# Logarithmic Comparison Theorems

F. J. Castro-Jiménez, D. Mond, L. Narváez-Macarro Departamento de Álgebra &
Instituto de Matemáticas (IMUS), Facultad de Matemáticas, Universidad de
Sevilla, Spain, e-mail<EMAIL_ADDRESS>Institute, University of
Warwick, Coventry CV4 7AL United Kingdom, e-mail:
<EMAIL_ADDRESS>de Álgebra & Instituto de Matemáticas
(IMUS), Facultad de Matemáticas, Universidad de Sevilla, Spain, e-mail
<EMAIL_ADDRESS>
###### Abstract

In this paper we study the comparison between the logarithmic and the
meromorphic de Rham complexes along a divisor in a complex manifold. We focus
on the case of free divisors, starting with the case of locally
quasihomogeneous divisors, and we explain how D-module theory can be used for
this comparison.

We dedicate this chapter to the memory of our colleagues Jim Damon and José
Luis Vicente Córdoba.

## Introduction

We survey a number of results known as comparison theorems, along the lines of
the comparison theorem of Grothendieck. Grothendieck’s Comparison Theorem
states that, for any hypersurface $D$ in a complex manifold $X$, the
cohomology of the meromorphic de Rham complex with respect to $D\subset X$
(i.e. $h^{q}(\Omega_{X}^{\bullet}(\star D))$ for $0\leq q\leq\dim X$)
coincides with $\mathbb{R}^{q}j_{*}\mathbb{C}_{U}$ where $j:U:=X\smallsetminus
D\rightarrow X$ is the inclusion and $\mathbb{C}_{U}$ is the constant sheaf
$\mathbb{C}$ on $U$. In effect, the hypercohomology of this complex is the
(topological) cohomology $H^{\bullet}(U;\mathbb{C})$. If $D$ is a normal
crossing divisor (NCD), this comparison result was proved by Atiyah and Hodge,
and Grothendieck’s proof reduces the general case to the case of a NCD by
using Hironaka’s resolution of singularities. When $D$ is an arrangement of
hyperplanes in $\mathbb{C}^{n}$, the Brieskorn complex $B^{\bullet}$ computes
the singular cohomology of the complement $U=\mathbb{C}^{n}\smallsetminus D$,
$H^{\bullet}(U;\mathbb{C})$; and this is also a comparison result. There is a
class of divisors, the so-called free divisors, introduced by Kyoji Saito, for
which a certain number of comparison results can be proven. We survey these
results and show how $\mathcal{D}$–module theory provides a way to deal with
them.

The content of the paper is as follows. In Sect. 1.1 we first recall
Grothendieck’s comparison theorem (see also Sect. 2.6), that there is a
canonical isomorphism

$h^{q}(\Omega_{X}^{\bullet}(\star
D))\rightarrow\mathbb{R}^{q}j_{*}\mathbb{C}_{U}$

for $0\leq q\leq\dim X$. If $X$ is Stein, then the global morphisms analogous
to the previous ones

$h^{q}(\Gamma(X,\Omega_{X}^{\bullet}(\star D)))\rightarrow
H^{q}(U;\mathbb{C})$

are isomorphisms. We then recall Brieskorn’s Theorem proving that if $D$ is a
finite union of hyperplanes in $X=\mathbb{C}^{n}$, with equations $h_{j}=0$,
then the cohomology of the complement $H^{q}(X\smallsetminus D;\mathbb{C})$
can be computed as the cohomology of the so called Brieskorn complex
$B^{\bullet}$, which is the $\mathbb{C}$-subalgebra of the exterior algebra
$\Gamma(X,\Omega_{X}^{\bullet}(\star D))$ generated by the forms
$\frac{dh_{j}}{h_{j}}$.

Kyoji Saito (see Sect. 1.3) introduced the notion of logarithmic meromorphic
form with respect to a divisor $D$ in a complex manifold $X$: a meromorphic
form $\omega$ has a logarithmic pole along the divisor $D$ if both $h\omega$
and $hd\omega$ are holomorphic, where $h$ is a local equation for $D$. Note in
particular that all of Brieskorn’s forms have logarithmic poles along an
arrangement of hyperplanes $D\subset\mathbb{C}^{n}$. Logarithmic meromorphic
forms form an $\mathcal{O}_{X}$-subcomplex of the meromorphic de Rham complex
$\Omega_{X}^{\bullet}(\star D)$. A natural question asks for the class of
divisors $D\subset X$ for which the inclusion

$\Omega_{X}^{\bullet}(\log D)\hookrightarrow\Omega_{X}^{\bullet}(\star D)$

is a quasi-isomorphism (i.e. when the two complexes have the same cohomology).
By analogy with Grothendieck’s comparison theorem, if the morphism
$\Omega_{X}^{\bullet}(\log D)\hookrightarrow\Omega_{X}^{\bullet}(\star D)$ is
a quasi-isomorphism we say that the divisor $D$ satisfies the logarithmic
comparison theorem (LCT), or that LCT holds for $D$. Theorem 1.3 states that
any locally quasihomogeneous free divisor $D\subset X$ satisfies LCT. In 1977
H. Terao conjectured (see Sect. 1.4) that LCT always holds for hyperplane
arrangements (which are, of course, locally quasihomogeneous), and this
conjecture was finally proved by Daniel Bath in [2].

A free divisor is linear if the module
$\operatorname{\mathcal{D}\mathit{er}}(-\log D)$ of logarithmic derivations
with respect to $D$ (as defined by K. Saito in [60], see Sect. 1.3) has a
basis of vector fields whose coefficients are linear forms. In Sect. 1.5 we
recall a number of results on linear free divisors and in particular the
theorem of [28], that LCT holds for “reductive” linear free divisors.

In Sect. §2 we introduce the background in $\mathcal{D}$–module theory that is
used in Sect. §3. This last section is devoted to state a $\mathcal{D}$–module
criterion for the Logarithmic Comparison Theorem for free divisors. First, and
following the paper of F.J. Calderón-Moreno [9], we consider the sheaf
$\mathcal{V}_{X}^{D}$ of logarithmic differential operators on $X$, with
respect to a free divisor $D\subset X$ and the logarithmic Spencer complex
${\mathrm{Sp}}^{\bullet}(\log D)$ which is a locally free resolution of the
$\mathcal{V}_{X}^{D}$–module $\mathcal{O}_{X}$. By using [12] one generalizes
the construction of this last complex to an arbitrary left
$\mathcal{V}_{X}^{D}$–module. This constructions are used to prove the main
theorem in this context. This theorem, proved by Calderón-Moreno and Narváez-
Macarro ([12, Cor. 4.2] and [55, Corollary 1.7.2]), states that a free divisor
$D\subset X$ satisfies the Logarithmic Comparison Theorem if and only if the
natural morphism

$\varrho:\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}(D)\longrightarrow\mathcal{O}_{X}(\star
D)$

is an isomorphism in the derived category of $\mathcal{D}_{X}$-modules (see
Theorem 3.3). Fixing a point $p\in D$, a reduced local equation $f$ of the
germ $(D,p)$ and a basis $\\{\delta_{1},\ldots,\delta_{n}\\}$ of
$\operatorname{\mathcal{D}\mathit{er}}(-\log D)_{p}$, the morphism
$\varrho_{p}$ is an isomorphism if and only if the complex
$\mathcal{D}_{X,p}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X,p}^{D}}\mathcal{O}_{X,p}(D)$
is concentrated in cohomological degree 0,
$\mathcal{D}_{X,p}f^{-1}=\mathcal{O}_{X,p}(\star D)$ and the
$\mathcal{D}_{X,p}$–annihilator of $f^{-1}\in\mathcal{O}_{X,p}(\star D)$
equals the left ideal
$\mathcal{D}_{X,p}(\delta_{1}+\alpha_{1},\ldots,\delta_{n}+\alpha_{n})$, where
$\delta_{i}(f)=\alpha_{i}f$ for $i=1,\ldots,n$, see Sect. 3.3. That gives a
$\mathcal{D}$–module criterion to test if LCT holds for a given free divisor
$D\subset X$. Finally we treat some examples to show how this criterion can be
applied in practice.

##  §1 Comparison Theorems

###  1.1 Grothendieck’s Comparison Theorem

To calculate the cohomology of a space with constant coefficients
$\mathbb{C}$, one can use a resolution of the constant sheaf $\mathbb{C}_{X}$
and then calculate the hypercohomology of the resolution. For example, on a
complex manifold the holomorphic Poincaré lemma (see e.g. [26, Ch. 2, Par. 2,
Ex. 2.5.1] – the proof for $C^{\infty}$ forms is easily adapted to the
holomorphic case) shows that the complex of holomorphic differential forms

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}_{X}=\Omega}^{0}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{{\Omega}^{1}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{\cdots}$
(1)

is exact, so that ${\Omega}^{\bullet}_{X}$ is a resolution of $\mathbb{C}_{X}$
by locally free $\mathcal{O}_{X}$-modules. If $X$ is in addition a Stein space
– for example, the complement of a divisor $D$ in $\mathbb{C}^{n}$, or a
convex open set in $\mathbb{C}^{n}$ – then all of the sheaves in the complex
are acyclic: $H^{j}(X,{\Omega}^{k}_{X})=0$ for $j>0$ and $k\geq 0$ (in effect,
this is the definition of ‘Stein space’). In this case the double complex with
which one calculates the hypercohomology is reduced to the complex of global
sections

$0{\ \rightarrow\ }{\Gamma}(X,\mathcal{O}_{X}){\ \rightarrow\
}{\Gamma}(X,{\Omega}^{1}_{X}){\ \rightarrow\ }\cdots$

and so one has the analytic de Rham theorem:

(1.1) Theorem. If $X$ is a Stein manifold then

$\textstyle{h^{q}({\Gamma}(X,{\Omega}^{\bullet}_{X}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{q}(X;\mathbb{C})}$
(2)

is an isomorphism.

Here if we view $H^{j}(X;\mathbb{C})$ as the singular cohomology group, then
the arrow is given by the integration of differential forms along singular
chains, which we will refer to as the de Rham morphism. Note that this theorem
already implies the non-trivial fact that the cohomology of a Stein manifold
vanishes above middle dimension. If $X$ is the complement of a divisor (a
hypersurface) $D\subset\mathbb{C}^{n}$, then ${\Omega}^{k}_{X}$ (or more
precisely $j_{*}{\Omega}^{k}_{X}$, where $j:X{\ \rightarrow\ }\mathbb{C}^{n}$
is inclusion) is equal to the sheaf of germs of holomorphic forms with
arbitrary singularities along $D$. This is strictly larger than the sheaf
${\Omega}^{k}(\star D)$ of meromorphic forms with poles along $D$, since it
places no restriction on the behaviour of the extension to $D$, and in
particular allows also forms with essential singularities along $D$. In [32,
Th. 2]111It is noted in [32] that the contents of this paper formed part of a
letter of the author to M. F. Atiyah, dated October 14, 1963, except for some
remarks dated November 1963 and July 1965., Alexander Grothendieck showed that
despite this difference, there are canonical isomorphism of sheaves of complex
vector spaces

$h^{q}\left({\Omega}^{\bullet}_{\mathbb{C}^{n}}(\star
D)\right)\simeq\mathbb{R}^{q}j_{*}\mathbb{C}_{\mathbb{C}^{n}\smallsetminus
D},\quad q=0,\dots,n,$ (3)

where $\mathbb{R}^{q}j_{*}$ is the $q$-th right derived functor of the left-
exact functor $j_{*}$, see e.g. [36, I.7]. The maps in (3) are given by the
composition of the adjunction

${\Omega}^{\bullet}_{\mathbb{C}^{n}}(\star D){\ \rightarrow\
}j_{*}j^{-1}{\Omega}^{\bullet}_{\mathbb{C}^{n}}(\star
D)=j_{*}{\Omega}^{\bullet}_{\mathbb{C}^{n}\smallsetminus D}$

with the inverse of Poincaré quasi-isomorphism
$\mathbb{C}_{\mathbb{C}^{n}\smallsetminus D}\stackrel{{\scriptstyle\sim}}{{{\
\rightarrow\ }}}{\Omega}^{\bullet}_{\mathbb{C}^{n}\smallsetminus D}$.

Taking into account that $\mathbb{C}^{n}$ is Stein, by [32, Corollary, page
97], one has $H^{j}(\mathbb{C}^{n},{\Omega}^{k}_{\mathbb{C}^{n}}(\star D))=0$
for $j>0$ and any $k$, and there are canonical isomorphisms of complex vector
spaces

$\textstyle{h^{q}({\Gamma}(\mathbb{C}^{n},{\Omega}^{\bullet}_{\mathbb{C}^{n}}(\star
D)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{q}(\mathbb{C}^{n}\smallsetminus
D;\mathbb{C})}$ (4)

for $q\geq 0$.

These maps are formal in the sense that they come from adjunction and Poincaré
lemma. A well known (but not obvious) fact, see Remark 1.1, is that these maps
coincide with the de Rham map given by integration of meromorphic forms along
singular chains composed with the isomorphism between singular cohomology with
coefficients in $\mathbb{C}$ and sheaf cohomology with coefficients in the
constant sheaf $\mathbb{C}_{\mathbb{C}^{n}\smallsetminus D}$ (see [58, Ch. 4;
Th. 4.14]).

The isomorphism in (3) can be stated as an isomorphism in the derived
category, which is of course valid on any complex manifold $X$ and any divisor
$D\subset X$, namely

${\Omega}^{\bullet}_{X}(\star
D)\simeq\mathbb{R}j_{*}\mathbb{C}_{X\smallsetminus D}.$ (5)

This is known as Grothendieck’s Comparison Theorem (see Theorem 2.6). The core
assertion is that one can ignore essential singularities along $D$.

To end this section, let us emphasize that Grothendieck’s proof of (5) used in
an essential way Hironaka’s resolution of singularities (in the complex
algebraic and the complex analytic settings), and became a model to follow for
the cohomological study of algebraic varieties over an arbitrary field. This
fact brought conferred upon the resolution of singularities in positive
characteristic a privileged place in the construction of cohomological
theories, to the extent that it appeared as a matter of substance in this
context. On the other hand, Grothendieck’s comparison theorem became a
completely new and unexpected way to understand regular singular points (at
infinity) of integrable connections on algebraic fiber bundles (see footnote
(13) in [32], dated July 1965, and the crucial work [21]), and more generally,
a conceptual way to incorporate regularity as a central notion in
$\mathcal{D}$-module theory (see [46]). Finally, $\mathcal{D}$-module theory
itself has provided new tools to understand Grothendieck’s comparison theorem
and to avoid resolution of singularities in its treatment [48, 49] (see also
[50]).

In this paper we would like: (1) to explain what the logarithmic comparison
means, drawing some analogies with Grothendieck’s comparison theorem; (2) to
survey the original proof of the Logarithmic Comparison Theorem for locally
quasihomogeneous free divisors (see Theorem 1.3); and (3) to explain how
$\mathcal{D}$-module theory provides a characterization of those free divisors
which satisfy the Logarithmic Comparison Theorem (see Sect. 3.3). We emphasise
that in all of them resolution of singularities does not play any rôle.

(1.2) Remark. For any topological space $X$, one denotes by
$H_{q}(X;\mathbb{C})$ the singular homology group of $X$ with values in
$\mathbb{C}$, for $q\geq 0$. Recall that the singular homology group
$H_{q}(X;\mathbb{C})$ is by definition the homology group
$h_{q}(C_{\bullet}(X),\partial)$ of the complex $(C_{\bullet}(X),\partial)$ of
singular chains of $X$ with coefficients in $\mathbb{C}$, where $\partial$
denotes the boundary map.

Recall also that the complex of singular cochains $(C^{\bullet}(X),\delta)$ is
defined as

$C^{\bullet}(X):=\operatorname{\rm
Hom}_{\mathbb{C}}(C_{\bullet}(X),\mathbb{C}),$

where $\delta$ is the coboundary map associated with the boundary map
$\partial$. The singular cohomology group $H^{q}(X;\mathbb{C})$, of $X$ with
values in $\mathbb{C}$ for $q\geq 0$, is by definition the cohomology group
$h^{q}(C^{\bullet}(X),\delta)$.

There exists a natural morphism

$H^{q}(X;\mathbb{C})\stackrel{{\scriptstyle\psi_{q}}}{{\longrightarrow}}H_{q}(X;\mathbb{C})^{*}:=\operatorname{\rm
Hom}_{\mathbb{C}}(H_{q}(X;\mathbb{C}),\mathbb{C}).$

For any $[\eta]\in H^{q}(X;\mathbb{C})$ and any $[\gamma]\in
H_{q}(X;\mathbb{C})$, one has

$\psi_{q}([\eta])([\gamma])=\eta(\gamma)$

where $[\,]$ means equivalence class in the corresponding (co)homology group.
The morphism $\psi_{q}$ is an isomorphism, for $q\geq 0$, since $\mathbb{C}$
is a field.

If $X$ is a complex manifold with $\dim X=n\geq 1$, the de Rham morphism

$dR_{q}:h^{q}(\Gamma(X,\Omega_{X}^{\bullet}))\longrightarrow
H^{q}(X;\mathbb{C})$

is defined (or induced) by integration of differential $q$-forms over singular
$q$-chains:

$dR_{q}([\omega])([\gamma]):=\int_{\gamma}\omega\in\mathbb{C}$

where $\omega$ is a $q$-form, $\gamma$ is a singular $q$-chain and $[\,]$
means equivalence class in the corresponding (co)homology group. Here, to be
precise, one needs to consider $C^{\infty}$ chains, and the point is that the
inclusion of the complex of $C^{\infty}$ singular chains of $X$ with
coefficients in $\mathbb{C}$ in $C_{\bullet}(X)$ is a quasi-isomorphism (see
[45, Ap. A, Th. 2.1]).

The de Rham theorem states that if $X$ is a Stein manifold then $dR_{q}$ is an
isomorphism for $q=0,\ldots,n$; see e.g. [45, App. A, Th. 3.1]. In the
$C^{\infty}$ category, the de Rham theorem holds in much greater generality,
for all paracompact differentiable manifolds, (see loc. cit.).

Recall that by the Poincaré Lemma the complex (1), i.e. the complex

$0\rightarrow\mathbb{C}_{X}\rightarrow\Omega_{X}^{\bullet},$

is exact. Since the category $\operatorname{\rm Mod}(\mathbb{C}_{X})$, of
sheaves of $\mathbb{C}$-vector spaces, has enough injectives (see e.g. [67,
Ex. 2.3.12]), there exists an injective resolution

$0\rightarrow\mathbb{C}_{X}\rightarrow I_{X}^{\bullet}$

and a natural morphism

$\Omega_{X}^{\bullet}\rightarrow I_{X}^{\bullet},$

unique up to homotopy equivalence, making the obvious diagram commute (see
[31, Rq. 3, Th. 2.4.1] and e.g. [67, 2.5]). So for any $q$, there is a natural
morphism

$\alpha_{q}:\,h^{q}(\Gamma(X,\Omega_{X}^{\bullet}))\rightarrow
h^{q}(\Gamma(X,I^{\bullet}_{X}))=H^{q}(X;\mathbb{C}_{X})$

where the last equality is just the definition of the sheaf cohomology of the
constant sheaf $\mathbb{C}_{X}$.

Moreover, as mentioned in the introduction, if $X$ Stein, then each
$\Omega^{q}_{X}$ is acyclic for the “global sections functor” $\Gamma(X;\,)$,
and hence the natural morphism $\alpha_{q}$ is an isomorphism (see [31, Rq. 3,
Th. 2.4.1] and e.g. [67, 2.5]).

Finally, for any complex manifold $X$ (in fact, for any locally contractible
topological space) let us consider, see e.g. [58, Ch. 4, Th. 4.14], the
complex

$0\rightarrow\mathbb{C}_{X}\rightarrow\widetilde{S}_{X}^{\bullet}$ (6)

where, for $q\geq 0$, one denotes by $\widetilde{S}_{X}^{q}$ the sheaf
associated to the presheaf ${S}_{X}^{q}$ of singular $q$–cochains on $X$.
Local contractibility means that each point has a fundamental system of
neighbourhoods which are contractible. In each of these, the complex of
singular cochains is exact. It follows that the complex of sheaves (6) is
exact. As before for the complex $\Omega^{\bullet}_{X}$ (but now with the
complex $\widetilde{S}_{X}^{\bullet}$ in its place), and again using the
injective resolution $I^{\bullet}_{X}$ of $\mathbb{C}_{X}$, for each $q$ there
is a natural morphism

$\beta_{q}:\,H^{q}(X;\mathbb{C})=h^{q}(\Gamma(X,\widetilde{S}_{X}^{\bullet}))\rightarrow
h^{q}(\Gamma(X,I^{\bullet}_{X}))=H^{q}(X;\mathbb{C}_{X})$

where the first equality is just the definition of the singular cohomology
$H^{q}(X;\mathbb{C})$ (see [58, Ch. 4, Prop. 4.12]). Since the sheaves
$\widetilde{S}_{X}^{q}$ are all flabby they are also acyclic for the functor
$\Gamma(X;\,)$ (see [58, Ch. 4, ex. 1.10, 2 and Prop. 3.3]). Hence the
morphism $\beta_{q}$ is an isomorphism for $q\geq 0$.

So, for any $q$ we have the following diagram of isomorphisms

${h^{q}(\Omega^{\bullet}(X))}$${H^{q}(X;\mathbb{C})}$${H^{q}(X;\mathbb{C}_{X}).}$$\scriptstyle
dR_{q}$$\scriptstyle\alpha_{q}$$\scriptstyle\beta_{q}$ (7)

It is “well-known” that the diagram commutes, up to a sign, although a
concrete reference for this result seems to be elusive.

###  1.2 The Brieskorn Complex

For a special class of divisors, namely hyperplane arrangements in affine
space $X=\mathbb{C}^{n}$, Brieskorn, in [6] generalising Arnol’d in [1], had
already given a way of calculating the cohomology of the complement. If $D$ is
the union of hyperplanes $H_{j}$ with equations $h_{j}=0$, $j=1,{\ldots},N$,
then the collection of meromorphic 1-forms $\frac{dh_{j}}{h_{j}}$ generates a
$\mathbb{C}$-subalgebra $B^{\bullet}$ of the exterior algebra
$\Gamma(X,\Omega_{X}^{\bullet}(\star D))$. Since
$d\left(\frac{dh_{j}}{h_{j}}\right)=0$, all exterior derivatives on
$B^{\bullet}$ are zero, and $B^{\bullet}$ is a subcomplex of the complex
$\Gamma(X,\Omega_{X}^{\bullet}(\star D))$ with derivative zero, known as the
Brieskorn complex. Brieskorn showed

(1.3) Theorem. The de Rham morphism $B^{p}{\ \rightarrow\
}H^{p}(U;\mathbb{C})$ is an isomorphism.$\Box$

Note that each of Brieskorn’s forms

$\omega_{j_{1},{\ldots},j_{k}}:=\frac{dh_{j_{1}}}{h_{j_{1}}}\wedge\cdots\wedge\frac{dh_{j_{k}}}{h_{j_{k}}}$

has at most a first order pole along $D$, so in this special case his result
is stronger than Grothendieck’s.

An earlier version of Brieskorn’s result, that it holds for a normal crossing
divisor (NCD)([34]) played an important role in Deligne’s mixed Hodge theory:
every quasiprojective variety $U$ has a projective compactification
$U\hookrightarrow X$ in which $D:=X\smallsetminus U$ is a NCD. Similarly,
every singular variety has a resolution whose exceptional divisor is once
again an NCD. Deligne used the poles to define the weight filtration on the
cohomology of $U$.

If $D=\bigcup_{j=1}^{N}\\{h_{j}=0\\}$ is a union of more general irreducible
divisors $\\{h_{j}=0\\}$, the analogous Brieskorn complex $B^{\bullet}$
generated by the closed forms $\frac{dh_{j}}{h_{j}}$ does not in general
calculate the cohomology of the complement: for example, if $D$ is
irreducible, this complex reduces to $0{\ \rightarrow\ }B^{0}{\ \rightarrow\
}B^{1}{\ \rightarrow\ }0$, whereas $X\smallsetminus D$ may have cohomology in
higher dimensions. Deligne and Dimca explored the relation between pole order
and the Hodge filtration in [22].

###  1.3 The Logarithmic Comparison Theorem

Kyoji Saito, in [60], introduced a fruitful generalisation of the Brieskorn
complex. A meromorphic form $\omega$ has a logarithmic pole along the divisor
$D$ if both $h{\omega}$ and $hd{\omega}$ are regular, where $h$ is a local
equation for $D$. Note in particular that all of Brieskorn’s forms have
logarithmic poles along $D$.

Denote the sheaf of germs of meromorphic $k$-forms with logarithmic poles by
$\Omega^{k}_{X}(\log D)$. Then $\Omega_{X}^{\bullet}(\log D)$ is again a
subcomplex of $\Omega_{X}^{\bullet}(\star D)$. In the same article Saito also
described the dual of $\Omega_{X}^{1}(\log D)$, namely the sheaf of
“logarithmic derivations”, $\operatorname{\mathcal{D}\mathit{er}}(-\log D)$,
whose stalk at $x\in D$ consists of germs of vector fields on $X$ which are
tangent to $D$ at its smooth points. The duality arises from the contraction
pairing

$\Omega_{X}^{1}(\log D)\times\operatorname{\mathcal{D}\mathit{er}}(-\log D){\
\rightarrow\
}\mathcal{O}_{X},\quad({\omega},\xi)\mapsto\iota_{\xi}(\omega)={\omega}(\xi).$

Saito showed that this is a perfect pairing, so that $\Omega_{X}^{1}(\log D)$
is also the dual of $\operatorname{\mathcal{D}\mathit{er}}(-\log D)$. Saito’s
interest was especially in the case where $D$ is a free divisor, that is,
where $\operatorname{\mathcal{D}\mathit{er}}(-\log D)$ (or, equivalently,
$\Omega_{X}^{1}(\log D)$) is a locally free sheaf of
$\mathcal{O}_{X}$-modules. Because the dual of any $\mathcal{O}_{X}$-module
has depth at least 2 when $0pt\,\mathcal{O}_{X}\geq 2$ (see e.g. [62], or [39,
Lemma 9.2]), it follows from this that every plane curve is a free divisor.
More interesting is the fact, proved by K. Saito, that the discriminant in the
base of a versal deformation of an isolated hypersurface singularity is a free
divisor. Looijenga generalised this by showing in [39, Corollary 6.13] that it
holds also for the discriminant in the base of a versal deformation of an
isolated complete intersection singularity (ICIS), and later authors ([65],
[7]) have extended this to the discriminants of a range of non-ICIS
singularities, to discriminants in quiver representation spaces [8] and more
generally in prehomogeneous vector spaces [29].

Since for any divisor $D\subset X$ one has a natural morphism

$\Omega_{X}^{\bullet}(\log D)\hookrightarrow\Omega_{X}^{\bullet}(\star D)$

one can ask for the class of divisors $D\subset X$ such that the previous
natural morphism is a quasi-isomorphism (or an isomorphism in the derived
category of sheaves of $\mathbb{C}_{X}$–vector spaces).

(1.4) Definition. If the morphism $\Omega_{X}^{\bullet}(\log
D)\hookrightarrow\Omega_{X}^{\bullet}(\star D)$ is a quasi-isomorphism we say
that the divisor $D$ satisfies the logarithmic comparison theorem (LCT), or
even that the LCT holds for $D$.

By Grothendieck’s comparison theorem, the LCT holds for a divisor $D$ if and
only if the morphism $h^{q}(\Omega_{X}^{\bullet}(\log
D))\rightarrow\mathbb{R}^{q}j_{*}\mathbb{C}_{U}$ is an isomorphism for $0\leq
q\leq\dim X$. Any NCD satisfies the LCT, after [21, Cap. II, Lemme 6.9]. The
result of Atiyah and Hodge cited above shows that every cohomology class on
the complement of the divisor can be represented by a logarithmic form, but
not that the logarithmic complex calculates this cohomology.

(1.5) Theorem. [Logarithmic Comparison Theorem, [15]] If $D\subset X$ is a
locally quasihomogeneous free divisor with complement $U$, then the de Rham
morphism

${\Omega}^{\bullet}_{X}(\log D){\ \rightarrow\ }\mathbb{R}j_{*}\mathbb{C}_{U}$
(8)

is a quasi-isomorphism.

Here locally quasihomogeneous means that at each point $p\in D$ there are
local coordinates on $X$ in which $D$ has a weighted homogeneous local
equation with all weights strictly positive – in other words, locally there is
a good $\mathbb{C}^{*}$-action centred at $p$. Divisors with this property are
also known as strongly quasihomogeneous, and as positive. Evidently hyperplane
arrangements have this property, and it also holds for the discriminants of
stable maps in Mather’s “nice dimensions”, because of the remarkable (and
unexplained) fact that in the nice dimensions, all stable germs are
quasihomogeneous in suitable coordinates (see [52, Section 7.4] for a
pedestrian proof of this). Local quasihomogeneity is used twice in the proof
of Theorem 1.3. First, it allows an inductive argument on the codimension of
strata.

(1.6) Lemma. Let $X$ be a complex manifold of dimension $n$, let $D$ be a
strongly quasihomogeneous divisor in $X$, and let $p\in D$. Then there is an
open neighbourhood $U$ of $p$ such that for each $q\in U\cap D$, with $q\neq
p$, the germ at $q$ of the pair $(X,D)$ is isomorphic to the germ at $0$ of a
product $(\mathbb{C}^{n-1}\times\mathbb{C},D_{0}\times\mathbb{C})$ where
$D_{0}$ is a strongly quasihomogeneous divisor.

To prove that (8) is an isomorphism at $p\in D$, one uses induction on the
dimension of the divisor. It is easy to see that if (8) holds for a a divisor
$D_{0}\subset\mathbb{C}^{n-1}$ then it holds for
$D_{0}\times\mathbb{C}\subset\mathbb{C}^{n-1}\times\mathbb{C}$. Lemma 1.3
therefore says that we may assume by induction that (8) holds at all points of
$D\cap U\smallsetminus\\{p\\}$. Note that the induction begins with the
divisor $0\subset\mathbb{C}$; it is well known that
$H^{1}(\mathbb{C}\smallsetminus\\{0\\};\mathbb{C})\simeq\mathbb{C}$, generated
by the logarithmic form $\frac{dz}{z}$. Thus in what follows we assume $n\geq
2$. The main body of the proof of 1.3 then consists in showing that (8) also
holds at $p$ (which for convenience we suppose is the point $(0,{\ldots},0)$),
by showing that for any sufficiently small polycylinder $V$ centred at $p$,
the de Rham morphism

$h^{p}(\Gamma(V,\Omega_{X}^{\bullet}(\log D)){\ \rightarrow\
}H^{p}(V\smallsetminus D;\mathbb{C})$

is an isomorphism. The argument involves two Čech-de Rham double complexes
associated with Stein covers $\\{V_{i}\\}$ of $V\smallsetminus\\{0\\}$, with
$V_{i}=V\smallsetminus\\{x_{i}=0\\}$, and
$\\{V_{i}^{\prime}=V_{i}\smallsetminus D\\}$ of $V\smallsetminus D$, and the
two standard spectral sequences associated to each one. The complexes are

$\displaystyle K^{p,q}$
$\displaystyle=\bigoplus_{i_{0}<i_{1}<\cdots<i_{q}}\Gamma\left(\bigcap_{j=0}^{q}V_{i_{j}},{\Omega}_{X}^{p}(\log
D)\right)$ (9) and $\displaystyle\tilde{K}^{p,q}$
$\displaystyle=\bigoplus_{i_{0}<i_{1}<\cdots<i_{q}}\Gamma\left(\bigcap_{j=0}^{q}V^{\prime}_{i_{j}},{\Omega}_{X}^{p}\right)$
(10)

with differentials $d$, the exterior derivative, and $\check{d}$, the Cech
differential. The inclusion $V_{i}^{\prime}\hookrightarrow V_{i}$ determines a
morphism $\rho_{0}:K^{p,q}{\ \rightarrow\ }\tilde{K}^{p,q}$ which commutes
with $d$ and $\check{d}$ and thus gives rise to morphisms of the associated
spectral sequences, which we denote by ${}^{\prime}\rho_{\ell}$ and
${}^{\prime\prime}\rho_{\ell}$, where the subindex $\ell$ refers to the page
of the spectral sequence. First spectral sequences Applying $d$ to (9) and
(10), we get the first page of the first spectral sequence associated to each
double complex:

${}^{\prime}E_{1}^{p,q}$
$\displaystyle=h^{p}(K^{\bullet,q})=\bigoplus_{i_{0}<i_{1}<\cdots<i_{q}}h^{p}\left(\Gamma\left(\bigcap_{j=0}^{q}V_{i_{j}},{\Omega}_{X}^{\bullet}(\log
D)\right)\right)$ and ${}^{\prime}\tilde{E}_{1}^{p,q}$
$\displaystyle=h^{p}(\tilde{K}^{\bullet,q})=\bigoplus_{i_{0}<i_{1}<\cdots<i_{q}}h^{p}\left(\Gamma\left(\bigcap_{j=0}^{q}V^{\prime}_{i_{j}},{\Omega}_{X}^{\bullet}\right)\right)=\bigoplus_{i_{0}<i_{1}<\cdots<i_{q}}H^{p}(\bigcap_{j=0}^{q}V^{\prime}_{i_{j}};\mathbb{C}).$

Since $0$ is excluded from all of the open sets in the covers, the induction
hypothesis implies that ${}^{\prime}\rho_{1}^{p,q}:{{}^{\prime}E}_{1}^{p,q}{\
\rightarrow\ }{{}^{\prime}\tilde{E}}_{1}^{p,q}$ is an isomorphism for all
$p,q$, and it follows that ${}^{\prime}\rho_{\infty}^{p,q}$ is also an
isomorphism. Thus, $\rho_{0}$ induces an isomorphism of the cohomology of the
total complexes of $K^{\bullet,\bullet}$ and $\tilde{K}^{\bullet,\bullet}$.
Second spectral sequences Because $\\{V_{i}\\}$ and $\\{V_{i}^{\prime}\\}$ are
Stein covers, applying $\check{d}$ gives, as first pages of the second
spectral sequences,

${}^{\prime\prime}E^{p,q}_{1}=H^{q}(V\smallsetminus\\{0\\},{\Omega}_{X}^{p}(\log
D))\quad\text{and}\quad^{\prime\prime}\tilde{E}^{p,q}_{1}=H^{q}(V\smallsetminus
D,{\Omega}_{X}^{p}).$

Because $D$ is a free divisor, all of the ${\Omega}_{X}^{p}(\log D)$, as well
as the ${\Omega}_{X}^{p}$, are free $\mathcal{O}_{X}$-modules, and thus since
$H^{q}(V\smallsetminus\\{0\\},\mathcal{O}_{X})$ is zero except for $q=0$ and
$q=n-1$ (see e.g. [39, (8.14)]),
$H^{q}(V\smallsetminus\\{0\\},{\Omega}_{X}^{p}(\log D))$ also vanishes except
in these dimensions. In other words

${}^{\prime\prime}\tilde{E}_{1}^{p,q}=0\text{\ \ except for }q=0\text{ and
}q=n-1.$

Now because $V\smallsetminus D$ is a Stein space and ${\Omega}_{X}^{p}$ is
coherent, ${}^{\prime\prime}\tilde{E}^{p,q}_{1}=H^{q}(V\smallsetminus
D,{\Omega}_{X}^{p})$ is equal to $0$ for all $q>0$, and the spectral sequence
${}^{\prime\prime}\tilde{E}$ collapses at $E_{2}$, with

$^{\prime\prime}\tilde{E}_{\infty}^{p,0}=H^{p}(V\smallsetminus D;\mathbb{C}),\
\ ^{\prime\prime}\tilde{E}^{p,q}_{\infty}=0\ \text{ if }q>0.$ (11)

We claim that ${}^{\prime\prime}E$ also collapses at $E_{2}$. In view of the
vanishing already remarked upon, it is enough to show that the complex
$(^{\prime\prime}E_{1}^{\bullet,q},d)$, i.e.

$0{\ \rightarrow\ }H^{n-1}(V\smallsetminus\\{0\\},{\Omega}_{X}^{0}(\log D)){\
\rightarrow\ }H^{n-1}(V\smallsetminus\\{0\\},{\Omega}_{X}^{1}(\log D)){\
\rightarrow\ }\cdots{\ \rightarrow\
}H^{n-1}(V\smallsetminus\\{0\\},{\Omega}_{X}^{t}(\log D)){\ \rightarrow\ }0$
(12)

is exact. This is the only point at which the argument departs from “general
nonsense”.

Each of the groups in (12) is generated by classes of the form
$c_{\alpha}{\omega}_{\alpha}$ where $c_{\alpha}\in
H^{n-1}(V\smallsetminus\\{0\\},\mathcal{O}_{X})$ and ${\omega}_{\alpha}$ is a
basis element of ${\Omega}_{X}^{p}(\log D)$ in $V$.

Recall that $(D,0)$ is assumed weighted homogeneous in $V$, with respect to
positive weights $w_{1},{\ldots},w_{t}$ for the coordinate functions. We have

$H^{n-1}(V\smallsetminus\\{0\\},\mathcal{O}_{X})=\frac{\Gamma(V\smallsetminus\bigcup_{i}\\{x_{i}=0\\},\mathcal{O}_{X})}{\sum_{j}\Gamma(V\smallsetminus\bigcup_{i\neq
j}\\{x_{i}=0\\},\mathcal{O}_{X})}.$

Each term in the numerator can be represented by a Laurent series in which all
exponents are negative; series in the denominator have all exponents negative,
bar one. An easy lemma shows that ${\Omega}_{X}^{p}(\log D)_{0}$ has a basis
consisting of forms of weighted degree strictly less than $\sum_{j}w_{j}$. It
follows that the complex (12) has zero weight-zero part. But since the
exterior derivative $d$ preserves weighted degrees, it follows that using the
Lie derivative with respect to the Euler form, one can construct a contracting
homotopy from the complex to its weight zero part. Thus, it is exact.

We are left with

$^{\prime\prime}E_{\infty}^{p,0}=h^{p}(\Gamma(V\smallsetminus\\{0\\},\Omega_{X}^{\bullet}(\log
D))=h^{p}(\Gamma(V,\Omega_{X}^{\bullet}(\log D)),\ \
^{\prime\prime}E_{\infty}^{p,q}=0\ \text{ if }q>0$ (13)

(since $tn>1$ and $\Omega^{p}(\log D)$ is locally free, $0$ is a removable
singularity). The theorem now follows from (11), (13) and the isomorphism of
the cohomology of the total complexes of $K^{\bullet,\bullet}$ and
$\tilde{K}^{\bullet,\bullet}$.

The importance of local quasihomogeneity for logarithmic comparison theorem
remains unclear. It was shown to be necessary for plane curves in [14], and
for free surfaces in 3-space in [30], but in higher dimensions, local weak
quasihomogeneity is sufficient, see [19], [55, Remark 1.7.4] and Theorem 3.3.

(1.7) Definition. A divisor $D$ is locally weakly quasihomogeneous [19, Def.
2.1] if at each point $p\in D$ there are local coordinates on $X$ in which $D$
has a weak weighted homogeneous local reduced equation $f=0$, that is: all the
weights are non negative and not all of them are 0, with $f$ of strictly
positive weight.

###  1.4 Conjecture of Terao

H. Terao conjectured in [63, Conjecture 3.1] that LCT always holds for
hyperplane arrangements. After partial results of Wiens and Yuzvinsky in [68],
the conjecture was finally proved by Daniel Bath in [2]. Bath’s proof uses
induction on the codimension of the flats of the arrangement, and a spectral
sequence argument similar to the argument of [15] given above.

###  1.5 Linear free divisors and logarithmic comparison revisited

A free divisor $D\subset\mathbb{C}^{n}$ is linear if $\operatorname{\rm
Der}(-\log
D)=\Gamma(\mathbb{C}^{n},\operatorname{\mathcal{D}\mathit{er}}(-\log D))$ has
a basis of vector fields whose coefficients are linear forms. Normal crossing
divisors are of course the simplest examples, and are in fact the only linear
free divisors among hyperplane arrangements. In [8] R.-O. Buchweitz and the
second author showed that the discriminant in the representation space of a
Dynkin quiver, with a root of the underlying diagram as dimension vector, is a
linear free divisor. All irreducible linear free divisors are classified,
though not under that name, in [61], where the operation of castling derives
them from four basic examples.

If $D\subset\mathbb{C}^{n}$ is a linear free divisor then the set of weight
zero vector fields in $\operatorname{\rm Der}(-\log D)$ is a Lie algebra of
dimension $n$; it is isomorphic to the Lie algebra of the Lie subgroup of
$G_{D}$ consisting of linear automorphisms of the pair $(\mathbb{C}^{n},D)$.
For linear free divisors, [28] gives an argument for the logarithmic
comparison theorem which is quite different from the proof of [15]. When D is
a linear free divisor, then it turns out that $\mathbb{C}^{n}\smallsetminus D$
is a single orbit of the identity component $G^{0}_{D}$ of the group $G_{D}$,
so $H^{*}(\mathbb{C}^{n}\smallsetminus D;\mathbb{C})$ is the cohomology of
$G^{0}_{D}/S_{p}$, where $S_{p}$ is the isotropy subgroup of a point $p$.
Since $G^{0}_{D}$ is path connected, the action of $S_{p}$ on
$H^{*}(G^{0}_{D};\mathbb{C})$ is trivial, so
$H^{*}(G^{0}_{D}/S_{p};D)=H^{*}(G^{0}_{D};\mathbb{C})$. Second, in this case
the weight zero subcomplex of
${\Gamma}(\mathbb{C}^{n},{\Omega}_{\mathbb{C}^{n}}^{\bullet}(\log D))$
coincides with the complex of Lie algebra cohomology, with complex
coefficients, of the Lie algebra ${\mathfrak{g}}_{D}$ of $G^{0}_{D}$. By an
argument using weighted homogeneity,
${\Gamma}(\mathbb{C}^{n},{\Omega}_{\mathbb{C}^{n}}^{\bullet}(\log D))$ is
chain homotopic to its weight zero subcomplex. Thus,
${\Gamma}(\mathbb{C}^{n},{\Omega}_{\mathbb{C}^{n}}^{\bullet}(\log D))$
coincides with the Lie algebra cohomology of ${\mathfrak{g}}_{D}$ with complex
coefficients. For compact connected Lie groups $G$, a well-known argument
shows that the Lie algebra cohomology coincides with the topological
cohomology of the group. For linear free divisors the group $G^{0}_{D}$ is
never compact, but the isomorphism also holds good for the larger class of
reductive groups, and for a significant class of linear free divisors,
including all those mentioned above, $G^{0}_{D}$ is indeed reductive.

###  1.6 A pairing $\mathbf{H^{p}(X\smallsetminus D)\times H^{q}(D){\
\rightarrow\ }H^{p+q}(D)}$

The paper [51] introduces a variant $\check{\Omega}^{k}_{D}$ of the module of
Kähler forms ${\Omega}^{k}_{D}$, defined by

$\check{\Omega}^{k}_{D}:=\frac{\Omega^{k}_{X}}{h\Omega_{X}^{k}(\log D)},$

where $h$ is a reduced equation for $D$. Note that
$\check{\Omega}^{0}_{D}={\Omega}^{0}_{D}=\mathcal{O}_{D}$. Since
$\frac{dh}{h}$ has a logarithmic pole and ${\Omega}_{X}^{k}(\log
D)\supset{\Omega}^{k}_{X}$, it follows that $h{\Omega}_{X}^{k}(\log
D)\supseteq dh\wedge{\Omega}^{k-1}_{X}+h{\Omega}^{k}_{X}$, so
$\check{\Omega}^{k}_{D}$ is a quotient of ${\Omega}^{k}_{D}$. If $D$ is a free
divisor then each $\check{\Omega}^{k}_{D}$ is a maximal Cohen-Macaulay
$\mathcal{O}_{D}$-module, and coincides with ${\Omega}^{k}_{D}$ at smooth
points of $D$. The $\check{\Omega}^{k}_{D}$ form a complex with respect to the
usual exterior derivative: if $\omega\in{\Omega}_{X}^{k}(\log D)_{x}$ then
$\frac{dh}{h}\wedge\omega\in{\Omega}_{X}^{k+1}(\log D)$, from which it follows
that $d(h\omega)\in h\Omega_{X}^{k+1}(\log D)$. We denote the exterior
derivative on this complex by $\check{d}$. If $D$ is quasihomogeneous then
$(\check{\Omega}^{\bullet}_{D},\check{d})$ is a resolution of $\mathbb{C}_{D}$
([51, Lemma 3.3]). Straightforward calculations show

  1. 1.

there is a well-defined pairing

${\Omega}_{X}^{k}(\log D)\times\check{\Omega}^{\ell}_{D}{\ \rightarrow\
}\check{\Omega}^{k+\ell}_{D},\ \ \text{defined by
}({\omega}_{k},{\omega}_{\ell})\mapsto{\omega}_{k}\wedge{\omega}_{\ell}.$ (14)

Note that there is no comparable wedge pairing

${\Omega}_{X}^{k}(\log D)\times{\Omega}^{\ell}_{D}{\ \rightarrow\
}{\Omega}^{k+\ell}_{D}.$

For such a pairing to be well defined, the wedge of
${\omega}_{k}\in{\Omega}_{X}^{k}(\log D)$ with
$\omega_{\ell}\in(dh\wedge{\Omega}^{\ell-1}_{X}+h{\Omega}^{\ell}_{X})$ would
have to lie in $dh\wedge{\Omega}^{k+\ell-1}_{X}+h{\Omega}^{k+\ell}_{X}$, and
in general this does not hold. For example, if $h=h_{1}h_{2}$, and we take
$\omega_{1}=\frac{dh_{1}}{h_{1}}\in\Omega_{X}^{1}(\log D)$ and
${\omega}_{1}^{\prime}=dh\in dh\wedge{\Omega}^{0}_{X}$, then

${\omega}_{1}\wedge{\omega}_{1}^{\prime}=\frac{dh_{1}}{h_{1}}\wedge\bigl{(}h_{1}dh_{2}+h_{2}dh_{1}\bigr{)}=dh_{1}\wedge
dh_{2}\notin\bigl{(}dh\wedge{\Omega}^{1}_{X}+h{\Omega}^{2}_{X}\bigr{)}.$

  2. 2.

The pairing (14) descends to a pairing on the homology of the two complexes,

$h^{k}(\Gamma(X,{\Omega}_{X}^{\bullet}(\log D))\times
h^{\ell}(\Gamma(X,\check{\Omega}^{\bullet}_{D})){\ \rightarrow\
}h^{k+\ell}(\Gamma(X,\check{\Omega}^{\bullet}_{D})).$

  3. 3.

When $D$ is locally quasihomogeneous then in view of the LCT and the exactness
of $\check{\Omega}^{\bullet}_{D}$, this gives a pairing

$H^{k}(X\smallsetminus D;\mathbb{C})\times H^{\ell}(D;\mathbb{C}){\
\rightarrow\ }H^{k+\ell}(D;\mathbb{C}).$

The properties of this pairing remain to be explored.

##  §2 $\mathcal{D}_{X}$–modules

In this section we recall some of the basics of $\mathcal{D}$–module theory.
We mainly follow [38], [47] and [27]. The section contains the necessary
terminology and results that are used in the subsequent sections.

###  2.1 Basic objects

Let $X$ be a complex manifold of dimension $n\geq 1$ and $\mathcal{O}_{X}$ (or
simply $\mathcal{O}$) be the sheaf of germs of holomorphic functions on $X$.
The sheaf of rings (or more precisely of $\mathbb{C}$–algebras)
$\mathcal{O}_{X}$ is coherent: this is Oka’s theorem [57], [42].

We denote by $\mathcal{D}_{X}$ (or simply by $\mathcal{D}$) the sheaf of
linear differential operators on $X$ with holomorphic coefficients [38, I,
§1], [27, Def. 3]. Sometimes we say operators (or differential operators) on
$X$ instead of linear differential operators on $X$ with holomorphic
coefficients. For any open set $U\subseteq X$, $\mathcal{D}(U)$ is a
$\mathcal{O}(U)$–algebra and so $\mathcal{D}$ is a sheaf of
$\mathcal{O}$–algebras.

If $(U;x_{1},\ldots,x_{n})$ is a chart in $X$ and $U\subset X$ is connected,
any operator $P\in\mathcal{D}(U)$ can be written in a unique way as a finite
sum

$P=\sum_{\alpha\in\mathbb{N}^{n}}c_{\alpha}\partial^{\alpha}=\sum_{\alpha\in\mathbb{N}^{n}}c_{\alpha}\partial_{1}^{\alpha_{1}}\cdots\partial_{n}^{\alpha^{n}}$

where $\alpha=(\alpha_{1},\ldots,\alpha_{n})$, $c_{\alpha}\in\mathcal{O}(U)$
and $\partial_{i}$ is the partial derivative $\frac{\partial}{\partial
x_{i}}$. For any $x\in X$, the stalk $\mathcal{D}_{X,x}$ is a non-commutative
$\mathcal{O}_{X,x}$–algebra, since $\partial_{i}x_{i}-x_{i}\partial_{i}=1$ for
$1\leq i\leq n$.

If $P\in\mathcal{D}(U)$ is non zero, the order of $P$ is the non negative
integer

${\mathrm{ord}}(P)=\max\\{|\alpha|\,;\,c_{\alpha}\not=0\\}$

where $|\alpha|=\alpha_{1}+\cdots+\alpha_{n}$. We also write
${\mathrm{ord}}(0)=-\infty$.

For $k\in\mathbb{N}$, we denote by $F^{k}(\mathcal{D})$ the subsheaf of
$\mathcal{D}$ whose sections can be written locally as operators of order less
than or equal to $k$. Each $F^{k}(\mathcal{D})$ is a sheaf of coherent
$\mathcal{O}$–modules and the family $F^{\bullet}:=(F^{k}(\mathcal{D}))_{k}$
is a discrete increasing exhaustive filtration of $\mathcal{D}$. One has
$F^{0}(\mathcal{D})=\mathcal{O}$ and
$F^{k}(\mathcal{D})F^{\ell}(\mathcal{D})=F^{k+\ell}(\mathcal{D})$ for
$k,\ell\geq 0$. The associated sheaf of graded rings (and more precisely, of
graded $\mathbb{C}$–algebras) is

$\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D}):=\bigoplus_{k\in\mathbb{N}}\frac{F^{k}(\mathcal{D})}{F^{k-1}(\mathcal{D})}$

where we write $F^{-1}(\mathcal{D})=\\{0\\}$.

If $P,Q$ are local sections in $\mathcal{D}$, the difference $[P,Q]:=PQ-QP$ is
called the commutator of $(P,Q)$. For $x\in X$, the stalk $\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D}_{x})\simeq\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D})_{x}$ is a commutative ring since
${\mathrm{ord}}([P,Q])\leq{\mathrm{ord}}(P)+{\mathrm{ord}}(Q)-1$. Thus,
$\operatorname{\rm gr}_{F^{\bullet}}(\mathcal{D})$ is a sheaf of commutative
$\mathbb{C}$–algebras.

The sheaves of rings $\mathcal{D}_{X}$ and $\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D}_{X})$ are coherent (see [38, I, Th. 3.2], [27,
Prop. 9]).

We denote

$\sigma_{k}:F^{k}(\mathcal{D})\rightarrow\frac{F^{k}(\mathcal{D})}{F^{k-1}(\mathcal{D})}$

the natural projection map (which is a morphism of $\mathcal{O}$–modules) and
we call it the $k^{th}$ symbol map.

If $P$ is an operator of order $k$ we simply write

$\sigma(P)=\sigma_{k}(P)$

and we call this element the principal symbol of $P$.

If $(x_{1},\ldots,x_{n})$ is a system of local coordinates around a point
$x\in X$, then $\partial_{i}=\frac{\partial}{\partial x_{i}}\in
F^{1}(\mathcal{D}_{x})$ and we write $\xi_{i}=\sigma(\partial_{i})$ for $1\leq
i\leq n$.

There is an isomorphism of $\mathcal{O}_{x}$–algebras

$\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D}_{x})\longrightarrow\mathcal{O}_{x}[\xi_{1},\ldots,\xi_{n}]$
(15)

see e.g. [38, I, §1], [27, Prop. 4]; the image of an element
$P+F^{k-1}(\mathcal{D}_{x})$, with $P\in F^{k}(\mathcal{D}_{x})$, being simply
$\sigma_{k}(P)$. Compare this morphism with (2.3). Since $\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D}_{x})$ is noetherian, a standard argument using
induction on the order of the operators, proves that $\mathcal{D}_{x}$ is a
left and right noetherian ring.

###  2.2 Filtrations

Let $\mathcal{M}$ be a left $\mathcal{D}$–module (i.e. a sheaf of left
$\mathcal{D}$–modules). A filtration of $\mathcal{M}$ is a collection,
$(\mathcal{M}_{k})_{k\in\mathbb{N}}$, of $\mathcal{O}$-submodules of
$\mathcal{M}$ such that:

  1. (i)

For any $k,\ell\in\mathbb{N}$ : $\mathcal{M}_{k}\subset\mathcal{M}_{k+1}$ and
$F^{\ell}(\mathcal{D})\mathcal{M}_{k}\subset\mathcal{M}_{k+\ell}$ .

  2. (ii)

$\mathcal{M}=\cup_{k\in\mathbb{N}}\mathcal{M}_{k}$.

To any filtration $\Gamma:=(\mathcal{M}_{k})_{k\in\mathbb{N}}$ of
$\mathcal{M}$ we associate its graded $\mathcal{O}$–module

$\operatorname{\rm
gr}_{\Gamma}(\mathcal{M}):=\bigoplus_{k\in\mathbb{N}}\frac{\mathcal{M}_{k}}{\mathcal{M}_{k-1}}$

with $\mathcal{M}_{-1}:=\\{0\\}$. More generally, $\operatorname{\rm
gr}_{\Gamma}(\mathcal{M})$ is a $\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D})$–module. A filtration $(\mathcal{M}_{k})_{k}$
of $\mathcal{M}$ is said to be good is the following two conditions hold

  1. 1.

For any $k\in\mathbb{N}$, $\mathcal{M}_{k}$ is a coherent
$\mathcal{O}_{X}$–module.

  2. 2.

There exist $k_{0}\in\mathbb{N}$ such that for any $\ell\in\mathbb{N}$,
$F^{\ell}(\mathcal{D})\mathcal{M}_{k_{0}}=\mathcal{M}_{k_{0}+\ell}$.

Any coherent $\mathcal{D}$–module $\mathcal{M}$ admits locally (i.e. after
restriction to sufficiently small open subsets in $X$) good filtrations (see
e.g. [38, I, p. 7], [27, Prop. 10]).

Let $\Gamma:=(\mathcal{M}_{k})_{k})$ be a filtration of $\mathcal{M}$. Then
$\Gamma$ is locally good (i.e. after restriction to sufficiently small open
subsets in $X$, $\Gamma$ is a good filtration) if and only if
$\operatorname{\rm gr}_{\Gamma}(\mathcal{M})$ is a coherent $\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D})$–module (see [38, I, Prop. 4.1], [27, Th. 1]).

Let $\mathcal{M}$ be a coherent left $\mathcal{D}_{X}$–module. Then there
exists a coherent sheaf of ideals $\mathcal{J}(\mathcal{M})$ of
$\operatorname{\rm gr}_{F^{\bullet}}(\mathcal{D}_{X})$ with the following
property: for any open subset $U\subset X$ such that the restriction
$\mathcal{M}_{U}$ admits a good filtration, one has
$\mathcal{J}(\mathcal{M})_{U}\simeq\sqrt{\operatorname{\rm
ann}_{\operatorname{\rm gr}(\mathcal{D}_{U})}(\operatorname{\rm
gr}(\mathcal{M}_{U}))}$, where $\operatorname{\rm
gr}(\mathcal{D}_{U})=\operatorname{\rm gr}_{F^{\bullet}}(\mathcal{D}_{U})$ and
$\operatorname{\rm ann}_{\operatorname{\rm
gr}(\mathcal{D}_{U})}(\operatorname{\rm gr}(\mathcal{M}_{U}))$ is the sheaf of
annihilating ideals of the coherent $\operatorname{\rm
gr}(\mathcal{D}_{U})$–module $\operatorname{\rm gr}(\mathcal{M}_{U})$ (see
[27, Prop.17; Rmk. 6]).

(2.1) Definition. Let $\mathcal{M}$ be a coherent $\mathcal{D}_{X}$–module.
The closed analytic subset of the cotangent bundle $T^{*}X$ defined by the
coherent sheaf of ideals $\mathcal{J}(\mathcal{M})$ is called the
characteristic variety of $\mathcal{M}$ and is denoted by $\text{\rm
Char}(\mathcal{M})$.

(2.2) Example. 1) The characteristic variety of the $\mathcal{D}_{X}$–module
$\\{0\\}$ is $\emptyset$, and $\text{\rm Char}(\mathcal{D}_{X})=T^{*}X$ and
then $\dim\text{\rm Char}(\mathcal{D}_{X})=2n=2\dim X$.
2) If $\mathcal{M}=\frac{\mathcal{D}_{X}}{\mathcal{I}}$ where
$\mathcal{I}\subset\mathcal{D}_{X}$ is a coherent sheaf of left ideals in
$\mathcal{D}_{X}$, the characteristic variety $\text{\rm Char}(\mathcal{M})$
is the closed analytic subset of $T^{*}X$ defined by the coherent sheaf of
ideals $\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{I})\subset\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D}_{X})$, since the filtration on the quotient
$\mathcal{M}$, induced by the $F^{\bullet}$ filtration on $\mathcal{D}_{X}$,
is a locally good filtration.
3) The $\mathcal{D}_{X}$–module $\mathcal{O}_{X}$ can be presented locally as
a quotient
$\mathcal{D}_{X,x}/\mathcal{D}_{X,x}(\partial_{1},\cdots,\partial_{n})$ where
$\\{\partial_{1},\ldots,\partial_{n}\\}$ is a basis of
$\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X,x})$. The graded ideal
$\operatorname{\rm gr}(\mathcal{D}_{X,x}(\partial_{1},\cdots,\partial_{n}))$
equals $\operatorname{\rm gr}(\mathcal{D})(\xi_{1},\ldots,\xi_{n})$ (recall
that $\xi_{i}$ is the principal symbol of $\partial_{i}$ (2.1)). Then
$\text{\rm Char}(\mathcal{O}_{X})=T^{*}_{X}X$, that is, the zero section of
the cotangent bundle $T^{*}X$, and $\dim\text{\rm Char}(\mathcal{O}_{X})=n$.

The characteristic variety $\text{\rm Char}(\mathcal{M})\subset T^{*}X$ of a
coherent $\mathcal{D}_{X}$–module is involutive (e.g. [43], [25], [27, App.
B]). Hence, if $\mathcal{M}$ is a non zero module, one has
$2n\geq\dim(\text{\rm Char}(\mathcal{M}))\geq n$. This is called Bernstein’s
inequality. It is proved in [3] for modules over the Weyl algebra.

(2.3) Definition. A coherent $\mathcal{D}_{X}$–module is said to be holonomic
if either $\mathcal{M}=\\{0\\}$ or $\dim(\text{\rm
Char}(\mathcal{M}))=\dim\,X$.

The $\mathcal{D}_{X}$–module $\mathcal{O}_{X}$ is holonomic while
$\mathcal{D}_{X}$ is not holonomic as $\mathcal{D}_{X}$–module. If
$\mathcal{M}=\frac{\mathcal{D}_{X}}{\mathcal{I}}$ where $\mathcal{I}$ is a
sheaf of non zero locally principal ideals (i.e. locally generated by a single
differential operator) then $\mathcal{M}$ is holonomic if and only if
$\dim\,X=1$, since $\dim\text{\rm
Char}(\frac{\mathcal{D}_{X}}{\mathcal{I}})=2n-1$. A central result in
$\mathcal{D}_{X}$–module theory is

(2.4) Theorem. [37] If $D$ is a hypersurface in a complex manifold $X$ then
the $\mathcal{D}_{X}$–module $\mathcal{O}_{X}(\star D)$ of meromorphic
functions in $X$ with poles on $D$ is holonomic.

We finish this subsection with a result that we use later on (see Sect. 2.5):

(2.5) Theorem. A non zero coherent $\mathcal{D}_{X}$–module $\mathcal{M}$ is
holonomic if and only if
$\operatorname{\mathcal{E}\\!\mathit{xt}}^{i}_{\mathcal{D}_{X}}(\mathcal{M},\mathcal{D}_{X})=0$
for $i\neq n$. In this case,
$\operatorname{\mathcal{E}\\!\mathit{xt}}^{n}_{\mathcal{D}_{X}}(\mathcal{M},\mathcal{D}_{X})$
is a coherent holonomic right $\mathcal{D}_{X}$–module.

The proof follows from [38, IV; Ths. 4.2.5, 4.2.6], see also [27, Ths. 7, 8].

###  2.3 The natural isomorphism $\operatorname{{\rm
Sym}}_{\mathcal{O}_{X}}(\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D}_{X})$

There is a natural injective morphism of $\mathcal{O}_{X}$–modules

$\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})\stackrel{{\scriptstyle\iota}}{{\rightarrow}}\operatorname{\rm
gr}^{(1)}_{F^{\bullet}}(\mathcal{D})\subset\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D})$

mapping a local section
$\delta\in\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})$ to
$\sigma(\delta)\in\operatorname{\rm gr}^{(1)}_{F^{\bullet}}(\mathcal{D})$.
Then, by the universal property of the symmetric algebra (see [5, §6, Prop.
2]) there exists a unique morphism of $\mathcal{O}_{X}$–algebras

$\kappa:\operatorname{{\rm
Sym}}_{\mathcal{O}_{X}}(\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X}))\rightarrow\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D}_{X}),$ (16)

extending the morphism
$\iota:\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})\rightarrow\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D}).$ Since $\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D}_{X})$ is a graded $\mathcal{O}_{X}$–algebra and
$\iota(\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X}))\subset\operatorname{\rm
gr}^{(1)}_{F^{\bullet}}(\mathcal{D})$, by [Rmk. 1, page 498, loc.cit.] the
morphism $\kappa$ is graded so that we have, for any $k\geq 0$ a morphism of
$\mathcal{O}_{X}$–modules

$\kappa_{k}:\operatorname{{\rm
Sym}}^{(k)}_{\mathcal{O}_{X}}(\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X}))\rightarrow\operatorname{\rm
gr}^{(k)}_{F^{\bullet}}(\mathcal{D}_{X}).$

By [5, §6, Th. 1] and since
$\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})$ is a locally free
$\mathcal{O}_{X}$–module of rank $n$, for any $x\in X$, $\operatorname{{\rm
Sym}}_{\mathcal{O}_{X,x}}(\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X,x}))$
is canonically isomorphic to the polynomial algebra
$\mathcal{O}_{X,x}[T_{1},\ldots,T_{n}]$, the canonical isomorphism being
obtained by mapping $\partial_{i}$ to $T_{i}$ for $1\leq i\leq n$. Locally,
for $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ and
$|\alpha|=\sum_{i}\alpha_{i}=k$, one has
$\kappa_{k}(\partial_{1}^{\alpha_{1}}\cdots\partial_{n}^{\alpha_{n}})=\sigma(\partial_{1}^{\alpha_{1}}\cdots\partial_{n}^{\alpha_{n}})$
where $\sigma(\,\,)$ means the principal symbol (see (2.1)). Then $\kappa_{k}$
is an isomorphism for any $k$ and $\kappa$ is a graded isomorphism. The
isomorphism $\kappa$ is the intrinsic version of the isomorphism (15).

###  2.4 The Spencer complex

Let $X$ be a complex manifold of dimension $n\geq 1$.

(2.6) Definition. [47, Chap I, (2.1)] The Spencer complex in $X$ is the
following complex of left $\mathcal{D}_{X}$–modules, denoted
${\mathrm{Sp}}^{\bullet}_{X}$ (or simply ${\mathrm{Sp}}^{\bullet}$):

$0\rightarrow\mathcal{D}\otimes_{\mathcal{O}_{X}}\bigwedge^{n}\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})\stackrel{{\scriptstyle\epsilon_{-n}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\epsilon_{-2}}}{{\longrightarrow}}\mathcal{D}\otimes_{\mathcal{O}_{X}}\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})\stackrel{{\scriptstyle\epsilon_{-1}}}{{\longrightarrow}}\mathcal{D}$

where the differential $\epsilon_{\bullet}$ is defined by

$\begin{split}\epsilon_{-1}(P\otimes\delta)&=P\delta,\\\
\epsilon_{-p}(P\otimes(\delta_{1}\wedge\cdots\wedge\delta_{p}))&=\sum_{i=1}^{p}(-1)^{i-1}P\delta_{i}\otimes(\delta_{1}\wedge\cdots\widehat{\delta_{i}}\cdots\wedge\delta_{p}))\,+\\\
&\sum_{1\leq i<j\leq
p}(-1)^{i+j}P\otimes([\delta_{i},\delta_{j}]\wedge\delta_{1}\wedge\cdots\widehat{\delta_{i}}\cdots\widehat{\delta_{j}}\cdots\wedge\delta_{p})\,\,\,(2\leq
p\leq n).\end{split}$

We denote by $\widetilde{{\mathrm{Sp}}^{\bullet}}_{X}$ (or simply
$\widetilde{{\mathrm{Sp}}^{\bullet}}$) the augmented complex

${\mathrm{Sp}}_{X}^{\bullet}\stackrel{{\scriptstyle\epsilon_{0}}}{{\longrightarrow}}\mathcal{O}_{X}\rightarrow
0$

where $\epsilon_{0}(P)=P(1)$.

(2.7) Remark. The complex ${\mathrm{Sp}}^{\bullet}_{X}$ is a locally free
resolution of the left $\mathcal{D}_{X}$–module $\mathcal{O}_{X}$. Since
$\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})$ is a locally free
$\mathcal{O}_{X}$–module, each left $\mathcal{D}_{X}$–module

${\mathrm{Sp}}_{X}^{p}:=\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})$

is locally free (or rank $\binom{n}{p}$).

To prove that ${\mathrm{Sp}}^{\bullet}_{X}$ is a resolution of
$\mathcal{O}_{X}$, or equivalently that
$\widetilde{{\mathrm{Sp}}^{\bullet}}_{X}$ is acyclic, we can proceed locally
at each point $x\in X$. If we choose local coordinates $(x_{1},\ldots,x_{n})$
on $X$ around $x$, the partial derivatives
$(\partial_{1},\ldots,\partial_{n})$ – where
$\partial_{i}=\frac{\partial}{\partial x_{i}}$ – form a basis of the free
$\mathcal{O}_{X}$–module
$\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})_{x}$. The differential
$\epsilon_{-p,x}$ can be written as

$\epsilon_{-p,x}(P\otimes(\partial_{i_{1}}\wedge\cdots\wedge\partial_{i_{p}}))=\sum_{j=1}^{p}(-1)^{j-1}P\partial_{i_{j}}\otimes(\partial_{i_{1}}\wedge\cdots\widehat{\partial_{i_{j}}}\cdots\wedge\partial_{i_{p}}).$

So, the complex $\widetilde{{\mathrm{Sp}}^{\bullet}}_{X,x}$ is a Koszul
complex (which is denoted by
$K(\partial_{1},\ldots,\partial_{n};\mathcal{D}_{X})$ in [44, I.2]. This
complex is exact (see e.g. [24]). We give here a proof of the acyclicity of
$\widetilde{{\mathrm{Sp}}^{\bullet}}_{X,x}$ based on [9, Th. 3.1.2; Prop.
4.1.3] (which proves a more general result and follows a suggestion of B.
Malgrange [44, I.2]).

We consider a discrete increasing filtration
$G^{\bullet}:=G^{\bullet}\left(\widetilde{{\mathrm{Sp}}^{\bullet}}_{X}\right)$
on the complex $\widetilde{{\mathrm{Sp}}^{\bullet}}_{X}$ (or more precisely on
the complex $\widetilde{{\mathrm{Sp}}^{\bullet}}_{X,x}$). The discrete
filtration $G^{\bullet}$ is compatible with the differentials and the
associated graded complex is exact. This implies that the complex
$\widetilde{{\mathrm{Sp}}^{\bullet}}_{X}$ is exact since the filtration
$G^{\bullet}$ is discrete (i.e. $G^{k}=0$ if $k<0$).

The definition of $G^{\bullet}$ is as follows: for $0\leq p\leq n$ and
$k\in\mathbb{N}$, write

$G^{k,-p}:=G^{k}\left(\mathcal{D}\otimes_{\mathcal{O}_{X}}\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})\right)=F^{k-p}(\mathcal{D})\otimes_{\mathcal{O}_{X}}\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})$

where $F^{\bullet}(\mathcal{D})$ is the order filtration in $\mathcal{D}$. We
also write $G^{k,1}:=G^{k}(\mathcal{O}_{X})=\mathcal{O}_{X}$ for all
$k\in\mathbb{N}$.

For each $k,p$ one has $\epsilon_{-p}(G^{k,-p})\subset G^{k,-p+1}$ since for
any $P\in F^{k-p}(\mathcal{D})$ one has $P\partial_{i}\in
F^{k-p+1}(\mathcal{D})$. Thus, $G^{k,\bullet}$ is a complex of
$\mathcal{O}_{X}$–modules for any $k$ and $(G^{k,\bullet})_{k}$ is a discrete
increasing exhaustive filtration (simply denoted by $G^{\bullet}$) of the
complex $\widetilde{{\mathrm{Sp}}^{\bullet}}_{X}$.

For each $0\leq p\leq n$, the family $(G^{k,-p})_{k}$ is a discrete increasing
exhaustive filtration of the $\mathcal{D}$–module
$\mathcal{D}\otimes\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})$
whose associated graded $\mathcal{O}_{X}$–module

$\operatorname{\rm
gr}_{G^{\bullet,-p}}\left(\mathcal{D}\otimes\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})\right)$

is naturally isomorphic to

$\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D})[-p]\otimes\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X}).$

The last isomorphism follows from the fact that
$\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})$ is a
locally free $\mathcal{O}_{X}$–module and therefore $\mathcal{O}_{X}$–flat.

Then the associated graded complex $\operatorname{\rm
gr}_{G^{\bullet}}(\widetilde{{\mathrm{Sp}}^{\bullet}})$ is

$0\rightarrow\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D})[-n]\otimes\bigwedge^{n}\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X})\stackrel{{\scriptstyle\tau_{-n}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\tau_{-1}}}{{\longrightarrow}}\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D})\stackrel{{\scriptstyle\tau_{0}}}{{\rightarrow}}\mathcal{O}_{X}\rightarrow
0$

where the differential $\tau_{\bullet}=\operatorname{\rm
gr}(\epsilon_{\bullet})$ is acting by

$\begin{split}\tau_{0}(Q)=Q_{0},\\\
\tau_{-1}(Q\otimes\partial_{i})&=Q\sigma(\partial_{i}),\\\
\tau_{-p}(Q\otimes(\partial_{i_{1}}\wedge\cdots\wedge\partial_{i_{p}}))&=\sum_{j=1}^{p}(-1)^{j-1}Q\sigma(\partial_{i_{j}})\otimes(\partial_{i_{1}}\wedge\cdots\widehat{\partial_{i_{j}}}\cdots\wedge\partial_{i_{p}}))\,\,\,(2\leq
p\leq n),\end{split}$

where the tensor product is taken over $\mathcal{O}_{X}$, $Q_{0}$ is the
$0-th$ homogenous component of $Q\in\operatorname{\rm
gr}_{F^{\bullet}}(\mathcal{D})$ and $\sigma(\,)$ is the principal symbol of
the corresponding differential operator (see (2.1)). So, the last complex is
nothing but the augmented Koszul complex with respect to the regular sequence
$\\{\sigma(\partial_{1}),\ldots,\sigma(\partial_{n})\\}$ in the commutative
ring $\operatorname{\rm gr}_{F^{\bullet}}(\mathcal{D})$. Hence this complex is
acyclic.

###  2.5 The de Rham complex of a $\mathcal{D}_{X}$–module

We follow here [47, Chap. I (2.6)]. Let $\mathcal{M}$ be a left
$\mathcal{D}_{X}$–module. The left exact functor

$\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}(\mathcal{O}_{X},*):\operatorname{\rm
Mod}(\mathcal{D}_{X})\longrightarrow\operatorname{\rm Mod}(\mathbb{C}_{X})$

can be derived to give a functor

$\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}(\mathcal{O}_{X},*):D^{b}(\mathcal{D}_{X})\longrightarrow
D^{b}(\mathbb{C}_{X}).$

Here $\operatorname{\rm Mod}(A_{X})$ stands for the category of (left)
$A_{X}$–modules and $D^{b}(A_{X})$ for the derived category of complexes of
$A_{X}$–modules with bounded cohomology.

(2.8) Definition. Let $\mathcal{M}$ be a left $\mathcal{D}_{X}$–module. The
complex
$\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}(\mathcal{O}_{X},\mathcal{M})$
in $D^{b}(\mathbb{C}_{X})$ is called the de Rham complex of $\mathcal{M}$. It
is denoted by $\operatorname{DR}(\mathcal{M})$.

Let $\mathcal{M}$ be a left $\mathcal{D}_{X}$–module. Since $\mathcal{M}$
carries an integrable connection, there is a natural morphism of sheaves of
$\mathbb{C}_{X}$–vector spaces

$\nabla:\mathcal{M}\longrightarrow\Omega^{1}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{M}$

given locally by

$\nabla(m)=\sum_{i=1}^{n}dx_{i}\otimes\partial_{i}(m).$

(2.9) Proposition. [47, Lemme (2.6.3)] For any left $\mathcal{D}_{X}$–module
$\mathcal{M}$, the complex $\operatorname{DR}(\mathcal{M})$ can be represented
by the complex

$\Omega^{\bullet}_{X}(\mathcal{M}):=0\rightarrow\mathcal{M}\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}\Omega^{1}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{M}\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}\Omega^{n}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{M}\rightarrow
0$

concentrated in degrees $[0,n]$, where $\nabla(\omega\otimes
m)=\omega\wedge\nabla(m)+(-1)^{\deg(\omega)}d\omega\otimes m.$

Proof. By Remark 2.4 the complex $\widetilde{{\mathrm{Sp}}}^{\bullet}$ is a
locally free resolution of the left $\mathcal{D}_{X}$–module
$\mathcal{O}_{X}$. Thus $\operatorname{DR}(\mathcal{M})$ is represented by
$\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}({\mathrm{Sp}}^{\bullet},\mathcal{M})$.
Moreover, for $p=0,\ldots,n$, the $\mathcal{O}_{X}$–modules
$\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X}),\mathcal{M})$
and $\Omega^{p}_{X}\otimes\mathcal{M}$ are isomorphic. And there exists a
natural quasi-isomorphism, in $D^{b}(\mathbb{C}_{X})$, from
$\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\bigwedge^{\bullet}\operatorname{\mathcal{D}\mathit{er}}(\mathcal{O}_{X}),\mathcal{M})$
to $\Omega^{\bullet}_{X}\otimes\mathcal{M}$. Q.E.D.

Following [47, Ch. 1, (4.1)], if $\mathcal{M}$ is a complex in the derived
category $D^{b}_{c}(\mathcal{D}_{X})$ (complexes of left
$\mathcal{D}_{X}$–modules, with bounded and coherent cohomology), the complex

$\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}(\mathcal{M},\mathcal{D}_{X})$

of right $\mathcal{D}_{X}$–modules has bounded and coherent cohomology.

(2.10) Definition. Let $\mathcal{M}$ be a complex in
$D^{b}_{c}(\mathcal{D}_{X})$. The dual of $\mathcal{M}$ is the complex
$\mathcal{M}^{*}$ in $D^{b}(\mathcal{D}_{X})_{c}$ defined by

$\mathcal{M}^{*}=\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{O}_{X}}(\Omega^{n}_{X},\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}(\mathcal{M},\mathcal{D}_{X}))[n]$

where $n=\dim X$.

The dual $\mathcal{M}^{*}$ is also denoted $\mathds{D}(\mathcal{M})$. The dual
$\mathcal{M}^{*}$ of a holonomic $\mathcal{D}_{X}$–module $\mathcal{M}$ is
also holonomic (see [47, Ch. I, (4.1)] and Th. 2.2). The dual
$\mathcal{O}_{X}^{*}$ is naturally isomorphic to $\mathcal{O}_{X}$ and there
is a natural isomorphism in the derived category $D^{b}(\mathbb{C}_{X})$

$\operatorname{DR}(\mathcal{M})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}(\mathcal{M}^{*},\mathcal{O}_{X}).$

The last complex is called the (holomorphic) solution complex of
$\mathcal{M}^{*}$.

###  2.6 Grothendieck’s Comparison Theorem revisited

The statement of (a version of) Grothendieck’s Comparison Theorem is as
follows (see Sect. 1.1):

(2.11) Theorem. ([32, Th. 2]) If $D$ is a divisor in the complex manifold $X$,
and $j:U:=X\smallsetminus D\hookrightarrow X$ is the inclusion, then the de
Rham morphism

$\Omega^{\bullet}_{X}(\star D)\longrightarrow{\mathbb{R}}j_{*}\mathbb{C}_{U}$
(17)

is a quasi-isomorphism.

(2.12) Remark. If $D$ is a normal crossing divisor, the result is due to
Atiyah-Hodge [34]. The proof of Grothendieck uses Hironaka’s resolution of
singularities to reduce the general case to the normal crossing divisor one.

Theorem 2 in [32] is more general. It holds for a reduced complex analytic
space $X$, an analytic closed subset $D\subset X$, assuming
$U:=X\smallsetminus D$ is non singular and dense in $X$, and that $U$ can be
defined locally by one equation.

Mebkhout ([47, Chap. 2, §2])222Following Mebkhout’s Ph.D. Thesis interprets
Grothendieck’s comparison theorem as the regularity of the
$\mathcal{D}_{X}$–module $\mathcal{O}_{X}$. This regularity is equivalent to
the fact that, in the derived category $D^{b}(\mathbb{C}_{X})$, the natural
morphism

$\operatorname{DR}(\mathcal{O}_{X}(\star
D))\rightarrow\operatorname{DR}(\mathbb{R}j_{*}j^{-1}\mathcal{O}_{X})$

is a quasi-isomorphism for any divisor $D$. This last morphism is the de Rham
morphism (17) since $\operatorname{DR}(\mathcal{O}_{X}(\star D))$ equals
$\Omega_{X}^{\bullet}(\star D)$ and
$\operatorname{DR}(\mathbb{R}j_{*}j^{-1}\mathcal{O}_{X})$ is quasi-isomorphic
to
$\mathbb{R}j_{*}j^{-1}\operatorname{DR}(\mathcal{O}_{X})=\mathbb{R}j_{*}j^{-1}\Omega^{\bullet}_{X}\simeq\mathbb{R}j_{*}\mathbb{C}_{U}$,
by Poincaré Lemma.

##  §3 Free divisors and logarithmic $D$–modules

###  3.1 The sheaf of logarithmic differential operators with respect to a
free divisor

Let $X$ be a complex analytic manifold of dimension $n$ and $D\subset X$ a
hypersurface. Let us denote $U=X\smallsetminus D$ and $j:U\hookrightarrow X$
the corresponding open immersion.

When $D$ is a free divisor, there is a nice sheaf of subrings of the sheaf
$\mathcal{D}_{X}$ of linear differential operators in $X$, namely the sheaf of
logarithmic differential operators with respect to $D$, denoted by
$\mathcal{V}^{D}_{X}$. It is the sheaf of subrings of $\mathcal{D}_{X}$ with
stalks

$\mathcal{V}^{D}_{X,x}=\\{P\in\mathcal{D}_{X,x}\ |\
P(\mathcal{J}_{x}^{j})\subset\mathcal{J}_{x}^{j}\ \forall j\geq 0\\}.$

Moreover, $\mathcal{V}_{X}$ is a sheaf of filtered rings, with the induced
filtration by the order filtration in $\mathcal{D}_{X}$, whose graded ring is
commutative and the canonical map

$\operatorname{{\rm
Sym}}_{\mathcal{O}_{X}}\operatorname{\mathcal{D}\mathit{er}}(-\log
D)\longrightarrow\operatorname{\rm gr}\mathcal{V}^{D}_{X}$ (18)

is an isomorphism of commutative graded $\mathcal{O}_{X}$-algebras (see [9,
Cor. 2.1.6]; compare with Sect. 2.3). As a consequence, $\mathcal{V}^{D}_{X}$
is generated by $\mathcal{O}_{X}$ and
$\operatorname{\mathcal{D}\mathit{er}}(-\log D)$, and $\mathcal{V}^{D}_{X}$ is
the enveloping algebra of $\operatorname{\mathcal{D}\mathit{er}}(-\log D)$
considered as a Lie algebroid (cf. [20, §(2.1)]). From there we deduce that
$\mathcal{V}^{D}_{X}$ is a left and right coherent sheaf of rings (one can
proceed as in the case of $\mathcal{D}_{X}$, Sect. 2.1, or as in [4, Th.
1.2.5]) and its stalk at each point of $X$ is a left and right Noetherian ring
of finite global homological dimension $\leq 2n$ (cf. [4, App. IV, Prop. 4.14
and Th. 5.1]).

###  3.2 The logarithmic Spencer complex

From now on, we will assume that $D\subset X$ is a free divisor in a complex
manifold $X$ of dimension $n\geq 1$.

(3.1) Definition. [9, Def. 3.1.1] The logarithmic Spencer complex associated
with $D\subset X$ is the following complex of left
$\mathcal{V}_{X}^{D}$–modules, denoted ${\mathrm{Sp}}^{\bullet}(\log D)$:

$0\rightarrow\mathcal{V}_{X}^{D}\otimes_{\mathcal{O}_{X}}\bigwedge^{n}\operatorname{\mathcal{D}\mathit{er}}(-\log
D)\stackrel{{\scriptstyle\epsilon_{-n}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\epsilon_{-2}}}{{\longrightarrow}}\mathcal{V}_{X}^{D}\otimes_{\mathcal{O}_{X}}\operatorname{\mathcal{D}\mathit{er}}(-\log
D)\stackrel{{\scriptstyle\epsilon_{-1}}}{{\longrightarrow}}\mathcal{V}_{X}^{D}$

where the differential $\epsilon_{\bullet}$ is defined by

$\begin{split}\epsilon_{-1}(P\otimes\delta)&=P\delta,\\\
\epsilon_{-p}(P\otimes(\delta_{1}\wedge\cdots\wedge\delta_{p}))&=\sum_{i=1}^{p}(-1)^{i-1}P\delta_{i}\otimes(\delta_{1}\wedge\cdots\widehat{\delta_{i}}\cdots\wedge\delta_{p}))\,+\\\
&\sum_{1\leq i<j\leq
p}(-1)^{i+j}P\otimes([\delta_{i},\delta_{j}]\wedge\delta_{1}\wedge\cdots\widehat{\delta_{i}}\cdots\widehat{\delta_{j}}\cdots\wedge\delta_{p})\,\,\,(2\leq
p\leq n).\end{split}$

We denote by $\widetilde{{\mathrm{Sp}}}^{\bullet}(\log D)$ the augmented
complex

${\mathrm{Sp}}^{\bullet}(\log
D)\stackrel{{\scriptstyle\epsilon_{0}}}{{\longrightarrow}}\mathcal{O}_{X}$

where $\epsilon_{0}(P)=P(1)$.

For a free divisor $D$ the complex ${\mathrm{Sp}}^{\bullet}(\log D)$ is a
locally free resolution of the left $\mathcal{V}_{X}^{D}$-module
$\mathcal{O}_{X}$ [9, Th. 3.1.2] (this is a particular case of Proposition
3.2).

(3.2) Remark. The logarithmic Spencer complex should be compared with the
Spencer complex (see Definition 2.4).

The logarithmic Spencer complex generalizes the one given in [23, App. A
(A.4)] for a normal crossing divisor $D\subset X$. Actually, this definition
is a sheaf version of the Rinehart complex of a Lie-Rinehart algebra (see
[59]).

A logarithmic connection (with respect to $D$) on an $\mathcal{O}_{X}$-module
$\mathcal{E}$ is a $\mathbb{C}$-linear map

$\nabla:\mathcal{E}{\ \rightarrow\ }\Omega^{1}_{X}(\log
D)\otimes_{\mathcal{O}_{X}}\mathcal{E}$

satisfying the Leibniz rule, i.e. $\nabla(ae)=da\otimes e+a\nabla(e)$ for any
holomorphic function $a$ and any local section $e$ of $\mathcal{E}$.

As in the classical case (see Sect. 2.5), such a $\nabla$ may be extended to a
family of $\mathbb{C}$-linear maps

$\nabla^{p}:\Omega_{X}^{p}(\log D)\otimes_{\mathcal{O}_{X}}\mathcal{E}{\
\rightarrow\ }\Omega_{X}^{p+1}(\log D)\otimes_{\mathcal{O}_{X}}\mathcal{E},$

with $\nabla^{p}(\alpha\otimes e)=d\alpha\otimes
e+(-1)^{p}\alpha\wedge\nabla(e)$ for any logarithmic $p$-form $\alpha$ and any
local section $e$ of $\mathcal{E}$.

We say that the logarithmic connection $\nabla$ is integrable if
$\nabla^{p+1}\circ\nabla^{p}=0$ for all $p\geq 0$. In such a case, the
logarithmic de Rham complex of $(\mathcal{E},\nabla)$ is by definition the
complex (of sheaves of complex vector spaces)

$\Omega^{\bullet}_{X}(\log D)(\mathcal{E}):=\mathcal{E}{\ \rightarrow\
}\Omega^{1}_{X}(\log D)\otimes_{\mathcal{O}_{X}}\mathcal{E}{\ \rightarrow\
}\Omega^{2}_{X}(\log D)\otimes_{\mathcal{O}_{X}}\mathcal{E}{\ \rightarrow\
}\cdots{\ \rightarrow\ }\Omega^{n}_{X}(\log
D)\otimes_{\mathcal{O}_{X}}\mathcal{E},$

where $\mathcal{E}$ is placed in degree $0$.

Any logarithmic connection $\nabla$ on an $\mathcal{O}_{X}$-module
$\mathcal{E}$ gives rise to an action of logarithmic vector fields
$\operatorname{\mathcal{D}\mathit{er}}(-\log D)$ on $\mathcal{E}$

$(\delta,e)\longmapsto\nabla_{\delta}(e)=\langle\delta,\nabla(e)\rangle$

for any logarithmic vector field $\delta$ and any local section $e$ of
$\mathcal{E}$, where $\langle\delta,\nabla(e)\rangle$ is induced by the
contraction of logarithmic 1-forms by logarithmic vector fields.

Obviously, the exterior derivative $d:\mathcal{O}_{X}{\ \rightarrow\
}\Omega^{1}_{X}(\log D)$ is a logarithmic connection which is integrable, and
$\Omega^{\bullet}_{X}(\log D)(\mathcal{O}_{X})=\Omega^{\bullet}_{X}(\log D)$.

When $D$ is free, the following result holds, by essentially the same proof as
in the classical case.

(3.3) Proposition. Assume that $D$ is a free divisor and let $\nabla$ be a
logarithmic connection on an $\mathcal{O}_{X}$-module $\mathcal{E}$. The
following properties are equivalent:

  1. (i)

$\nabla$ is integrable.

  2. (ii)

$\nabla_{[\delta,\delta^{\prime}]}=[\nabla_{\delta},\nabla_{\delta^{\prime}}]$
for all logarithmic vector fields $\delta,\delta^{\prime}$.

Since the sheaf of logarithmic differential operators $\mathcal{V}_{X}^{D}$ is
the enveloping algebra of the Lie algebroid
$\operatorname{\mathcal{D}\mathit{er}}(-\log D)$ provided that $D$ is a free
divisor, we obtain the following corollary.

(3.4) Corollary. Assume that $D$ is a free divisor and let $\mathcal{E}$ be an
$\mathcal{O}_{X}$-module. The following data are equivalent:

  1. (a)

An integrable logarithmic connection $\nabla$ on $\mathcal{E}$.

  2. (b)

A structure of left $\mathcal{V}_{X}^{D}$-module on $\mathcal{E}$ extending
its $\mathcal{O}_{X}$-module structure.

Moreover, the action of a logarithmic derivation $\delta$ on $\mathcal{E}$ in
(b) is given by $\delta\cdot e=\nabla_{\delta}(e)$ for each local section $e$
of $\mathcal{E}$.

From now on, we assume that $D$ is a free divisor.

Any locally free $\mathcal{O}_{X}$-module of finite rank endowed with an
integrable logarithmic connection will be called an ILC, for short. Examples
of ILCs are the invertible $\mathcal{O}_{X}$-modules $\mathcal{O}_{X}(kD)$,
$k\in\mathbb{Z}$, which carry an obvious structure of left
$\mathcal{V}_{X}^{D}$-module.

We define the logarithmic Spencer complex of an arbitrary left
$\mathcal{V}_{X}^{D}$-module in the following way.

(3.5) Definition. [12] Let $\mathcal{E}$ be a left
$\mathcal{V}_{X}^{D}$-module. The logarithmic Spencer complex of $\mathcal{E}$
(with respect to $D\subset X$) is the following complex of left
$\mathcal{V}_{X}^{D}$–modules, denoted ${\mathrm{Sp}}^{\bullet}(\log
D)(\mathcal{E})$:

$0\rightarrow\mathcal{V}_{X}^{D}\otimes_{\mathcal{O}_{X}}\bigwedge^{n}\operatorname{\mathcal{D}\mathit{er}}(-\log
D)\otimes_{\mathcal{O}_{X}}\mathcal{E}\stackrel{{\scriptstyle\epsilon_{-n}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\epsilon_{-2}}}{{\longrightarrow}}\mathcal{V}_{X}^{D}\otimes_{\mathcal{O}_{X}}\operatorname{\mathcal{D}\mathit{er}}(-\log
D)\otimes_{\mathcal{O}_{X}}\mathcal{E}\stackrel{{\scriptstyle\epsilon_{-1}}}{{\longrightarrow}}\mathcal{V}_{X}^{D}\otimes_{\mathcal{O}_{X}}\mathcal{E}$

where the differential $\epsilon_{\bullet}$ is defined by

$\begin{split}\epsilon_{-1}(P\otimes\delta\otimes e)=P\delta\otimes
e-P\otimes\delta e,&\\\
\epsilon_{-p}(P\otimes(\delta_{1}\wedge\cdots\wedge\delta_{p}))\otimes e&=\\\
\sum_{i=1}^{p}(-1)^{i-1}P\delta_{i}\otimes(\delta_{1}\wedge\cdots\widehat{\delta_{i}}\cdots\wedge\delta_{p}))\otimes
e\,-\,\sum_{i=1}^{p}(-1)^{i-1}P\otimes(\delta_{1}\wedge\cdots\widehat{\delta_{i}}\cdots\wedge\delta_{p}))\otimes\delta_{i}e\,+&\\\
\sum_{1\leq i<j\leq
p}(-1)^{i+j}P\otimes([\delta_{i},\delta_{j}]\wedge\delta_{1}\wedge\cdots\widehat{\delta_{i}}\cdots\widehat{\delta_{j}}\cdots\wedge\delta_{p})\otimes
e\,\,\,(2\leq p\leq n).&\end{split}$

We denote by $\widetilde{{\mathrm{Sp}}}^{\bullet}(\log D)(\mathcal{E})$ the
augmented complex

${\mathrm{Sp}}^{\bullet}(\log
D)(\mathcal{E})\stackrel{{\scriptstyle\epsilon_{0}}}{{\longrightarrow}}\mathcal{E}$

where $\epsilon_{0}(P\otimes e)=Pe$.

Note that for $\mathcal{E}=\mathcal{O}_{X}$ we have
${\mathrm{Sp}}^{\bullet}(\log D)(\mathcal{O}_{X})={\mathrm{Sp}}^{\bullet}(\log
D)$.

(3.6) Proposition. For any ILC $\mathcal{E}$, the logarithmic Spencer complex
of $\mathcal{E}$ (with respect to $D\subset X$) ${\mathrm{Sp}}^{\bullet}(\log
D)(\mathcal{E})$ is a locally free resolution of $\mathcal{E}$.

The proof of this proposition is similar to the proof in Remark 2.4. Namely,
we consider the discrete increasing filtration
$G^{\bullet}:=G^{\bullet}\left(\widetilde{{\mathrm{Sp}}}^{\bullet}(\log
D)(\mathcal{E})\right)$ on the complex
$\widetilde{{\mathrm{Sp}}}^{\bullet}(\log D)(\mathcal{E})$ given, for $0\leq
p\leq n$ and $k\in\mathbb{N}$, by

$G^{k,-p}:=G^{k}\left(\mathcal{V}_{X}^{D}\otimes_{\mathcal{O}_{X}}\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(-\log
D)\otimes_{\mathcal{O}_{X}}\mathcal{E}\right)=F^{k-p}(\mathcal{V}_{X}^{D})\otimes_{\mathcal{O}_{X}}\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(-\log
D)\otimes_{\mathcal{O}_{X}}\mathcal{E}$

where $F^{\bullet}(\mathcal{V}_{X}^{D})$ is the filtration induced by the
order filtration in $\mathcal{D}$. We also write
$G^{k,1}:=G^{k}(\mathcal{E})=\mathcal{E}$ for all $k\in\mathbb{N}$. The
associated graded complex turns out to be (locally) the tensor product over
$\mathcal{O}_{X}$ of $\mathcal{E}$ and the augmented Koszul complex with
respect to
$\\{\sigma(\delta_{1}),\dots,\sigma(\delta_{n})\\}\subset\operatorname{\rm
gr}\mathcal{V}_{X}^{D}$, $\\{\delta_{1},\dots,\delta_{n}\\}$ being a local
$\mathcal{O}_{X}$-basis of $\operatorname{\mathcal{D}\mathit{er}}(-\log D)$,
which is exact since $\mathcal{E}$ is locally free over $\mathcal{O}_{X}$ and
$\operatorname{\rm gr}\mathcal{V}_{X}^{D}$ is (locally) a polynomial ring over
$\mathcal{O}_{X}$ in the variables
$\\{\sigma(\delta_{1}),\dots,\sigma(\delta_{n})\\}$ (see the isomorphism
(18)).

By using the logarithmic Spencer resolution 3.2, as in the case of
$\mathcal{D}_{X}$ (see Sect. 2.4), we obtain a canonical isomorphism of
complexes of sheaves of complex vector spaces

$\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{V}_{X}^{D}}(\operatorname{\rm
Sp}^{\bullet}(\log D),\mathcal{E})\simeq\Omega^{\bullet}_{X}(\log
D)(\mathcal{E}),$

for any left $\mathcal{V}_{X}^{D}$-module $\mathcal{E}$, and so an isomorphism
in the derived category
$\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{V}_{X}^{D}}(\mathcal{O}_{X},\mathcal{E})\simeq\Omega^{\bullet}_{X}(\log
D)(\mathcal{E})$. This is completely similar to Proposition 2.5.

###  3.3 A $\mathcal{D}$–module criterion for LCT

Grothendieck’s comparison theorem (see (1.1) and (2.6)) tells us that the
natural map $\Omega^{\bullet}_{X}(\star D){\ \rightarrow\
}\mathbb{R}j_{*}\mathbb{C}_{X\smallsetminus D}$ obtained by composition of the
adjunction map $\Omega^{\bullet}_{X}(\star D){\ \rightarrow\
}\mathbb{R}j_{*}j^{-1}\Omega^{\bullet}_{X}(\star
D)=\mathbb{R}j_{*}\Omega^{\bullet}_{X\smallsetminus
D}=j_{*}\Omega^{\bullet}_{X\smallsetminus D}$ with the inverse of the induced
map by the Poincaré quasi-isomorphism $\mathbb{C}_{X\smallsetminus D}{\
\rightarrow\ }\Omega^{\bullet}_{X\smallsetminus D}$ is an isomorphism in the
derived category of sheaves of complex vector spaces. Definition 1.3 tells us
that LCT holds for $D$ if the natural morphism

$\Omega^{\bullet}_{X}(\log D)\hookrightarrow\Omega^{\bullet}_{X}(\star D)$
(19)

is a quasi-isomorhism. Let us explain how this map can be interpreted in terms
of $\mathcal{D}$–module theory in the case of free divisors.

On one hand, we have canonical isomorphisms (in the derived category)

$\Omega^{\bullet}_{X}(\log
D)\simeq\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{V}_{X}^{D}}(\mathcal{O}_{X},\mathcal{O}_{X})\simeq\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}\left(\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X},\mathcal{O}_{X}\right).$

By taking $\mathcal{D}_{X}$-duals $\mathds{D}(-)$ (see Def. 2.5) we obtain

$\displaystyle\displaystyle\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}\left(\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X},\mathcal{O}_{X}\right)\simeq\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}\left(\mathds{D}\,\mathcal{O}_{X},\mathds{D}\left(\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}\right)\right)\simeq$
$\displaystyle\displaystyle\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}\left(\mathcal{O}_{X},\mathds{D}\left(\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}\right)\right)=\operatorname{DR}\left(\mathds{D}\left(\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}\right)\right),$

and by using the duality formula in [12, Cor. 3.1.2] (see also [56, Th.
(A.32)]), we obtain

$\mathds{D}\left(\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}\right)\simeq\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}(D),$

and so

$\Omega^{\bullet}_{X}(\log
D)\simeq\cdots\simeq\operatorname{DR}\left(\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}(D)\right).$
(20)

On the other hand we have another canonical isomorphism (see Proposition 2.5)

$\Omega^{\bullet}_{X}(\star D)=\Omega^{\bullet}_{X}(\mathcal{O}_{X}(\star
D))\simeq\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}(\mathcal{O}_{X},\mathcal{O}_{X}(\star
D))=\operatorname{DR}(\mathcal{O}_{X}(\star D)).$ (21)

The point is that the inclusion
$\mathcal{O}_{X}(D)\subset\mathcal{O}_{X}(\star D)$ induces a canonical left
$\mathcal{D}_{X}$-linear map

$\varrho:\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}(D)\longrightarrow\mathcal{O}_{X}(\star
D),$ (22)

and we have the following theorem ([12, Cor. 4.2] and [55, Cor. 1.7.2]):

(3.7) Theorem. Under the above hypotheses, the following properties hold:

  1. (1)

The logarithmic comparison map (19) corresponds to the map (22) under the
$\operatorname{DR}$ functor and the canonical isomorphisms (20) and (21).

  2. (2)

The following properties are equivalent:

    1. (2-1)

The logarithmic comparison theorem holds for $D$, i.e. the map (19) is a
quasi-isomorphism.

    2. (2-2)

The map (22) is an isomorphism (in the derived category of
$\mathcal{D}_{X}$-modules).

    3. (2-3)

The canonical map

$j_{!}\mathbb{C}_{X\smallsetminus D}\longrightarrow\Omega^{\bullet}_{X}(\log
D)(\mathcal{O}_{X}(-D))$

is a quasi-isomorphism.

(3.8) Remark. (a) Let us notice that property (2-3) above comes from taking
Grothendieck-Verdier duals. Namely,
$\left(\mathbb{R}j_{*}\mathbb{C}_{X\smallsetminus D}\right)^{\vee}\simeq
j_{!}\mathbb{C}_{X\smallsetminus D}$ and

$\displaystyle\displaystyle\left(\Omega^{\bullet}_{X}(\log
D)\right)^{\vee}\simeq\left(\operatorname{DR}\left(\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}(D)\right)\right)^{\vee}\stackrel{{\scriptstyle(\star)}}{{\simeq}}\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}\left(\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}(D),\mathcal{O}_{X}\right)\simeq$
$\displaystyle\displaystyle\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{V}_{X}^{D}}\left(\mathcal{O}_{X}(D),\mathcal{O}_{X}\right)\simeq\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{V}_{X}^{D}}\left(\mathcal{O}_{X}^{*},\mathcal{O}_{X}(D)^{*}\right)\simeq$
$\displaystyle\displaystyle\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{V}_{X}^{D}}\left(\mathcal{O}_{X},\mathcal{O}_{X}(-D)\right)\simeq\Omega^{\bullet}_{X}(\log
D)(\mathcal{O}_{X}(-D)),$

where we have used Mebkhout local duality formula in $(\star)$ (see [47, Chap.
I, (4.3)]; see also [54]).

(b) Let us also notice that the complex $\Omega^{\bullet}_{X}(\log
D)(\mathcal{O}_{X}(-D))$ is a subcomplex of $\Omega^{\bullet}_{X}$, since
locally $\Omega^{\bullet}_{X}(\log
D)(\mathcal{O}_{X}(-D))=f\,\Omega^{\bullet}_{X}(\log D)$ with $f=0$ a reduced
local equation of $D$, and $f\,\Omega^{p}_{X}(\log D)\subset\Omega^{p}_{X}$
for $p=0,\dots,n$. Moreover,

$\displaystyle\displaystyle\Omega^{\bullet}_{X}(\log
D)(\mathcal{O}_{X}(-D))\simeq\cdots\simeq\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}\left(\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}(D),\mathcal{O}_{X}\right)\simeq$
$\displaystyle\displaystyle\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}\left(\mathds{D}\,\mathcal{O}_{X},\mathds{D}\left(\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}(D)\right)\right)\simeq\mathbb{R}\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{D}_{X}}\left(\mathcal{O}_{X},\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}\right)=\operatorname{DR}\left(\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}\right)$

and we can prove that the inclusion $\Omega^{\bullet}_{X}(\log
D)(\mathcal{O}_{X}(-D))\hookrightarrow\Omega^{\bullet}_{X}$ comes from the map
of $\mathcal{D}_{X}$-modules

$\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}\longrightarrow\mathcal{O}_{X},\quad
P\otimes a\longmapsto P(a),$ (23)

by applying the $\operatorname{DR}(-)$ functor.

(3.9) Remark. Let us understand in concrete terms the significance of the map
(22) being an isomorphism. First of all, (22) is an isomorphism if and only if
it is so stalkwise, and obviously $\varrho_{p}$ is an isomorphism for each
$p\in X\smallsetminus D$. Let us take a point $p\in D$, a reduced local
equation $f\in\mathcal{O}_{X,p}$ of $D$ and a basis
$\delta_{1},\dots,\delta_{n}\in\operatorname{\mathcal{D}\mathit{er}}(-\log
D)_{p}$ with $\delta_{i}(f)=\alpha_{i}f$, $\alpha_{i}\in\mathcal{O}_{X,p}$,
$i=1,\dots,n$. On the other hand, $\mathcal{O}_{X,p}(D)$ is generated as
$\mathcal{V}^{D}_{X,p}$-module by $f^{-1}$ and the kernel of

$P\in\mathcal{V}^{D}_{X,p}\longmapsto P(f^{-1})\in\mathcal{O}_{X,p}(D)$

is the left ideal generated by
$\delta_{1}+\alpha_{1},\dots,\delta_{n}+\alpha_{n}$, i.e.

$\mathcal{O}_{X,p}(D)\simeq\mathcal{V}^{D}_{X,p}/\mathcal{V}^{D}_{X,p}\langle\delta_{1}+\alpha_{1},\dots,\delta_{n}+\alpha_{n}\rangle.$

Consequently the map
$\varrho_{p}:\mathcal{D}_{X,p}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}^{D}_{X,p}}\mathcal{O}_{X,p}(D)\longrightarrow\mathcal{O}_{X,p}(\star
D)$ is an isomorphism (in the derived category of $\mathcal{D}_{X,p}$-modules)
if and only if the following properties hold:

  1. (i)

The complex
$\mathcal{D}_{X,p}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}^{D}_{X,p}}\mathcal{O}_{X,p}(D)$
is exact in cohomological degrees $\neq 0$, and

  2. (ii)

$\mathcal{O}_{X,p}(\star
D)\simeq\mathcal{D}_{X,p}/\mathcal{D}_{X,p}\langle\delta_{1}+\alpha_{1},\dots,\delta_{n}+\alpha_{n}\rangle$.

The isomorphism in (ii) comes from the map

$P\in\mathcal{D}_{X,p}\longmapsto P(f^{-1})\in\mathcal{O}_{X,p}(\star D),$

and so property (ii) is equivalent to

  1. (ii-1)

$\mathcal{O}_{X,p}(\star D)$ is generated as $\mathcal{D}_{X,p}$-module by
$f^{-1}$, and

  2. (ii-2)

The $\mathcal{D}_{X,p}$-annihilator of $f^{-1}\in\mathcal{O}_{X,p}(\star D)$
is the left ideal generated by
$\delta_{1}+\alpha_{1},\dots,\delta_{n}+\alpha_{n}$.

Notice that property (ii-2) is equivalent to the fact that the
$\mathcal{D}_{X,p}$-annihilator of $f^{-1}$ is generated by order $1$
operators. From [64] we know that the last property implies that the
$b$-function of the germ $f$ has no integer roots strictly less than $-1$, and
so property (ii-2) implies property (ii-1) (by the Bernstein functional
equation). We conclude that the map $\varrho_{p}$ is an isomorphism if and
only if the following properties hold:

  1. (i)

The complex
$\mathcal{D}_{X,p}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}^{D}_{X,p}}\mathcal{O}_{X,p}(D)$
is exact in cohomological degrees $\neq 0$, and

  2. (ii-2)

The $\mathcal{D}_{X,p}$-annihilator of $f^{-1}\in\mathcal{O}_{X,p}(\star D)$
is the left ideal generated by
$\delta_{1}+\alpha_{1},\dots,\delta_{n}+\alpha_{n}$.

Let us recall the following definition [16].

(3.10) Definition. A free divisor $D\subset X$ is called Spencer if the
complex
$\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}$
is concentrated in cohomological degree $0$ and
$\mathcal{D}_{X}\otimes_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}$ is holonomic.

If the the LCT holds for $D$, then $\mathds{D}(\mathcal{O}_{X}(\star
D))\simeq\mathds{D}\left(\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}(D)\right)\simeq\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}$,
and so $D$ is Spencer and $\mathds{D}(\mathcal{O}_{X}(\star
D))\simeq\mathcal{D}_{X}\otimes_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}$ (since
both $\mathcal{O}_{X}(\star D)$ and its dual are holonomic).

To check whether the complex
$\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}$
is concentrated in cohomological degree $0$, we can use the logarithmic
Spencer complex ${\mathrm{Sp}}^{\bullet}(\log D)$, which is a locally free
resolution of $\mathcal{O}_{X}$ as a left $\mathcal{V}_{X}^{D}$-module. So we
have to check whether the complex
$\mathcal{D}_{X}\otimes_{\mathcal{V}_{X}^{D}}{\mathrm{Sp}}^{\bullet}(\log D)$
is concentrated in cohomological degree $0$ or not.

For this, we can consider again the discrete increasing filtration
$G^{\bullet}:=G^{\bullet}\left(\mathcal{D}_{X}\otimes_{\mathcal{V}_{X}^{D}}{\mathrm{Sp}}^{\bullet}(\log
D)\right)$ on the complex
$\mathcal{D}_{X}\otimes_{\mathcal{V}_{X}^{D}}{\mathrm{Sp}}^{\bullet}(\log D)$
given by, for $0\leq p\leq n$ and $k\in\mathbb{N}$,

$G^{k,-p}:=G^{k}\left(\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(-\log
D)\right)=F^{k-p}(\mathcal{D}_{X})\otimes_{\mathcal{O}_{X}}\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(-\log
D)$

where $F^{\bullet}(\mathcal{D}_{X})$ is the filtration by the order filtration
in $\mathcal{D}$. The associated graded complex turns out to be (locally) the
Koszul complex with respect to
$\\{\sigma(\delta_{1}),\dots,\sigma(\delta_{n})\\}\subset\operatorname{\rm
gr}\mathcal{D}_{X}$, $\\{\delta_{1},\dots,\delta_{n}\\}$ being a local
$\mathcal{O}_{X}$-basis of $\operatorname{\mathcal{D}\mathit{er}}(-\log D)$,
but in general, this is not a regular sequence333Remember that
$\\{\sigma(\delta_{1}),\dots,\sigma(\delta_{n})\\}$ is a regular sequence in
$\operatorname{\rm gr}\mathcal{V}_{X}^{D}$ since this ring is (locally) a
polynomial ring in the variables
$\\{\sigma(\delta_{1}),\dots,\sigma(\delta_{n})\\}$ with coefficients in
$\mathcal{O}_{X}$. and so $\operatorname{\rm
gr}_{G^{\bullet}}\left(\mathcal{D}_{X}\otimes_{\mathcal{V}_{X}^{D}}{\mathrm{Sp}}^{\bullet}(\log
D)\right)$ is not concentrated in cohomological degree $0$. This fact
motivates the following definition [9, Def. 4.1.1].

(3.11) Definition. We say that a free divisor $D\subset X$ is Koszul at a
point $p\in D$ if for some (and hence for any) local basis
$\\{\delta_{1},\dots,\delta_{n}\\}$ of
$\operatorname{\mathcal{D}\mathit{er}}(-\log D)_{p}$, the symbols
$\\{\sigma(\delta_{1}),\dots,\sigma(\delta_{n})\\}$ form a regular sequence in
$\operatorname{\rm gr}\mathcal{D}_{X,p}$; and we say that $D$ is Koszul if it
so at any point $p\in D$.

Let us notice that the Koszul property for a free divisor $D\subset X$ is
equivalent to saying that the complex $\operatorname{\rm
gr}\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\operatorname{\rm
gr}\mathcal{V}_{X}^{D}}\mathcal{O}_{X}$ is concentrated in cohomological
degree $0$. In that case, the module $\operatorname{\rm
gr}\mathcal{D}_{X}\otimes_{\operatorname{\rm
gr}\mathcal{V}_{X}^{D}}\mathcal{O}_{X}$ has automatically dimension $n=\dim
X$.

Any Koszul free divisor $D$ is Spencer since, from the very definition, the
complex
$\mathcal{D}_{X}\otimes_{\mathcal{V}_{X}^{D}}{\mathrm{Sp}}^{\bullet}(\log D)$
is concentrated in cohomological degree $0$ (its graded complex with respect
to $G^{\bullet}$ is concentrated in cohomological degree $0$), and
$\mathcal{D}_{X}\otimes_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}$ is holonomic
since it is locally presented as
$\mathcal{D}_{X,p}/\mathcal{D}_{X,p}\langle\delta_{1},\dots,\delta_{n}\rangle$,
where $\\{\delta_{1},\dots,\delta_{n}\\}$ is a basis of
$\operatorname{\mathcal{D}\mathit{er}}(-\log D)_{p}$, and the quotient of
$\operatorname{\rm gr}\mathcal{D}_{X,p}$ by the ideal
$\langle\sigma(\delta_{1}),\dots,\sigma(\delta_{n})\rangle$ has dimension $n$.
Moreover, in this case $\operatorname{\rm
gr}\left(\mathcal{D}_{X}\otimes_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}\right)\simeq\operatorname{\rm
gr}\mathcal{D}_{X}\otimes_{\operatorname{\rm
gr}\mathcal{V}_{X}^{D}}\mathcal{O}_{X}$, $\\{\delta_{1},\dots,\delta_{n}\\}$
is an involutive basis of the ideal
$\mathcal{D}_{X,p}\langle\delta_{1},\dots,\delta_{n}\rangle$ with respect to
the order filtration and the characteristic variety of
$\mathcal{D}_{X}\otimes_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}$ is (locally)
given by $\sigma(\delta_{1})=\cdots=\sigma(\delta_{n})=0$.

Actually, the condition for a free divisor to be Koszul is equivalent to the
fact that the logarithmic stratification of $D$ [60, §3] is locally finite, or
equivalently, that any logarithmic stratum of $D$ is holonomic in the sense of
loc. cit. (see [28, Theorem (7.4)]). In particular, any plane curve and any
free hyperplane arrangement is a Koszul free divisor.

On the other hand, we know that any locally quasihomogeneous free divisor is
Koszul [10]444Actually, it is easy to see that for any locally
quasihomogeneous divisor, free or not, the logarithmic stratification of $D$
is locally finite.. Furthermore, the roots of the $b$-function of a reduced
local equation $f=0$ of any locally quasihomogeneous free divisor are
symmetric with respect to $-1$ (see [11, Theorem 5.6], [56, Corollary (4.2)])
and the $\mathcal{D}[s]$-annhilator of $f^{s}$ is generated by order one
operators [11, Corollary 5.8]. From there we deduce a purely algebraic proof
of Theorem 1.3, based on Theorem 3.3 (see [56, Corollary (4.5) and Remark
(4.6)]).

(3.12) Remark. Notice also that, for any hyperplane arrangement, free or not,
with equation $f=0$, reduced or not, the $\mathcal{D}_{X,p}$–annihilator of
$f^{-1}\in\mathcal{O}_{X,p}(*D)$ if generated by operators of order 1 [66, Th.
5.3]. As mentioned before, see Sect. 1.4, LCT holds for any hyperplane
arrangement [2].

If a free divisor $D\subset\mathbb{C}^{n}$ is Spencer with a polynomial
defining equation $f=0$, then by [17, Crit. 3.1 and 4.1], $D$ satisfies LCT if
and only if the annihilating ideal in $\mathcal{D}$ of $1/f$ is generated by
operators of order 1; that is, if condition (ii.2) in Remark 3.3 holds. In
[17, Rk. 5.8] examples are given of three free divisors in $\mathbb{C}^{3}$,
defined by quasi-homogeneous polynomials (with strictly positive weights),
that do not satisfy LCT and hence, by Th. 1.3, they are not locally quasi-
homogeneous. That gives a negative answer to a question proposed in [11, Prob.
6.5].

(3.13) Remark. The sheaf $\check{\Omega}^{\bullet}_{D}$ introduced in [51]
(see 1.6) is the quotient of $\Omega^{\bullet}_{X}$ by the subcomplex
$\Omega^{\bullet}_{X}(\log D)(\mathcal{O}_{X}(-D))$, and so there is a
commutative diagram of complexes of sheaves of $\mathbb{C}$-vector spaces

${0}$${j_{!}\mathbb{C}_{X\smallsetminus
D}}$${\mathbb{C}_{X}}$${\mathbb{C}_{D}}$${0}$${0}$${\Omega^{\bullet}_{X}(\log
D)(\mathcal{O}_{X}(-D))}$${\Omega^{\bullet}_{X}}$${\check{\Omega}^{\bullet}_{D}}$${0}$adj.$\scriptstyle{\lambda}$q-i$\scriptstyle{\overline{\lambda}}$incl.
(24)

with exact rows, where the vertical arrows are the natural ones. The bottom
row comes from the triangle

$\mathcal{D}_{X}\stackrel{{\scriptstyle\mathbb{L}}}{{\otimes}}_{\mathcal{V}_{X}^{D}}\mathcal{O}_{X}\longrightarrow\mathcal{O}_{X}\longrightarrow
K^{\bullet}\stackrel{{\scriptstyle+1}}{{\longrightarrow}}$

by applying the $\operatorname{DR}(-)$ functor.

The middle vertical arrow in (24) is a quasi-isomorphism by Poincaré lemma,
and $\lambda$ is a quasi-isomorphism if and only if $\overline{\lambda}$ is a
quasi-isomorphism. So, we can add another equivalent property to (2) in
Theorem 3.3:

  1. (2-4)

The canonical map $\mathbb{C}_{D}\longrightarrow\check{\Omega}^{\bullet}_{D}$
is a quasi-isomorphism.

We deduce a new proof of Theorem 1.3 by using [51, Lemma 3.3], where it is
proven that the map $\overline{\lambda}$ is a quasi-isomorphism for any
locally quasihomogeneous divisor (not necessarily free).

In fact, a similar argument shows that LCT holds for any locally weakly
quasihomogeneous free divisor and a posteriori any such free divisor is
Spencer (see Definition 1.3) (see [19, Remark 3.11] and [55, Remark 1.7.4],
although the statement in [19, Remark 3.11] that any logarithmic differential
form has a non negative weight is wrong). Namely, we have the following.

(3.14) Theorem. If $D\subset X$ is a locally weakly quasihomogeneous free
divisor, then $D$ satisfies the logarithmic comparison theorem.

Proof. By Theorem 3.3 (2-3), we need to prove that the complex
$\Omega^{\bullet}_{X}(\log D)(\mathcal{O}_{X}(-D))_{p}$ is exact for any $p\in
D$. Since $D$ is locally weakly quasihomogeneous, there are local coordinates
$x_{1},\dots,x_{r},x_{r+1},\dots,x_{n}$ centered at $p$, $r>0$, weights
$(w_{1},\dots,w_{r},0,\dots,0)$, $w_{i}>0$ for all $i=1,\dots,r$, and a
reduced local equation $f$ of $D$ at $p$ which is quasihomogeneous in these
coordinates with $\operatorname{\rm wt}(f)>0$.

By the standard argument (see Lemma 3.3 in [51]), the complex

$\Omega^{\bullet}_{X}(\log
D)(\mathcal{O}_{X}(-D))_{p}=f\Omega^{\bullet}_{X}(\log D)_{p}$

is homotopic to its weight zero subcomplex, and the theorem is a consequence
of the following lemma. Q.E.D.

(3.15) Lemma. Let $f$ be a quasihomogeneous polynomial in
$\mathbb{C}\\{x_{r+1},\dots,x_{n}\\}[x_{1},\dots,x_{r}]$, of strictly positive
weight with respect to weights $(w_{1},\dots,w_{r},0,{\ldots},0)$ with
$w_{1},{\ldots},w_{r}$ all strictly positive. Then, for any $p>0$ and any non-
zero quasihomogeneous logarithmic $p$-form $\omega$, we have
$\operatorname{\rm wt}\omega>-\operatorname{\rm wt}f$.

Proof. Since $f\omega$ must be holomorphic, we know that $\operatorname{\rm
wt}\omega\geq-\operatorname{\rm wt}f$. We may assume
${\partial}f/{\partial}x_{1}\neq 0$ (i.e. $f$ effectively depends on $x_{1}$).

Suppose that $\omega$ is a non-zero logarithmic $p$-form, and that
$\operatorname{\rm wt}\omega=-\operatorname{\rm wt}f$. Then, $\alpha=f\omega$
is a non-zero holomorphic $p$-form with $\operatorname{\rm wt}\alpha=0$. That
means that $\alpha$ only depends on the variables $x_{r+1},\dots,x_{n}$ of
weight 0:

$\alpha=\sum_{r+1\leq i_{1}<{\ldots}<i_{p}\leq
n}\alpha_{i_{1},{\ldots},i_{p}}(x_{r+1},\dots,x_{n})dx_{i_{1}}\wedge{\ldots}\wedge
dx_{i_{p}}.$

Reordering the variables, we may assume $\alpha_{r+1,{\ldots},r+p}\neq 0$.

Since $\omega$ is logarithmic, the $p+1$-form $df\wedge\omega$ is holomorphic
(this follows from the fact that $f\omega$ and $fd\omega$ are holomorphic).
The coefficient of $dx_{1}\wedge dx_{r+1}\wedge{\ldots}\wedge dx_{r+p}$ in
$df\wedge\omega$, which must be holomorphic, is

$\frac{\alpha_{r+1,{\ldots},r+p}}{f}\frac{{\partial}f}{{\partial}x_{1}}.$

So $f$ divides $\alpha_{r+1,{\ldots},r+p}\frac{{\partial}f}{{\partial}x_{1}}$
in the ring of power series
$\mathbb{C}\\{x_{1},\dots,x_{r},x_{r+1},\dots,x_{n}\\}$. However, $f$ and
${\partial}f/{\partial}x_{1}$ are quasihomogeneous polynomials in
$\mathbb{C}\\{x_{r+1},\dots,x_{n}\\}[x_{1},\dots,x_{r}]$. By equating
homogeneous parts, we deduce that $f$ also divides
$\alpha_{r+1,{\ldots},r+p}{\partial}f/{\partial}x_{1}$ in
$\mathbb{C}\\{x_{r+1},\dots,x_{n}\\}[x_{1},\dots,x_{r}]$. Since
$\operatorname{\rm wt}({\partial}f/{\partial}x_{1})=\operatorname{\rm
wt}(f)-\operatorname{\rm wt}(x_{1})$, this is impossible. Q.E.D.

A natural question is to characterize free divisors for which LCT holds. This
has been done in [14] for plane curves, and more generally in [64, Cor. 1.8]
and [56, Th. (4.7)] for Koszul free divisors. For general free divisors the
following conjecture remains open [14, Conjecture 1.4]:

(3.16) Conjecture. Let $D\subset X$ a free divisor. If LCT holds for $D$, then
$D$ is strongly Euler homogeneous, i.e. for each $p\in D$ there is a local
reduced equation of $D$ at $p$ and a germ of vector field $\chi$ at $p$,
singular at $p$, such that $\chi(f)=f$.

In [18, §3, 4] the authors provide an infinite family of free divisors for
which LCT does not hold. Any divisor of this family is defined by a polynomial

$f_{p,q}=(x_{1}^{p}-x_{2}^{q})\prod_{i=3}^{n}(x_{1}x_{i}+x_{2})$

for $n\geq 3$ and $3\leq p<q$.

###  3.4 The logarithmic Spencer complex revisited

Recall that, see (3.2), for a free divisor $D$ in a complex manifold $X$, the
logarithmic Spencer complex ${\mathrm{Sp}}^{\bullet}(\log D)$ is a locally
free resolution of the left $\mathcal{V}_{X}^{D}$-module $\mathcal{O}_{X}$ [9,
Th. 3.1.2]. We are going to describe the complex
$\mathcal{D}_{X}\otimes_{\mathcal{V}_{X}^{D}}{\mathrm{Sp}}^{\bullet}(\log D)$
in terms of a (local) basis $\\{\delta_{1},\ldots,\delta_{n}\\}$ of the free
$\mathcal{O}_{X}$–module $\operatorname{\mathcal{D}\mathit{er}}(-\log D)$. We
can write

$[\delta_{i},\delta_{j}]=\sum_{k=1}^{n}\alpha_{k}^{ij}\delta_{k}$

for $1\leq i<j\leq n$ where $\alpha_{k}^{ij}\in\mathcal{O}_{X}$ for
$k=1,\ldots,n$. We also have

$\mathcal{D}_{X}\otimes_{\mathcal{V}_{X}^{D}}{\mathrm{Sp}}^{p}(\log
D)=\mathcal{D}_{X}\otimes_{\mathcal{V}_{X}^{D}}{\mathcal{V}_{X}^{D}}\otimes_{\mathcal{O}_{X}}\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(-\log
D)\simeq\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(-\log
D)$

for $p=0,\ldots,n$, and we also denote $\epsilon_{-p}$ the differential of
this complex of left $\mathcal{D}_{X}$–modules. We denote this complex by
${\mathrm{Sp}}^{\bullet}_{\mathcal{D}_{X}}(\log D)$.

For $p\in\mathbb{N}$ we denote
$\Lambda_{p}:=\\{(i_{1},\ldots,i_{p})\in\mathbb{N}^{p}\,|\,1\leq
i_{1}<i_{2}<\ldots<i_{p}\leq n\\}$. The free left $\mathcal{D}_{X}$–module
$\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(-\log
D)\simeq\bigoplus_{{\underline{{\bf
i}}}\in\Lambda_{p}}\mathcal{D}_{X}\,e_{\underline{{\bf i}}}$ has rank
$\binom{n}{p}$ with basis $\\{e_{\underline{{\bf i}}}\,|{\underline{{\bf
i}}}\in\Lambda_{p}\\}$, for $p=0,\ldots,n$. An isomorphism from
$\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}\bigwedge^{p}\operatorname{\mathcal{D}\mathit{er}}(-\log
D)$ onto $\bigoplus_{{\underline{{\bf
i}}}\in\Lambda_{p}}\mathcal{D}_{X}\,e_{\underline{{\bf i}}}$ maps

$1\otimes\delta_{i_{1}}\wedge\cdots\wedge\delta_{i_{p}}\,\mapsto\,e_{{\underline{{\bf
i}}}}.$

For $p\in\mathbb{N}$ and ${\underline{{\bf i}}}\in\Lambda_{p}$ we fix the
following notations:

  * •

The support of ${\underline{{\bf i}}}$ is $\operatorname{\rm
supp}({\underline{{\bf i}}})=\\{i_{1},\ldots,i_{p}\\}$.

  * •

$\overline{\operatorname{\rm supp}({\underline{{\bf
i}}})}=\\{1,\ldots,n\\}\,\,\smallsetminus\,\,\operatorname{\rm
supp}({\underline{{\bf i}}})$.

  * •

${\underline{{\bf
i}}}(\widehat{k}):=(i_{1},\ldots,i_{k-1},i_{k+1},\ldots,i_{p})\in\Lambda_{p-1},$
for $1\leq k\leq p$.

  * •

${\underline{{\bf i}}}(\widehat{k,\ell}):={\underline{{\bf
i}}}(\widehat{k})(\widehat{\ell-1})\in\Lambda_{p-2},$ $1\leq k<\ell\leq p$.

  * •

$\sigma(q;{\underline{{\bf i}}}):=\max\\{j\in\\{1,\ldots,p\\}\,|\,i_{j}<q\\}$
and $\sigma(q;{\underline{{\bf i}}})=0$ if $q<i_{1}$.

  * •

${\underline{{\bf
i}}}(\widecheck{q})=(i_{1},\ldots,i_{k-1},q,i_{k+1},\ldots,i_{p})\in\Lambda_{p+1}$
for $k=\sigma(q;{\underline{{\bf i}}})$ and
$q\in\\{1,\ldots,n\\}\smallsetminus\\{i_{1},\ldots,i_{p}\\}$.

The complex $({\mathrm{Sp}}^{\bullet}_{\mathcal{D}_{X}}(\log
D),\,\epsilon_{-\bullet})$ can be written as the complex
$(\bigoplus_{{\underline{{\bf
i}}}\in\Lambda_{\bullet}}\mathcal{D}_{X}\,e_{\underline{{\bf
i}}},\,\widetilde{\epsilon_{-\bullet}})$ where

$\widetilde{\epsilon_{-p}}:\bigoplus_{{\underline{{\bf
i}}}\in\Lambda_{p}}\mathcal{D}_{X}\,e_{\underline{{\bf
i}}}\longrightarrow\bigoplus_{{\underline{{\bf
j}}}\in\Lambda_{p-1}}\mathcal{D}_{X}\,e_{\underline{{\bf j}}}$

is the morphism of left $\mathcal{D}_{X}$–modules defined by

$\widetilde{\epsilon_{-p}}(e_{\underline{{\bf
i}}})=\sum_{k=1}^{p}(-1)^{k-1}\delta_{i_{k}}e_{{\underline{{\bf
i}}}(\widehat{k})}+\sum_{1\leq\ell<m\leq
p}(-1)^{\ell+m}\sum_{q\in\overline{\operatorname{\rm supp}({\underline{{\bf
i}}}(\widehat{\ell,m}))}}(-1)^{\sigma(q;{\underline{{\bf
i}}}(\widehat{\ell,m}))}\alpha^{i_{\ell}i_{m}}_{q}e_{{\underline{{\bf
i}}}(\widehat{\ell,m})(\widecheck{q})}.$

Notice that the complex $(\bigoplus_{{\underline{{\bf
i}}}\in\Lambda_{\bullet}}\mathcal{D}_{X}\,e_{\underline{{\bf
i}}},\,\widetilde{\epsilon_{-\bullet}})$ can be viewed as a non commutative
version of a Koszul complex.

Let us write down an example for $n=3$.

One has:

$\begin{split}\epsilon_{-1}(e_{i})&=\delta_{i}{\mbox{ for }}i=1,2,3,\\\
\epsilon_{-2}(e_{ij})&=\delta_{i}e_{j}-\delta_{j}e_{i}-\sum_{q=1}^{3}\alpha_{q}^{ij}e_{q}\,{\mbox{
for }}1\leq i<j\leq 3,\\\
\epsilon_{-3}(e_{123})&=\delta_{1}e_{23}-\delta_{2}e_{13}+\delta_{3}e_{12}+(\alpha_{1}^{13}+\alpha_{2}^{23})e_{12}+(-\alpha_{1}^{12}+\alpha_{3}^{23})e_{13}+(-\alpha_{2}^{12}-\alpha_{3}^{13})e_{23}\\\
&=(\delta_{3}+\alpha_{1}^{13}+\alpha_{2}^{23})e_{12}+(-\delta_{2}-\alpha_{1}^{12}+\alpha_{3}^{23})e_{13}+(\delta_{1}-\alpha_{2}^{12}-\alpha_{3}^{13})e_{23}.\end{split}$

###  3.5 The annihilator ideal of $1/f$ for a reduced equation $f$ of a normal
crossing divisor in $\mathbb{C}^{n}$

Write $\mathcal{D}=\mathcal{D}_{\mathbb{C}^{n},0}$ and $f=x_{1}\cdots x_{r}$
for $1\leq r\leq n$. One has the following equality

$Ann_{\mathcal{D}}(1/f)=\mathcal{D}(\partial_{r+1},\ldots,\partial_{n},x_{1}\partial_{1}+1,\ldots,x_{r}\partial_{r}+1).$

Since the inclusion $A_{n}:=A_{n}(\mathbb{C})\subset\mathcal{D}$ is flat, it
is enough to prove a similar equality for the Weyl algebra $A_{n}$ instead of
$\mathcal{D}$. The inclusion

$A_{n}(\partial_{r+1},\ldots,\partial_{n},x_{1}\partial_{1}+1,\ldots,x_{r}\partial_{r}+1)\subset\operatorname{\rm
Ann}_{A_{n}}\left(\frac{1}{f}\right)$

is obvious since each generator of the first ideal annihilates $1/f$. To prove
the opposite inclusion, let us assume first $1=r=n$ and write $x=x_{1}$ and
$\partial=\partial_{x}$. First of all, any $P=P(x,\partial)\in A_{1}$ can be
written as

$P=Q(x\partial+1)+R+S$

for unique $Q\in A_{1}$, $R:=R(x)\in\mathbb{C}[x]$ and
$S:=S(\partial)\in\mathbb{C}[\partial](\partial)$. If $P(\frac{1}{x})=0$ and
$S(\partial)$ is not zero with order $s\geq 1$, then
$\frac{R}{x}+S(\partial)(\frac{1}{x})=0$ which is a contradiction since the
order of the pole of the rational function
$\frac{R}{x}+S(\partial)(\frac{1}{x})$ is $s+1$. Then $S=0$ and then $R=0$ and
$P\in A_{1}(x\partial+1)$. The general case $1\leq r\leq n$ can be reduced to
$r=n$ by taking, for any operator $P$ annihilating $\frac{1}{x_{1}\cdots
x_{r}},$ the remainder of the division of $P$ by
$\partial_{r+1},\ldots,\partial_{n}$. Finally, for the case $r=n$ we proceed
by division of an operator $P$ annihilating $\frac{1}{x_{1}\cdots x_{n}},$
with respect to $\partial_{1}x_{1},\ldots,\partial_{n}x_{n}$.

The $\mathcal{D}$–module criterion for LCT gives another proof that LCT holds
for normal crossing divisors. Indeed, any such divisor $D\equiv(x_{1}\cdots
x_{r}=0)\subset\mathbb{C}^{n}$ is Koszul free and then the complex
$\mathcal{D}\otimes_{\mathcal{V}}{\mathrm{Sp}}^{\bullet}(\log D)$ is
concentrated in cohomological degree $0$ and
$\mathcal{D}\otimes_{\mathcal{V}}\mathcal{O}$ is locally presented as the
quotient
$\mathcal{D}/\mathcal{D}(\partial_{r+1},\ldots,\partial_{n},x_{1}\partial_{1}+1,\ldots,x_{r}\partial_{r}+1)$,
since
$\\{x_{1}\partial_{1},\ldots,x_{r}\partial_{r},\partial_{r+1},\ldots,\partial_{n}\\}$
is a basis of $\operatorname{\rm Der}(-\log D)$. Finally, one has

$\frac{\mathcal{D}}{\mathcal{D}(\partial_{r+1},\ldots,\partial_{n},x_{1}\partial_{1}+1,\ldots,x_{r}\partial_{r}+1)}\simeq\mathcal{D}f^{-1}=\mathcal{O}(\star
D).$

The last equality follows from the well-know fact that the (global) Bernstein-
Sato polynomial of $f=x_{1}\cdots x_{r}$ is $(s+1)^{r}$.

Acknowledgements

The authors thank the editors of the Handbook of Geometry and Topology of
Singularities for the invitation to contribute this survey, and the referee
and Alberto Castaño for very constructive comments.

FCJ acknowledges support from PID2020-117843GB-I00/AEI/10.13039/501100011033.
LNM acknowledges support from PID2020-114613GB-I00/AEI/10.13039/501100011033

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