The theory of logarithmic differential forms and its applications. Torsion differentials and residues

A.G.Aleksandrov^{†}

Abstract: This paper deals with the notion of logarithmic differential forms founded by Kyoji Saito. Relations between the properties of logarithmic differential forms and torsion holomorphic differentials on a singular hypersurface are established. Making use of the theory of logarithmic and regular meromorphic differential forms, a natural extension of the classical Poincaré residue to the case of singular hypersurfaces is obtained. A variant of Grothendieck duality for nonnormal reduced hypersurfaces is described. The logarithmic de Rham complex is studied and its cohomology groups for hypersurfaces with isolated singularities are computed.

Introduction

A general notion of logarithmic differential forms appeared in 1975. It was introduced by Kyoji Saito in his famous preprinted notes (see [27]). In these notes he established the basic properties of such forms and considered applications to computing the Gauss-Manin connection associated with the versal deformation of a simple hypersurface singularities of types A_{2} and A_{3}. For unclear and unknown reasons, this preprint was reproduced only two years later in 1977 in a volume of the RIMS-publication series [28], which is not accessible to many of those interested in the subject. In 1980, K.Saito published, at last, a paper [29] containing some part of the material of the mentioned preprint. Among other things, in this paper the general notion and a clear description of the basic properties of logarithmic differential forms and vector fields are given. Somewhat later, after the RIMSpublication of Saito in 1978, the same notion of logarithmic differential forms was introduced in the book by Ph.Griffiths and J.Harris [18], with no references to the previous work. It should be remarked that in their book, they used this notion only in the case of hypersurfaces which are normal crossings of smooth ones. In fact, this case had been also treated in earlier works by P.Deligne [11], J.Steenbrink [32], F.Pham [25], R.Gérard and A.Levelt [15], and others.

†Partially supported by the Russian Foundation of Fundamental Research (RFFR), Grant No. 99-01-01067

2000 AMS Mathematics Subject Classification: Primary 32S25, 14F40, 14F10, 14J17; Secondary 32S22, 32S45, 58K65, 14F25

Key words and phrases: logarithmic differential forms; hypersurface singularity; torsion differentials; regular meromorphic differential forms; Poincaré residue; logarithmic residue; duality; de Rham cohomology

However, at present time it is the general notion of logarithmic differential forms which is exploited fruitfully in many fields of modern mathematics. Among them, one can mention the following:

algebraic geometry (the cohomology theory of algebraic varieties and Hodge theory [12], [10], [32], etc.),

topology and geometry (the theory of arrangements of real and complex hyperplanes [24], [8], the fundamental group of the complement of a singular hypersurface [21], etc.),

the theory of singularities, the deformation theory and the theory of Gauss-Manin connexion [28], [30], [7], etc.,

the theory of D-modules, the microlocal analysis, the theory of differential equations [11], [25], the theory of flat coordinate systems [31], etc.,

complex analysis (the theory of Abel’s integrals [17], Torelli theorems, the theory of primitive forms and their periods [18], etc.),

the theory of special functions (generalized hypergeometric functions [14], etc.), mathematical and theoretical physics (the theory of Frobenius varieties and the

topological field theory [13], [23], etc.)

Of course, this list is quite incomplete and can be easily extended by the specialists in related fields of mathematics.

These notes deal with the theory of logarithmic differential forms and with some of the less known applications of this theory. In section 1 we recall the basic notations, definitions and properties of logarithmic differential forms along a reduced hypersurface in a complex analytic manifold. In section 2 we consider some properties of logarithmic differential forms in relation to properties of the torsion holomorphic differentials on a singular hypersurface. In the next two sections we recall the definition of a Poincaré residue, consider its properties, and discuss its relation to a logarithmic residue introduced by K.Saito in [27] and [29]. Following D.Barlet [9], we then apply the theory of regular meromorphic differential forms to the special case of hypersurfaces. Among other things, we obtain a highly elegant description of these modules on an arbitrary singular hypersurface D in terms of the residue of logarithmic differential forms. In the section 5, a variant of the Grothendieck duality for non-normal reduced hypersurfaces is described. In the final section we study the logarithmic de Rham complex and compute its cohomology for a hypersurface with isolated singularities.

Acknowledgment. These notes have been prepared for my talk at the MPIM seminar on March 9, 2000. The aim of the talk was to acquaint the participants with the nice and useful theory of logarithmic differential forms used now in many fields of modern mathematics. It is a pleasure to thank the Max-Planck-Institut für Mathematik for its financial assistance and hospitality during the preparation of the notes.

1. Logarithmic differential forms

Let U be an open subset of C^{m}, and let D be a hypersurface defined by an equation h(z) = 0, where h(z) = h(z_{1}, …, z_{m}) is a holomorphic function in U, and z_{1}, …, z_{m} is a system of coordinates. Suppose that D is reduced, that is, h(z) has no multiple factors. The following basic definition is due to K.Saito (see [27], [29]).

Definition 1.1. A meromorphic differential q-form ω, q ≥ 0, on U is called logarithmic (along a divisor D) if ω and its differential dω have poles along D at worst of the first order. It means that hω and hdω are holomorphic differential forms on U.

In practical computations, the second condition is usually replaced by the condition “dh∧ω is a holomorphic differential form on U”; both conditions are equivalent, in view of the identity d(hω) = dh ∧ ω + h · dω.

Let S be an m-dimensional complex manifold, and let Ω_{S}^{•} = (Ω_{S}^{q}, d)_{q=0,1,...} be the de Rham complex of germs of holomorphic differential forms on S, whose terms, locally at the point 0 ∈ S, are defined as follows:

Ω_{S,0}^{q} = O_{S,0}⟨…, dz_{i_{1}} ∧ … ∧ dz_{i_{q}}, …⟩, (i_{1}, …, i_{q}) ∈ [1, m].

Let D be a reduced hypersurface of S, and let h_{x} = 0 be an equation of D, locally at the point x ∈ D. A meromorphic q-form ω is logarithmic along D at x, if h_{x}ω and h_{x}dω are holomorphic. We denote the O_{S,x}-module of germs of logarithmic q-form at x and the corresponding sheaf of logarithmic differential q-form on S by Ω_{S,x}^{q}(log D) and Ω_{S}^{q}(log D), respectively. Thus, the O_{S}-module Ω_{S}^{1}(log D) is a submodule of Ω_{S}^{1}(⋆D), consisting of all the “differential forms with polar singularities along D.” Obviously, the sheaves Ω_{S}^{q}(log D) and Ω_{S}^{q} coincide off the divisor D, for all q ≥ 0. By definition,

Ω_{S,x}^{0}(log D) ≅ Ω_{S,x}^{0} ≅ O_{S,x}, Ω_{S,x}^{m}(log D) ≅ \frac{1}{h_{x}} Ω_{S,x}^{m}.

In what follows, when we consider the local situation the point x will be taken to be 0 for simplicity. We shall also assume that U is an open subset of C^{m} containing the origin.

Example 1.2. Suppose D ⊂ U be a hyperplane or, more generally, a smooth hypersurface defined by the equation z_{1} = 0. Then

Ω_{S,0}^{1}(log D) ≅ O_{S,0}⟨\frac{dz_{1}}{z_{1}} , dz_{2}, …, dz_{m}⟩

is a free O_{S,0}-module of rank m, generated by the forms dz_{1}/z_{1}, dz_{2}, …, dz_{m}. Moreover,

Ω_{S,0}^{q}(log D) ≅ ⋀^{q} Ω_{S,0}^{1}(log D), 1 ≤ q ≤ m.

Example 1.3. More generally, let us consider the case when D is the union of k ≤ m coordinates hyperplanes in S = C^{m}. In other words, D is a normal crossing. It is the case considered in many works published before Saito’s preprint [27]. Then the defining equation of D is: h = z_{1} ⋯ z_{k} = 0, and an easy calculation shows that

Ω_{S,0}^{1}(log D) ≅ O_{S,0}⟨\frac{dz_{1}}{z_{1}} , …, \frac{dz_{k}}{z_{k}} , dz_{k+1}, …, dz_{m}⟩,

and for all 1 ≤ q ≤ m there are the following isomorphisms

Ω_{S,0}^{q}(log D) ≅ ⋀^{q} Ω_{S,0}^{1}(log D).

Thus, Ω_{S,0}^{q}(log D) is a free O_{S,0}-module of rank \binom{m}{q}.

The following statement is a direct consequence of the basic definition (see [4], or [5], §1).

Claim 1.4. Let D ⊂ U be a reduced hypersurface defined by the equation h = 0. Then for any q ≥ 1 there exists a natural isomorphism of O_{S,0}-modules

Ω_{S,0}^{q} ⋂ ((dh/h) ∧ Ω_{S,0}^{q−1}) ≅ dh ∧ Ω_{S,0}^{q−1}(log D).

Proof. Let us remark at first that there is a natural inclusion

Ω_{S,0}^{q} ⋂ ((dh/h) ∧ Ω_{S,0}^{q−1}) ↪ dh ∧ Ω_{S,0}^{q−1}(log D).

For, if an element ω ∈ Ω_{S,0}^{q} belongs to the O_{S,0}-module on the left side, then it is of the form ω = (dh/h) ∧ η for some η ∈ Ω_{S,0}^{q−1}. Hence, by definition,

(η/h) ∈ Ω_{S,0}^{q−1}(log D) ⇒ ω ∈ dh ∧ Ω_{S,0}^{q−1}(log D),

and we obtain the desirable left arrow. On the other hand, h · Ω_{S,0}^{q−1}(log D) ⊆ Ω_{S,0}^{q−1}. Multiplication by ∧dh induces the map

dh ∧ Ω_{S,0}^{q−1}(log D) ⟶ \frac{dh}{h} ∧ Ω_{S,0}^{q−1}.

Obviously this gives us the inverse map to the first inclusion. This completes the proof of Claim.

The following reformulation of the basic definition is also due to K.Saito (see [29], (1.1), iii)).

Lemma 1.5. Let ω be a meromorphic q-form on U, q ≥ 0, and let D ⊂ U be a hypersurface as above. Then ω is logarithmic along D if and only if there exist a holomorphic function g defining a hypersurface V ⊂ U, a holomorphic (q −1)-form ξ and a holomorphic q-form η on U such that

a) dim_{C} D ∩ V ≤ m − 2,

b) gω = \frac{dh}{h} ∧ ξ + η.

Proof. For simplicity let us consider the case m = 2. Suppose that ω is a logarithmic q-form. Then we have

ω = \frac{a_{1}dz_{1} + a_{2}dz_{2}}{h} , dh ∧ ω = \frac{h′_{1}a_{2} − h′_{2}a_{1}}{h} dz_{1} ∧ dz_{2} =^{def} b(z)dz_{1} ∧ dz_{2},

where a_{1}, a_{2} and b(z) are holomorphic, and h′_{i} = ∂h/∂z_{i}, i = 1, 2. Further,

h′_{1}ω = \frac{h′_{1}a_{1}dz_{1} + h′_{1}a_{2}dz_{2}}{h} =

\frac{h′_{1}a_{1}dz_{1} + h′_{2}a_{1}dz_{2}}{h} + \frac{h′_{1}a_{2} − h′_{2}a_{1}}{h} dz_{2} = \frac{dh}{h} ∧ a_{1} + b(z)dz_{2}.

It means that

h′_{1}ω = \frac{dh}{h} ∧ a_{1} + b(z)dz_{2}.

There are analogous presentations for h′_{2}ω, and hence for any g ∈ J(h), where J(h) = (h′_{1}, h′_{2}) is the Jacobian ideal of h. Since D is reduced, there is a function g ∈ J(h) defining a non zero divisor in O_{D}/(h) as required in the condition a).

The converse is simpler. Indeed, the equation b) implies that

hω = dh ∧ \frac{ξ}{g} + \frac{η}{g} ,

that is, hω and dh ∧ ω are holomorphic at least in codimension ≥ 2, hence, in virtue of the Riemann extension theorem, they are holomorphic everywhere. This completes the proof when m = 2. The general case can be considered analogously.

Corollary 1.6.

i) Ω_{S,0}^{q}(log D), q = 0, 1, …, m, are O_{S,0}-modules of finite type,

ii) ⊕_{q=0}^{m} Ω_{S,0}^{q}(log D) is an O_{S,0}-exterior algebra closed under the exterior differentiation.

As a consequence, we see that Ω_{S}^{q}(log D), q = 0, 1, …, m, are coherent sheaves of O_{S}-modules.

2. Torsion differentials

In this section we consider simple relations between logarithmic differential forms and torsion holomorphic differentials on a hypersurface with singularities.

Given a commutative ring A with the total ring of fractions F, and an A-module N of finite type, we call the kernel of the canonical map

ι: N → N ⊗_{A} F

the torsion submodule Tors N of N. Thus the torsion submodule consists of all the elements of N which are killed by the nonzero elements of A.

By definition, set O_{D,0} = O_{S,0}/(h)O_{S,0}, and

Ω_{D,0}^{q} = Ω_{S,0}^{q}/(h · Ω_{S,0}^{q} + dh ∧ Ω_{S,0}^{q−1}), q ≥ 1.

Then Ω_{D,0}^{q} is the O_{D,0}-module of germs of holomorphic differential forms on the hypersurface D at the point 0 ∈ D. This module is usually called the module of Kähler regular differentials of the q-th order. The standard differentiation d induces the action on Ω_{D,0}^{q} denoted by the same symbol. Thus, the de Rham complex of sheaves of germs of holomorphic differential forms on the whole D is well defined:

Ω_{D}^{•} = (Ω_{D}^{q}, d)_{q=0,1,...}.

Example 2.1. Suppose S = C^{2} and consider the hypersurface D defined by the equation h = x^{2} − y^{3} = 0. It is a curve with a cusp singularity at the origin. Easy calculations show that

Ω_{S,0}^{1}(log D) ≅ O_{S,0}⟨\frac{dh}{h}, \frac{2ydx − 3xdy}{h} ⟩, Ω_{S,0}^{2}(log D) ≅ O_{S,0}⟨\frac{dx ∧ dy}{h} ⟩

are free O_{S,0}-modules of rank 2 and 1, respectively.

Note that the numerator of the second generator of Ω_{S,0}^{1}(log D), the differential 1-form θ = 2ydx − 3xdy, represents an element of the torsion submodule Tors Ω_{D,0}^{1} ⊂ Ω_{D,0}^{1}. Indeed, in our case

A = O_{D,0} ≅ C⟨t^{2}, t^{3}⟩, N = Ω_{D,0}^{1}, F = C(t),

and the mapping ι is defined by the normalization map x = t^{3}, y = t^{2} of the curve D. Thus,

ι(θ) = ι(2ydx − 3xdy) = 2t^{2}dt^{3} − 3t^{3}dt^{2} = 6t^{4}dt − 6t^{4}dt = 0,

that is, θ ∈ Ker(ι) ≅ Tors Ω_{D,0}^{1}.

Equivalently, take a non-zero element x ∈ O_{D,0}. One then obtains

x · θ = 2xydx − 3x^{2}dy = ydh − 3hdy ≡ 0

in Ω_{D,0}^{1} = Ω_{S,0}^{1}/(h · Ω_{S,0}^{1} + dh ∧ O_{S,0}). Further calculations show (cf. [34]) that

Tors Ω_{D,0}^{1} ≅ O_{D,0}⟨θ⟩ ≅ C⟨θ, y · θ⟩ .

In the terminology of local cohomology theory Tors Ω_{D,0}^{1} ≅ H_{{0}}^{0}(Ω_{D,0}^{1}), so the torsion submodule is an initial term of the standard long exact sequence:

0 ⟶ H_{{0}}^{0}(Ω_{D,0}^{1}) ⟶ Ω_{D,0}^{1} ⟶ j_{∗}j^{∗}Ω_{D,0}^{1} ⟶ H_{{0}}^{1}(Ω_{D,0}^{1}) ⟶ 0,

where j: D \ {0} ⟶ D is the canonical embedding of the non-singular part of D.

Analogously, one can consider a plane curve with a node which is a particular case of Example 1.3.

Example 2.2. Suppose m = 2, that is S = C^{2}, and the hypersurface D is defined by the equation h = xy = 0. It is a curve with a node singularity at the origin. Then

Ω_{S,0}^{1}(log D) ≅ O_{S,0}⟨\frac{dx}{x}, \frac{dy}{y} ⟩, Ω_{S,0}^{2}(log D) ≅ ⋀^{2} Ω_{S,0}^{1}(log D) ≅ O_{S,0}⟨\frac{dx ∧ dy}{xy} ⟩

are again free O_{S,0}-modules of rank 2 and 1, respectively. In this case there is a similar presentation of Ω_{S,0}^{1}(log D) with θ = ydx − xdy :

Ω_{S,0}^{1}(log D) ≅ O_{S,0}⟨\frac{dx}{x}, \frac{dy}{y} ⟩ ≅ O_{S,0}⟨\frac{dh}{h}, \frac{θ}{h}⟩,

where h = xy and θ ∈ Tors Ω_{D,0}^{1}. Taking a non-zero element x ∈ O_{D,0} = O_{S,0}/(xy) we obtain in Ω_{D,0}^{1} two such identities:

x · θ = xydx − x^{2}dy = −xdh + 2hdx ≡ 0.

In this case, Tors Ω_{D,0}^{1} ≅ O_{D,0}⟨θ⟩ ≅ C⟨θ⟩ .

These observations lead us to the following statement (cf. [4]).

Proposition 2.3. For q = 1, …, m, there are exact sequences of O_{S,0}-modules

0 ⟶ Ω_{S,0}^{q−1}/h · Ω_{S,0}^{q−1}(log D) ⟶^{∧dh} Ω_{S,0}^{q}/h · Ω_{S,0}^{q} ⟶ Ω_{D,0}^{q} ⟶ 0,

0 ⟶ Ω_{S,0}^{q}/dh ∧ Ω_{S,0}^{q−1}(log D) ⟶^{·h} Ω_{S,0}^{q}/dh ∧ Ω_{S,0}^{q−1} ⟶ Ω_{D,0}^{q} ⟶ 0,

0 ⟶ Ω_{S,0}^{q} + \frac{dh}{h} ∧ Ω_{S,0}^{q−1} ⟶ Ω_{S,0}^{q}(log D) ⟶^{·h} Tors Ω_{D,0}^{q},

where we denote by ∧dh and by ·h the homomorphisms of exterior and usual multiplication, respectively. Moreover, when D is irreducible at 0 ∈ D then the third exact sequence will be exact from the right:

0 ⟶ Ω_{S,0}^{q} + \frac{dh}{h} ∧ Ω_{S,0}^{q−1} ⟶ Ω_{S,0}^{q}(log D) ⟶^{·h} Tors Ω_{D,0}^{q} ⟶ 0.

Proof. Consider a differential q-form ω ∈ Ω_{S,0}^{q−1} which represents a certain element of the quotient Ω_{S,0}^{q−1}/h · Ω_{S,0}^{q−1}(log D) and suppose that dh ∧ ω = h · η, η ∈ Ω_{S,0}^{q}. We set ω̃ = ω/h. It is obvious that h · ω̃ and dh ∧ ω̃ are holomorphic, hence ω̃ ∈ Ω_{S,0}^{q−1}(log D) by definition. Thus the first sequence is exact from the left. Evidently it is exact from the right too. In the same way, one can easily prove the exactness of the second sequence.

The exactness from the left of the third one follows by definition. Besides, in virtue of the basic Lemma 1.5, it is clear that Im (·h) ⊆ Tors Ω_{D,0}^{q} :

gω = \frac{dh}{h} ∧ ξ + η ⇒ ghω = dh ∧ ξ + hη ≡ 0.

Now suppose that D is irreducible so that O_{D} has no zero divisors. Take an element ω ∈ Tors Ω_{D,0}^{q}. This means that there is a non-zero element g ∈ O_{D,0} such that g · ω = 0. We will denote by g and ω their representatives in O_{S,0} and Ω_{S,0}^{q}, respectively. Then we have

g · ω = dh ∧ ξ + h · η, ξ ∈ Ω_{S,0}^{q−1}, η ∈ Ω_{S,0}^{q}.

Since g is not a zero divisor, the condition b) of the Lemma 1.5 is satisfied. This implies that ω/h = ω̃ ∈ Ω_{S,0}^{q}(log D), that is ω ∈ Im (·h).

Remark 2.4. When D is smooth, then Tors Ω_{D,0}^{q} = 0, q ≥ 0, and from the above Proposition we get isomorphisms of the Example 2.1 :

Ω_{S,0}^{q}(log D) ≅ Ω_{S,0}^{q} + \frac{dh}{h} ∧ Ω_{S,0}^{q−1}, q ≥ 0.

In fact, it is well-known from the singularity theory (see, for example, [16]) that

Tors Ω_{D,0}^{q} = 0, 0 < q < ℭ,

where ℭ = codim (Sing D, D). In particular, when D is a normal hypersurface the third exact sequence of Proposition 2.3 yields the following isomorphism:

Ω_{S,0}^{1}(log D) ≅ Ω_{S,0}^{1} + \frac{dh}{h}O_{S,0}

since ℭ ≥ 2 (cf. [27], [28]). This is clearly true when the hypersurface D has isolated singularities only and dim D ≥ 2.

Corollary 2.5. Suppose that D is irreducible at 0 ∈ D. Then there is a long exact sequence of O_{S,0}-modules

0 ⟶ dh ∧ Ω_{S,0}^{q−1}(log D) ⟶ Ω_{S,0}^{q} ⊕ \frac{dh}{h} ∧ Ω_{S,0}^{q−1} ⟶

⟶ Ω_{S,0}^{q}(log D) ⟶^{·h} Tors Ω_{D,0}^{q} ⟶ 0.

Proof. This is an immediate consequence of Proposition 2.3 and Claim 1.4.

Remark 2.6. It is well-known that torsion differentials play important role in the description of the geometry and topology of singularities. For example, in the case of an isolated singularity, the torsion modules are finite dimensional vector spaces. For a quasihomogeneous singularity of a curve we have:

dim_{C}Tors Ω_{D,0}^{1} = μ(D),

where μ(D) is the Milnor number of the singularity. The latter is the basic topological invariant of singularities. A similar remark concerns hypersurfaces or, more generally, complete intersections with an isolated singularity of arbitrary dimension.

The above Corollary is very useful in computing the torsion modules in the case when Ω_{S,0}^{q}(log D) is a free O_{S,0}-module. Indeed, Corollary 2.5 gives us an O_{S,0}-free resolution of the torsion module.

Following P.Cartier a divisor D ⊂ S is called a Saito-divisor, or sometimes, a free divisor if the O_{S}-module Ω_{S}^{q}(log D) for some q ≥ 1 is locally free. For example, the discriminants of the minimal versal deformations of isolated hypersurface or complete intersections singularities are free Saito divisors. In fact, the theory of such type divisors is very interesting and rich. Here we only note that for a quasihomogeneous free Saito divisor it is possible to compute all its Milnor numbers μ_{i}, i = 1, …, dim D, in terms of the homology groups of a Lie algebra consisting of vector fields logarithmic along D and annihilating the defining ideal of D (see details in [2]).

3. The Poincaré residue map

We start with the definition of the classical Poincaré residue following [17]. In fact, this notion appears in the works of Georges de Rham (1954) and Jean Leray (1959) and was inspired by the ideas of H. Poincaré who in 1887 introduced the notion of 1-form-residue of any rational 2-form in C^{2}.

Let ω be a meromorphic differential m-form on an m-dimensional complex analytic manifold S with a polar divisor D ⊂ S. Thus, locally we have a representation:

ω = \frac{f(z)dz_{1} ∧ … ∧ dz_{m}}{h(z)} .

By definition, the Poincaré residue rés_{D}(ω) is a meromorphic (m − 1)-form on D whose singularities are contained in the singular locus Sing D ⊂ D. To define this form explicitly, let us note that at each point x ∈ D \ Sing D one of the derivatives does not vanish:

\frac{∂h}{∂z_{i}} (x) ≠ 0.

Hence there is a locally defined expression

rés_{D}(ω) = (−1)^{m−i} \frac{f(x)dz_{1} ∧ … ∧ d̂z_{i} ∧ … ∧ dz_{m}}{\frac{∂h}{∂z_{i}} (x)} |_{D}.

It is not difficult to verify that this restriction depends neither on the index i nor on the local coordinates and on a defining equation of D. Moreover, the Poincaré residue rés_{D}(ω) is holomorphic on the complement S \ D. When D is smooth, one can take h(z) = z_{m}, and then

rés_{D}(\frac{f(x)dz_{1} ∧ … ∧ dz_{m}}{z_{m}} ) = f(x)dz_{1} ∧ … ∧ dz_{m−1}

coincides with the usual residue.

For brevity, the Poincaré residue rés_{D}(ω) will be denoted by rés(ω). When D ⊂ S is a nonsingular hypersurface, the Poincaré residue induces an exact sequence

0 ⟶ Ω_{S}^{m} ⟶ Ω_{S}^{m}(D) \underrightarrow{rés} Ω_{D}^{m−1} ⟶ 0 ,

where Ω_{S}^{m}(D) denotes the set of meromorphic forms on S having a pole of the first order along the divisor D. In particular we obtain that the germ of every holomorphic (m − 1)-form on D is a Poincaré residue. It is obvious that this is true globally when the first cohomology group vanishes: H^{1}(S, Ω_{S}^{m}) = 0.

More generally, one can apply the above construction to meromorphic differential q-forms on S with a smooth polar divisor D ⊂ S for any 1 ≤ q ≤ m, and write down locally for any differential q-form ω with poles of the first order the following presentation:

ω = \frac{dh}{h} ∧ ξ + η,

where ξ and η are holomorphic. Then the restriction of ξ to D is a holomorphic form which is independent of the choice of a local equation h = 0 of D, and is denoted by rés(ω) as before. As a result one gets (see [26]) an exact sequence

0 ⟶ Ω_{S}^{q} ⟶ Ω_{S}^{q}(D) \underrightarrow{rés} Ω_{D}^{q−1} ⟶ 0, 1 ≤ q ≤ m,

which is equivalent, since the divisor D is a smooth hypersurface, to the sequence

0 ⟶ Ω_{S}^{q} ⟶ Ω_{S}^{q}(log D) \underrightarrow{rés} Ω_{D}^{q−1} ⟶ 0 .

However, an extension of the notion of Poincaré residue to the case when the divisor D has singularities requires more delicate considerations. Here we only note that already de Rham and Leray remarked that the above definition could be applied to the case when the singular locus Sing D of the divisor D consists of isolated double quadratic points only, or when the divisor D is the union of smooth hypersurfaces in general position. It should be also remarked that influenced by Leray’s works M.Herrera, P.Liberman and their followers, have developed the theory of the residue-currents.

We shall describe another approach, based on the notions of logarithmic and regular meromorphic differential forms, which leads to a natural generalization of the Poincaré residue to the general case of a singular divisor D.

4. A logarithmic residue

Now recall another important definition due to K.Saito [28], [29]. Let D ⊂ U be a hypersurface, and let the sheaf M_{D} be the O_{D}-Module of germs of meromorphic functions on D. Definition 4.1 (see [29], (2.2)). The (logarithmic) residue morphism is a homomorphism of O_{S}-modules

res.: Ω_{S}^{q}(log D) ⟶ M_{D} ⊗_{O_{D}} Ω_{D}^{q−1},

defined locally as follows. By the basic Lemma 1.5, for any ω ∈ Ω_{S,0}^{q}(log D) there exist a holomorphic function g defining a hypersurface V ⊂ U ⊂ S, a holomorphic (q − 1)-form ξ and a holomorphic q-form η in U such that

a) dim_{C} D ∩ V ≤ m − 2,

b) gω = \frac{dh}{h} ∧ ξ + η.

Then res. ω is given by the germ of the meromorphic (q −1)-form \frac{1}{g}·ξ in the module

M_{D,0} ⊗_{O_{D,0}} Ω_{D,0}^{q−1}.

Claim 4.2 (see [29], (2.5)). Let D ⊂ S be a hypersurface. Then for any q ≥ 1 there exists an exact sequence of O_{S}-modules

0 ⟶ Ω_{S}^{q} ⟶ Ω_{S}^{q}(log D) ⟶^{res.} M_{D} ⊗_{O_{D}} Ω_{D}^{q−1}.

Proof. Making use of the presentation of logarithmic forms as in the definition of the symbol res. above, we have

res. ω = 0 ⇔ gω ∈ Ω_{S,0}^{q} ⇔ ω ∈ Ω_{S,0}^{q}.

This completes the proof. In particular, for q = 1 we have

0 ⟶ Ω_{S}^{1} ⟶ Ω_{S}^{1}(log D) ⟶^{res.} M_{D} ≅ M_{D},

where D̃ is the normalization of D, that is, π: D̃ ⟶ D. K.Saito remarked, that in the case when D is the normal crossing outside of a subvariety of codimension at least 2, then the image Im (res.) contains π_{∗}(O_{D}) consisting of so-called weakly holomorphic function on D (see [29], (2.8)), that is, meromorphic functions, whose preimage becomes holomorphic on the normalization. Some interesting properties of the class of weakly holomorphic functions are investigated in [33].

Remark 4.3. By this way we can consider the image of the logarithmic residue res. Ω_{S}^{q}(log D) as an O_{D}-Module. Indeed, the definition of logarithmic forms implies that

h · (Ω_{S,0}^{q}(log D)/Ω_{S,0}^{q}) = 0.

Hence the multiplication by h· annihilates Im (res.).

Following D.Barlet [9], let us consider the O_{D}-modules of regular meromorphic differential forms ω_{D}^{q}, q ≥ 0, on the hypersurface D, making use of the following local representation:

0 ⟶ ω_{D,0}^{q} ⟶^{C} Ext_{O_{S,0}}^{1}(O_{D,0}, Ω_{S,0}^{q+1}) ⟶^{∧dh} Ext_{O_{S,0}}^{1}(O_{D,0}, Ω_{S,0}^{q+2}), q ≥ 0,

where ω_{D,0}^{q} ⊂ j_{∗}j^{∗}Ω_{D,0}^{q} and C is induced by the multiplication on the fundamental class of D in S. Thus C(v) corresponds to a Čech cocycle w/h such that w = v∧dh. It is easy to see, that there is an isomorphism

ω_{D,0}^{m−1} ≅ Ext_{O_{S,0}}^{1}(O_{D,0}, Ω_{S,0}^{m}).

Hence ω_{D,0}^{m−1} is the Grothendieck dualizing module of the (m − 1)-dimensional germ of the hypersurface (D, 0).

The following theorem shows that the logarithmic residue introduced by K.Saito is highly suitable for a good extension or a generalization of the classical notion of the Poincaré residue to the case of singular hypersurfaces.

Theorem 4.4 (see [5], §4). Let D ⊂ S be a reduced hypersurface. Then for any q ≥ 0 there are an isomorphism of O_{D}-modules

ω_{D}^{q} ≅ res. Ω_{S}^{q+1}(log D),

and an exact sequence

0 ⟶ Ω_{S}^{q+1} ⟶ Ω_{S}^{q+1}(log D) ⟶^{res.} ω_{D}^{q} ⟶ 0 .

Proof. It is sufficient to verify both statements locally. Let K.(h) be the usual Koszul complex associated with h

0 ⟶ O_{S,0}e_{0} ⟶^{d_{0}} O_{S,0} ⟶^{d_{−1}} O_{D,0} ⟶ 0 ,

where K_{1}(h) = O_{S,0}e_{0}, K_{0}(h) = O_{S,0} and d_{0}(e_{0}) = h, d_{−1}(1) = 1. Then we have the following piece of the dual exact sequence

⋯ ⟶ Hom_{O_{S,0}}(K_{0}(h), Ω_{S,0}^{q+1}) ⟶^{d^{0}} Hom_{O_{S,0}}(K_{1}(h), Ω_{S,0}^{q+1}) ⟶

⟶ Ext_{O_{S,0}}^{1}(O_{D,0}, Ω_{S,0}^{q+1}) ⟶ 0.

This means that any element of Ext_{O_{S,0}}^{1}(O_{D,0}, Ω_{S,0}^{q+1}) can be presented as a Čech 0-cochain (more explicitly, a 0-cocycle)

ν/h ∈ Hom_{O_{S,0}}(K_{1}(h), Ω_{S,0}^{q+1}) ≅ C_{(1)}^{0}(Ω_{S,0}^{q+1}),

where ν ∈ Ω_{S,0}^{q+1}.

Choose now an element ν ∈ Ω_{S,0}^{q+1} such that

\frac{ν}{h} ∧ dh ∈ Ext_{O_{S,0}}^{1}(O_{D,0}, Ω_{S,0}^{q+2}),

corresponds to the trivial element (see the definition of ω_{D,0}^{q}). That is, ν ∧ dh/h is defined by an element of

d^{0}(Hom_{O_{S,0}}(K_{0}(h), Ω_{S,0}^{q+2})).

This means that ν∧dh = h·η for some form η ∈ Ω_{S,0}^{q+2}. From the first exact sequence in Proposition 2.3, it follows that ν ∈ h · Ω_{S,0}^{q+1}(log D). Hence, ν̃ = C^{−1}(ν/h). Set in the definition v = ν̃, w = ν; it follows that C(ν̃) corresponds to a Čech cocycle ν/h such that ν = ν̃ ∧ dh. This implies ν/h = ν̃ ∧ dh/h, and res. (ν/h) = ν̃. Thus, for any element ν̃ ∈ ω_{D}^{q} there is a preimage under the logarithmic residue map represented by ν/h.

Remark 4.5. In the case when D is a normal hypersurface, the sheaves ω_{D}^{q} have the following well-known interpretation:

ω_{D}^{q} ≅ j_{∗}j^{∗}Ω_{D}^{q},

where j: D \ Sing D ⟶ D is the canonical embedding. More precisely, in such a case for any q ≥ 0 there is an exact sequence of local cohomology groups with supports in Z = Sing D :

0 ⟶ H_{Z}^{0}(Ω_{D}^{q}) ⟶ Ω_{D}^{q} ⟶ ω_{D}^{q} ⟶ H_{Z}^{1}(Ω_{D}^{q}) ⟶ 0.

It should be remarked also that in general ω_{D,0}^{0}, as well as res. Ω_{S,0}^{1}(log D), contains all the germs of locally bounded meromorphic functions on D (see [9], §2 ex.i)). Moreover, these two sets coincide if the local fundamental group of the complement S \ D is abelian (see [27]).

Example 4.6 (see [9], end of §1). In the above Example 2.1 of a cusp singularity, it is easy to compute that res. Ω_{S,0}^{1}(log D) and ω_{D,0}^{0} are generated by 1 and y/x which one can think of as elements belonging to M_{S,0} or j_{∗}j^{∗}Ω_{D,0}^{0} ≅ j_{∗}j^{∗}O_{D,0} respectively, while ω_{D,0}^{1} and res. Ω_{S,0}^{2}(log D) are generated by dy/2x = dx/3y^{2} which also belongs to M_{S,0} ⊗ Ω_{S,0}^{1} or to j_{∗}j^{∗}Ω_{D,0}^{1}.

5. Duality

Following [2], §5 we now explain that there is a natural extension of Grothendieck duality for isolated singularities to the case of non-normal hypersurfaces. The main idea is to use localization of the germ of such a hypersurface, relative to the ideal which defines its singular locus.

More precisely, let X be a reduced irreducible Cohen-Macaulay germ of an ndimensional complex variety, O = O_{X} its (dual) analytic algebra and let Z ⊂ X be any subgerm defined by the ideal J ⊂ O. Suppose that Z is pure-dimensional and dim Z = d. In other words, the ideal J is unmixed (i.e. without embedded prime ideals) and its height is equal to n − d ≥ 1. We denote by {℘_{1}, …, ℘_{h}} the set of all the minimal prime divisors of the ideal J . Then every ideal ℘_{i}, i = 1, …, h,

corresponds to some d-dimensional irreducible subgerm Z_{i} ⊂ X, hence Z_{red} = ⋃_{i=1}^{h} Z_{i}

is the irreducible decomposition of the support Z.

Further let L = O \ ⋃_{i=1}^{h} ℘_{i} ⊂ O be the multiplicative subset of the local ring

O. Then the localization L^{−1}O of the ring O by this multiplicative system is an (n − d)-dimensional reduced semilocal rings (and a C-algebra at the same time) with maximal ideals 𝔐_{i} equal to L^{−1}℘_{i}, i = 1, …, h, and besides L^{−1}O is a Cohen-Macaulay ring as well as O. We shall call L^{−1}O the semilocal ring associated with the reducible germ Z and denote it by O_{Z}.

Under our assumptions, all the fields C(Z_{i}) ≅ O_{℘_{i}}/𝔐_{i}, i = 1, …, h, have the same transcendence degree over C, namely n − d. Let x̃_{i} = {x̃_{i_{1}}, …, x̃_{i_{n−d}}} be any transcendental basis of C(Z_{i})/C and let x_{i} = {x_{i_{1}}, …, x_{i_{n−d}}} be any system of representatives of x̃_{i} in O_{℘_{i}} (cf. [20], §1). It is known that there exists a field of representatives T_{i} for C(Z_{i}) in the completion Ô_{℘_{i}}. Thus, C(x_{i_{1}}, …, x_{i_{n−d}}) belongs to T_{i}. Throughout this section we shall assume that T_{i} ⊂ O_{℘_{i}}. For instance, this is true for a quasi-homogeneous germ Z. In addition, we will suppose (but this is not

essential) that O is a domain. Hence, we have the equality L^{−1}O = ⋂_{i=1}^{h} O_{℘_{i}} in the

field of fractions of O, and one can define the field of representatives T = ∩T_{i} for the semilocal ring O. Theorem 5.1 (cf. [20], (2.3)). Let Z ⊂ X be a Cohen-Macaulay germ of pure dimension d. Under the above assumptions, there exist F-functorial isomorphisms in the category of coherent Ô_{Z}-modules:

Exti ; Ô_{Z}(F̂_{Z}, ω̂_{Z}) ≅ Hom_{T} (H_{Z}^{d−i}(F), Ω_{T /C}^{n−d}), i = 0, 1, …, d.

Here we denote the Grothendieck dualizing module of the germ X by ω = ω_{X}^{n}, the completion of O_{Z} with respect to the radical ℜ = ∩ 𝔐_{i} by Ô_{Z}, F̂_{Z} = F ⊗_{O} Ô_{Z} (just the same ω̂_{Z}), the local cohomology functor with supports in Z_{red} by H_{Z}^{∗}(−), and the module of meromorphic differential (n − d)-forms on T /C by Ω_{T /C}^{n−d}.

Proof. Under our assumptions this Theorem can be proved as in [20], (2.2) taking into consideration the following. Instead of the algebraically geometric objects of [20] we can consider in a similar way, analytic objects, and we can take the system of regular parameters (in the semilocal ring O_{Z}) which belongs to the radical ℜ ⊂ O_{Z}, as well as the corresponding system in the case of an irreducible germ Z.

Example 5.2. Suppose that X = (D, 0) is a germ of graded Saito singularity of dimension 2 defined by a quasi-homogeneous polynomial h. Set Z = Sing D ⊂ D. Then codim (Z, D) = 1 and Z is a Cohen-Macaulay germ (see [3], §2). Let F = Ω_{D}^{1} be the O_{D}-module of holomorphic differential 1-forms on D. We have

ω = ω_{D}^{n} ≅ O_{D}(dz/dh), H_{Z}^{0}(Ω_{D}^{1}) ≅ Tors Ω_{D}^{1}, Ext_{O_{D}}^{1}(Ω_{D}^{1}, O_{D}) ≅ T^{1}(D),

where T^{1}(D) is the space of the first cotangent cohomology of D. We remark that T^{1}(D) and Q_{h} = O_{D}/J(h) are canonically isomorphic up to a shift of grading by deg h, where J(h) denotes the Jacobian ideal generated by the partial derivatives of quasihomogeneous polynomial h. Theorem 5.1 states that there exists a T -linear perfect pairing of graded infinite dimensional C-vector spaces:

Tors Ω_{D}^{1} × T^{1}(D) ⟶^{Res^{T}} Ω_{T /C}^{1}.

Thus, one can consider this pairing as an analog of Grothendieck’s duality for ndimensional hypersurface isolated singularities between finite dimensional C-vector spaces (cf. also [6], Cor. 3):

Tors Ω_{D}^{n} × T^{1}(D) ⟶^{Res} C .

For example (see [3], [5]), consider the discriminant D of the versal deformation ϕ : V ⟶ S of the simple hypersurface singularity of type A_{3}. In this case S ≅ (C^{3}, 𝔬) and the subgerm D ⊂ S is defined by the polynomial h of degree 12, so that (see [27])

h = \frac{1}{48}(3ζ^{3} − ξ^{2}), where ξ = \frac{8}{3}z_{2}^{3} + 36z_{3}^{2} − 96z_{2}z_{4}, ζ = \frac{4}{3}z_{2}^{2} + 16z_{4},

and the free O_{S}-module Ω_{S}^{1}(log D) is generated by three logarithmic differential forms dh/h, θ_{1}/h and θ_{2}/h, where

ϑ_{1} = (32z_{2}z_{3}z_{4} − 9z_{3}^{3})dz_{2} + (6z_{2}z_{3}^{2} + 64z_{4}^{2} − 16z_{2}^{2}z_{4})dz_{3} − (4z_{2}^{2}z_{3} + 48z_{3}z_{4})dz_{4} ;

ϑ_{2} = (8z_{2}^{2}z_{4} − 3z_{2}z_{3}^{2} − 32z_{4}^{2})dz_{2} + (2z_{2}^{2}z_{3} + 24z_{3}z_{4})dz_{3} + (16z_{2}z_{4} − 4z_{2}^{3} − 18z_{3}^{2})dz_{4}.

Using the above notations, one obtains that

℘_{1} = (z_{3}, z_{2}^{2} − 4z_{4}), T_{1} ≅ C(z_{2}), ℘_{2} = (ξ, ζ), T_{2} ≅ C(\sqrt{z_{2}}),

T = T_{1} ∩ T_{2} ≅ C(z_{2}), ω ≅ O_{D}(dz/dh) ≅ O_{D}(−3).

Making use of Proposition 2.3, we get that Tors Ω_{D}^{1} can be generated as a C[z_{2}]- module by the following set of 1-forms:

{ θ_{2}, θ_{1}, z_{3}θ_{2}, z_{4}θ_{2}, z_{3}z_{4}θ_{2}}

which have degrees 10, 11, 13, 14, 17, respectively. On the other hand, Q_{h} = O_{D}/J(h) is generated as a C[z_{2}]-module by the elements

{ 1, z_{3}, z_{4}, z_{3}^{2}, z_{3}z_{4} }

and their degrees (as elements of T^{1}(D)) are equal to −12, −9, −8, −6, −5, respectively. Let us denote by C_{D} the class of the 2-form dz/dh in L^{−1}ω, where L = O \ {℘_{1} ∪ ℘_{2}} ⊂ O and ω ≅ O_{D}(dz/dh).

Since O_{Z} = L^{−1}O is an one-dimensional reduced semilocal Cohen-Macaulay ring, we can apply [20], (1.7), (2.1), and calculate

Res^{T} (⟨z_{3}z_{4}θ_{2}, 1⟩ ∧ C_{D}) = Res^{T} (z_{3}z_{4}θ_{2} ∧ C_{D}) = const · dz_{2} ∈ Ω_{T /C}^{1},

and so on. Moreover, we can also write out the Poincaré series

P (Tors Ω_{D}^{1}; t) = t^{10} · (1 + t + t^{3} + t^{4} + t^{7})/(1 − t^{2}) := P(t)/(1 − t^{2}),
P (Q_{h}; t) = (1 + t^{3} + t^{4} + t^{6} + t^{7})/(1 − t^{2}),

P (T^{1}(D); t) = t^{−12} · P (Q_{h}; t) := R(t)/(1 − t^{2}), P (C[z_{2}]) = 1/(1 − t^{2}).

The existence of the above-mentioned T -linear perfect pairing in the case we have just considered means that t^{5} · P(t^{−1}) = R(t), 5 = − deg(ω) + deg(dz_{2}).

6. Logarithmic de Rham complex

One can easily verify that the usual exterior differential d defined on the de Rham complex Ω_{S}^{•} on a complex manifold S induces an action on Ω_{S}^{q}(log D). So the logarithmic de Rham complex

Ω_{S}^{•}(log D) = (Ω_{S}^{q}(log D), d)_{q=0,1,...}

is well defined. In this connection, there arises the problem of calculating the cohomology groups of logarithmic de Rham complex H^{p}(S, Ω_{S}^{•}(log D)), as well as the corresponding cohomology group sheaves H^{p}(Ω_{S}^{•}(log D)) whose stalk at a point x ∈ S is defined as follows:

H^{p}(Ω_{S}^{•}(log D))_{x}: = H_{DR}^{p}(Ω_{S,x}^{•}(log D)).

For example, we get dim_{C}H_{DR}^{0}(Ω_{S,0}^{•}(log D)) = 1. Furthermore, as a direct consequence of the holomorphic Poincaré lemma one obtains (see [26]) that in the case of a smooth hypersurface D the cohomology sheaves of Ω_{S}^{•}(log D) satisfy

H^{0}(Ω_{S}^{•}(log D)) = C_{S}, H^{1}(Ω_{S}^{•}(log D)) = C_{D}, H^{q}(Ω_{S}^{•}(log D)) = 0, q ≥ 2.

This implies that the subcomplex Ω_{S}^{•}(log D) of the complex Ω_{S}^{•}(⋆D) of meromorphic forms on S with poles along D of any order has the same cohomology sheaves as the full complex. Hence the hypercohomology groups of Ω_{S}^{•}(log D) give us the complex cohomology of the complement S \ D. Proposition 6.1. Let D ⊂ S be a reduced hypersurface. Then there exists an exact sequence of complexes of O_{S}-modules

0 ⟶ Ω_{S}^{•} ⟶ Ω_{S}^{•}(log D) ⟶^{res.} ω_{D}^{•−1} ⟶ 0

which induces a natural isomorphism

H_{DR}^{p}(Ω_{S}^{•}(log D)) ≅ H_{DR}^{p−1}(ω_{D}^{•})

for any p ≥ 1, that is, there is a quasi-isomorphism of complexes with shifting grading by −1.

Proof. It is easy to see that the map res. commutes with the exterior differential d (see [29], (2.6)), and we get the first assertion from the Theorem 4.4 above. Let us apply the cohomology functor to this exact sequence. The acyclicity of the de Rham complex in positive dimensions on S gives us the isomorphism

H_{DR}^{p}(Ω_{S}^{•}(log D)) ≅ H_{DR}^{p−1}(ω_{D}^{•}), p ≥ 2,

and the long exact sequence

0 ⟶ H_{DR}^{1}(Ω_{S}^{•}(log D)) ⟶ H_{DR}^{1}(ω_{D}^{•−1}) ⟶

⟶ H_{DR}^{0}(Ω_{S}^{•}) ⟶ H_{DR}^{0}(Ω_{S}^{•}(log D)) ⟶ 0.

But the last two terms of that sequence have the same dimension one. This completes the proof.

Remark 6.2. When D is any reduced hypersurface germ and h is the number of irreducible components of D, then dim_{C}H_{DR}^{0}(ω_{D,0}^{•}) = h (see [19], (4.1)), and the long exact sequence from the proof of Proposition 6.1 yields dim_{C}H_{DR}^{1}(Ω_{S,0}^{•}(log D)) = h (cf. [28], §1).

Corollary 6.3. Assume that the singular locus of D consists of quotient singularities by finite groups only. Then

H_{DR}^{p}(Ω_{S}^{•}(log D)) = 0, p ≥ 2.

Proof. Under our assumptions the complex ω_{D}^{•} is exact (see [19], (4.3)).

Corollary 6.4. Suppose that D ⊂ S is a germ of a hypersurface with an isolated singularity, dim D = n ≥ 2, μ and τ are the Milnor number and the Tjurina number of D, respectively. Then

dim_{C}H_{DR}^{1}(Ω_{S}^{•}(log D)) = 1, H_{DR}^{p}(Ω_{S}^{•}(log D)) = 0 for any 1 < p < n,
dim_{C}H_{DR}^{n+1}(Ω_{S}^{•}(log D)) − dim_{C}H_{DR}^{n}(Ω_{S}^{•}(log D)) = μ − τ.

Proof. This corollary is a direct consequence of the above Proposition and [19], (4.6).

Corollary 6.5. Suppose in addition that D is a quasi-homogeneous germ, defined by a polynomial, of degree d ∈ Z relative to the variables z_{i} of weights w_{i}, i = 1, …, m, respectively. Then

dim_{C}H_{DR}^{n+1}(Ω_{S}^{•}(log D)) = dim_{C}H_{DR}^{n}(Ω_{S}^{•}(log D)) =
= res_{t=0}t^{−d−1} · Π(t^{d} − t^{w_{i}})/(t^{w_{i}} − 1).

Proof. From [19], (4.8) it follows that H_{DR}^{n}(ω_{D,0}^{•}) ≅ (ω_{D,0}^{n})_{0}, where the subscript 0 denotes the zero-homogeneous piece of the graded C-algebra ω_{D,0}^{n}. Analogously, since Tors (Ω_{D,0}^{n})_{0} = (Ω_{D,0}^{n})_{0} = 0, from the exact sequence

0 ⟶ H_{{0}}^{0}(Ω_{D,0}^{n}) ⟶ Ω_{D,0}^{n} ⟶ ω_{D,0}^{n} ⟶ H_{{0}}^{1}(Ω_{D,0}^{n}) ⟶ 0

we get (ω_{D,0}^{n})_{0} ≅ (H_{{0}}^{1}(Ω_{D,0}^{n}))_{0}. The Poincaré series of H_{{0}}^{1}(Ω_{D,0}^{n}) has been written down (see [1]). The second equality follows from the well-known isomorphism

H_{{0}}^{1}(Ω_{D,0}^{n}) ≅ H_{{0}}^{1}(Ω_{D,0}^{n−1}).

In fact, there are other interesting relations between logarithmic and regular meromorphic differential forms, especially in the case of the free Saito divisors.

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