential forms. Then, we recall a result of G.-M. Greuel on torsion modules.

Let be an open neighborhood of the origin in . Let be the sheaf on of holomorphic functions and the stalk at of the sheaf .

### 2.1 Logarithmic residues

Let be a holomorphic function defined on . Let denote the hypersurface defined by .

###### Definition 2.1.

Let be a meromorphic differential -form on , which may have poles only along . The form is a logarithmic differential form along if it satisfies the following equivalent four conditions:

1. and are holomorphic on .

2. and are holomorphic on .

3. There exist a holomorphic function and a holomorphic -form and a holomorphic -form on , such that:

4. There exists an -dimensional analytic set such that the germ of at any point belongs to where denotes the module of germs of holomorphic -forms on  at .

For the equivalence of the condition above, see [34]. Let denote the sheaf of logarithmic -forms along . Let be the sheaf on of meromorphic functions, let be the sheaf on of holomorphic -forms defined to be

 ΩqS=ΩqX/(fΩqX+df∧Ωq−1X).
###### Definition 2.2.

The residue map is defined as follows: For , by definition, there exist , and such that

1. , and

2. .

Then the residue of is defined to be in .

Note that it is easy to see that the image sheaf of the residue map of the subsheaf of is equal to :

 res(dff∧Ωq−1X+ΩqX)=Ωq−1X∣∣\raisebox−3.01pt[4.3pt][−8.6pt]$S$.

See also [34] for details on logarithmic residues. The concept of residues for logarithmic differential forms can be actually regarded as a natural generalization of the classical Poincaré residue.

### 2.2 Barlet sheaf and torsion differential forms

In 1978, by using results of F. El Zein on fundamental classes, D. Barlet introduced in [5] the notion of the sheaf of regular meromorphic differential forms in a quite general setting. He showed that for the case , the sheaf coincides with the Grothendieck dualizing sheaf and can also be defined in the following manner.

###### Definition 2.3.

Let be a hypersurface in . Let be the Grothendieck dualizing sheaf . Then, the sheaf of regular meromorphic differential forms , on is defined to be

 ωqS=HomOS(Ωn−1−qS,ωn−1S).

In 1990, A.G. Aleksandrov [2] obtained the following result.

###### Theorem 2.4.

For any , there is an isomorphism of modules

 res(ΩqX(logS))≅ωq−1S.

See [2] or [3] for the proof.

Let denote the sheaf of torsion differential -forms of .

###### Example 2.5.

Let be an open neighborhood of the origin in Let and . Then, for stalk at the origin of the sheaves of logarithmic differential forms, we have

 Ω1X,O(logS)≅OX,O(dff,βf),Ω2X,O(logS)≅OX,O(dx∧dyf),

where is the stalk at the origin of the sheaf of holomorphic functions and . The differential form , as an element of , is a torsion. The differential form is also a torsion. Since the defining function is quasi-homogeneous, the dimension of the vector space is equal to the Milnor number of  [18, 47]. Therefore we have .

In 1988 [1], A.G. Aleksandrov studied logarithmic differential forms and residues and proved in particular the following.

###### Theorem 2.6.

Let be a hypersurface in . For , there exists an exact sequence of sheaves of modules,

 0⟶dff∧Ωq−1X+ΩqX⟶ΩqX(logS)⋅f⟶Tor(ΩqS)⟶0.

The result above yields the following observation: plays a key role to study the structure of .

### 2.3 Vanishing theorem

In 1975, in his study [13] on Gauss–Manin connections G.-M. Greuel proved the following results on torsion differential forms.

###### Theorem 2.7.

Let be a hypersurface in with an isolated singularity at . Then,

1. , .

2. is a skyscraper sheaf supported at the origin .

3. The dimension, as a vector space over , of the torsion module is equal to , the Tjurina number of the hypersurface at the origin defined to be

 τ(f)=dimC(OX,O/(f,∂f∂x1,∂f∂x2,…,∂f∂xn)),

where is the ideal in generated by .

Note that the first result was obtained by U. Vetter in [46] and the last result above is a generalization of a result of O. Zariski [47]. G.-M. Greuel obtained much more general results on torsion modules. See [13, Proposition 1.11, p. 242].

Assume that the hypersurface 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 section, the following:

1. , ,

2. .

Accordingly we have the following.

###### Proposition 2.8.

Let be a hypersurface in with an isolated singularity at . Then, , holds.

###### Proof.

Since , , the result of A.G. Aleksandrov presented in the last section yields the result. ∎

## 3 Description via logarithmic residues

In this section, we recall results given in [40] to show that torsion differential forms can be described in terms of non-trivial logarithmic vector fields. We also recall basic ideas and the framework for computing non-trivial logarithmic vector fields. As an application, we give a method for computing logarithmic residues.

### 3.1 Logarithmic vector fields

A vector field on with holomorphic coefficients is called logarithmic along the hypersurface , if the holomorphic function is in the ideal generated by in . Let denote the sheaf of modules on of logarithmic vector fields along  [34].

Let . For a holomorphic vector field , let denote the inner product of by .

###### Proposition 3.1.

Let be a hypersurface with an isolated singularity at the origin. Then, is isomorphic to , more precisely

 Ωn−1X,O(logS)={iv(ωX)f∣∣∣ v∈DerX,O(−logS)}

holds.

###### Proof.

Let and set . Then, is a holomorphic differential form. Therefore, the meromorphic differential form is logarithmic if and only if is a holomorphic differential -form. Since , we have . Hence, the condition above means is in the ideal generated by . This completes the proof. ∎

A germ of logarithmic vector field generated over by

 f∂∂xi,i=1,2,…,n,∂f∂xj∂∂xi−∂f∂xi∂∂xj,1≤i

is called trivial.

###### Lemma 3.2.

Let be a germ of a logarithmic vector field. Then, the following conditions are equivalent:

1. belongs to ,

2. is a trivial vector field.

###### Proof.

The logarithmic differential form is in if and only if the numerator is in . The last condition is equivalent to the triviality of the vector field , which completes the proof. ∎

For , let denote the Kähler differential form in defined by , that is, is the equivalence class in of .

The lemma above amount to say that, for logarithmic vector fields , is a non-zero torsion differential form in if and only if is a non-trivial logarithmic vector field.

We say that germs of two logarithmic vector fields are equivalent, denoted by , if is trivial. Let denote the quotient by the equivalence relation . (See [39].)

Now consider the following map

 Θ: DerX,O(−logS)/∼⟶Ωn−1X,O/(fΩn−1X,O+df∧Ωn−2X,O)

defined to be where is the equivalence class in of . It is easy to see that the map is well-defined. We arrive at the following description of the torsion module.

###### Theorem 3.3 ([40]).

The map

 Θ: DerX,O(−logS)/∼⟶Tor(Ωn−1S)

is an isomorphism.

### 3.2 Polar method

In [39], 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 [27, 39] we recall some basics and give a description of non-trivial logarithmic vector fields.

Let be a hypersurface with an isolated singularity. In what follows, we assume that is a regular sequence and the common locus is the origin . See [19] for an algorithm of testing zero-dimensionality of varieties at a point.

Let denote the ideal quotient, in the local ring , of , by . We have the following.

###### Lemma 3.4.

Let be a germ of holomorphic function in . Then, the following are equivalent:

1. .

2. There exists a germ of logarithmic vector field in such that

 v=a(x)∂∂x1+a2(x)∂∂x2+⋯+an−1(x)∂∂xn−1+an(x)∂∂xn,

where .

Note that in [24, 27], by utilizing local cohomology and Grothendieck local duality, an effective method of computing a set of generators over the local ring of the module of logarithmic vector fields is given. See the next section.

###### Lemma 3.5.

Assume that is a regular sequence. Let be a logarithmic vector fields in of the form

 v′=a2(x)∂∂x2+a3(x)∂∂x3+⋯+an(x)∂∂xn.

Then, is trivial.

Lemmas 3.4 and 3.5 immediately yield the following.

###### Proposition 3.6.

Let be a regular sequence. Let  be a germ of logarithmic vector field along of the form

 v=a1(x)∂∂x1+a2(x)∂∂x2+⋯+an−1(x)∂∂xn−1+an(x)∂∂xn.

Then, the following conditions are equivalent:

1. is trivial,

2. .

Therefore, we have the following.

###### Theorem 3.7 ([39]).

is isomorphic to

 ((f,∂f∂x2,∂f∂x3,…,∂f∂xn):(∂f∂x1))/(f,∂f∂x2,∂f∂x3,…,∂f∂xn).

To be more precise, let be a basis as a vector space of the quotient

 ((f,∂f∂x2,∂f∂x3,…,∂f∂xn):(∂f∂x1))/(f,∂f∂x2,∂f∂x3,…,∂f∂xn).

Then the corresponding logarithmic vector fields,

 v=a(x)∂∂x1+a2(x)∂∂x2+⋯+an−1(x)∂∂xn−1+an(x)∂∂xn,a(x)∈A

give rise to a basis of .

### 3.3 Local cohomology and duality

In this section, we briefly recall some basics on local cohomology and Grothendieck local duality. We give an outline for computing non-trivial logarithmic vector fields. We refer to [40] for details.

Let denote the local cohomology supported at the origin of the sheaf of holomorphic -forms. Then, the stalk and the local cohomology are mutually dual as locally convex topological vector spaces.

The duality is given by the point residue pairing:

 Res{O}(∗,∗): OX,O×Hn{O}(ΩnX)⟶C.

Let denote the set of local cohomology classes in that are annihilated by , :

 WΓ(f)={φ∈Hn{O}(ΩnX)∣∣∣φ∈Hn{O}(ΩnX)fφ=∂f∂x2φ=⋯=∂f∂xnφ=0}.

Then, a complex analytic version of Grothendieck local duality on residue implies that the pairing

 OX,O/(f,∂f∂x2,∂f∂x3,…,∂f∂xn)×WΓ(f)⟶C

is non-degenerate.

Let and denote the Milnor number of

and that of a hyperplane section

of , where is the restriction of to the hyperplane . Then, the classical Lê–Teissier formula [17, 43] and the Grothendieck local duality imply the following:

 dimCWΓ(f)=μ(f)+μ(f|Hx1).

Let be a map defined by and let be the image of the map :

 WΔ(f)={∂f∂x1∗φ∣∣∣∂f∂x1∗φφ∈WΓ(f)}.

Let be the annihilator in of the set of local cohomology classes. We have the following.

.

###### Proof.

See [20, 39, 41]. ∎

Recall that the ideal quotient is coefficient ideal w.r.t.  of logarithmic vector fields along . The lemma above says that the coefficient ideal can be described in terms of local cohomology .

Let be the kernel of the map . By definition we have

Since the pairing

 OX,O/(f,∂f∂x1∂f∂x2,∂f∂x3,…,∂f∂xn)×WT(f)⟶C

is non-degenerate by Grothendieck local duality, is equal to

 τ=dimC(OX,O/(f,∂f∂x1∂f∂x2,∂f∂x3,…,∂f∂xn)),

the Tjurina number.

From the exactness of the sequence

 0⟶WT(f)⟶WΓ(f)⟶WΔ(f)⟶0,

we have

 dimCWΔ(f)=μ(f)−τ(f)+μ(f|Hx1).

The argument above also implies the following.

###### Corollary 3.9 ([39]).
 dimC(DerX,O(−logS)/∼)=τ.

Notice that the dimension of that measures the way of vanishing of coefficients of logarithmic vector fields depends on the choice of a system of coordinates, or a hyperplane. In order to analyze complex analytic properties of logarithmic vector fields, as we observed in [39], it is important to select an appropriate system of coordinates or a generic hyperplane. We return to this issue afterwards at the end of this section.

Now let be the sheaf of algebraic local cohomology and let

 HΓ(f)={ϕ∈Hn[O](OX)∣∣∣ϕ∈Hn[O](OX)fϕ=∂f∂x2ϕ=⋯=∂f∂xnϕ=0}, HΔ(f)={∂f∂x1ϕ∣∣∣∂f∂x1ϕϕ∈HΓ(f)}.

Then, the following holds

 WΓ(f)={ϕ⋅ωX|ϕ∈HΓ(f)},WΔ(f)={ϕ⋅ωX|ϕ∈HΔ(f)}.

In [41], algorithms for computing algebraic local cohomology classes and some relevant algorithms are given. Accordingly, are computable. Note also that a standard basis of the ideal quotient can be computed by using in an efficient manner [41].

Now we present an outline of a method for constructing a basis, as a vector space, of the quotient space .

We fix a term ordering on and its inverse term ordering on the local ring .

1. Compute a basis of .

2. Compute a monomial basis of the quotient space , with respect to , by using .

3. Compute of each and compute a basis of .

4. Compute a standard basis of the ideal by using .

5. Compute the normal form of for .

6. Compute a basis , as a vector space, of ,.

Then, we have the following:

 SpanC(A)≅((f,∂f∂x2,∂f∂x3,…,∂f∂xn):(∂f∂x1))/(f,∂f∂x2,∂f∂x3,…,∂f∂xn).

Note that, by utilizing algorithms given in [22], the method proposed above can be extended to treat parametric cases, the case where the input data contain parameters.

In order to obtain non-trivial logarithmic vector fields, it is enough to do the following.

For each , compute , such that

 a(x)∂f∂x1+a2(x)∂f∂x2+⋯+an−1(x)∂f∂xn−1+an(x)∂f∂xn−b(x)f(x)=0.

Then,

 a(x)∂∂x1+a2(x)∂∂x2+⋯+an−1(x)∂∂xn−1+an(x)∂∂xn,a(x)∈A

gives rise to the desired set of non-trivial logarithmic vector fields.

The step above can be executed efficiently by using an algorithm described in [21]. See also [40] for details.

Before ending this section, we turn to the issue on the genericity. For this purpose, let us recall a result of B. Teissier on this subject.

Let be a non-zero vector and let denote the corresponding point in the projective space . We identify the hyperplane

 Hp′={(x1,x2,…,xn)∈Cn|p′1x1+p′2x2+⋯+p′nxn=0}

with the point in . In [43, 44], B. Teissier introduced an invariant as

 μ(n−1)(f)=min[p′]∈Pn−1μ(f|Hp′),

where is the restriction of to and is the Milnor number at the origin of the hyperplane section of . He also proved that the set

 U={[p′]∈Pn−1|μ(f|Hp′)=μ(n−1)(f)}

is a Zariski open dense subset of .

Accordingly, in order to obtain good representations of logarithmic vector fields, it is desirable to use a generic system of coordinate or a generic hyperplane that satisfies the condition .

In a previous paper [25], methods for computing limiting tangent spaces were studied and an algorithm of computing , was given. In [23, 26], more effective algorithms for computing were given. Utilizing the results in [23, 26], an effective method for computing logarithmic vector fields that takes care of the genericity condition is designed in [27, 40]. See also [42] for related results.

### 3.4 Regular meromorphic differential forms

Now we are ready to consider a method for computing regular meromorphic differential forms. For simplicity, we first consider a 3-dimensional case. Assume that a non-trivial logarithmic vector field  is given:

 v=a1(x)∂∂x1+a2(x)∂∂x2+a3(x)∂∂x3.

Let and , where . We have . We introduce differential forms  and  as

 ξ=−a2(x)dx3+a3(x)dx2,η=b(x)dx2∧dx3.

Let . Then, the following holds

 g(x)β=df∧ξ+f(x)η.

Accordingly, the logarithmic differential form satisfies

 g(x)ω=dff∧ξ+η.

We may assume that the coordinate system is generic [27] and satisfies the condition , of in Definition 2.1.

Since we have, by definition, the following:

 res(βf)=ξ