Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues
Logarithmic differential forms and logarithmic vector fields associated with 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 d...
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| description | Logarithmic differential forms and logarithmic vector fields associated with 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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| first_indexed | 2026-03-16T13:08:32Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 019, 21 pages
Computing Regular Meromorphic Differential Forms
via Saito’s Logarithmic Residues
Shinichi TAJIMA a and Katsusuke NABESHIMA b
a) Graduate School of Science and Technology, Niigata University,
8050, Ikarashi 2-no-cho, Nishi-ku Niigata, Japan
E-mail: tajima@emeritus.niigata-u.ac.jp
b) Graduate School of Technology, Industrial and Social Sciences, Tokushima University,
2-1, Minamijosanjima-cho, Tokushima, Japan
E-mail: nabeshima@tokushima-u.ac.jp
Received July 24, 2020, in final form February 05, 2021; Published online February 27, 2021
https://doi.org/10.3842/SIGMA.2021.019
Abstract. Logarithmic differential forms and logarithmic vector fields associated to a hyper-
surface 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. Alek-
sandrov, 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.
Key words: logarithmic vector field; logarithmic residue; torsion module; local cohomology
2020 Mathematics Subject Classification: 32S05; 32A27
Dedicated to Kyoji Saito
on the occasion of his 77th birthday
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 A2 and A3 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]
This paper is a contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji
Saito for his 77th birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Saito.html
mailto:tajima@emeritus.niigata-u.ac.jp
mailto:nabeshima@tokushima-u.ac.jp
https://doi.org/10.3842/SIGMA.2021.019
https://www.emis.de/journals/SIGMA/Saito.html
2 S. Tajima and K. Nabeshima
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 previ-
ous 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 loga-
rithmic 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 X be an open neighborhood of the origin O in Cn. Let OX be the sheaf on X of holo-
morphic functions and OX,O the stalk at O of the sheaf OX .
2.1 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 2.1. 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) fω and fdω are holomorphic on X.
(ii) fω and df ∧ ω are holomorphic on X.
(iii) There exist a holomorphic function g(x) and a holomorphic (q− 1)-form ξ and a holomor-
phic q-form η on X, such that:
Computing Regular Meromorphic Differential Forms via Saito’s Logarithmic Residues 3
(a) dimC(S ∩ {x ∈ X | g(x) = 0}) ≤ n− 2,
(b) gω =
df
f
∧ ξ + η.
(iv) There exists an (n − 2)-dimensional analytic set A ⊂ S such that the germ of ω at any
point p ∈ S − A belongs to df
f ∧ Ωq−1
X,p + Ωq
X,p, where Ωq
X,p denotes the module of germs
of holomorphic q-forms on X at p.
For the equivalence of the condition above, see [34]. Let Ωq
X(logS) denote the sheaf of loga-
rithmic q-forms along S. Let MS be the sheaf on S of meromorphic functions, let Ωq
S be the
sheaf on S of holomorphic q-forms defined to be
Ωq
S = Ωq
X/
(
fΩq
X + df ∧ Ωq−1
X
)
.
Definition 2.2. The residue map res : Ωq
X(logS) −→ MS ⊗OX Ωq−1
S is defined as follows:
For ω ∈ Ωq
S(logS), by definition, there exist g, ξ and η such that
(a) dimC(S ∩ {x ∈ X | g(x) = 0}) ≤ n− 2, and
(b) gω =
df
f
∧ ξ + η.
Then the residue of ω is defined to be res(ω) = ξ
g
∣∣∣
S
in MS ⊗OX Ωq−1
S .
Note that it is easy to see that the image sheaf of the residue map res of the subsheaf
df
f ∧ Ωq−1
X + Ωq
X of Ωq
X(logS) is equal to Ωq−1
X
∣∣∣
S
:
res
(
df
f
∧ Ωq−1
X + Ωq
X
)
= Ωq−1
X
∣∣∣
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 ωqS of regular meromorphic differential forms in a quite general setting.
He showed that for the case q = n−1, the sheaf ωn−1
S coincides with the Grothendieck dualizing
sheaf and ωqS can also be defined in the following manner.
Definition 2.3. Let S be a hypersurface in X ⊂ Cn. Let ωn−1
S be the Grothendieck dua-
lizing sheaf Ext1
OX
(
OS ,Ωn
X
)
. Then, the sheaf of regular meromorphic differential forms ωqS ,
q = 0, 1, . . . , n− 2 on S is defined to be
ωqS = HomOS
(
Ωn−1−q
S , ωn−1
S
)
.
In 1990, A.G. Aleksandrov [2] obtained the following result.
Theorem 2.4. For any q ≥ 0, there is an isomorphism of OS modules
res
(
Ωq
X(logS)
) ∼= ωq−1
S .
See [2] or [3] for the proof.
Let Tor(Ωq
S) denote the sheaf of torsion differential q-forms of Ωq
S .
4 S. Tajima and K. Nabeshima
Example 2.5. Let X be an open neighborhood of the origin O in C2. Let f(x, y) = x2 − y3
and S = {(x, y) ∈ X | f(x, y) = 0}. Then, for stalk at the origin of the sheaves of logarithmic
differential forms, we have
Ω1
X,O(logS) ∼= OX,O
(
df
f
,
β
f
)
, Ω2
X,O(logS) ∼= OX,O
(
dx ∧ dy
f
)
,
where OX,O is the stalk at the origin of the sheaf OX of holomorphic functions and β = 2ydx−
3xdy. The differential form β, as an element of Ω1
S = Ω1
X/
(
OXdf + fΩ1
X
)
, is a torsion. The dif-
ferential form yβ is also a torsion. Since the defining function f is quasi-homogeneous, the
dimension of the vector space Tor
(
Ω1
S
)
is equal to the Milnor number µ = 2 of S [18, 47].
Therefore we have Tor
(
Ω1
S
) ∼= OX,O(β) ∼= C(β, yβ).
In 1988 [1], A.G. Aleksandrov studied logarithmic differential forms and residues and proved
in particular the following.
Theorem 2.6. Let S = {x ∈ X | f(x) = 0} be a hypersurface in X ⊂ Cn. For q = 0, 1, . . . , n,
there exists an exact sequence of sheaves of OX modules,
0 −→ df
f
∧ Ωq−1
X + Ωq
X −→ Ωq
X(logS)
·f−→ Tor
(
Ωq
S
)
−→ 0.
The result above yields the following observation: Tor
(
Ωq
S
)
plays a key role to study the
structure of res
(
Ωq
X(logS)
)
.
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 S = {x ∈ X | f(x) = 0} be a hypersurface in X with an isolated singularity
at O ∈ Cn. Then,
(i) Tor
(
Ωq
S
)
= 0, q = 0, 1, . . . , n− 2.
(ii) Tor
(
Ωn−1
S
)
is a skyscraper sheaf supported at the origin O.
(iii) The dimension, as a vector space over C, of the torsion module Tor
(
Ωn−1
S
)
is equal to τ(f),
the Tjurina number of the hypersurface S at the origin defined to be
τ(f) = dimC
(
OX,O
/(
f,
∂f
∂x1
,
∂f
∂x2
, . . . ,
∂f
∂xn
))
,
where
(
f, ∂f∂x1 ,
∂f
∂x2
, . . . , ∂f∂xn
)
is the ideal in OX,O generated by f, ∂f∂x1 ,
∂f
∂x2
, . . . , ∂f∂xn .
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 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 section, the following:
(i) Ωq
X,O(logS) = df
f ∧ Ωq−1
X,O + Ωq
X,O, q = 1, 2, . . . , n− 2,
(ii) 0 −→ df
f ∧ Ωn−2
X,O + Ωn−1
X,O −→ Ωn−1
X,O(logS)
·f−→ Tor
(
Ωn−1
S
)
−→ 0.
Accordingly we have the following.
Computing Regular Meromorphic Differential Forms via Saito’s Logarithmic Residues 5
Proposition 2.8. Let S = {x ∈ X | f(x) = 0} be a hypersurface in X with an isolated sin-
gularity at O ∈ Cn. Then, ωqS = Ωq
X , q = 0, 1, . . . , n− 3 holds.
Proof. Since res
(
Ωq
X(logS)
)
= Ωq−1
X
∣∣∣
S
, q = 1, 2, . . . , n − 2, 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 descri-
bed 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 v onX 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 OX . Let DerX(− logS)
denote the sheaf of modules on X of logarithmic vector fields along S [34].
Let ωX = dx1 ∧dx2 ∧ · · · ∧dxn. For a holomorphic vector field v, let iv(ωX) denote the inner
product of ωX by v.
Proposition 3.1. Let S = {x ∈ X | f(x) = 0} be a hypersurface with an isolated singularity at
the origin. Then, Ωn−1
X,O(logS) is isomorphic to DerX,O(− logS), more precisely
Ωn−1
X,O(logS) =
{
iv(ωX)
f
∣∣∣∣ v ∈ DerX,O(− logS)
}
holds.
Proof. Let β = iv(ωX), and set ω = β
f . Then, fω = β is a holomorphic differential form.
Therefore, the meromorphic differential n−1 form ω is logarithmic if and only if df∧ βf is a holo-
morphic differential n-form. Since df ∧ β = df ∧ iv(ωX) = v(f)ωX , we have df ∧ β
f = v(f)
f ωX .
Hence, the condition above means v(f) is in the ideal (f) ⊂ OX,O generated by f . This completes
the proof. �
A germ of logarithmic vector field v generated over OX,O by
f
∂
∂xi
, i = 1, 2, . . . , n,
∂f
∂xj
∂
∂xi
− ∂f
∂xi
∂
∂xj
, 1 ≤ i < j ≤ n,
is called trivial.
Lemma 3.2. Let v be a germ of a logarithmic vector field. Then, the following conditions are
equivalent:
(i) ω =
iv(ωX)
f
belongs to
df
f
∧ Ωn−2
X,O + Ωn−1
X,O,
(ii) v is a trivial vector field.
Proof. The logarithmic differential form ω = iv(ωX)
f is in Ωn−1
X,O + df
f ∧ Ωn−2
X,O if and only if the
numerator iv(ωX) is in fΩn−1
X,O + df ∧ Ωn−2
X,O. The last condition is equivalent to the triviality
of the vector field v, which completes the proof. �
6 S. Tajima and K. Nabeshima
For β ∈ Ωn−1
X,O, let [β] denote the Kähler differential form in Ωn−1
S,O defined by β, that is, [β] is
the equivalence class in Ωn−1
X,O/
(
fΩn−1
X,O + df ∧ Ωn−2
X,O
)
of β.
The lemma above amount to say that, for logarithmic vector fields v, [iv(ωX)] is a non-zero
torsion differential form in Tor
(
Ωn−1
S,O
)
if and only if v is a non-trivial logarithmic vector field.
We say that germs of two logarithmic vector fields v, v′ ∈ DerX,O(− logS) are equivalent,
denoted by v ∼ v′, if v − v′ is trivial. Let DerX,O(− logS)/∼ denote the quotient by the
equivalence relation ∼. (See [39].)
Now consider the following map
Θ: DerX,O(− logS)/∼ −→ Ωn−1
X,O/
(
fΩn−1
X,O + df ∧ Ωn−2
X,O
)
defined to be Θ([v]) = [iv(ωX)], where [v] is the equivalence class in DerX,O(− logS)/∼ of v.
It is easy to see that the map Θ 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−1
S
)
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 S = {x ∈ X | f(x) = 0} be a hypersurface with an isolated singularity. In what follows, we
assume that f, ∂f∂x2 ,
∂f
∂x3
, . . . , ∂f∂xn is a regular sequence and the common locus V
(
f, ∂f∂x2 ,
∂f
∂x3
, . . . ,
∂f
∂xn
)
∩X is the origin O. See [19] for an algorithm of testing zero-dimensionality of varieties at
a point.
Let
(
f, ∂f∂x2 ,
∂f
∂x3
, . . . , ∂f∂xn
)
:
( ∂f
∂x1
)
denote the ideal quotient, in the local ring OX,O, of
(
f, ∂f∂x2 ,
∂f
∂x3
, . . . , ∂f∂xn
)
by
( ∂f
∂x1
)
. We have the following.
Lemma 3.4. Let a(x) be a germ of holomorphic function in OX,O. Then, the following are
equivalent:
(i) a(x) ∈
(
f,
∂f
∂x2
,
∂f
∂x3
, . . . ,
∂f
∂xn
)
:
(
∂f
∂x1
)
.
(ii) There exists a germ of logarithmic vector field v in DerX,O(− logS) such that
v = a(x)
∂
∂x1
+ a2(x)
∂
∂x2
+ · · ·+ an−1(x)
∂
∂xn−1
+ an(x)
∂
∂xn
,
where a2(x), . . . , an(x) ∈ OX,O.
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 OX,O of the module of logarithmic
vector fields is given. See the next section.
Lemma 3.5. Assume that f, ∂f∂x2 ,
∂f
∂x3
, . . . , ∂f∂xn is a regular sequence. Let v′ be a logarithmic
vector fields in DerX,O(− logS) of the form
v′ = a2(x)
∂
∂x2
+ a3(x)
∂
∂x3
+ · · ·+ an(x)
∂
∂xn
.
Then, v′ is trivial.
Computing Regular Meromorphic Differential Forms via Saito’s Logarithmic Residues 7
Lemmas 3.4 and 3.5 immediately yield the following.
Proposition 3.6. Let f, ∂f∂x2 ,
∂f
∂x3
, . . . , ∂f∂xn be a regular sequence. Let v be a germ of logarithmic
vector field along S of the form
v = a1(x)
∂
∂x1
+ a2(x)
∂
∂x2
+ · · ·+ an−1(x)
∂
∂xn−1
+ an(x)
∂
∂xn
.
Then, the following conditions are equivalent:
(i) v is trivial,
(ii) a1(x) ∈
(
f, ∂f∂x2 ,
∂f
∂x3
, . . . , ∂f∂xn
)
.
Therefore, we have the following.
Theorem 3.7 ([39]). DerX,O(− logS)/∼ is isomorphic to((
f,
∂f
∂x2
,
∂f
∂x3
, . . . ,
∂f
∂xn
)
:
(
∂f
∂x1
))/(
f,
∂f
∂x2
,
∂f
∂x3
, . . . ,
∂f
∂xn
)
.
To be more precise, let A 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 DerX,O(− logS)/∼.
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.
LetHn{O}
(
Ωn
X
)
denote the local cohomology supported at the origin O of the sheaf Ωn
X of holo-
morphic n-forms. Then, the stalk OX,O and the local cohomology Hn{O}
(
Ωn
X
)
are mutually dual
as locally convex topological vector spaces.
The duality is given by the point residue pairing:
Res{O}(∗, ∗) : OX,O ×Hn{O}
(
Ωn
X
)
−→ C.
Let WΓ(f) denote the set of local cohomology classes in Hn{O}
(
Ωn
X
)
that are annihilated by f ,
∂f
∂x2
, ∂f∂x3 , . . . ,
∂f
∂xn
:
WΓ(f) =
{
ϕ ∈ Hn{O}
(
Ωn
X
) ∣∣∣∣ fϕ =
∂f
∂x2
ϕ = · · · = ∂f
∂xn
ϕ = 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.
8 S. Tajima and K. Nabeshima
Let µ(f) and µ(f |Hx1 ) denote the Milnor number of f and that of a hyperplane section f |Hx1
of f , where f |Hx1 is the restriction of f to the hyperplane Hx1 = {x ∈ X |x1 = 0}. Then, the
classical Lê–Teissier formula [17, 43] and the Grothendieck local duality imply the following:
dimCWΓ(f) = µ(f) + µ(f |Hx1 ).
Let γ : WΓ(f) −→ WΓ(f) be a map defined by γ(ϕ) = ∂f
∂x1
∗ ϕ and let WΓ(f) be the image of
the map γ:
W∆(f) =
{
∂f
∂x1
∗ ϕ
∣∣∣∣ϕ ∈WΓ(f)
}
.
Let AnnOX,O(W∆(f)) be the annihilator in OX,O of the set W∆(f) of local cohomology classes.
We have the following.
Lemma 3.8 ([39]). AnnOX,O(W∆(f)) =
(
f, ∂f∂x2 ,
∂f
∂x3
, . . . , ∂f∂xn
)
:
( ∂f
∂x1
)
.
Proof. See [20, 39, 41]. �
Recall that the ideal quotient
(
f, ∂f∂x2 ,
∂f
∂x3
, . . . , ∂f∂xn
)
:
( ∂f
∂x1
)
is coefficient ideal w.r.t. ∂
∂x1
of logarithmic vector fields along S. The lemma above says that the coefficient ideal can be de-
scribed in terms of local cohomology W∆(f).
Let WT (f) be the kernel of the map γ. By definition we have
WT (f) =
{
ϕ ∈ Hn{O}
(
Ωn
X
) ∣∣∣∣ fϕ =
∂f
∂x1
ϕ =
∂f
∂x2
ϕ = · · · = ∂f
∂xn
ϕ = 0
}
.
Since the pairing
OX,O
/(
f,
∂f
∂x1
∂f
∂x2
,
∂f
∂x3
, . . . ,
∂f
∂xn
)
×WT (f) −→ C
is non-degenerate by Grothendieck local duality, dimC(WT (f)) 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) −→W∆(f) −→ 0,
we have
dimCW∆(f) = µ(f)− τ(f) + µ(f |Hx1 ).
The argument above also implies the following.
Corollary 3.9 ([39]).
dimC
(
DerX,O(− logS)/∼
)
= τ.
Computing Regular Meromorphic Differential Forms via Saito’s Logarithmic Residues 9
Notice that the dimension of W∆(f) that measures the way of vanishing of coefficients of loga-
rithmic 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 Hn
[O](OX) = lim
k→∞
ExtnOX
(
OX,O/(x1, x2, . . . , xn)k,OX
)
be the sheaf of algebraic local
cohomology and let
HΓ(f) =
{
φ ∈ Hn
[O](OX)
∣∣∣∣ fφ =
∂f
∂x2
φ = · · · = ∂f
∂xn
φ = 0
}
,
H∆(f) =
{
∂f
∂x1
φ
∣∣∣∣φ ∈ HΓ(f)
}
.
Then, the following holds
WΓ(f) = {φ · ωX |φ ∈ HΓ(f)}, W∆(f) = {φ · ωX |φ ∈ H∆(f)}.
In [41], algorithms for computing algebraic local cohomology classes and some relevant algo-
rithms are given. Accordingly, HΓ(f), H∆(f) are computable. Note also that a standard basis
of the ideal quotient
(
f, ∂f∂x2 ,
∂f
∂x3
, . . . , ∂f∂xn
)
:
( ∂f
∂x1
)
can be computed by using H∆(f) 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
((
f, ∂f∂x2 ,
∂f
∂x3
, . . . , ∂f∂xn
)
:
( ∂f
∂x1
))
/
(
f, ∂f∂x2 ,
∂f
∂x3
, . . . , ∂f∂xn
)
.
We fix a term ordering � on Hn
[O](OX) and its inverse term ordering �−1 on the local
ring OX,O.
Step 1: Compute a basis ΦΓ(f) of HΓ(f).
Step 2: Compute a monomial basis MΓ(f) of the quotient space OX,O/
(
f, ∂f∂x2 ,
∂f
∂x3
, . . . , ∂f∂xn
)
,
with respect to �−1, by using ΦΓ(f).
Step 3: Compute ∂f
∂xn
φ of each φ ∈ ΦΓ(f) and compute a basis Φ∆(f) of H∆(f).
Step 4: Compute a standard basis SB of the ideal AnnOX,O(H∆(f)) by using Φ∆(f).
Step 5: Compute the normal form NF�−1
(
xλs(x)
)
of xλs(x) for xλ ∈MΓ(f), s(x) ∈ SB.
Step 6: Compute a basis A, as a vector space, of SpanC
{
NF�−1(xλs(x)) |xλ ∈ MΓ(f),
s(x) ∈ SB
}
.
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 a(x) ∈ A, compute a2(x), a3(x), . . . , an(x), b(x) ∈ OX,O, 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.
10 S. Tajima and K. Nabeshima
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 p′ = (p′1, p
′
2, . . . , p
′
n) be a non-zero vector and let [p′] denote the corresponding point
in the projective space Pn−1. We identify the hyperplane
Hp′ =
{
(x1, x2, . . . , xn) ∈ Cn | p′1x1 + p′2x2 + · · ·+ p′nxn = 0
}
with the point [p′] in Pn−1. In [43, 44], B. Teissier introduced an invariant µ(n−1)(f) as
µ(n−1)(f) = min
[p′]∈Pn−1
µ(f |Hp′ ),
where f |Hp′ is the restriction of f to Hp′ and µ(f |Hp′ ) is the Milnor number at the origin O
of the hyperplane section f |Hp′ of f . He also proved that the set
U =
{
[p′] ∈ Pn−1 |µ(f |Hp′ ) = µ(n−1)(f)
}
is a Zariski open dense subset of Pn−1.
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 Hp′ that satisfies the condition
µ(f |Hp′ ) = µ(n−1)(f).
In a previous paper [25], methods for computing limiting tangent spaces were studied and
an algorithm of computing µ(f |Hp′ ), p
′ ∈ Pn−1 was given. In [23, 26], more effective algorithms
for computing µ(n−1) were given. Utilizing the results in [23, 26], an effective method for compu-
ting 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 v is given:
v = a1(x)
∂
∂x1
+ a2(x)
∂
∂x2
+ a3(x)
∂
∂x3
.
Let v(f) = b(x)f(x) and β = iv(ωX), where ωX = dx1 ∧ dx2 ∧ dx3. We have β = a1(x)dx2 ∧
dx3 − a2(x)dx1 ∧ dx3 + a3(x)dx1 ∧ dx2. We introduce differential forms ξ and η as
ξ = −a2(x)dx3 + a3(x)dx2, η = b(x)dx2 ∧ dx3.
Let g(x) = ∂f
∂x1
. Then, the following holds
g(x)β = df ∧ ξ + f(x)η.
Accordingly, the logarithmic differential form ω = β
f satisfies
g(x)ω =
df
f
∧ ξ + η.
We may assume that the coordinate system (x1, x2, x3) is generic [27] and g(x) satisfies the
condition (a), (b) of (iii) in Definition 2.1.
Computing Regular Meromorphic Differential Forms via Saito’s Logarithmic Residues 11
Since g(x) = ∂f
∂x1
, we have, by definition, the following:
res
(
β
f
)
=
ξ
∂f
∂x1
∣∣∣∣∣
S
.
Notice that the differential form ξ above is directly defined from the coefficients of the logarithmic
vector field v.
Proposition 3.10. Let S = {x ∈ X | f(x) = 0} be a hypersurface with an isolated singularity
at the origin O ∈ X ⊂ Cn. Assume that the coordinate system (x1, x2, . . . , xn) is generic so that(
f, ∂f∂x2 ,
∂f
∂x3
, . . . , ∂f∂xn
)
is a regular sequence and g(x) = ∂f
∂x1
satisfies the condition (a), (b) of (iii)
in Definition 2.1. Let
v = a1(x)
∂
∂x1
+ a2(x)
∂
∂x2
+ · · ·+ an(x)
∂
∂xn
be a germ of non-trivial logarithmic vector field along S. Let v(f) = b(x)f(x), β = iv(ωX).
Let ξ, η denote the differential form defined to be
ξ = −a2(x)dx3 ∧ dx4 ∧ · · · ∧ dxn + a3(x)dx2 ∧ dx4 ∧ · · · ∧ dxn − · · ·
+ (−1)(n+1)an(x)dx2 ∧ dx3 ∧ · · · ∧ dxn−1,
η = b(x)dx2 ∧ dx3 ∧ · · · ∧ dxn.
Then,
g(x)
β
f
=
df
f
∧ ξ + η and res
(
β
f
)
=
ξ
∂f
∂x1
∣∣∣∣∣
S
hold.
Note that, in 1984, M. Kersken [16] obtained related results on regular meromorphic differ-
ential forms. The statement in Proposition 3.10 above is a refinement a result of M. Kersken.
Theorem 3.11. Let S = {x ∈ X | f(x) = 0} be a hypersurface with an isolated singularity at the
origin O ∈ X ⊂ Cn. Let V = {v1, v2, . . . , vτ} be a set of non-trivial logarithmic vector fields such
that the class [v1], [v2], . . . , [vτ ] constitute a basis of the vector space DerX,O(− logS)/∼, where τ
stands for the Tjurina number of f . Let ξ1, ξ2, . . . , ξτ be the differential forms correspond to
v1, v2, . . . , vτ defined in Proposition 3.10.
Then, any logarithmic residue in res
(
Ωn−1(logS)
)
, or a regular meromorphic differential
form γ in ωn−2
S can be represented as
γ =
(
1
∂f
∂x1
(c1ξ1 + c2ξ2 + · · ·+ cτξτ )
)∣∣∣∣
S
+ α,
where ci ∈ C, i = 1, 2, . . . , τ , and α ∈ Ωn−2
X
∣∣
S
.
4 Examples
In this section, we give examples of computation for illustration. Data is an extraction from [40].
Let f0(z, x, y) = x3 + y3 + z4 and let ft(z, x, y) = f0(z, x, y) + txyz2, where t is a deformation
parameter. We regard z as the first variable. Then, f0 is a weighted homogeneous polynomial
with respect to a weight vector (3, 4, 4) and ft is a µ-constant deformation of f0, called U12
12 S. Tajima and K. Nabeshima
singularity. The Milnor number µ(ft) of U12 singularity is equal to 12. In contrast, the Tjurina
number τ(ft) depends on the parameter t. In fact, if t = 0, then τ(f0) = 12 and if t 6= 0, then
τ(ft) = 11. In the computation, we fix a term order �−1 on OX,O which is compatible with the
weight vector (3, 4, 4).
We consider these two cases separately.
Example 4.1 (weighted homogeneous U12 singularity). Let f0(z, x, y) = x3 + y3 + z4. Then,
µ(f0) = τ(f0) = 12. The monomial basis M with respect to the term ordering �−1 of the
quotient space OX,O/(f0,
∂f0
∂x ,
∂f0
∂y ) is
M =
{
xiyjzk | i = 0, 1, j = 0, 1, k = 0, 1, 2, 3
}
.
The standard basis Sb of the ideal quotient
(
f0,
∂f0
∂x ,
∂f0
∂y
)
:
(∂f0
∂z
)
is
Sb =
{
x2, y2, z
}
.
The normal form in OX,O/
(
f0,
∂f0
∂x ,
∂f0
∂y
)
of x2, y2 and z are
NF�−1
(
x2
)
= NF�−1
(
y2
)
= 0, NF�−1(z) = z.
Therefore, A = {xiyjzk | i = 0, 1, j = 0, 1, k = 1, 2, 3}. Notice that A consists of 12 elements.
It is easy to see that the Euler vector field
v = 4x
∂
∂x
+ 4y
∂
∂y
+ 3z
∂
∂z
that corresponds to the element z ∈ A is a non-trivial logarithmic vector field. Therefore, the
torsion module of the hypersurface S0 =
{
(x, y, z) |x3 + y3 + z4 = 0
}
is given by
Tor
(
Ω2
S0
)
=
{
xiyjzkiv(ωX) | i = 0, 1, j = 0, 1, k = 1, 2, 3
}
,
where ωX = dz ∧ dx ∧ dy.
Let ξ = −4xdy+4ydx. Then res
( iv(ωX)
f
)
= ξ
4z3
∣∣∣
S
. Computation of other logarithmic residues
are same.
The following is also an extraction from [40].
Example 4.2 (semi quasi-homogeneous U12 singularity). Let f(x, y, z) = x3 + y3 + z4 + txyz2,
t 6= 0. Then, µ(f) = 12, τ(f) = 11 and µ(f |Hz) = 4. We have dimCHΓ(f) = 16, dimCH∆(f) = 5.
Let � be a term ordering on H3
[O](OX) which is compatible with the weight vector (4, 4, 3).
A basis ΦΓ(f) of HΓ(f) is given by{[
1
xyz
]
,
[
1
xyz2
]
,
[
1
x2yz
]
,
[
1
xy2z
]
,
[
1
xyz3
]
,
[
1
x2yz2
]
,
[
1
xy2z2
]
,
[
1
x2y2z
]
,
[
1
xyz4
]
,[
1
x2yz3
]
− t
3
[
1
xy3z
]
,
[
1
xy2z3
]
− t
3
[
1
x 3yz
]
,
[
1
x2y2z2
]
,
[
1
x2yz4
]
− t
3
[
1
xy3z2
]
,[
1
xy2z4
]
− t
3
[
1
x3yz2
]
,
[
1
x2y2z3
]
− t
3
[
1
x4yz
]
− t
3
[
1
xy4z
]
− t
3
[
1
xyz5
]
,[
1
x2y2z4
]
− t
3
[
1
x4yz2
]
− t
3
[
1
xy4z2
]
− t
3
[
1
xyz6
]}
.
Computing Regular Meromorphic Differential Forms via Saito’s Logarithmic Residues 13
The monomial basis M with respect to the term ordering �−1 of the quotient OX,O/
(
f,
∂f
∂x ,
∂f
∂y
)
is
M =
{
xiyjzk | i = 0, 1, j = 0, 1, k = 0, 1, 2, 3
}
.
A basis Φ∆(f) of H∆(f) is given by{[
1
xyz
]
,
[
1
xyz2
]
,
[
1
x2yz
]
,
[
1
xy2z
]
,
[
1
x2y2z
]
+
t
6
[
1
xyz3
]}
.
We see from this data that the standard basis of the ideal quotient
(
f, ∂f∂x ,
∂f
∂y
)
:
(∂f
∂z
)
in the
local ring OX,O is
Sb =
{
z2 − t
6
xy, xz, yz, x2, y2
}
.
From Sb and M, we have
A =
{
z2 − t
6
xy, xz, yz, z3, xz2, yz2, xyz, xz3, yz3, xyz2, xyz3
}
.
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.
(i) Let a = 6z2 − txy. Then,
v =
d1
27 + t3z2
∂
∂x
+
d2
27 + t3z2
∂
∂y
+
(
6z2 − txy
) ∂
∂z
is a non-trivial logarithmic vector field, where
d1 = 216xz − 6t2y2z − 2t4x2yz, d2 = 216yz + 24t2x2z + 10t3yz3 − 2t4xy2z.
(ii) Let a = xz. Then,
v =
d1
27 + t3z2
∂
∂x
+
d2
27 + t3z2
∂
∂y
+ xz
∂
∂z
is a non-trivial logarithmic vector field, where
d1 = 36x2 − 6yz2 − 6t2xy2, d2 = 36xy + 2t2x3 − 4t2y3 − 2t2z4.
We omit the other nine cases. As described in Theorem 3.11, regular meromorphic differential
forms can be constructed directly from these data.
5 Brieskorn formula
In 1970, B. Brieskorn studied the monodromy of Milnor fibration and developed the theory
of Gauss–Manin connection [7]. He proved the regularity of the connection and proposed an alge-
braic framework for computing the monodromy via Gauss–Manin connection. He gave in par-
ticular 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 log-
arithmic vector fields. We present an alternative method for computing non-trivial logarithmic
vector fields. The resulting algorithm can be used as a basic tool for studying Gauss–Manin
connections. We also present some examples for illustration.
14 S. Tajima and K. Nabeshima
5.1 Brieskorn lattice and Gauss–Manin connection
We briefly recall some basics on Brieskorn lattice and Brieskorn formula. We refer to [6, 7, 37].
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 Cn. Let
H ′0 = Ωn−1
X,O/
(
df ∧ Ωn−2
X,O + dΩn−2
X,O
)
, H ′′0 = Ωn
X,O/df ∧ dΩn−2
X,O.
Then, df ∧H ′0 ⊂ H ′′0 . A map D : df ∧H ′0 −→ H ′′0 is defined as follows:
D(df ∧ ϕ) = [dϕ], ϕ ∈ Ωn−1
X,O.
Let ϕ =
∑n
i=1(−1)i+1hi(x)dx1 ∧ dx2 ∧ · · · ∧ dxi−1 ∧ dxi+1 ∧ · · · ∧ dxn. Then
df ∧ ϕ =
(
n∑
i=1
hi(x)
∂f
∂xi
)
ωX ,
where ωX = dx1 ∧ dx2 ∧ · · · ∧ dxn. Therefore in terms of the coordinate we have the following,
known as Brieskorn formula
D(df ∧ ϕ) =
(
n∑
i
∂hi
∂xi
)
ωX .
Example 5.1. Let f(x, y) = x2 − y3 and S = {(x, y) ∈ X | f(x, y) = 0} where X ⊂ C2 is
an open neighborhood of the origin O. The Jacobi ideal J of f is
(
x, y2
)
⊂ OX,O and M = {1, y}
is a monomial basis of the quotient OX,O/J . Let τ denote the Tjurina number. Then, since f
is a weighted homogeneous polynomial, we have τ = µ = 2 (see Example 2.5).
Let v = 1
6
(
3x ∂
∂x + 2y ∂
∂y
)
be the Euler vector field. Then, v is logarithmic along S.
Let β = iv(ωX). Then, β = 1
6(3xdy − 2ydx). Since v(f) = f , we have df ∧ β = fωX , where
ωX = dx ∧ dy. By Brieskorn formula, we have
D(fωX) = D(df ∧ β) =
5
6
ωX .
Note that the formula above is equivalent d
( β
fλ
)
= 0, with λ = 5
6 .
Likewise, for yβ, we have df ∧ (yβ) = f(x, y)yωX and
D(f(x, y)yωX) = D(df ∧ (yβ)) =
7
6
yωX ,
which is equivalent to d
( yβ
fλ
)
= 0, with λ = 7
6 .
Since Df = fD + 1 as operators, we have
fD(ωX) = −1
6
ωX , fD(yωX) =
1
6
yωX .
Notice that β, yβ are non-zero torsion differential forms in Ω1
S and v, yv are non-trivial loga-
rithmic vector fields along S. Note also that yv(f) = yf . Notably, Brieskorn formula described
in terms of differential forms can be rewritten in terms of non-trivial logarithmic vector fields v
and yv which satisfy v(f) = f and yv(f) = yf respectively.
Let S = {x ∈ X | f(x) = 0} be the hypersurface with an isolated singularity at the origin
O ∈ X defined by f . Consider, for instance, a trivial vector field v′ = ∂f
∂x2
∂
∂x1
− ∂f
∂x1
∂
∂x2
. Since
v′(f) = 0 and ∂
∂x1
( ∂f
∂x2
)
+ ∂
∂x2
(
− ∂f
∂x1
)
= 0 hold, we have a trivial relation D((0 · ωX) = 0 · ωX .
It is easy to see in general that, from a trivial vector field Brieskorn formula only gives the trivial
relation.
The observation above leads the following.
Computing Regular Meromorphic Differential Forms via Saito’s Logarithmic Residues 15
Proposition 5.2. Let S = {x ∈ X | f(x) = 0} be a hypersurface with an isolated singularity at
the origin O ∈ X, where X ⊂ Cn. Let
v = a1(x)
∂
∂x1
+ a2(x)
∂
∂x2
+ · · ·+ an(x)
∂
∂xn
be a germ of non-trivial logarithmic vector field along S. Let v(f) = b(x)f(x) Then,
D(f(x)b(x)ωX) =
(
n∑
i=1
∂ai
∂xi
)
ωX
holds, where ωX = dx1 ∧ dx2 ∧ · · · ∧ dxn.
Proof. Let β = iv(ωX). Since df ∧ β = v(f)ωX , we have df ∧ β =
(∑n
i=1 ai(x) ∂f∂xi
)
ωX . Since
v(f) = b(x)f(x), Brieskorn formula implies the result. �
Notice that the action of Df on b(x)ωX in the formula above is completely written in terms
of non-trivial logarithmic vector field v such that v(f) = b(x)f . To the best of our knowledge,
this simple observation has not been explicitly stated in literature on Gauss–Manin connections.
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
OX,O
/( ∂f
∂x1
,
∂f
∂x2
, . . . ,
∂f
∂xn
)
.
Step 2: Compute a standard basis Sb of the ideal quotient(
∂f
∂x1
,
∂f
∂x2
, . . . ,
∂f
∂xn
)
: (f).
Step 3: Compute a basis B of the vector space by using Sb and M((
∂f
∂x1
,
∂f
∂x2
, . . . ,
∂f
∂xn
)
: (f)
)/( ∂f
∂x1
,
∂f
∂x2
, . . . ,
∂f
∂xn
)
.
Step 4: For each b(x) ∈ B, compute a logarithmic vector field along S such that
v(f) = b(x)f(x).
The method above computes a basis of non-trivial logarithmic vector fields. Each step can
be effectively executable, as in [40], by utilizing algorithms described in [20, 21, 22, 41].
Note that, the number of non-trivial logarithmic vector fields in the output is equals to the
Tjurina number τ(f). See also [18].
Let
v = a1(x)
∂
∂x1
+ a2(x)
∂
∂x2
+ · · ·+ an(x)
∂
∂xn
be a germ of non-trivial logarithmic vector field along S, such that v(f) = b(x)f(x). Then from
Proposition 5.2, we have
D(f(x)b(x)ωX) =
(
n∑
i=1
∂ai
∂xi
)
ωX .
Therefore, the proposed method can be used as a basic procedure for computing a connection
matrix of Gauss–Manin connection.
One of the advantages of the proposed method lies in the fact that the resulting algorithm
also can handle parametric cases.
16 S. Tajima and K. Nabeshima
5.2 Examples
Let us recall that x3 + y7 + txy5 is the standard normal form of semi quasi-homogeneous E12
singularity. The weight vector is (7, 3) and the weighted degree of the quasi-homogeneous part
is equal to 21 and the weighted degree of the upper monomial txy5 is equal to 22. We examine
here, by contrast, the case where the weighted degree of an upper monomial is bigger than 22.
Example 5.3. Let f(x, y) = x3 + y7 + txy6, where t is a parameter. Notice that the polyno-
mial f is not weighted homogeneous. The weighted degree of the upper monomial txy6 is equal
to 25, which is bigger than that of txy5. Accordingly f is a quasi homogeneous function. The
Milnor number µ is equal to 12.
Let HJ denote the set of local cohomology classes in H2
[0,0](OX) that are killed by the Jacobi
ideal J =
(∂f
∂x ,
∂f
∂y
)
:
HJ =
{
ψ ∈ H2
[0,0](OX)
∣∣∣∣ ∂f∂xψ =
∂f
∂y
ψ = 0
}
.
Then, by using an algorithm given in [22, 41], a basis as a vector space of HJ is computed as[
1
xy
]
,
[
1
xy2
]
,
[
1
xy3
]
,
[
1
x2y
]
,
[
1
xy4
]
,
[
1
x2y2
]
,
[
1
xy5
]
,
[
1
x2y3
]
,
[
1
xy6
]
,
[
1
x2y4
]
,
[
1
x2y5
]
,[
1
x2y6
]
− 6
7
t
[
1
xy7
]
+
2
7
t2
[
1
x3y
]
,
where [ ] stands for Grothendieck symbol.
It is easy to see that every local cohomology classes in HJ is killed by f , that is f · ϕ = 0,
∀ϕ ∈ HJ . Therefore, f is in the ideal
(∂f
∂x ,
∂f
∂y
)
⊂ OX,O.
Therefore, by a classical result of K. Saito [31], f is in fact quasi-homogeneous. The Tjurina
number τ is equal to the Milnor number µ = 12. A monomial basis M of OX,O/
(∂f
∂x ,
∂f
∂y
)
is
M =
{
1, y, y2, x, y3, xy, y4, xy2, y5, xy3, xy4, xy5
}
.
Since a standard basis Sb of
(∂f
∂x ,
∂f
∂y
)
: (f) is {1}, a basis B of the vector space
((∂f
∂x ,
∂f
∂y
)
:
(f)
)
/
(∂f
∂x ,
∂f
∂y
)
is equal to M that consists of τ = 12 elements.
By using an algorithm given in [21], we compute a logarithmic vector field which plays the
role of Euler vector field. The result of computation is the following:
v =
d1
3
(
49 + 12t3y4
) ∂
∂x
+
d2
3
(
49 + 12t3y4
) ∂
∂y
,
where
d1 = 49x+ 8t2y5 + 12t3xy4, d2 = 21y − 4tx+ 4t3y5.
The vector field v enjoys v(f) = f . Note also that for the case t = 0, we have
v =
1
21
(
7x
∂
∂x
+ 3y
∂
∂y
)
.
We emphasize here the fact that, the algorithm in [27] for computing logarithmic vector fields
can handle parametric cases. Since v(f) = f holds, the other non-trivial logarithmic vector fields
can be obtained from v. In fact, for xiyj ∈M , we have xiyjv(f) = xiyjf .
Therefore, thanks to Brieskorn formula, Gauss–Manin connection can be determined explici-
tly by using these non-trivial logarithmic vector fields,
Computing Regular Meromorphic Differential Forms via Saito’s Logarithmic Residues 17
Remark 5.4. Recall that, according to Grothendieck local duality theorem, the vector space HJ
can be regarded as a dual space to OX,O/J . Since these local cohomology classes given above
constitute a dual basis of the monomial basis M of the quotient space OX,O/J , the normal form
of a holomorphic function w.r.t. OX,O/J can be computed by using the basis of HJ in an efficient
manner, without using division algorithms [41].
Therefore the use of local cohomology classes in reduction steps allows us to design an effective
procedure for computing the connection matrix of Gauss–Manin connection.
J. Scherk studied in [35] the following case.
Example 5.5. Let f(x, y) = x5 + x2y2 + y5. Then, the Milnor number µ(f) is equal to 11
and the Tjurina number τ(f) is equal to 10. A monomial basis M of OX,O/
(∂f
∂x ,
∂f
∂y
)
is M ={
1, x, x2, x3, x4, x5, xy, y, y2, y3, y4
}
. A standard basis Sb of the ideal quotient
(∂f
∂x
∂f
∂y
)
: (f) is
{x, y}. A basis B of the vector space
((∂f
∂x ,
∂f
∂y
)
: (f)
)
/
(∂f
∂x ,
∂f
∂y
)
is
B =
{
x, x2, x3, x4, x5, xy, y, y2, y3, y4
}
.
Since Sb ∩ B = {x, y}, we first compute non-trivial logarithmic vector fields associated to x
and y.
(i) For b(x, y) = x, we have
v =
d1
5(4− 25xy)
∂
∂x
+
d2
5(4− 25xy)
∂
∂y
,
where d1 = 4x2 − 25x3y − 5y3, d2 = 6xy − 25x2y2.
Since v(f) = xf , by a direct computation, we have for instance
D(f(x, y)xωX) =
(
7
10
x− 3× 25
16
y4
)
ωX mod
(
∂f
∂x
,
∂f
∂y
)
.
Since xiv(f) = xi+1f , i = 1, 2, 3, 4 and yv(f) = xyf hold, we can compute the action
of Df on xi+1ωX and xyωX by using the vector field v above.
(ii) For b(x, y) = y, we have
v =
d1
5(4− 25xy)
∂
∂x
+
d2
5(4− 25xy)
∂
∂y
,
where d1 = 6xy − 25x2y2, d2 = 4y2 − 25xy3 − 5x3 and
D(f(x, y)yωX) =
(
7
10
y − 3× 25
16
x4
)
ωX mod
(
∂f
∂x
,
∂f
∂y
)
.
Since the vector field v above satisfies v(f) = yf , we also have yjv(f) = yj+1f , j = 1, 2, 3.
We can use these relations to compute the action of Df on yj+1ωX , j = 1, 2, 3. In this way,
we obtain τ = 10 fundamental relations.
Since the Milnor number µ is equal to 11, these 10 relations are not enough to compute
a connection matrix of the Gauss–Manin connection. We have to compute the saturation.
Now recall the classical result on integral closure due to J. Briano̧n and H. Skoda [38]. From
the Briano̧n–Skoda theorem, we see that the function f2 is in the ideal J =
(∂f
∂x ,
∂f
∂y
)
. In [35],
18 S. Tajima and K. Nabeshima
J. Scherk computed the following relation explicitly and exploited it as the starting point for
computing D(f2ωX) and D(fD(fωX)):
25(4− 25xy)f2 =
{(
20x− 125x2y
)
f + 4x3y2 − 5xy5 − 25x4y3
}∂f
∂x
+
{(
20y − 125xy2
)
f + 6x2y3 − 25x3y4
}∂f
∂y
.
Here we propose a slightly different approach. By using an algorithm given in [28], we can
compute the following integral dependence relation
25(4− 25xy)f2 = 10x
(
∂f
∂x
)
f+10y
(
∂f
∂y
)
f+d2,0
(
∂f
∂x
)2
+d1,1
(
∂f
∂x
)(
∂f
∂y
)
+d0,2
(
∂f
∂y
)2
,
where
d2,0 = 2x2 − 25x3y − 10y3, d1,1 = 11xy − 50x2y2, d0,2 = 2y2 − 25xy3 − 10x3.
Compare to the relation used by Scherk, the integral dependence relation given above repre-
sent much more precise relations between f2, f
(∂f
∂x
)
, f
(∂f
∂y
)
,
(∂f
∂x
)2
,
(∂f
∂x
)(∂f
∂y
)
,
(∂f
∂y
)2
. Thanks
to this property, the use of the integral dependence relation, or the integral equation leads
an effective method for computing D
(
f2ωX
)
and D(fD(fωX)).
Note that in [28], we consider integral dependence relations in the context of symbolic com-
putation and introduced a concept of generalized integral dependence relations. From this
point of view relations obtained from non-trivial logarithmic vector fields can be interpreted as
generalized integral dependence relations. These relations can also be computed by using the
algorithms described in [28].
Let f(x) be a holomorphic function defined on X ⊂ Cn. Assume that the degree of integral
equation, or the integral number of f over the Jacobi ideal in the local ring OX,O is equal to two.
Let
f(x)2 +
n∑
i=1
ai(x)f(x)
∂f
∂xi
(x) +
∑
j≥i
ai,j(x)
∂f
∂xi
(x)
∂f
∂xj
(x) = 0
be the integral equation of f . Then, from the Brieskorn formula, we have
D(f(x)2ωX) = −D
{
n∑
i=1
(
ai(x)f(x) +
∑
j≥i
ai,j(x)
∂f
∂xj
(x)
)
∂f
∂xi
(x)ωX
}
= −
{
n∑
i=1
∂
∂xi
(
ai(x)f(x) +
∑
j≥i
ai,j(x)
∂f
∂xj
(x)
)}
ωX
which is equal to
−
{
n∑
i=1
ai(x)
∂f
∂xi
(x) +
∑
j≥i
∂ai,j
∂xi
(x)
∂f
∂xj
+R(x)
}
ωX ,
where
R(x) =
( n∑
i=1
∂ai
∂xi
(x)
)
f(x) +
∑
j≥i
ai,j(x)
∂2f
∂xi∂xj
(x).
Computing Regular Meromorphic Differential Forms via Saito’s Logarithmic Residues 19
If, there exist holomorphic functions ci(x), i = 1, 2, . . . , n such that
R(x) =
n∑
i=1
ci(x)
∂f
∂xi
(x),
then, we have for instance the following relation that can be used as a starting point of the
computation of a saturation
D2(f(x)2ωX) = −
{
n∑
i=1
(
∂ai
∂xi
(x) +
∂ci
∂xi
(x)
)
+
∑
j≥i
∂2ai,j
∂xj∂xi
(x)
}
ωX .
Computing Gauss–Manin connections is a quite difficult problem [11, 14, 35, 36, 45]. We ex-
pect that the approach presented in this paper provides a method to reduce difficulty to some
extent.
Acknowledgements
This work has been partly supported by JSPS Grant-in-Aid for Scientific Research (C)
(18K03320 and 18K03214).
References
[1] Aleksandrov A.G., A de Rahm complex of nonisolated singularities, Funct. Anal. Appl. 22 (1988), 131–133.
[2] Aleksandrov A.G., Nonisolated hypersurface singularities, in Theory of Singularities and its Applications,
Adv. Soviet Math., Vol. 1, Amer. Math. Soc., Providence, RI, 1990, 211–246.
[3] Aleksandrov A.G., Logarithmic differential forms, torsion differentials and residue, Complex Var. Theory
Appl. 50 (2005), 777–802.
[4] Aleksandrov A.G., Tsikh A.K., Théorie des résidus de Leray et formes de Barlet sur une intersection complète
singulière, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), 973–978.
[5] Barlet D., Le faisceau ω′X sur un espace analytique X de dimension pure, in Fonctions de plusieurs variables
complexes, III (Sém. François Norguet, 1975–1977), Lecture Notes in Math., Vol. 670, Springer, Berlin, 1978,
187–204.
[6] Brasselet J.P., Sebastiani M., Brieskorn and the monodromy, J. Singul. 18 (2018), 84–104.
[7] Brieskorn E., Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math. 2 (1970),
103–161.
[8] Brunella M., Some remarks on indices of holomorphic vector fields, Publ. Mat. 41 (1997), 527–544.
[9] Corrêa M., da Silva Machado D., Residue formulas for logarithmic foliations and applications, Trans. Amer.
Math. Soc. 371 (2019), 6403–6420, arXiv:1611.01203.
[10] Corrêa M., da Silva Machado D., GSV-index for holomorphic Pfaff systems, Doc. Math. 25 (2020), 1011–
1027, arXiv:1611.09376.
[11] Douai A., Très bonnes bases du réseau de Brieskorn d’un polynôme modéré, Bull. Soc. Math. France 127
(1999), 255–287.
[12] Granger M., Schulze M., Normal crossing properties of complex hypersurfaces via logarithmic residues,
Compos. Math. 150 (2014), 1607–1622, arXiv:1109.2612.
[13] Greuel G.M., Der Gauss–Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten,
Math. Ann. 214 (1975), 235–266.
[14] Guimaraes A.G., Polinômio de Bernstein–Sato de uma hipersuperficie com singularidade isolada, Ph.D. The-
sis, ICMC-USP, São Carlos, 2002.
[15] Kersken M., Der Residuenkomplex in der lokalen algebraischen und analytischen Geometrie, Math. Ann.
265 (1983), 423–455.
[16] Kersken M., Reguläre Differentialformen, Manuscripta Math. 46 (1984), 1–25.
https://doi.org/10.1007/BF01077604
https://doi.org/10.1070/sm1990v065n02abeh001164
https://doi.org/10.1080/02781070500128313
https://doi.org/10.1080/02781070500128313
https://doi.org/10.1016/S0764-4442(01)02166-8
https://doi.org/10.1007/BFb0064400
https://doi.org/10.5427/jsing.2018.18f
https://doi.org/10.1007/BF01155695
https://doi.org/10.5565/PUBLMAT_41297_17
https://doi.org/10.1090/tran/7584
https://doi.org/10.1090/tran/7584
https://arxiv.org/abs/1611.01203
https://doi.org/10.25537/dm.2020v25.1011-1027
https://arxiv.org/abs/1611.09376
https://doi.org/10.24033/bsmf.2348
https://doi.org/10.1112/S0010437X13007860
https://arxiv.org/abs/1109.2612
https://doi.org/10.1007/BF01352108
https://doi.org/10.1007/BF01455946
https://doi.org/10.1007/BF01185193
20 S. Tajima and K. Nabeshima
[17] Lê D.T., Calcul du nombre de cycles évanouissants d’une hypersurface complexe, Ann. Inst. Fourier (Greno-
ble) 23 (1973), 261–270.
[18] Michler R., Torsion of differentials of hypersurfaces with isolated singularities, J. Pure Appl. Algebra 104
(1995), 81–88.
[19] Nabeshima K., Tajima S., Testing zero-dimensionality of varieties at a point, Math.Comput.Sci., to appear,
arXiv:1903.12365.
[20] Nabeshima K., Tajima S., Computing Tjurina stratifications of µ-constant deformations via parametric local
cohomology systems, Appl. Algebra Engrg. Comm. Comput. 27 (2016), 451–467.
[21] Nabeshima K., Tajima S., Solving extended ideal membership problems in rings of convergent power se-
ries via Gröbner bases, in Mathematical Aspects of Computer and Information Sciences, Lecture Notes in
Comput. Sci., Vol. 9582, Springer, Cham, 2016, 252–267.
[22] Nabeshima K., Tajima S., Algebraic local cohomology with parameters and parametric standard bases for
zero-dimensional ideals, J. Symbolic Comput. 82 (2017), 91–122, arXiv:1508.06724.
[23] Nabeshima K., Tajima S., Computing µ∗-sequences of hypersurface isolated singularities via parametric
local cohomology systems, Acta Math. Vietnam. 42 (2017), 279–288.
[24] Nabeshima K., Tajima S., Computation methods of logarithmic vector fields associated to semi-weighted
homogeneous isolated hypersurface singularities, Tsukuba J. Math. 42 (2018), 191–231.
[25] Nabeshima K., Tajima S., A new method for computing the limiting tangent space of an isolated hypersurface
singularity via algebraic local cohomology, in Singularities in Generic Geometry, Adv. Stud. Pure Math.,
Vol. 78, Math. Soc. Japan, Tokyo, 2018, 331–344.
[26] Nabeshima K., Tajima S., Alternative algorithms for computing generic µ∗-sequences and local Euler ob-
structions of isolated hypersurface singularities, J. Algebra Appl. 18 (2019), 1950156, 13 pages.
[27] Nabeshima K., Tajima S., Computing logarithmic vector fields and Bruce–Roberts Milnor numbers via local
cohomology classes, Rev. Roumaine Math. Pures Appl. 64 (2019), 523–540.
[28] Nabeshima K., Tajima S., Generalized integral dependence relations, in Mathematical Aspects of Computer
and Information Sciences, Lecture Notes in Computer Science, Vol. 11989, Springer, Cham, 2020, 48–63.
[29] Pol D., On the values of logarithmic residues along curves, Ann. Inst. Fourier (Grenoble) 68 (2018), 725–766,
arXiv:1410.2126.
[30] Pol D., Characterizations of freeness for equidimensional subspaces, J. Singul. 20 (2020), 1–30,
arXiv:1512.06778.
[31] Saito K., Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math. 14 (1971), 123–142.
[32] Saito K., Calcul algébrique de la monodromie, Astérisque 7 (1973), 195–211.
[33] Saito K., On the uniformization of complements of discriminant loci, in Hyperfunctions and Linear Partial
Differential Equations, RIMS Kôkyûroku, Vol. 287, Res. Inst. Math. Sci. (RIMS), Kyoto, 1977, 117–137.
[34] Saito K., Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo
Sect. IA Math. 27 (1980), 265–291.
[35] Scherk J., On the Gauss–Manin connection of an isolated hypersurface singularity, Math. Ann. 238 (1978),
23–32.
[36] Scherk J., On the pole order and Hodge filtrations of isolated hypersurface singularities, Canad. Math. Bull.
36 (1993), 368–372.
[37] Schulze M., Algorithms for the Gauss–Manin connection, J. Symbolic Comput. 32 (2001), 549–564.
[38] Skoda H., Briançon J., Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point
de Cn, C. R. Acad. Sci. Paris Sér. A 278 (1974), 949–951.
[39] Tajima S., On polar varieties, logarithmic vector fields and holonomic D-modules, in Recent Development
of Micro-Local Analysis for the Theory of Asymptotic Analysis, RIMS Kôkyûroku Bessatsu, Vol. 40, Res.
Inst. Math. Sci. (RIMS), Kyoto, 2013, 41–51.
[40] Tajima S., Nabeshima K., An algorithm for computing torsion differential forms associated to an isolated
hypersurface singularity, Math.Comput.Sci., to appear.
[41] Tajima S., Nakamura Y., Nabeshima K., Standard bases and algebraic local cohomology for zero dimensional
ideals, in Singularities – Niigata–Toyama 2007, Adv. Stud. Pure Math., Vol. 56, Math. Soc. Japan, Tokyo,
2009, 341–361.
https://doi.org/10.5802/aif.491
https://doi.org/10.5802/aif.491
https://doi.org/10.1016/0022-4049(94)00117-2
https://doi.org/10.1007/s11786-020-00484-y
https://arxiv.org/abs/1903.12365
https://doi.org/10.1007/s00200-016-0289-4
https://doi.org/10.1007/978-3-319-32859-1_22
https://doi.org/10.1016/j.jsc.2017.01.003
https://arxiv.org/abs/1508.06724
https://doi.org/10.1007/s40306-016-0198-4
https://doi.org/10.21099/tkbjm/1554170422
https://doi.org/10.2969/aspm/07810331
https://doi.org/10.1142/S0219498819501561
https://doi.org/10.1007/978-3-030-43120-4_6
https://doi.org/10.5802/aif.3176
https://arxiv.org/abs/1410.2126
https://doi.org/10.5427/jsing.2020.20a
https://arxiv.org/abs/1512.06778
https://doi.org/10.1007/BF01405360
https://doi.org/10.1007/BF01351450
https://doi.org/10.4153/CMB-1993-050-9
https://doi.org/10.1006/jsco.2001.0482
https://doi.org/10.1007/s11786-020-00486-w
https://doi.org/10.2969/aspm/05610341
Computing Regular Meromorphic Differential Forms via Saito’s Logarithmic Residues 21
[42] Tajima S., Shibuta T., Nabeshima K., Computing logarithmic vector fields along an ICIS germ via Matlis
duality, in Computer Algebra in Scientific Computing, Lecture Notes in Computer Science, Vol. 12291,
Springer, Cham, 2020, 543–562.
[43] Teissier B., Cycles évanescents, sections planes et conditions de Whitney, Astérisque 7 (1973), 285–362.
[44] Teissier B., Variétés polaires. I. Invariants polaires des singularités d’hypersurfaces, Invent. Math. 40 (1977),
267–292.
[45] van Straten D., The spectrum of hypersurface singularities, arXiv:2002.00519.
[46] Vetter U., Äußere Potenzen von Differentialmoduln reduzierter vollständiger Durchschnitte, Manuscripta
Math. 2 (1970), 67–75.
[47] Zariski O., Characterization of plane algebroid curves whose module of differentials has maximum torsion,
Proc. Nat. Acad. Sci. USA 56 (1966), 781–786.
https://doi.org/10.1007/978-3-030-60026-6_32
https://doi.org/10.1007/BF01425742
https://arxiv.org/abs/2002.00519
https://doi.org/10.1007/BF01168480
https://doi.org/10.1007/BF01168480
https://doi.org/10.1073/pnas.56.3.781
1 Introduction
2 Logarithmic differential forms and residues
2.1 Logarithmic residues
2.2 Barlet sheaf and torsion differential forms
2.3 Vanishing theorem
3 Description via logarithmic residues
3.1 Logarithmic vector fields
3.2 Polar method
3.3 Local cohomology and duality
3.4 Regular meromorphic differential forms
4 Examples
5 Brieskorn formula
5.1 Brieskorn lattice and Gauss–Manin connection
5.2 Examples
References
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| id | nasplib_isofts_kiev_ua-123456789-211169 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T13:08:32Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Tajima, Shinichi Nabeshima, Katsusuke 2025-12-25T13:21:00Z 2021 Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues. Shinichi Tajima and Katsusuke Nabeshima. SIGMA 17 (2021), 019, 21 pages 1815-0659 2020 Mathematics Subject Classification: 32S05; 32A27 arXiv:2007.09950 https://nasplib.isofts.kiev.ua/handle/123456789/211169 https://doi.org/10.3842/SIGMA.2021.019 Logarithmic differential forms and logarithmic vector fields associated with 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. This work has been partly supported by JSPS Grant-in-Aid for Scientific Research (C) (18K03320 and 18K03214). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues Article published earlier |
| spellingShingle | Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues Tajima, Shinichi Nabeshima, Katsusuke |
| title | Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues |
| title_full | Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues |
| title_fullStr | Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues |
| title_full_unstemmed | Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues |
| title_short | Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues |
| title_sort | computing regular meromorphic differential forms via saito's logarithmic residues |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211169 |
| work_keys_str_mv | AT tajimashinichi computingregularmeromorphicdifferentialformsviasaitoslogarithmicresidues AT nabeshimakatsusuke computingregularmeromorphicdifferentialformsviasaitoslogarithmicresidues |