# Quadratic relations between periods of connections

Javier Fresán  CMLS, École polytechnique, 91128 Palaiseau cedex, France
<EMAIL_ADDRESS>http://javier.fresan.perso.math.cnrs.fr ,
Claude Sabbah  CMLS, CNRS, École polytechnique, Institut Polytechnique de
Paris, 91128 Palaiseau cedex, France<EMAIL_ADDRESS>http://www.math.polytechnique.fr/perso/sabbah and  Jeng-Daw Yu  Department of
Mathematics, National Taiwan University, Taipei 10617, Taiwan<EMAIL_ADDRESS>http://homepage.ntu.edu.tw/ jdyu/

###### Abstract.

We prove the existence of quadratic relations between periods of meromorphic
flat bundles on complex manifolds with poles along a divisor with normal
crossings under the assumption of “goodness”. In dimension one, for which
goodness is always satisfied, we provide methods to compute the various
pairings involved. In an appendix, we recall some classical results needed for
the proofs.

###### Key words and phrases:

Period pairing, Poincaré-Verdier duality, quadratic relation, rapid-decay
homology and cohomology, moderate homology and cohomology, real blow-up, good
meromorphic flat bundle

###### 2010 Mathematics Subject Classification:

32G20, 34M35

This research was partly supported by the PICS project TWN 8094 from CNRS. The
research of J.F. was also partly supported by the grant ANR-18-CE40-0017 of
Agence Nationale de la Recherche.

###### Contents

  1. 1 Introduction
  2. 2 Pairings for bundles with flat connection and quadratic relations
    1. 2.a Setting and notation
    2. 2.b De Rham formalism for quadratic relations
    3. 2.c Middle quadratic relations
    4. 2.d Betti formalism for quadratic relations
    5. 2.e Period realizations
    6. 2.f Quadratic relations for good meromorphic flat bundles
  3. 3 Algebraic computation of de Rham duality in dimension one
    1. 3.a Setting, notation, and objectives
    2. 3.b Čech computation of de Rham cohomologies
    3. 3.c Pairings
    4. 3.d Computations on the real blow-up space
    5. 3.e Computation of Betti period pairings
    6. 3.f Quadratic relations in dimension one
  4. A Twisted singular chains
    1. A.a Presheaves and sheaves
    2. A.b Homotopically fine sheaves
    3. A.c Homotopy operator for singular chains
    4. A.d Singular chains with coefficients in a sheaf
    5. A.e Piecewise smooth and simplicial chains
    6. A.f The dual chain complex with coefficient in a sheaf
  5. B Poincaré lemma for currents
    1. B.a Functions with moderate growth and rapid decay along ${\partial M}$
    2. B.b Poincaré lemma for currents
    3. B.c Poincaré lemma for current with moderate growth and rapid decay
    4. B.d Dolbeault lemmas
  6. C Remarks on Verdier duality
    1. C.a The duality isomorphism
    2. C.b Poincaré-Verdier duality

##  1\. Introduction

Let $U$ be an affine open set of the complex projective line. For an algebraic
vector bundle $V$ on $U$ endowed with an algebraic connection $\nabla$ and a
nondegenerate pairing compatible with $\nabla$, there exists a natural pairing
$\mathsf{P}$, called the _period pairing_ , between the first de Rham
cohomology space $\mathrm{H}^{1}_{\mathrm{dR}}(U,V)$ and the first homology
space $\mathrm{H}_{1}(U^{\mathrm{an}},V^{\nabla})$—also known as the twisted
homology space— on the associated analytic space $U^{\mathrm{an}}$ with
coefficients in the local system $V^{\nabla}$ of horizontal sections of
$(V,\nabla)$. This pairing is obtained by integrating $1$-forms with values in
$V$ against $1$-cycles twisted by $V^{\nabla}$, by means of the given pairing
$V\otimes
V\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{O}(U)$.
There also exists a similar pairing $\mathsf{P}^{\mathrm{c}}$ between the de
Rham cohomology with compact support
$\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)$ and the Borel-Moore twisted
homology
$\mathrm{H}_{1}^{\scriptscriptstyle\mathrm{BM}}(U^{\mathrm{an}},V^{\nabla})$.
The latter pairing is less used than the former because the expression of de
Rham classes with compact support is not as simple as that of de Rham classes
with no support condition. Moreover, since Borel-Moore cycles may have
boundary in $\mathbb{P}^{1}\smallsetminus U$, this pairing needs a
regularization procedure to be expressed.

In the case when $\nabla$ has regular singularities at infinity, both pairings
$\mathsf{P}$ and $\mathsf{P}^{\mathrm{c}}$ are known to be nondegenerate, as
well as the de Rham duality pairing

$\mathsf{S}:\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)\otimes\mathrm{H}^{1}_{\mathrm{dR}}(U,V)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}$

and the intersection pairing

$\mathsf{B}:\mathrm{H}_{1}^{\scriptscriptstyle\mathrm{BM}}(U^{\mathrm{an}},V^{\nabla})\otimes\mathrm{H}_{1}(U^{\mathrm{an}},V^{\nabla})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}.$

Poincaré duality, regarded as an isomorphism between cohomology and homology,
implies a relation between the matrices of these parings:

$(2\pi\mathsf{i})\mathsf{B}=\mathsf{P}\cdot\mathsf{S}^{-1}\cdot{}^{t}\mathsf{P}^{\mathrm{c}}.$

In many interesting examples, both cohomologies
$\mathrm{H}^{1}_{\mathrm{dR}}(U,V)$ and
$\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)$ resp.homologies
$\mathrm{H}_{1}(U^{\mathrm{an}},V^{\nabla})$ and
$\mathrm{H}_{1}^{\scriptscriptstyle\mathrm{BM}}(U^{\mathrm{an}},V^{\nabla})$
coincide, and there is no distinction between $\mathsf{P}$ and
$\mathsf{P}^{\mathrm{c}}$. This relation is then regarded as a family of
polynomial relations of degree two between the entries of the period matrix
$\mathsf{P}$. If $V$ and $V^{\nabla}$ are defined respectively over subfields
$\boldsymbol{K}$ and ${\boldsymbol{k}}$ of $\mathbf{C}$, the pairings
$\mathsf{Q}$ and $\mathsf{S}$ take values in these fields and such relations
take the form of an equality between a quadratic expression of the periods
with coefficients in $\boldsymbol{K}$ and an element in
$(2\pi\mathsf{i}){\boldsymbol{k}}$. In a series of papers, Matsumoto et al.
[12, 31, 32, 27] have developed this technique and have used it to produce
quadratic relations between classical special functions, seealso [22, 23]. The
aforementioned articles show many interesting examples where these quadratic
relations can be effectively computed. This procedure can be extended in
higher dimension, giving rise to examples related to hyperplane arrangements.
We refer to the book [2] where many examples are treated with details.

If one relaxes the condition of regular singularities for the connection
$\nabla$, the period pairings $\mathsf{P},\mathsf{P}^{\mathrm{c}}$ may be
degenerate because they pair spaces of possible different dimensions. In [38],
the authors have extended the quadratic relations to connections, with
irregular singularities at some points of $\mathbb{P}^{1}\smallsetminus U$,
obtained by adding a rational $1$-form to the trivial connection
${\mathrm{d}}$ on the trivial bundle $V=\mathscr{O}_{U}$. The twisted cycles
have to be replaced with twisted cycles with rapid decay, which happen to
coincide with those introduced by Bloch-Esnault [5]. These are supported by
chains with boundary in $\mathbb{P}^{1}\smallsetminus U$ that reach these
boundary points in suitable directions so that the corresponding period
integrals converge. It is then convenient to work with the topological space
obtained by completing $U^{\mathrm{an}}$ with the space of directions at each
point of $\mathbb{P}^{1}\smallsetminus U$, that is, the real oriented blow-up
space of $\mathbb{P}^{1}$ at each point of $\mathbb{P}^{1}\smallsetminus U$.
Correspondingly, the de Rham cohomologies with and without support conditions
are computed by means of the $C^{\infty}$ de Rham cohomology with rapid decay
and moderate growth respectively on this blow-up space, in a way very similar
to that developed in [38] (seealso [41]).

The main purpose of this article is to establish the existence of quadratic
relations that take into account rapid decay and moderate growth properties on
the real oriented blow-up space, and to give some methods to compute them for
any differential equation on a Riemann surface, without any assumption on the
kind of singularities it may have. We then obtain quadratic relations with
enough generality in order to analyze, say, quadratic relations between Bessel
moments, which are periods of symmetric powers of the Bessel differential
equation [21]. Treating this example was the main motivation for developing
the theory in this direction, and the reader will find in loc.cit. a much
detailed example of the various notions explained in the present article.

As for the case with regular singularities, these quadratic relations also
exist in higher dimensions and may be interesting when considering period
pairings for irregular GKZ systems or hyperplane arrangements twisted by
exponentials of rational functions for example. The general rule is to replace
forms with “no support condition” (resp.“Borel-Moore” cycles) by forms
(resp.cycles) with “moderate growth”, and similarly “compact support” by
“rapid decay”.

#### Contents and organization of the article

In Section 2, we consider the case of a differentiable manifold with corners
and a differentiable vector bundle on it endowed with an integrable connection
having poles along the boundary. No analyticity property is needed, but an
assumption on the behaviour near the boundary of the de Rham complexes of
differential forms seems to be essential (Assumptions 2.1). Corollaries 2.14
and 2.24 provide quadratic relations in a very general form. We emphasize
_middle quadratic relations_ since they tend to be the most non-trivial part
of quadratic relations. We revisit the period pairings introduced by Bloch and
Esnault [5] in the form considered by Hien [24, 25, 26]. Applications to
period pairings similar to those of loc.cit. in any dimension are given in
Section 2.f. The main idea, taken from [24, 25, 26], is to define local period
pairings in the sheaf-theoretical sense, prove that they are perfect, and get
for free the perfectness of the global period pairings by applying the
Poincaré-Verdier duality theorem. In that case, the manifold with corners is
nothing but the real oriented blow-up of a complex manifold along a divisor
with normal crossings. Provided Assumptions 2.1 are satisfied, these ideas
apply to the general framework for proving quadratic relations between
periods.

Section 3 focuses on the case of meromorphic connections on Riemann surfaces.
A formula “à la Čech” for computing the de Rham pairing is provided by Theorem
3.14. In the case of rank-one bundles with meromorphic connection, this result
already appears in Deligne’s notes [16]. We also give a formula for computing
period matrices (Proposition 3.19) that goes back, for connections with
regular singularities, at least to [19, Rem. 2.16]. Quadratic relations in the
sense of Matsumoto et al. are obtained in (3.22). The reader will find these
notions illustrated in the example treated in [21].

The appendix recalls classical results which are taken from the Séminaire
Cartan [7, 8], the books of de Rham [13, 14], of Malgrange [39], and of
Kashiwara and Schapira [28]. Let us emphasize that a similar approach has
already been used by Kita and Yoshida [33] in order to define and compute the
Betti intersection pairing for ordinary twisted homology (seealso [2, Chap.
2]). We improve their results by extending them to connections of any rank and
possibly with irregular singularities.

Needless to say, the results of this article are not essentially new, but we
have tried to give them with enough generality, rigor, and details so that
they can be used in various situations without reproving the intermediate
steps.

##  2\. Pairings for bundles with flat connection and quadratic relations

###  2.a. Setting and notation

We consider a $C^{\infty}$ manifold $M$ of real dimension $m$ with corners,
that we assume to be connected and orientable. The boundary of $M$ is denoted
by ${\partial M}$, and the inclusion of the interior
$M^{\circ}=M\smallsetminus{\partial M}$ by $j\colon
M^{\circ}\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}M$.
In the neighbourhood of each point of ${\partial M}$, the pair $(M,{\partial
M})$ is diffeomorphic to
$(\mathbf{R}_{+}{}^{p},\partial(\mathbf{R}_{+}{}^{p}))\times\mathbf{R}^{m-p}$
for some integer $p\in[1,m]$. The sheaf $\mathscr{C}^{\infty}_{M}$ of
$C^{\infty}$ functions (resp.$\mathfrak{Db}_{M}$ of distributions) on $M$ is
locally the sheaf-theoretic restriction to
$\mathbf{R}_{+}{}^{p}\times\mathbf{R}^{m-p}$ of the sheaf of $C^{\infty}$
functions (resp.distributions) on $\mathbf{R}^{m}$. We will also consider the
sheaf $\mathscr{C}_{M}^{\infty}(*)$ of $C^{\infty}$ functions on $M^{\circ}$
having at most poles along ${\partial M}$. The de Rham complex
$(\mathscr{E}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}_{M},{\mathrm{d}})$
of $C^{\infty}$ differential forms on $M$ is a resolution of the constant
sheaf $\mathbf{C}_{M}$. The complex of currents
$(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}},\partial)$, denoted
homologically, can be regarded cohomologically as
$(\mathfrak{Db}_{M}^{m-{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}})$.
Recall (seeAppendix B) that a current of homological index $q$ is a linear
form on test differential forms of degree $q$. Then the inclusion of complexes
$(\mathscr{E}^{m-{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}_{M},{\mathrm{d}})\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}(\mathfrak{Db}_{M}^{m-{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}})$
is a quasi-isomorphism. In other words,
$(\mathfrak{Db}_{M}^{m-{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}})$
is a resolution of $\mathbf{C}_{M}[m]$. We refer the reader to Appendix B for
the notation and results concerning currents with moderate growth. The results
in this section are a mere adaptation of those contained in de Rham’s book
[13, 14] (seeChapter IV in [14]). In order to simplify the discussion, and
since we are only interested in this case, we assume throughout this section
that $M$ is moreover _compact_.

###  2.b. De Rham formalism for quadratic relations

#### Setting and the main assumption

We keep the assumptions and notations as in Section 2.a. Let
$(\mathscr{V},\nabla)$ be a locally free $\mathscr{C}_{M}^{\infty}(*)$-module
of finite rank on $M$ endowed with a flat connection $\nabla$ and a _flat non-
degenerate_ pairing

$\left\langle{{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\right\rangle:\mathscr{V}\otimes\mathscr{V}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{C}_{M}^{\infty}(*),$

i.e., compatible with the connections. It induces a self-duality
$\lambda:\mathscr{V}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathscr{V}^{\scriptscriptstyle\vee}$
which factorizes
$\left\langle{{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\right\rangle$
as the composition

$\mathscr{V}\otimes\mathscr{V}\mathchoice{\xrightarrow{\kern
4.0pt1\otimes\lambda\kern 3.0pt}}{\stackrel{{\scriptstyle
1\otimes\lambda}}{{\longrightarrow}}}{}{}\mathscr{V}\otimes\mathscr{V}^{\scriptscriptstyle\vee}\mathchoice{\xrightarrow{\kern
4.0pt({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\mkern 1.0mu|\mkern
1.0mu{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})\kern
3.0pt}}{\stackrel{{\scriptstyle({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\mkern
1.0mu|\mkern
1.0mu{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})}}{{\longrightarrow}}}{}{}\mathscr{C}_{M}^{\infty}(*),$

where $({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\mkern 1.0mu|\mkern
1.0mu{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$ denotes the natural
duality pairing. We can define the de Rham complexes of $(\mathscr{V},\nabla)$
with respectively $\mathscr{C}_{M}^{\infty,\mathrm{rd}}$ and
$\mathscr{C}_{M}^{\infty,\mathrm{mod}}$ coefficients (seeDefinition B.1).
Since $\nabla$ has only poles as possible singularities along ${\partial M}$,
the complexes

$\operatorname{DR}^{\mathrm{rd}}(\mathscr{V},\nabla)=(\mathscr{E}_{M}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\otimes\mathscr{V},\nabla)\quad\text{and}\quad\operatorname{DR}^{\mathrm{mod}}(\mathscr{V},\nabla)=(\mathscr{E}_{M}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\otimes\mathscr{V},\nabla)$

are well-defined. The de Rham cohomologies with rapid decay
$\mathrm{H}_{{\mathrm{dR}},\mathrm{rd}}^{r}(M,\mathscr{V})$ and the de Rham
cohomologies with moderate growth
$\mathrm{H}_{{\mathrm{dR}},\mathrm{mod}}^{r}(M,\mathscr{V})$ are defined to be
the hypercohomologies of these respective complexes, which are equal to the
cohomologies of the global sections. We make the following assumptions.

######  Assumption 2.1.

  1. (1)

The de Rham complexes $\operatorname{DR}^{\mathrm{rd}}(\mathscr{V},\nabla)$
and $\operatorname{DR}^{\mathrm{mod}}(\mathscr{V},\nabla)$ have cohomology
sheaves concentrated in degree $0$, denoted respectively by
$\mathscr{V}^{\mathrm{rd}}$ and $\mathscr{V}^{\mathrm{mod}}$.

  2. (2)

The natural pairings defined by means of
$\left\langle{{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\right\rangle:\mathscr{V}\otimes\mathscr{V}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{C}^{\infty}_{M}(*)$
and of the wedge product:

$\displaystyle\mathsf{Q}_{\mathrm{rd},\mathrm{mod}}:\operatorname{DR}^{\mathrm{rd}}(\mathscr{V},\nabla)\otimes\operatorname{DR}^{\mathrm{mod}}(\mathscr{V},\nabla)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathscr{E}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}_{M},{\mathrm{d}}),$
$\displaystyle\mathsf{Q}_{\mathrm{mod},\mathrm{rd}}:\operatorname{DR}^{\mathrm{mod}}(\mathscr{V},\nabla)\otimes\operatorname{DR}^{\mathrm{rd}}(\mathscr{V},\nabla)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathscr{E}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}_{M},{\mathrm{d}})$

are perfect (seeExample C.9), i.e., the pairings induced on the
$\mathscr{H}^{0}$ are perfect:

$\displaystyle\mathscr{V}^{\mathrm{rd}}\otimes\mathscr{V}^{\mathrm{mod}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}j_{!}\mathbf{C}_{M^{\circ}},$
$\displaystyle\mathscr{V}^{\mathrm{mod}}\otimes\mathscr{V}^{\mathrm{rd}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}j_{!}\mathbf{C}_{M^{\circ}}.$

One can drop the assumption on the existence of a non-degenerate pairing
$\left\langle{{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\right\rangle$
and use $({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\mkern 1.0mu|\mkern
1.0mu{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$ instead. One can
easily adapt the results of this section by distinguishing between
$\mathscr{V}$ and $\mathscr{V}^{\scriptscriptstyle\vee}$.

#### The de Rham pairings

Verdier duality implies (Proposition C.4) that the natural pairings

(2.2)
$\displaystyle\mathrm{H}^{m-r}(M,\mathscr{V}^{\mathrm{rd}})\otimes\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{mod}})$
$\displaystyle\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}^{m}_{\mathrm{c}}(M^{\circ},\mathbf{C})\simeq\mathbf{C},$
$\displaystyle\mathrm{H}^{m-r}(M,\mathscr{V}^{\mathrm{mod}})\otimes\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{rd}})$
$\displaystyle\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}^{m}_{\mathrm{c}}(M^{\circ},\mathbf{C})\simeq\mathbf{C}$

are non-degenerate. We will interpret these pairings in terms of the de Rham
cohomologies $\mathrm{H}_{{\mathrm{dR}},\mathrm{rd}}^{m-r}(M,\mathscr{V})$ and
$\mathrm{H}_{{\mathrm{dR}},\mathrm{mod}}^{r}(M,\mathscr{V})$ to which these
cohomologies can respectively be identified by means of Assumption 2.1(1). We
note that the diagram

(2.3) $\begin{array}[]{c}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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commutes in the following sense. For any rapid-decay forms
$\omega_{1}^{\mathrm{rd}}$ and $\omega_{2}^{\mathrm{rd}}$ with coefficients in
$\mathscr{V}$, we let $\omega_{1}^{\mathrm{mod}},\omega_{2}^{\mathrm{mod}}$ be
the same forms considered as forms with moderate growth. Commutativity means
that for any such $\omega_{1}^{\mathrm{rd}},\omega_{2}^{\mathrm{rd}}$ the
pairings satisfy

$\mathsf{Q}_{\mathrm{rd},\mathrm{mod}}(\omega_{1}^{\mathrm{rd}},\omega_{2}^{\mathrm{mod}})=\mathsf{Q}_{\mathrm{mod},\mathrm{rd}}(\omega_{1}^{\mathrm{mod}},\omega_{2}^{\mathrm{rd}}).$

For any $r\geqslant 0$, let $\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}}$ be
the global de Rham pairing

$\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}}:\Gamma(M,\mathscr{E}_{M}^{\mathrm{rd},m-r}\otimes\mathscr{V})^{\nabla}\otimes\Gamma(M,\mathscr{E}_{M}^{\mathrm{mod},r}\otimes\mathscr{V})^{\nabla}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}$

obtained by restricting to horizontal sections the composition of
$\Gamma(M,\mathsf{Q}_{\mathrm{rd},\mathrm{mod}})$ with integration
$\int_{M}:\mathscr{E}_{M}^{\mathrm{rd},m}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}$.
Define $\mathsf{Q}^{m-r}_{\mathrm{mod},\mathrm{rd}}$ similarly. Since rapid
decay forms vanish on ${\partial M}$, it follows from Stokes formula that
$\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}}$
(resp.$\mathsf{Q}^{m-r}_{\mathrm{mod},\mathrm{rd}}$) vanishes when one of the
terms is a coboundary. We thus obtain from the perfectness of (2.2):

######  Corollary 2.4.

Under Assumptions 2.1, the induced pairing

$\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}}:\mathrm{H}_{{\mathrm{dR}},\mathrm{rd}}^{m-r}(M,\mathscr{V})\otimes\mathrm{H}_{{\mathrm{dR}},\mathrm{mod}}^{r}(M,\mathscr{V})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}$

is perfect, and so is $\mathsf{Q}^{m-r}_{\mathrm{mod},\mathrm{rd}}$.∎

#### Twisted currents with rapid decay and moderate growth

We consider the following complexes of currents with coefficients in
$\mathscr{V}$ (see(B.3) for the boundary operator):

  * $\scriptscriptstyle\bullet$

twisted currents with rapid decay

$(\mathfrak{C}^{\mathrm{rd}}_{M,{\scriptscriptstyle\bullet}}(\mathscr{V}),\partial)=(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{mod}},\partial)\otimes\operatorname{DR}^{\mathrm{rd}}(\mathscr{V},\nabla)\simeq(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{mod}},\partial)\otimes\mathscr{V}^{\mathrm{rd}},$

  * $\scriptscriptstyle\bullet$

twisted currents with moderate growth

$(\mathfrak{C}^{\mathrm{mod}}_{M,{\scriptscriptstyle\bullet}}(\mathscr{V}),\partial)=(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{mod}},\partial)\otimes\operatorname{DR}^{\mathrm{mod}}(\mathscr{V},\nabla)\simeq(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{mod}},\partial)\otimes\mathscr{V}^{\mathrm{mod}}.$

We define respectively the de Rham homology with rapid decay and with moderate
growth with coefficients in $\mathscr{V}$ as

$\displaystyle\mathrm{H}_{{\mathrm{dR}},r}^{\mathrm{rd}}(M,\mathscr{V})=\mathrm{H}_{r}\bigl{(}\Gamma(M,\mathfrak{C}^{\mathrm{rd}}_{M,{\scriptscriptstyle\bullet}}(\mathscr{V}))\bigr{)},$
$\displaystyle\mathrm{H}_{{\mathrm{dR}},r}^{\mathrm{mod}}(M,\mathscr{V})=\mathrm{H}_{r}\bigl{(}\Gamma(M,\mathfrak{C}^{\mathrm{mod}}_{M,{\scriptscriptstyle\bullet}}(\mathscr{V}))\bigr{)}$

(recall that $M$ is compact). For example, when
$(\mathscr{V},\nabla)=(\mathscr{C}_{M}^{\infty}(*),{\mathrm{d}})$, rapid decay
and moderate de Rham homologies are the homology and the Borel-Moore homology
of $M^{\circ}$ (seeRemark B.13).

#### The Poincaré isomorphisms

For the sake of simplicity, we will write the complex of currents
cohomologically via the relation
$\mathfrak{C}_{M,q}^{\mathrm{mod}}=\mathfrak{C}_{M}^{\mathrm{mod},m-q}$, so
that we have a quasi-isomorphism
$\mathbf{C}_{M}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(\mathfrak{C}_{M}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\partial)$
(seeProposition B.10). This leads to Poincaré isomorphisms in the bounded
derived category $\mathsf{D}^{\mathrm{b}}(\mathbf{C}_{M})$, making the
following diagram commutative:

(2.5) $\begin{array}[]{c}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
29.89449pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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42.57422pt\raise 6.9611pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 83.21567pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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101.20761pt\raise-29.49998pt\hbox{\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
81.10953pt\raise-41.27774pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{C}_{M}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\otimes\operatorname{DR}^{\mathrm{mod}}(\mathscr{V},\nabla)=\mathfrak{C}_{M}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}(\mathscr{V}).}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{array}$

By taking hypercohomologies (recall that $M$ is compact) and switching back to
homology on the right-hand side, we thus obtain global Poincaré isomorphisms

(2.6)
$\begin{split}\eucal{P}_{r}^{\mathrm{rd}}(\mathscr{V}):\mathrm{H}^{m-r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V})&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{H}_{{\mathrm{dR}},r}^{\mathrm{rd}}(M,\mathscr{V}),\\\
\eucal{P}_{r}^{\mathrm{mod}}(\mathscr{V}):\mathrm{H}^{m-r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V})&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{H}_{{\mathrm{dR}},r}^{\mathrm{mod}}(M,\mathscr{V}).\end{split}$

#### De Rham period pairings

We define the de Rham period pairing

(2.7)
$\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{\mathrm{dR}}:(\mathfrak{C}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}_{M}(\mathscr{V}),\partial)\otimes\operatorname{DR}^{\mathrm{mod}}(\mathscr{V},\nabla)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathfrak{C}_{M}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\partial)$

as follows: for a $q$-dimensional current $T_{q}=T^{m-q}$ with moderate
growth, an $r$-form $\eta^{r}$ with rapid decay, an $s$-form $\omega^{s}$ with
moderate growth and local sections $v,w$ of $\mathscr{V}$, the pairing of the
local sections $T_{q}\otimes\eta^{r}\otimes v$ and $\omega^{s}\otimes w$ is
the current

$\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{\mathrm{dR}}(T_{q}\otimes\eta^{r}\otimes
v,\omega^{s}\otimes w)=T_{q}\otimes(\langle
v,w\rangle\cdot\eta^{r}\wedge\omega^{s})$

of dimension $q-r-s$. Compatibility with differentials follows from (B.3). It
has rapid decay since $\langle v,w\rangle$ has at most a pole (and hence
moderate growth), and thus $\langle v,w\rangle\cdot\eta^{r}\wedge\omega^{s}$
has rapid decay. Note that, if $T_{q}$ is $C^{\infty}$, then we recover
$\mathsf{Q}_{\mathrm{rd},\mathrm{mod}}$.

Similarly we define the period pairing

(2.8)
$\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{\mathrm{dR}}:(\mathfrak{C}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}_{M}(\mathscr{V}),\partial)\otimes\operatorname{DR}^{\mathrm{rd}}(\mathscr{V},\nabla)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathfrak{C}_{M}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\partial).$

Then the following (cohomological) diagram commutes:

(2.9) $\begin{array}[]{c}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
59.92899pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 102.60811pt\raise 0.0pt\hbox{\hbox{\kern
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102.60811pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{DR}^{\mathrm{rd}}(\mathscr{V},\nabla)\otimes\operatorname{DR}^{\mathrm{mod}}(\mathscr{V},\nabla)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
102.81644pt\raise-41.27774pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-34.44434pt\raise-20.63887pt\hbox{{}\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern-34.44434pt\raise-5.5pt\hbox{\hbox{\kern
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102.81644pt\raise-41.27774pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{$\textstyle{(\mathscr{E}_{M}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\partial)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
113.06206pt\raise-20.63887pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
121.0065pt\raise-20.63887pt\hbox{{}\hbox{\kern 0.0pt\raise
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0.0pt\raise-3.0861pt\hbox{$\scriptstyle{\eucal{P}^{\mathrm{rd}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 121.0065pt\raise-5.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\end{array}$

as well as a similar diagram for $\mathsf{Q}_{\mathrm{mod},\mathrm{rd}}$. For
each $r\geqslant 0$, let
$\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{{\mathrm{dR}},r}$ and
$\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{{\mathrm{dR}},r}$ be the global period
pairings

$\displaystyle\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{{\mathrm{dR}},r}:\Gamma(M,\mathfrak{C}_{M}^{\mathrm{rd},m-r}(\mathscr{V}))^{\partial}\otimes\Gamma(M,\mathscr{E}_{M}^{\mathrm{mod},r}\otimes\mathscr{V})^{\nabla}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma(M,\mathfrak{C}_{M}^{\mathrm{rd},m})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}$
$\displaystyle\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{{\mathrm{dR}},r}:\Gamma(M,\mathfrak{C}_{M}^{\mathrm{mod},m-r}(\mathscr{V}))^{\partial}\otimes\Gamma(M,\mathscr{E}_{M}^{\mathrm{rd},r}\otimes\mathscr{V})^{\nabla}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma(M,\mathfrak{C}_{M}^{\mathrm{rd},m})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}$

obtained by evaluating on the constant function $1$ the zero-dimensional
rapid-decay current provided by
$\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{\mathrm{dR}}$ and
$\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{\mathrm{dR}}$ respectively. Then,
according to Stokes formula,
$\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{{\mathrm{dR}},r}$ and
$\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{{\mathrm{dR}},r}$ vanish if one of the
terms is a boundary, and hence induce pairings

$\displaystyle\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{{\mathrm{dR}},r}:\mathrm{H}_{{\mathrm{dR}},r}^{\mathrm{rd}}(M,\mathscr{V})\otimes\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}$
$\displaystyle\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{{\mathrm{dR}},r}:\mathrm{H}_{{\mathrm{dR}},r}^{\mathrm{mod}}(M,\mathscr{V})\otimes\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}.$

It is then clear that the following diagram commutes (and similarly for
$\mathsf{Q}^{r}_{\mathrm{mod},\mathrm{rd}}$):

(2.10) $\begin{array}[]{c}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
57.46936pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
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0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-34.44434pt\raise-20.72081pt\hbox{{}\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern-34.44434pt\raise-6.03333pt\hbox{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
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We then deduce from Corollary 2.4:

######  Corollary 2.11.

The period pairings $\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{{\mathrm{dR}},r}$
and $\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{{\mathrm{dR}},m-r}$ are perfect.∎

#### De Rham intersection pairing

We can apply once more the Poincaré isomorphisms to (2.9), and get the
morphism $\mathsf{B}_{\mathrm{dR}}^{\mathrm{mod},\mathrm{rd}}$
(resp.$\mathsf{B}_{\mathrm{dR}}^{\mathrm{rd},\mathrm{mod}}$) making the
following (cohomological) diagram commute:

$\textstyle{\mathfrak{C}_{M}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}(\mathscr{V})\otimes\operatorname{DR}^{\mathrm{rd}}(\mathscr{V},\nabla)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{\mathrm{dR}}}$$\scriptstyle{\eucal{P}^{\mathrm{rd}}(\mathscr{V})}$$\scriptstyle{\wr}$$\textstyle{(\mathfrak{C}_{M}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\partial)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{C}_{M}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}(\mathscr{V})\otimes\mathfrak{C}_{M}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}(\mathscr{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{B}_{\mathrm{dR}}^{\mathrm{mod},\mathrm{rd}}}$$\textstyle{(\mathfrak{C}_{M}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\partial)}$

and therefore also the following:

(2.12) $\begin{array}[]{c}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{H}_{{\mathrm{dR}},m-r}^{\mathrm{mod}}(M,\mathscr{V})\otimes\mathrm{H}_{{\mathrm{dR}},r}^{\mathrm{rd}}(\mathscr{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
68.50851pt\raise-34.64163pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
108.86678pt\raise-41.81107pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
108.86678pt\raise-41.81107pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{$\textstyle{\mathbf{C}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{array}$

For cohomology classes
$[\eta^{r}]\in\mathrm{H}^{m-r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V})$ and
$[\omega^{m-r}]\in\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V})$,
we have by definition

$\mathsf{Q}_{\mathrm{rd},\mathrm{mod}}^{r}([\eta^{r}],[\omega^{m-r}])=\mathsf{B}_{{\mathrm{dR}},m-r}^{\mathrm{rd},\mathrm{mod}}(\eucal{P}^{\mathrm{rd}}_{m-r}([\eta^{r}]),\eucal{P}^{\mathrm{mod}}_{r}([\omega^{m-r}])).$

######  Corollary 2.13.

The de Rham intersection pairings
$\mathsf{B}_{{\mathrm{dR}},r}^{\mathrm{rd},\mathrm{mod}},\mathsf{B}_{{\mathrm{dR}},r}^{\mathrm{mod},\mathrm{rd}}$
are perfect for each $r\geqslant 0$.

###### Proof.

This follows from the perfectness of the period pairings (seeCorollary 2.11).
∎

#### Quadratic relations

We assume that the cohomology and homology vector spaces we consider are
finite-dimensional and we fix bases of these spaces. We denote by the same
letter a pairing and its matrix with respect to the corresponding bases, and
we use the matrix notation in which, for a pairing $\mathsf{A}$, we also
denote $\mathsf{A}(x\otimes y)$ by ${}^{t}x\cdot\mathsf{A}\cdot y$, and the
transpose of the matrix $\mathsf{A}$ by ${}^{t}\mathsf{A}$.

######  Corollary 2.14 (Quadratic relations).

The matrices of the pairings satisfy the following relations:

${}^{t}\mathsf{B}_{{\mathrm{dR}},m-r}^{\mathrm{mod},\mathrm{rd}}=\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{{\mathrm{dR}},r}\circ(\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}})^{-1}\circ{}^{t}\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{{\mathrm{dR}},m-r}.$

###### Proof.

In terms of matrices, the commutations (2.10) and (2.12) read

${}^{t}\eucal{P}_{r}^{\mathrm{rd}}(\mathscr{V})\cdot\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{{\mathrm{dR}},r}=\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}},\quad\mathsf{B}_{{\mathrm{dR}},m-r}^{\mathrm{mod},\mathrm{rd}}\cdot\eucal{P}_{r}^{\mathrm{rd}}(\mathscr{V})=\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{{\mathrm{dR}},m-r},$

and, by eliminating $\eucal{P}_{r}^{\mathrm{rd}}(\mathscr{V})$, we obtain the
sought for relation. ∎

See Remark 2.17 for a justification of the terminology “quadratic relations”.

######  Remark 2.15 (Quadratic relations in presence of $\pm$-symmetry).

Let us assume that
$\left\langle{{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\right\rangle$
is either symmetric or skew-symmetric, a property that we call _$\pm$
-symmetric._ Let us denote by

${}^{t}\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}}:\mathrm{H}_{{\mathrm{dR}},\mathrm{mod}}^{r}(M,\mathscr{V})\otimes\mathrm{H}_{{\mathrm{dR}},\mathrm{rd}}^{m-r}(M,\mathscr{V})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}$

the pairing defined (with obvious notation) by

${}^{t}\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}}(h^{r},h^{m-r})=\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}}(h^{m-r},h^{r}),$

and let ${}^{t}\mathsf{Q}^{m-r}_{\mathrm{mod},\mathrm{rd}}$ be defined
similarly. If $\eta^{m-r}$ is a rapid decay $(m-r)$-form and $\omega^{r}$ an
$r$-form with moderate growth, and if $v,w$ are local sections of
$\mathscr{V}$, we have

$\displaystyle\mathsf{Q}_{\mathrm{rd},\mathrm{mod}}^{m-r}(\eta^{m-r}\otimes
v,\omega^{r}\otimes w)$ $\displaystyle=\langle
v,w\rangle\cdot\eta^{m-r}\wedge\omega^{r}$
$\displaystyle=\pm(-1)^{r(m-r)}\langle w,v\rangle\omega^{r}\wedge\eta^{m-r}$
$\displaystyle=\pm(-1)^{r(m-r)}\mathsf{Q}_{\mathrm{mod},\mathrm{rd}}^{r}(\omega^{r}\otimes
w,\eta^{m-r}\otimes v).$

Passing to cohomology, we find

${}^{t}\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}}=\pm(-1)^{r(m-r)}\mathsf{Q}_{\mathrm{mod},\mathrm{rd}}^{r},\qquad{}^{t}\mathsf{Q}^{m-r}_{\mathrm{mod},\mathrm{rd}}=\pm(-1)^{r(m-r)}\mathsf{Q}_{\mathrm{rd},\mathrm{mod}}^{r}.$

For the de Rham intersection pairing, we similarly find

${}^{t}\mathsf{B}^{\mathrm{mod},\mathrm{rd}}_{{\mathrm{dR}},m-r}=\pm(-1)^{r(m-r)}\mathsf{B}_{{\mathrm{dR}},r}^{\mathrm{rd},\mathrm{mod}}.$

As a consequence, the quadratic relations read

$\pm(-1)^{r(m-r)}\mathsf{B}_{{\mathrm{dR}},r}^{\mathrm{rd},\mathrm{mod}}=\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{{\mathrm{dR}},r}\circ(\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}})^{-1}\circ{}^{t}\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{{\mathrm{dR}},m-r}.$

###  2.c. Middle quadratic relations

For each degree $r$, we set

$\displaystyle\mathrm{H}^{\mathrm{mid}}_{{\mathrm{dR}},r}(M,\mathscr{V})$
$\displaystyle=\operatorname{im}\bigl{[}\mathrm{H}^{\mathrm{rd}}_{{\mathrm{dR}},r}(M,\mathscr{V})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}^{\mathrm{mod}}_{{\mathrm{dR}},r}(M,\mathscr{V})\bigr{]},$
$\displaystyle\mathrm{H}^{m-r}_{{\mathrm{dR}},{\mathrm{mid}}}(M,\mathscr{V})$
$\displaystyle=\operatorname{im}\bigl{[}\mathrm{H}^{m-r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}^{m-r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V})\bigr{]},$

where the maps are induced by the natural inclusion
$\mathscr{C}_{M}^{\infty,\mathrm{rd}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{C}_{M}^{\infty,\mathrm{mod}}$.
From (2.3) one deduces that $\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}}$ and
$\mathsf{Q}^{m-r}_{\mathrm{mod},\mathrm{rd}}$ induce the same _non-degenerate_
pairing

$\mathsf{Q}^{m-r}_{{\mathrm{mid}}}:\mathrm{H}_{{\mathrm{dR}},{\mathrm{mid}}}^{m-r}(M,\mathscr{V})\otimes\mathrm{H}_{{\mathrm{dR}},{\mathrm{mid}}}^{r}(M,\mathscr{V})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}.$

Similarly, according to (2.5), $\eucal{P}_{r}^{\mathrm{rd}}(\mathscr{V})$ and
$\eucal{P}_{r}^{\mathrm{mod}}(\mathscr{V})$ induce the same isomorphism

$\eucal{P}_{r}^{\mathrm{mid}}(\mathscr{V}):\mathrm{H}_{{\mathrm{dR}},{\mathrm{mid}}}^{m-r}(M,\mathscr{V})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}_{{\mathrm{dR}},r}^{\mathrm{mid}}(M,\mathscr{V}).$

It follows that $\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{{\mathrm{dR}},r}$ and
$\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{{\mathrm{dR}},r}$ induce the same
_non-degenerate_ period pairing

$\mathsf{P}^{{\mathrm{mid}}}_{{\mathrm{dR}},r}:\mathrm{H}_{{\mathrm{dR}},r}^{\mathrm{mid}}(M,\mathscr{V})\otimes\mathrm{H}_{{\mathrm{dR}},{\mathrm{mid}}}^{m-r}(M,\mathscr{V})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}.$

In the same way, $\mathsf{B}^{\mathrm{rd},\mathrm{mod}}_{{\mathrm{dR}},r}$ and
$\mathsf{B}^{\mathrm{mod},\mathrm{rd}}_{{\mathrm{dR}},r}$ induce the same
_non-degenerate_ intersection pairing

$\mathsf{B}^{{\mathrm{mid}}}_{{\mathrm{dR}},r}:\mathrm{H}_{{\mathrm{dR}},r}^{\mathrm{mid}}(M,\mathscr{V})\otimes\mathrm{H}_{{\mathrm{dR}},{m-r}}^{\mathrm{mid}}(M,\mathscr{V})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}.$

We conclude:

######  Corollary 2.16 (Middle quadratic relations).

These pairings are non-degenerate and their matrices satisfy the following
relations:

${}^{t}\mathsf{B}_{{\mathrm{dR}},m-r}^{{\mathrm{mid}}}=\mathsf{P}^{{\mathrm{mid}}}_{{\mathrm{dR}},r}\circ(\mathsf{Q}^{m-r}_{{\mathrm{mid}}})^{-1}\circ{}^{t}\mathsf{P}^{{\mathrm{mid}}}_{{\mathrm{dR}},m-r}.$

######  Remark 2.17 ($\pm$-Symmetry and quadratic relations).

Under the symmetry assumption of Remark 2.15, the following relations hold:

${}^{t}\mathsf{B}_{{\mathrm{dR}},m-r}^{{\mathrm{mid}}}=\pm(-1)^{r(m-r)}\mathsf{B}^{{\mathrm{mid}}}_{{\mathrm{dR}},r},\quad{}^{t}\mathsf{Q}^{r}_{{\mathrm{mid}}}=\pm(-1)^{r(m-r)}\mathsf{Q}_{{\mathrm{mid}}}^{m-r},$

and the relations of Corollary 2.16 read

$\pm(-1)^{r(m-r)}\mathsf{B}^{{\mathrm{mid}}}_{{\mathrm{dR}},r}=\mathsf{P}^{{\mathrm{mid}}}_{{\mathrm{dR}},r}\circ(\mathsf{Q}^{m-r}_{{\mathrm{mid}}})^{-1}\circ{}^{t}\mathsf{P}^{{\mathrm{mid}}}_{{\mathrm{dR}},m-r}.$

If $r=m-r$, we can regard these formulas as _algebraic relations of degree
two_ on the entries of the matrix
$\mathsf{P}^{{\mathrm{mid}}}_{{\mathrm{dR}},r}$, with coefficients in the
entries of $\mathsf{B}_{{\mathrm{dR}},r}^{{\mathrm{mid}}}$ and
$\mathsf{Q}^{m-r}_{{\mathrm{mid}}}$. This justifies the name “quadratic
relations” which, strictly speaking, only applies to the middle periods in
middle dimension.

######  Caveat 2.18 (On the notation ${\mathrm{mid}}$).

The terminology “middle quadratic relation” and the associated notation
${\mathrm{mid}}$ could be confusing, as it is usually associated with the
notion of intermediate (or middle) extension of a sheaf across a divisor.
Here, we use it in the naive sense above, and we do not claim in general (that
is, in the setting of Section 2.f below) any precise relation with the usual
notion of intermediate extension. However, we will check that both notions
coincide in complex dimension one (see(3.1)).

###  2.d. Betti formalism for quadratic relations

In the topological setting we replace the complex of currents
$(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{mod}},\partial)$ with
the complex of sheaves of relative singular chains $(\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}},\partial)$. We will consider the latter
sheaves with coefficients in $\mathbf{C}$ since we will compare them with
sheaves of currents. We refer to Appendix A for basic properties of this
complex, and we recall (seee.g.[25, p. 14]) that $(\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}},\partial)$, when regarded cohomologically by
setting $\eucal{C}_{M,{\partial
M}}^{m-{\scriptscriptstyle\bullet}}=\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}}$, is a homotopically fine resolution of
$\mathbf{C}_{M}$. Proposition A.17 enables us to replace the complex of
sheaves of singular chains with that of piecewise smooth singular chains, that
we will denote in the same way.

#### Betti period pairings

Integration along a piecewise smooth singular chain in $M$ of a test form with
rapid decay along ${\partial M}$ defines a morphism of chain complexes
$(\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}},\partial)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{mod}},\partial)$:
indeed, if $c$ is a $q$-dimensional chain and $\omega$ a form of degree $q$ on
$M^{\circ}$ with rapid decay, $\int_{c}\omega=0$ if $c$ is supported in
${\partial M}$; compatibility with $\partial$ follows from the Stokes formula.
This morphism is a quasi-isomorphism since it induces an isomorphism on the
unique non-zero homology sheaf of both complexes; it yields a morphism between
two resolutions of $\mathbf{C}_{M}$ taken in homological degree $m$.

We introduce (see Section A) the following complexes of singular chains:

  * $\scriptscriptstyle\bullet$

$(\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}}(\mathscr{V}^{\mathrm{rd}}),\partial)=(\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}},\partial)\otimes\mathscr{V}^{\mathrm{rd}}$:
chains with coefficients in $\mathscr{V}$ and rapid decay,

  * $\scriptscriptstyle\bullet$

$(\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}}(\mathscr{V}^{\mathrm{mod}}),\partial)=(\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}},\partial)\otimes\mathscr{V}^{\mathrm{mod}}$:
chains with coefficients in $\mathscr{V}$ and moderate growth.

We also set

$\displaystyle\mathrm{H}_{r}^{\mathrm{rd}}(M,\mathscr{V})$
$\displaystyle=\mathrm{H}_{r}\bigl{(}\Gamma(\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}}(\mathscr{V}^{\mathrm{rd}}),\partial)\bigr{)}$
and $\displaystyle\mathrm{H}_{r}^{\mathrm{mod}}(M,\mathscr{V})$
$\displaystyle=\mathrm{H}_{r}\bigl{(}\Gamma(\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}}(\mathscr{V}^{\mathrm{mod}}),\partial)\bigr{)}.$

We thus have quasi-isomorphisms of chain complexes

(2.19) $\begin{split}(\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}}(\mathscr{V}^{\mathrm{rd}}),\partial)&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{rd}}(\mathscr{V}),\partial),\\\
(\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}}(\mathscr{V}^{\mathrm{mod}}),\partial)&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{mod}}(\mathscr{V}),\partial),\end{split}$

which induce isomorphisms

(2.20)
$\mathrm{H}_{r}^{\mathrm{rd}}(M,\mathscr{V})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{H}_{{\mathrm{dR}},r}^{\mathrm{rd}}(M,\mathscr{V})\quad\text{and}\quad\mathrm{H}_{r}^{\mathrm{mod}}(M,\mathscr{V})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{H}_{{\mathrm{dR}},r}^{\mathrm{mod}}(M,\mathscr{V}).$

Composing (2.7) and (2.8) with these morphisms gives back the period pairings
as those defined in [25], that we denote by
$\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{r}$ and
$\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{r}$. In particular, these Betti period
pairings are perfect. Notice that, by means of these identifications, the
Poincaré-de Rham isomorphisms (2.6) correspond to the Poincaré isomorphisms,
denoted similarly:

(2.21)
$\begin{split}\eucal{P}_{r}^{\mathrm{rd}}(\mathscr{V}):\mathrm{H}^{m-r}(M,\mathscr{V}^{\mathrm{rd}})&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{H}_{r}(M,\mathscr{V}^{\mathrm{rd}}),\\\
\eucal{P}_{r}^{\mathrm{mod}}(\mathscr{V}):\mathrm{H}^{m-r}(M,\mathscr{V}^{\mathrm{mod}})&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{H}_{r}(M,\mathscr{V}^{\mathrm{mod}}).\end{split}$

According to Propositions A.14 and A.17, the homology spaces
$\mathrm{H}_{r}^{\mathrm{rd}}(M,\mathscr{V})$
(resp.$\mathrm{H}_{r}^{\mathrm{mod}}(M,\mathscr{V})$) can be computed as the
homology of the chain complexes

$(\widetilde{\eucal}{C}^{\mathrm{sm}}_{M,{\partial
M},{\scriptscriptstyle\bullet}}(M,\mathscr{V}^{\mathrm{rd}})),\partial)\quad(\text{resp.}\kern
3.00003pt(\widetilde{\eucal}{C}^{\mathrm{sm}}_{M,{\partial
M},{\scriptscriptstyle\bullet}}(M,\mathscr{V}^{\mathrm{mod}})),\partial))$

of piecewise smooth singular chains with coefficients in
$\mathscr{V}^{\mathrm{rd}}$ (resp.$\mathscr{V}^{\mathrm{mod}}$).

On the one hand, let
$\sigma:\Delta_{r}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M$
be a piecewise smooth singular simplex (that we can assume not contained in
${\partial M}$ since we only consider relative simplices), and let $v$ be a
section of $\mathscr{V}^{\mathrm{rd}}$ in the neighbourhood of $|\sigma|$, so
that $\sigma\otimes v$ belongs to $\widetilde{\eucal}{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}}^{\mathrm{sm}}(M,\mathscr{V}^{\mathrm{rd}})$.
On the other hand, note that, by using a partition of unity, any section in
$\Gamma(M,\mathscr{E}_{M}^{\mathrm{mod},r}\otimes\mathscr{V})$ is a sum of
terms $\omega^{r}\otimes w$, where $w$ is a section of $\mathscr{V}$ in the
neighbourhood of the support of the $C^{\infty}$ $r$-form $\omega^{r}$. Then
$\langle v,w\rangle$ is a $C^{\infty}$ function on the open set where both
sections are defined and has rapid decay along ${\partial M}$, so that
$\langle v,w\rangle\cdot\omega^{r}$ is a $C^{\infty}$-form with rapid decay,
and hence integrable along $\sigma$. We have

$\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{r}(\sigma\otimes v,\omega^{r}\otimes
w)=\int_{\sigma}\langle
v,w\rangle\cdot\omega^{r}=\int_{\Delta_{r}}\sigma^{*}\langle
v,w\rangle\cdot\sigma^{*}\omega^{r}.$

This interpretation makes the link between the approaches in [5] and [25]. A
similar interpretation holds for $\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{r}$:
now $\langle v,w\rangle$ has only moderate growth, but $\omega^{r}$ has rapid
decay, so that $\langle v,w\rangle\cdot\omega^{r}$ remains a $C^{\infty}$-form
with rapid decay.

#### Existence of a ${\boldsymbol{k}}$-structure

Let us fix a subfield ${\boldsymbol{k}}$ of $\mathbf{C}$. The sheaves
$\eucal{C}_{M,{\partial M},{\scriptscriptstyle\bullet}}$ are now considered
with coefficients in $\mathbf{Q}$. We will make explicit the
${\boldsymbol{k}}$-structure on the various cohomology groups occurring in the
de Rham model, provided that a ${\boldsymbol{k}}$-structure exists on the
underlying sheaves. If we are given a ${\boldsymbol{k}}$-structure
$(\mathscr{V}^{\nabla})_{\boldsymbol{k}}$ of $\mathscr{V}^{\nabla}$, we enrich
Assumptions 2.1 as follows.

######  Assumption 2.22.

There exist subsheaves
$\mathscr{V}_{\boldsymbol{k}}^{\mathrm{rd}}\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}\mathscr{V}_{\boldsymbol{k}}^{\mathrm{mod}}\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}j_{*}(\mathscr{V}^{\nabla})_{\boldsymbol{k}}$
of ${\boldsymbol{k}}$-vector spaces of $j_{*}\mathscr{V}^{\nabla}$ and a
nondegenerate pairing
$\left\langle{{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\right\rangle_{\boldsymbol{k}}$
on $(\mathscr{V}^{\nabla})_{\boldsymbol{k}}$ giving rise to
$\mathscr{V}^{\mathrm{rd}}\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}\mathscr{V}^{\mathrm{mod}}\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}j_{*}\mathscr{V}^{\nabla}$
and
$\left\langle{{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\right\rangle$
after tensoring with $\mathbf{C}$.

As a consequence, the perfect pairings
$\mathscr{H}^{0}\mathsf{Q}_{\mathrm{rd},\mathrm{mod}}$ and
$\mathscr{H}^{0}\mathsf{Q}_{\mathrm{mod},\mathrm{rd}}$ in Assumption 2.1(2)
are defined over ${\boldsymbol{k}}$:

$\mathscr{V}^{\mathrm{rd}}_{\boldsymbol{k}}\otimes\mathscr{V}^{\mathrm{mod}}_{\boldsymbol{k}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}j_{!}{\boldsymbol{k}}_{M^{\circ}},\quad\mathscr{V}^{\mathrm{mod}}_{\boldsymbol{k}}\otimes\mathscr{V}^{\mathrm{rd}}_{\boldsymbol{k}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}j_{!}{\boldsymbol{k}}_{M^{\circ}}.$

In particular, $\mathscr{H}^{0}$ of the commutative diagram (2.3) is defined
over ${\boldsymbol{k}}$. Furthermore, the de Rham isomorphism induces a
${\boldsymbol{k}}$-structure on the de Rham cohomology groups, by setting e.g.

$\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V})_{\boldsymbol{k}}=\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{rd}}_{\boldsymbol{k}})$

via the de Rham isomorphism
$\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{rd}})$.
The pairings of Corollary 2.4 are defined over ${\boldsymbol{k}}$ and
correspond to Poincaré-Verdier duality over ${\boldsymbol{k}}$.

The ${\boldsymbol{k}}$-structures on rapid decay and moderate growth cycles
are obtained by means of the complexes $(\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}},\partial)\otimes\mathscr{V}^{\mathrm{rd}}_{\boldsymbol{k}}$
and $(\eucal{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}},\partial)\otimes\mathscr{V}^{\mathrm{mod}}_{\boldsymbol{k}}$.
In other words, we set

$\mathrm{H}_{r}^{\mathrm{rd}}(M,\mathscr{V})_{\boldsymbol{k}}=\mathrm{H}_{r}(M,\mathscr{V}^{\mathrm{rd}}_{\boldsymbol{k}}),\
\text{etc.}$

The Poincaré isomorphisms (2.21) are then defined over ${\boldsymbol{k}}$.

######  Proposition 2.23.

Under Assumptions 2.1 and 2.22, let $(v_{i})_{i}$ be a basis of
$\mathrm{H}^{m-r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V})$ and let
$(v_{i}^{\scriptscriptstyle\vee})_{i}$ be the
$\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}}$-dual basis of
$\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V})$. The
${\boldsymbol{k}}$-vector space
$\mathrm{H}^{m-r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V})_{\boldsymbol{k}}$
is the ${\boldsymbol{k}}$-subspace of
$\mathrm{H}^{m-r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V})$ consisting of
the elements

$\sum_{i}\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{{\mathrm{dR}},r}(\alpha,v_{i}^{\scriptscriptstyle\vee})\cdot
v_{i},$

where $\alpha$ runs in
$\mathrm{H}_{r}(M,\mathscr{V}^{\mathrm{rd}}_{\boldsymbol{k}})$. A similar
assertion holds for
$\mathrm{H}^{m-r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V})_{\boldsymbol{k}}$.

###### Proof.

Let $e\in\mathrm{H}^{m-r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V})$ be an
element written as

$e=\sum_{i}\mathsf{Q}_{\mathrm{rd},\mathrm{mod}}^{m-r}(e,v_{i}^{\scriptscriptstyle\vee})\cdot
v_{i}.$

Let us set
$\alpha=\eucal{P}_{r}^{\mathrm{rd}}(e)\in\mathrm{H}_{{\mathrm{dR}},r}^{\mathrm{rd}}(M,\mathscr{V})$.
According to (2.10), we can rewrite $e$ as

$e=\sum_{i}\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{{\mathrm{dR}},r}(\alpha,v_{i}^{\scriptscriptstyle\vee})\cdot
v_{i}.$

The class $e$ belongs to
$\mathrm{H}^{m-r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V})_{\boldsymbol{k}}$
if and only if, when regarded as in
$\mathrm{H}^{m-r}(M,\mathscr{V}^{\mathrm{rd}})$, it belongs to
$\mathrm{H}^{m-r}(M,\mathscr{V}^{\mathrm{rd}}_{\boldsymbol{k}})$,
equivalently, its image $\alpha$ by the Poincaré isomorphism
$\eucal{P}_{r}^{\mathrm{rd}}(\mathscr{V})$ belongs to
$\mathrm{H}_{r}(M,\mathscr{V}^{\mathrm{rd}}_{\boldsymbol{k}})$. The assertion
of the proposition follows from the above identifications. ∎

#### Middle quadratic relations

The Betti intersection pairing

$\mathsf{B}^{\mathrm{rd},\mathrm{mod}}_{r}:\mathrm{H}_{r}^{\mathrm{rd}}(M,\mathscr{V})\otimes\mathrm{H}_{m-r}^{\mathrm{mod}}(M,\mathscr{V})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}$

is defined from the de Rham intersection pairing
$\mathsf{B}^{\mathrm{rd},\mathrm{mod}}_{{\mathrm{dR}},r}$ (seeCorollary 2.13)
by composing it with the isomorphisms (2.20). It is thus a _perfect pairing_.
A similar result holds for $\mathsf{B}^{\mathrm{mod},\mathrm{rd}}_{r}$.

From Corollary 2.16 we obtain immediately (and of course similarly for the
rapid decay and moderate growth analogues):

######  Corollary 2.24.

The middle Betti period pairings satisfy the quadratic relations of Corollary
2.16, where the de Rham intersection pairing
$\mathsf{B}^{\mathrm{mid}}_{{\mathrm{dR}},r}$ is replaced with the Betti
intersection pairing $\mathsf{B}^{\mathrm{mid}}_{r}$. That is,

${}^{t}\mathsf{B}^{\mathrm{mid}}_{m-r}=\mathsf{P}^{{\mathrm{mid}}}_{r}\circ(\mathsf{Q}_{\mathrm{mid}}^{m-r})^{-1}\circ{}^{t}\mathsf{P}^{{\mathrm{mid}}}_{m-r}.$

Under the symmetry assumption of Remark 2.17, it reads

$\pm(-1)^{r(m-r)}\mathsf{B}^{{\mathrm{mid}}}_{r}=\mathsf{P}^{{\mathrm{mid}}}_{r}\circ(\mathsf{Q}_{\mathrm{mid}}^{m-r})^{-1}\circ{}^{t}\mathsf{P}^{{\mathrm{mid}}}_{m-r}.$

#### Computation of the Betti intersection pairing

Let us assume that $(M,{\partial M})$ is endowed with a simplicial
decomposition $\mathscr{T}$ such that the sheaves
$\mathscr{V}^{\mathrm{rd}},\mathscr{V}^{\mathrm{mod}}$ satisfy Assumption A.18
and are locally constant on $M^{\circ}$. Then we can replace the complex
$(\widetilde{\eucal}{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}}^{\mathrm{sm}}(M,\mathscr{V}^{\mathrm{rd}})),\partial)$,
respectively the complex $(\widetilde{\eucal}{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}}^{\mathrm{sm}}(M,\mathscr{V}^{\mathrm{mod}})),\partial)$,
with the corresponding simplicial complex $(\widetilde{\eucal}{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}}^{\scriptscriptstyle\triangle}(M,\mathscr{V}^{\mathrm{rd}})),\partial)$,
respectively $(\widetilde{\eucal}{C}_{M,{\partial
M},{\scriptscriptstyle\bullet}}^{\scriptscriptstyle\square}(M,\mathscr{V}^{\mathrm{mod}})),\partial)$
(seeSection A.f).

The Betti intersection pairing can easily be computed in the framework of
simplicial chain complexes under this assumption. Indeed, choose an
orientation for each simplex (it is natural to assume that the maximal-
dimensional simplices have the orientation induced by that of $M$) and an
orientation for each dual cell $\overline{D}(\sigma)$. Let $\sigma_{r}\otimes
v$ be a simplex of dimension $r$ with coefficient $v$ in
$\mathscr{V}^{\mathrm{rd}}$ ($\sigma_{r}\not\subset{\partial M}$) and let
$\overline{D}(\sigma^{\prime}_{r})\otimes w$ be a cell of codimension $r$ with
coefficient $w$ in $\mathscr{V}^{\mathrm{mod}}$ for some simplex
$\sigma^{\prime}_{r}$ of $\mathscr{T}$ (possibly
$\sigma^{\prime}_{r}\subset{\partial M}$). We regard them as currents with
rapid decay and moderate growth respectively, according to (2.19).

######  Proposition 2.25 (Computation of the Betti intersection pairing).

With these assumptions,

$\mathsf{B}^{\mathrm{rd},\mathrm{mod}}_{r}(\sigma_{r}\otimes
v,\overline{D}(\sigma^{\prime}_{r})\otimes w)=\begin{cases}0&\text{if
}\sigma^{\prime}_{r}\neq\sigma_{r},\\\ \varepsilon(\sigma_{r})\cdot\langle
v,w\rangle(\widehat{\sigma}_{r})&\text{if
}\sigma^{\prime}_{r}=\sigma_{r},\end{cases}$

where $\widehat{\sigma}_{r}$ is the barycenter of $\sigma_{r}$ and
$\varepsilon(\sigma_{r})=\pm 1$ is the orientation change between
$\sigma_{r}\times\overline{D}(\sigma_{r})$ and $M$.

###### Proof.

By definition, we have

(2.26) $\mathsf{B}^{\mathrm{rd},\mathrm{mod}}_{r}(\sigma_{r}\otimes
v,\overline{D}(\sigma^{\prime}_{r})\otimes w)=\langle
v,w\rangle\cdot\mathsf{B}_{{\mathrm{dR}},r}(\sigma_{r},\overline{D}(\sigma^{\prime}_{r})),$

where $\mathsf{B}_{{\mathrm{dR}},r}$ corresponds to de Rham’s definition of
the Kronecker index (see[14, §20]).

For any $r$-dimensional simplex $\sigma^{\prime}_{r}$ of $\mathscr{T}$,
$\sigma^{\prime}_{r}$ is the only $r$-dimensional simplex of $\mathscr{T}$
physically intersected by the cell $\overline{D}(\sigma^{\prime}_{r})$
(seeLemma A.22). Hence, given $\sigma_{r}\\!\not\subset\\!{\partial M}$, we
have
$\mathsf{B}_{{\mathrm{dR}},r}(\sigma_{r},\overline{D}(\sigma^{\prime}_{r}))\\!=\\!0$
unless $\sigma^{\prime}_{r}=\sigma_{r}$ and we are reduced to computing (2.26)
in that case. This computation, done by de Rham (see[14, p. 85–86]), is
recalled in the appendix (Proposition B.7). The current
$\mathsf{B}_{{\mathrm{dR}},r}(\sigma_{r},\overline{D}(\sigma_{r}))$ is
supported on $\widehat{\sigma}_{r}=\sigma_{r}\cap\overline{D}(\sigma_{r})$ and
has coefficient $\varepsilon(\sigma_{r})$, so that

$\langle
v,w\rangle\cdot\mathsf{B}_{{\mathrm{dR}},r}(\sigma_{r},\overline{D}(\sigma_{r}))=\langle
v,w\rangle(\widehat{\sigma}_{r})\cdot\mathsf{B}_{{\mathrm{dR}},r}(\sigma_{r},\overline{D}(\sigma_{r}))=\varepsilon(\sigma_{r})\langle
v,w\rangle(\widehat{\sigma}_{r}).\qed$

###  2.e. Period realizations

We keep the setting and notation of Section 2.d. The abelian category
$\mathsf{Per}_{\boldsymbol{k}}$ of _${\boldsymbol{k}}$ -period structures_ has
objects consisting of pairs of finite dimensional vector spaces
$(V_{\mathbf{C}},V_{\boldsymbol{k}})$ together with an isomorphism
$\mathrm{per}:V_{\mathbf{C}}\simeq\mathbf{C}\otimes_{\boldsymbol{k}}V_{\boldsymbol{k}}$.
The morphisms are the natural ones. It is convenient to consider this category
from a dual point of view. Namely, we consider the category
$\mathsf{Per}_{\boldsymbol{k}}^{\scriptscriptstyle\vee}$ whose objects are
triples $(V_{\mathbf{C}},V_{\boldsymbol{k}},\mathsf{P})$, where
$\mathsf{P}:V_{\mathbf{C}}\otimes(V_{\boldsymbol{k}}\otimes\mathbf{C})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}$
is a nondegenerate pairing, and whose morphisms are the natural ones. The
duality functor
$V_{\boldsymbol{k}}\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}V_{\boldsymbol{k}}^{\scriptscriptstyle\vee}$
induces an equivalence
$T:\mathsf{Per}_{\boldsymbol{k}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathsf{Per}_{\boldsymbol{k}}^{\scriptscriptstyle\vee}$
by means of the tautological pairing
$({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\mkern 1.0mu|\mkern
1.0mu{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}):V_{\boldsymbol{k}}\otimes
V_{\boldsymbol{k}}^{\scriptscriptstyle\vee}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\boldsymbol{k}}$.

Let us consider
$(\mathscr{V},\nabla,\left\langle{{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\right\rangle)$
satisfying Assumptions 2.1 and 2.22. It defines moderate and rapid decay
period structures:

$\displaystyle\bigl{(}\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V}),\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{mod}}_{\boldsymbol{k}}),\mathrm{per}_{\mathrm{mod}}\bigr{)},$
$\displaystyle\bigl{(}\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V}),\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{rd}}_{\boldsymbol{k}}),\mathrm{per}_{\mathrm{rd}}\bigr{)},$

where
$\mathrm{per}_{\mathrm{mod}}:\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{mod}}_{\boldsymbol{k}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V})$
is composed from the natural isomorphism
$\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{mod}}_{\boldsymbol{k}})\otimes\mathbf{C}\simeq\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{mod}})$
and the inverse of the de Rham isomorphism
$\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{mod}})$,
and $\mathrm{per}_{\mathrm{rd}}$ is defined similarly.

######  Lemma 2.27.

There is a natural isomorphism

$\bigl{(}\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V}),\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{rd}}_{\boldsymbol{k}}),\mathrm{per}_{\mathrm{rd}}\bigr{)}\simeq\bigl{(}\mathrm{H}^{m-r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V}),\mathrm{H}^{m-r}(M,\mathscr{V}^{\mathrm{mod}}_{\boldsymbol{k}}),\mathrm{per}_{\mathrm{mod}}\bigr{)}^{\scriptscriptstyle\vee}.$

###### Proof.

This follows from the compatibility of the pairing $\mathsf{Q}$ and the
Poincaré-Verdier duality pairing via the de Rham isomorphism. ∎

######  Proposition 2.28.

There are isomorphisms of ${\boldsymbol{k}}$-period structures

$\displaystyle
T\bigl{(}\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V}),\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{mod}}_{\boldsymbol{k}}),\mathrm{per}_{\mathrm{mod}}\bigr{)}$
$\displaystyle\simeq\bigl{(}\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V}),\mathrm{H}_{r}^{\mathrm{rd}}(M,\mathscr{V})_{\boldsymbol{k}},\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{r}\bigr{)},$
$\displaystyle
T\bigl{(}\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V}),\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{rd}}_{\boldsymbol{k}}),\mathrm{per}_{\mathrm{rd}}\bigr{)}$
$\displaystyle\simeq\bigl{(}\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{rd}}(M,\mathscr{V}),\mathrm{H}_{r}^{\mathrm{mod}}(M,\mathscr{V})_{\boldsymbol{k}},\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{r}\bigr{)}.$

###### Proof.

We will check the first case for example. The pairing corresponding to the
dual presentation of the left-hand side amounts, according to the Poincaré-
Verdier duality pairing

$\mathrm{H}^{m-r}(M,\mathscr{V}^{\mathrm{rd}}_{\boldsymbol{k}})\otimes\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{mod}}_{\boldsymbol{k}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\boldsymbol{k}},$

to the period pairing

$\mathsf{Q}_{\boldsymbol{k}}^{m-r}:\mathrm{H}^{m-r}(M,\mathscr{V}^{\mathrm{rd}}_{\boldsymbol{k}})\otimes\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C},$

which corresponds

  * $\scriptscriptstyle\bullet$

either to the complexified Poincaré-Verdier pairing via the de Rham
isomorphism

$\mathrm{H}^{r}_{{\mathrm{dR}},\mathrm{mod}}(M,\mathscr{V})\\!\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\\!\mathrm{H}^{r}(M,\mathscr{V}^{\mathrm{mod}}),$

  * $\scriptscriptstyle\bullet$

or to the pairing $\mathsf{Q}^{m-r}_{\mathrm{rd},\mathrm{mod}}$ after
complexifying the first term and applying the corresponding de Rham
isomorphism.

Let us recall that the Poincaré isomorphism (2.6) is defined over
${\boldsymbol{k}}$ (see(2.21)), so that, applying it to the first term, we
obtain the period pairing $\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{r}$. ∎

###  2.f. Quadratic relations for good meromorphic flat bundles

We now consider the complex setup. Let $X$ be a compact complex manifold of
complex dimension $n$, let $D$ be a divisor with normal crossings, and let
$(\mathscr{V},\nabla)$ be a coherent $\mathscr{O}_{X}(*D)$-module with
integrable connection, endowed with a non-degenerate flat pairing
$\left\langle{{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\right\rangle:\mathscr{V}\otimes\mathscr{V}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{O}_{X}(*D)$.
Let
$\varpi:\widetilde{X}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$
denote the oriented real blow-up of $X$ along the irreducible components of
$D$. In a local chart of $X$ with coordinates $x_{1},\dots,x_{n}$ where $D$ is
defined by the equation $x_{1}\cdots x_{\ell}=0$, $\widetilde{X}$ is the space
of polar coordinates
$(r_{1},e^{\mathsf{i}\theta_{1}},\dots,r_{\ell},e^{\mathsf{i}\theta_{\ell}},x_{\ell+1},\dots,x_{n})$
and the $i$-th component of $\varpi$ ($1\leqslant i\leqslant\ell$) sends
$(r_{i},e^{\mathsf{i}\theta_{i}})$ to $x_{i}=r_{i}\cdot
e^{\mathsf{i}\theta_{i}}$ (seee.g.[51, §8.2] for the global construction of
$(\widetilde{X},\varpi)$). We set
$\widetilde{D}=\varpi^{-1}(D)=\partial\widetilde{X}$, and
$(\widetilde{X},\widetilde{D})$ will play the role of $(M,{\partial M})$ in
the settings 2.a.

There is a natural Cauchy-Riemann operator acting on the sheaf of $C^{\infty}$
functions on $\widetilde{X}$ and one defines the sheaves
$\mathscr{A}_{\widetilde{X}},\mathscr{A}_{\widetilde{X}}^{\mathrm{rd}},\mathscr{A}_{\widetilde{X}}^{\mathrm{mod}}$
respectively as the subsheaves of
$\mathscr{C}_{\widetilde{X}}^{\infty},\mathscr{C}_{\widetilde{X}}^{\infty,\mathrm{rd}},\mathscr{C}_{\widetilde{X}}^{\infty,\mathrm{mod}}$
annihilated by the Cauchy-Riemann operator (seee.g.[50]). Setting
$\widetilde{\mathscr{V}}=\mathscr{A}_{\widetilde{X}}\otimes_{\varpi^{-1}\mathscr{O}_{X}}\varpi^{-1}\mathscr{V}$,
the connection $\nabla$ lifts to a connection
$\widetilde{\mathscr{V}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\varpi^{-1}\Omega^{1}_{X}\otimes_{\varpi^{-1}\mathscr{O}_{X}}\widetilde{\mathscr{V}}$.
There is thus the corresponding notion of holomorphic de Rham complexes with
growth conditions, that we denote by
$\operatorname{DR}^{\mathrm{rd}}(\widetilde{\mathscr{V}},\nabla)$ and
$\operatorname{DR}^{\mathrm{mod}}(\widetilde{\mathscr{V}},\nabla)$.

######  Remark 2.29.

By the Dolbeault lemmas recalled in Appendix B, the complex
$\operatorname{DR}^{\mathrm{rd}}(\widetilde{\mathscr{V}},\nabla)$ is quasi-
isomorphic to its $C^{\infty}$ analogue, and the same property holds for the
complex $\operatorname{DR}^{\mathrm{mod}}(\widetilde{\mathscr{V}},\nabla)$.
Hence there is no risk of confusion. On the other hand, moderate and rapid
decay de Rham hypercohomologies of $\widetilde{\mathscr{V}}$ on
$\widetilde{X}$ can be identified with analytic de Rham hypercohomologies on
$X$ of $\mathscr{V}$ and of the dual $\mathscr{D}_{X}$-module
$\mathscr{V}(!D)$ respectively (see[45, Chap. 5]). Under these
identifications, the pairing
$\widetilde{\mathsf{S}}^{r}_{\mathrm{rd},\mathrm{mod}}$ defined in Convention
2.31 below induces a pairing $\mathsf{S}^{r}$ between
$\mathrm{H}^{r}(X,\operatorname{DR}(\mathscr{V}(!D))$ and
$\mathrm{H}^{2n-r}(X,\operatorname{DR}(\mathscr{V}))$. Furthermore, if $X$ is
projective,
$(\mathscr{V},\nabla,\langle{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\rangle)$
is the analytification of an
$\mathscr{O}_{X^{\mathrm{alg}}}(*D^{\mathrm{alg}})$-module with connection
$(\mathscr{V}^{\mathrm{alg}},\nabla^{\mathrm{alg}})$ equipped with a
nondegenerate pairing
$\langle{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\rangle$
(this follows from GAGA and the definition of a connection as in [15,
§I.2.3]). Setting $U=X^{\mathrm{alg}}\smallsetminus D^{\mathrm{alg}}$ and
$V=\mathscr{V}^{\mathrm{alg}}|_{U}$, then in each degree $r$ the pairing
$\mathsf{S}^{r}$ is the analytification of a pairing

$\mathsf{S}^{r}:\mathrm{H}_{{\mathrm{dR}},{\mathrm{c}}}^{r}(U,V)\otimes_{\mathbf{C}}\mathrm{H}_{{\mathrm{dR}}}^{2n-r}(U,V)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C},$

which is thus nondegenerate. In Section 3, where $X$ is algebraic of dimension
one, we provide an algebraic computation of $\mathsf{S}^{1}$.

#### The goodness property

Let $x$ be a point in $D$, and let $(x_{1},\dots,x_{n})$ be local complex
coordinates of $X$ centered at $x$ such that the germ $D_{x}$ is given by
$x_{1}\cdots x_{\ell}=0$ ($\ell\leqslant n$). Let $\mathscr{V}_{\widehat{x}}$
be the formal germ of $\mathscr{V}$ at $x$, which is an
$\mathscr{O}_{\widehat{x}}(*D)$-module of finite type with the integrable
connection induced by $\nabla$. We say that $(\mathscr{V},\nabla)$ is _good_
at $x$ if

  1. (a)

there exists a finite ramification

$\rho_{x}:(x^{\prime}_{1},\dots,x^{\prime}_{\ell},x_{\ell+1},\dots,x_{n})\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}(x_{1}=x^{\prime
p}_{1},\dots,x_{\ell}=x^{\prime p}_{\ell},x_{\ell+1},\dots,x_{n})$

such that the pullback $\rho_{x}^{*}(\mathscr{V}_{\widehat{x}},\nabla)$
decomposes as the direct sum of free
$\mathscr{O}_{\widehat{x}^{\prime}}(*D^{\prime})$-modules of finite rank with
an integrable connection of the form
$\nabla={\mathrm{d}}\varphi+\nabla^{\mathrm{reg}}$, where
$\nabla^{\mathrm{reg}}$ has a regular singularities along
$D^{\prime}=\rho_{x}^{-1}D$ and $\varphi$ belongs to a finite family
$\Phi_{x}\subset\mathscr{O}_{X^{\prime},x}(*D^{\prime})/\mathscr{O}_{X^{\prime},x}$,

  2. (b)

the finite family $\Phi_{x}$ is _good_ , that is, for any $\varphi\neq\psi$ in
$\Phi_{x}\cup\\{0\\}$, the divisor of the meromorphic function $\varphi-\psi$
is negative.111Goodness usually involves $\Phi_{x}$ rather than
$\Phi_{x}\cup\\{0\\}$. This stronger condition is needed for Assumptions 2.1
to be satisfied.

We say that $(\mathscr{V},\nabla)$ is _good_ if it is good at any $x\in X$. In
particular, $\mathscr{V}$ is locally $\mathscr{O}_{X}(*D)$-free. Two important
results should be emphasized.

  1. (1)

If $n=\dim X=1$, then any $(\mathscr{V},\nabla)$ is good. This is known as the
Levelt-Turrittin theorem, which has various different proofs (seee.g.[53, 35,
47, 3, 18, 40, 52, 1]).

  2. (2)

If $n\geqslant 2$, goodness may fail, but there exists a finite sequence of
complex blow-ups over a sufficiently small analytic neighbourhood of $x$, or
over $X$ if $X$ is quasi-projective, that are isomorphisms away from $D$ and
such that the pullback of $(\mathscr{V},\nabla)$ is good along the pullback of
$D$, which can be made with normal crossings. This is the Kedlaya-Mochizuki
theorem (see[50] for $n=2$ and rank $\leqslant 5$, [43] for $n=2$ in the
projective case, [29] for $n=2$ in the local formal and analytic cases, [44]
for $n\geqslant 2$ in the projective case, [30] for $n\geqslant 2$ in the
local formal and analytic cases, and also in the projective case).

######  Theorem 2.30.

If $(\mathscr{V},\nabla)$ is good, then Assumptions 2.1 and A.18 hold for the
rapid decay and moderate growth de Rham complexes of
$(\widetilde{\mathscr{V}},\nabla)$.

######  Convention 2.31.

In the case of complex manifolds, in order to make formulas independent of the
choice of a square root $\mathsf{i}$ of $-1$, we replace integration
$\int_{X}$ of a form of maximal degree with the trace
$\operatorname{tr}_{X}=(1/2\pi\mathsf{i})^{n}\cdot\int_{X}$. The global de
Rham pairing $\mathsf{Q}^{r}_{\mathrm{mid}}$ in degree $r$ on $\widetilde{X}$
will thus be replaced with the normalized de Rham pairing, e.g.

$\widetilde{\mathsf{S}}^{r}_{\mathrm{rd},\mathrm{mod}}=(1/2\pi\mathsf{i})^{n}\mathsf{Q}^{r}_{\mathrm{rd},\mathrm{mod}},$

which pairs
$\mathrm{H}^{r}(\widetilde{X},\operatorname{DR}^{\mathrm{rd}}(\widetilde{\mathscr{V}}))$
and
$\mathrm{H}^{2n-r}(\widetilde{X},\operatorname{DR}^{\mathrm{mod}}(\widetilde{\mathscr{V}}))$
and is thus perfect. We define its “middle” version
$\widetilde{\mathsf{S}}^{r}_{\mathrm{mid}}$ as in Section 2.c, that we can
interpret in terms hypercohomologies computed on $X$ as in Remark 2.29, in
which case we denote it by $\mathsf{S}^{r}_{\mathrm{mid}}$.

Let us for example consider the quadratic relations in the middle dimension
$n$ if
$\left\langle{{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\right\rangle$
is $\pm$-symmetric.

######  Corollary 2.32.

Assume that $X$ is compact. The quadratic relations for the Betti period
pairings

$\pm(-2\pi\mathsf{i})^{n}\,\mathsf{B}^{\mathrm{mid}}_{n}=\mathsf{P}^{{\mathrm{mid}}}_{n}\circ(\mathsf{S}_{\mathrm{mid}}^{n})^{-1}\circ{}^{t}\mathsf{P}^{{\mathrm{mid}}}_{n}$

hold for a good meromorphic flat bundle on $(X,D)$, and the Betti intersection
pairing can be computed with a suitable simplicial decomposition of
$(\widetilde{X},\widetilde{D})$ as in Proposition 2.25.

###### Proof of Theorem 2.30.

Let us start with Assumption 2.1(1). In dimension one, the rapid decay case is
the Hukuhara-Turrittin asymptotic expansion theorem (see[53] and [40, App.
1]). In dimension $\geqslant 2$, the proof is essentially a consequence of the
work of Majima [36] on asymptotic analysis (seealso [48, App.], [44, Chap.
20]). The rapid decay case is proved in [48] and the moderate case in [24,
App.] with the assumption $n\leqslant 2$, but the argument extends to any
$n\geqslant 2$ (see[25, Prop. 1]); both cases are also proved in [45, Prop.
5.1.3]. The proof that Assumption 2.1(2) holds is obtained by means of the
local description below (seee.g.the proof in dimension one of Proposition
3.18).

Due to the aforementioned theorems, the local structure of

$\mathscr{H}^{0}\operatorname{DR}^{\mathrm{rd}}(\widetilde{\mathscr{V}})=\mathscr{V}^{\mathrm{rd}}\quad\text{and}\quad\mathscr{H}^{0}\operatorname{DR}^{\mathrm{mod}}(\widetilde{\mathscr{V}})=\mathscr{V}^{\mathrm{mod}}$

is easy to understand. Let $\Delta(\varepsilon)$ denote a disc of radius
$\varepsilon$. Near $x\in D$ as above, the real blow-up $\widetilde{X}$ is
diffeomorphic to the product
$(S^{1})^{\ell}\times[0,\varepsilon)^{\ell}\times\Delta(\varepsilon)^{n-\ell}$
for some $\varepsilon>0$. In this model, denoting by $(r_{1},\dots,r_{\ell})$
the coordinates on $[0,\varepsilon)^{\ell}$, $\widetilde{D}$ is described as
$r_{1}\cdots r_{\ell}=0$, giving it the structure of a real analytic set. The
structure of manifold with corners is then clear.

From the set $\Phi_{x}$ of exponential factors defined after some ramification

$\rho_{x}:(x^{\prime},x^{\prime\prime})=(x^{\prime}_{1},\dots,x^{\prime}_{\ell},x_{\ell+1},\dots,x_{n})\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}(x^{\prime
a_{1}}_{1},\dots,x^{\prime a_{\ell}}_{\ell},x_{\ell+1},\dots,x_{n}),$

one constructs a local real semi-analytic stratification of $\widetilde{D}$:
any meromorphic function $\varphi\neq 0$ in $\Phi_{x}$ can be written in the
ramified coordinates $(x^{\prime},x^{\prime\prime})$ as
$\varphi=u_{\varphi}\prod_{i=1}^{\ell}x_{i}^{\prime-
q_{i}(\varphi)}(1+h(x^{\prime},x^{\prime\prime}))$, with $q_{i}(\varphi)$ non-
negative integers, $u_{\varphi}=u_{\varphi}(x^{\prime\prime})$ holomorphic and
invertible, and $h(x^{\prime},x^{\prime\prime})$ holomorphic. Then we can
write
$u_{\varphi}(x^{\prime\prime})=r_{\varphi}(x^{\prime\prime})e^{\mathsf{i}a_{\varphi}(x^{\prime\prime})}$,
where $r_{\varphi},a_{\varphi}$ are real analytic functions of their
arguments, and

$\varphi\circ\rho_{x}=r_{\varphi}\prod_{i=1}^{\ell}r_{i}^{\prime-
q_{i}(\varphi)}\cdot\exp\bigl{[}\mathsf{i}(a_{\varphi}-\textstyle\sum_{i}q_{i}(\varphi)\theta^{\prime}_{i})\bigr{]}\cdot(1+h(r^{\prime}e^{\mathsf{i}\theta^{\prime}},x^{\prime\prime})).$

We consider the sets defined by

$\cos\bigl{(}a_{\varphi}-\textstyle\sum_{i}q_{i}(\varphi)\theta^{\prime}_{i}\bigr{)}=0,$

and their projections $S_{\varphi}$ to $\widetilde{D}$, which are real semi-
analytic subsets of $\widetilde{D}$ (the multi-dimensional Stokes directions);
the desired stratification is any real semi-analytic stratification compatible
with all these subsets. We also denote by $S_{\varphi}^{\leqslant 0}$
(resp.$S_{\varphi}^{>0}$) the corresponding subsets defined with $\leqslant 0$
(resp.$>0$).

In the neighbourhood of any $\widetilde{x}\in\varpi^{-1}(x)$,
$\mathscr{V}^{\mathrm{rd}}|_{\widetilde{D}}$ decomposes as
$\mathop{\textstyle\bigoplus}\displaylimits_{\varphi}\mathscr{V}^{\mathrm{rd}}_{\varphi}$,
where $\mathscr{V}^{\mathrm{rd}}_{\varphi}{}$ is zero if $\varphi=0$ and
otherwise constant on $S_{\varphi}^{>0}$ and zero on $S_{\varphi}^{\leqslant
0}$. A similar property holds for
$\mathscr{V}^{\mathrm{mod}}|_{\widetilde{D}}$, with the only change that
$\mathscr{V}^{\mathrm{mod}}_{\varphi}$ is constant if $\varphi=0$. In
particular, $\mathscr{V}^{\mathrm{rd}}$ and $\mathscr{V}^{\mathrm{mod}}$ are
locally $\mathbf{R}$-constructible in the sense of [28, Def. 8.4.3] (we can
use a local stratification as above to check local
$\mathbf{R}$-constructibility). In this local setting,
$\mathscr{V}^{\mathrm{rd}}$ and $\mathscr{V}^{\mathrm{mod}}$, together with
any simplicial structure compatible with any real semi-analytic stratification
with respect to which they are $\mathbf{R}$-constructible,222It is standard
that such a simplicial structure exists, seee.g.[28, Prop. 8.2.5]. satisfy
thus Assumption A.18.

In order to conclude that Assumption A.18 holds in the global setting, we note
that the pair $(\widetilde{X},\widetilde{D})$ comes equipped with a real semi-
analytic structure which induces the previous one in each local chart adapted
to $D$, and that $\mathscr{V}^{\mathrm{rd}}$ and $\mathscr{V}^{\mathrm{mod}}$
are $\mathbf{R}$-constructible, since $\mathbf{R}$-constructibility is a local
property on $\widetilde{D}$, according to [28, Th. 8.4.2]. Then the same
conclusion as in the local setting holds. ∎

##  3\. Algebraic computation of de Rham duality in dimension one

In this section, we restrict to the case of complex dimension one and we make
explicit in algebraic terms the results obtained in Section 2.f.

###  3.a. Setting, notation, and objectives

Let $X$ be a connected smooth complex projective curve, let $D$ be a _non-
empty_ finite set of points in $X$, and let us define $j:U=X\smallsetminus
D\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}X$
as the complementary inclusion, so that $U$ is affine. Depending on the
context, we work in the Zariski topology or the analytic topology on $U$ and
$X$, and hence $\mathscr{O}_{U}$ will have the corresponding meaning. We
denote by $\mathscr{O}_{\widehat{D}}$ the structure sheaf of the formal
neighbourhood of $D$ in $X$, so that
$\mathscr{O}_{\widehat{D}}=\mathop{\textstyle\bigoplus}\displaylimits_{x\in
D}\mathscr{O}_{\widehat{x}}$ and
$\mathscr{O}_{\widehat{x}}\simeq\mathbf{C}[\\![z]\\!]$ for some local
coordinate $z$ centered at $x$. We fix an affine neighbourhood $U_{D}$ of $D$
in $X$. We set $U_{D}^{\circ}=U_{D}\smallsetminus D=U\cap U_{D}$ and we denote
by $\iota_{D}$ the restriction functor
$\mathsf{Mod}(\mathscr{O}(U))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathsf{Mod}(\mathscr{O}(U_{D}^{\circ}))$
attached to the inclusion
$U_{D}^{\circ}\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}U$.
We set $U_{\widehat{D}}$ to be the completion of $U_{D}$ at $D$ and we define
$U_{\widehat{D}}^{\circ}$ by
$\mathscr{O}(U_{\widehat{D}}^{\circ})=\mathscr{O}(U_{D}^{\circ})\otimes_{\mathscr{O}(U_{D})}\mathscr{O}(U_{\widehat{D}})$.
We similarly denote by $\iota_{\widehat{D}}$ the restriction functor
$\mathsf{Mod}(\mathscr{O}(U))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathsf{Mod}(\mathscr{O}(U_{\widehat{D}}^{\circ}))$.

We consider a locally free sheaf $V$ on $U$ endowed with a connection $\nabla$
together with the associated $\mathscr{O}_{X}(*D)$-module with connection
$j_{*}V$ (one also finds the notation $j_{+}V$ in the literature) that we
denote by $\mathscr{V}$. It is thus a left module on the sheaf
$\mathscr{D}_{X}$ of algebraic differential operators on $X$, and thereby
endowed with an action of meromorphic vector fields $\Theta_{X}(*D)$ with
poles along $D$.

On the other hand, let $(V^{\scriptscriptstyle\vee},\nabla)$ be the dual
bundle with connection and set
$\mathscr{V}^{\scriptscriptstyle\vee}=j_{*}V^{\scriptscriptstyle\vee}$. The
dual $\mathscr{D}_{X}$-module
$\boldsymbol{D}(\mathscr{V}^{\scriptscriptstyle\vee})$ is a holonomic
$\mathscr{D}_{X}$-module that we denote by $\mathscr{V}(!D)$ (one also finds
the notation $j_{!}V$ or $j_{\dagger}V$ in the literature). There exists a
natural morphism
$\mathscr{V}(!D)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{V}$
whose kernel $K$ and cokernel $C$ are supported on $D$ and whose image is
denoted by $\mathscr{V}(!{*}D)$.

We deduce natural morphisms between the associated algebraic de Rham complexes

$\operatorname{DR}\mathscr{V}(!D)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{DR}\mathscr{V}(!{*}D)\quad\text{and}\quad\operatorname{DR}\mathscr{V}(!{*}D)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{DR}\mathscr{V}.$

We denote the hypercohomologies on $X$ of $\operatorname{DR}\mathscr{V}(!D)$,
$\operatorname{DR}\mathscr{V}(!{*}D)$, and $\operatorname{DR}\mathscr{V}$
respectively by $\mathrm{H}^{*}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)$,
$\mathrm{H}^{*}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)$, and
$\mathrm{H}^{*}_{{\mathrm{dR}}}(U,V)$.

We note that $\operatorname{DR}K$ and $\operatorname{DR}C$ have non-zero
cohomology in degree one at most and are supported on $D$, whence exact
sequences

$\begin{gathered}\boldsymbol{H}^{1}(D,\operatorname{DR}K)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0\\\
0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{H}^{1}(D,\operatorname{DR}C).\end{gathered}$

In particular, we obtain an identification

(3.1)
$\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)=\operatorname{im}[\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)\rightarrow\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V)].$

Moreover, when working in the analytic topology, one can define the analytic
de Rham complex $\operatorname{DR}^{\mathrm{an}}\mathscr{V}$, which has
constructible cohomology. Since $X$ is compact, it can be deduced from the
GAGA theorem that the natural morphism
$\operatorname{DR}\mathscr{V}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{DR}^{\mathrm{an}}\mathscr{V}$
induces an isomorphism in hypercohomology (and similarly for
$\mathscr{V}(!D)$). This enables us to identify the algebraic de Rham
cohomology with the analytic one. We will not use the exponent ‘$\mathrm{an}$’
when the context is clear and we will use this GAGA theorem without mentioning
it explicitly.

If $(V,\nabla)=(\mathscr{O}_{U},{\mathrm{d}})$, one knows various forms of
$\operatorname{DR}(\mathscr{O}_{X}(!D))$. Namely,
$\operatorname{DR}(\mathscr{O}_{X}(!D))$ is quasi-isomorphic to one of the
following complexes:

  * $\scriptscriptstyle\bullet$

$\mathrm{Cone}\bigl{(}\operatorname{DR}\mathscr{O}_{X}(*D)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\iota_{D}\operatorname{DR}\mathscr{O}_{\widehat{D}}\bigr{)}[-1]$,

  * $\scriptscriptstyle\bullet$

$\mathscr{O}_{X}(-D)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{1}_{X}$,

and, for the sake of computing cohomology, one can replace these complexes
with their analytic counterparts. Another form proves useful, obtained by
means of the rapid decay de Rham complex defined on the real-oriented blow-up
$\widetilde{X}$ of $X$ at each point of $D$ (the one-dimensional version of
the rapid decay de Rham complex considered in Section 2.f, seeSection 3.d for
details).

The first goal of this section is to explain similar presentations for
$\operatorname{DR}\mathscr{V}(!D)$ and, correspondingly, the presentation of
$\operatorname{DR}\mathscr{V}$ in terms of the moderate de Rham complex on
$\widetilde{X}$. Furthermore, we express the natural de Rham pairing between
these complexes in terms of the analytic de Rham pairing
$\widetilde{\mathsf{Q}}_{\mathrm{rd},\mathrm{mod}}$ on the real blow-up space
$\widetilde{X}$ at $D$. This approach is reminiscent of that of [45] in any
dimension (seeCor. 5.2.7 from loc.cit. and Remark 2.29).

Theorem 3.14 gives a residue formula for this de Rham pairing $\mathsf{S}$. It
is a generalization to higher rank of a formula already obtained by Deligne
[16] (seealso [17]) in rank one.

Once this material is settled, we translate into this setting the general
results on quadratic relations obtained in Section 2.f, with the de Rham
cohomology (resp.with compact support) instead of moderate (resp.rapid decay)
cohomology. Also, we focus here our attention on the middle extension
(co)homology.

###  3.b. Čech computation of de Rham cohomologies

Recall that, since $U$ is affine, the de Rham cohomology of $(V,\nabla)$ is
computed as the cohomology of the complex

$\Gamma(U,V)\mathchoice{\xrightarrow{\kern 4.0pt\nabla\kern
3.0pt}}{\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}}{}{}\Gamma(U,\Omega^{1}_{U}\otimes
V).$

Hence, any element of $\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V)$ is represented by
an element of $\Gamma(U,\Omega^{1}_{U}\otimes V)$ modulo the image of
$\nabla$.

We will compute the de Rham cohomology in terms of a Čech complex relative to
the covering $(U_{D},U)$, whose differential is denoted by $\delta$. So for a
sheaf $F$ of $\mathscr{O}_{X}$-modules and $(f,\varphi)\in F(U_{D})\oplus
F(U)$, we have $\delta(f,\varphi)=\iota_{D}\varphi-f\in F(U_{D}^{\circ})$. We
will write $\mathscr{V}_{D}=\Gamma(U_{D},\mathscr{V})$ and implicitly identify
$\Gamma(U_{D}^{\circ},\iota_{D}V)$ with $\mathscr{V}_{D}$.

Replacing $X$ with the formal neighbourhood $\widehat{D}$ of $D$ in $X$ gives
rise to
$\mathscr{V}_{\widehat{D}}=\mathscr{O}_{\widehat{D}}\otimes_{\mathscr{O}_{X}}\mathscr{V}$,
which is an $\mathscr{O}_{\widehat{D}}(*D)$-module with connection, and hence
endowed with an action of the formal vector fields $\Theta_{\widehat{D}}$.

######  Proposition 3.2.

The complex $\operatorname{DR}\mathscr{V}(!D)$ is quasi-isomorphic to the
$(-1)$-shifted cone of the natural morphism
$\operatorname{DR}\mathscr{V}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{DR}\mathscr{V}_{\widehat{D}}$,
that is, to the simple complex associated with the double complex

$\textstyle{j_{*}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\widehat{D}}}$$\scriptstyle{\nabla}$$\textstyle{\mathscr{V}_{\widehat{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\nabla}}$$\textstyle{j_{*}(\Omega^{1}_{U}\otimes
V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\widehat{D}}}$$\textstyle{\Omega^{1}_{\widehat{D}}\otimes\mathscr{V}_{\widehat{D}}.}$

The complex $\boldsymbol{R}\Gamma(X,\operatorname{DR}\mathscr{V}(!D))$ is
quasi-isomorphic to the simple complex associated with the double complex

$\textstyle{\Gamma(U,V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\widehat{D}}}$$\scriptstyle{\nabla}$$\textstyle{\mathscr{V}_{\widehat{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\nabla}}$$\textstyle{\Gamma(U,\Omega^{1}_{U}\otimes
V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\widehat{D}}}$$\textstyle{\Omega^{1}_{\widehat{D}}\otimes\mathscr{V}_{\widehat{D}}.}$

###### Proof.

Any holonomic $\mathscr{D}_{X}$-module $N$ is endowed with a canonical
exhaustive increasing filtration by coherent $\mathscr{D}_{X}(\log
D)$-submodules $N_{\scriptscriptstyle\bullet}$ indexed by $\mathbf{Z}$, called
the _Kashiwara-Malgrange filtration_ (seee.g.[49, Chap. I, Prop. 6.1.2]) such
that

  * $\scriptscriptstyle\bullet$

$N_{-k}=\mathscr{O}(-kD)N_{0}$ for any $k\geqslant 0$, and
$N_{k+1}=N_{k}+\Theta_{X}N_{k}$ for any $k\geqslant 1$,

  * $\scriptscriptstyle\bullet$

for each $k\in\mathbf{Z}$, the eigenvalues of $\operatorname{Res}\nabla$ on
$N_{k}/N_{k-1}$ at each point of $D$ have their real part in $[k,k+1)$.

For each $k$, the connection $\nabla$ defines a connection
$\nabla:N_{k}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{1}_{X}\otimes
N_{k+1}$. The above properties imply that, for $k\neq 0$, the induced morphism
on graded spaces
$\nabla:\operatorname{gr}_{k}N\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{1}_{X}\otimes\operatorname{gr}_{k+1}N$
is bijective, so in particular the natural inclusion morphism is a quasi-
isomorphism:

(3.3) $\\{N_{0}\mathchoice{\xrightarrow{\kern 4.0pt\nabla\kern
3.0pt}}{\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}}{}{}\Omega^{1}_{X}\otimes
N_{1}\\}\overset{\sim}{\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}\operatorname{DR}N.$

For example, the following also holds:

  * $\scriptscriptstyle\bullet$

If $N_{|U}=V$, then the natural morphism
$N_{k}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{V}_{k}$
is an isomorphism for any $k\leqslant 0$.

  * $\scriptscriptstyle\bullet$

The formation of the Kashiwara-Malgrange filtration is compatible with
tensoring with $\mathscr{O}_{D}$ and $\mathscr{O}_{\widehat{D}}$ and, for any
$k$, $N/N_{k}\simeq N_{D}/N_{D,k}\simeq N_{\widehat{D}}/N_{\widehat{D},k}$.

  * $\scriptscriptstyle\bullet$

For $\mathscr{V}=j_{*}V$, we have
$\mathscr{V}_{k}=\mathscr{O}(kD)\mathscr{V}_{0}$ for any $k\in\mathbf{Z}$.

  * $\scriptscriptstyle\bullet$

Among all $N$’s such that $N_{|U}=V$, the $\mathscr{D}_{X}$-module
$\mathscr{V}(!D)$ is characterized by the property that
$\nabla:\operatorname{gr}_{0}N\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{1}_{X}\otimes\operatorname{gr}_{1}N$
is bijective.

By the third point, the left-hand complex in (3.3) for $\mathscr{V}$ reads

$\\{\mathscr{V}_{0}\mathchoice{\xrightarrow{\kern 4.0pt\nabla\kern
3.0pt}}{\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}}{}{}\Omega^{1}_{X}(D)\otimes\mathscr{V}_{0}\\},$

so (3.3) amounts to the quasi-isomorphism

$\\{\mathscr{V}_{0}\mathchoice{\xrightarrow{\kern 4.0pt\nabla\kern
3.0pt}}{\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}}{}{}\Omega^{1}_{X}(D)\otimes\mathscr{V}_{0}\\}\overset{\sim}{\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}\operatorname{DR}\mathscr{V}.$

On the other hand, the last point implies that the inclusion of complexes

$\\{\mathscr{V}(!D)_{-1}\mathchoice{\xrightarrow{\kern 4.0pt\nabla\kern
3.0pt}}{\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}}{}{}\Omega^{1}_{X}\otimes\mathscr{V}(!D)_{0}\\}\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}\operatorname{DR}\mathscr{V}(!D)$

is a quasi-isomorphism. The left-hand complex reads
$\\{\mathscr{V}_{-1}\mathchoice{\xrightarrow{\kern 4.0pt\nabla\kern
3.0pt}}{\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}}{}{}\Omega^{1}_{X}\otimes\mathscr{V}_{0}\\}$
according to the second point, so we obtain a quasi-isomorphism

(3.4) $\\{\mathscr{V}_{-1}\mathchoice{\xrightarrow{\kern 4.0pt\nabla\kern
3.0pt}}{\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}}{}{}\Omega^{1}_{X}(D)\otimes\mathscr{V}_{-1}\\}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{DR}\mathscr{V}(!D).$

The Čech resolution of the complex on the left is the simple complex
associated with the double complex

$\textstyle{\mathscr{V}_{D,-1}\oplus
j_{*}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{\nabla}$$\textstyle{\mathscr{V}_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla}$$\textstyle{\bigl{(}\Omega^{1}_{U_{D}}(D)\otimes\mathscr{V}_{D,-1}\bigr{)}\oplus
j_{*}\bigl{(}\Omega^{1}_{U}\otimes
V\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{\Omega^{1}_{U_{D}}(D)\otimes\mathscr{V}_{D}.}$

Since $\mathscr{V}_{D,-1}\subset\mathscr{V}_{D}$, this complex is quasi-
isomorphic to

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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 64.9233pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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0.0pt\raise
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64.9233pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{V}_{D}/\mathscr{V}_{D,-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
85.82631pt\raise-20.53665pt\hbox{{}\hbox{\kern 0.0pt\raise
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85.82631pt\raise-29.43555pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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41.74484pt\raise-41.0733pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\kern
24.67302pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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32.0816pt\raise 5.95139pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 66.3122pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
66.3122pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{V}_{\widehat{D}}/\mathscr{V}_{\widehat{D},-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
86.86778pt\raise-21.02332pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
86.86778pt\raise-30.40887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{j_{*}(\Omega^{1}_{U}\otimes
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26.74794pt\raise-36.09525pt\hbox{{}\hbox{\kern 0.0pt\raise
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41.74484pt\raise-42.04663pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
41.74484pt\raise-42.04663pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{$\textstyle{\Omega^{1}_{\widehat{D}}(D)\otimes(\mathscr{V}_{\widehat{D}}/\mathscr{V}_{\widehat{D},-1}).}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{array}$

Reversing the reasoning, it is also isomorphic to the Čech complex

$\begin{array}[]{c}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
62.37927pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{$\textstyle{\mathscr{V}_{\widehat{D},-1}\oplus
j_{*}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
56.20673pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\delta}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 96.66818pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.89441pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
96.66818pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{V}_{\widehat{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
105.64041pt\raise-20.9472pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\widehat{\nabla}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
105.64041pt\raise-30.25665pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\bigl{(}\Omega^{1}_{\widehat{D}}(D)\otimes\mathscr{V}_{\widehat{D},-1}\bigr{)}\oplus
j_{*}\bigl{(}\Omega^{1}_{U}\otimes
V\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
69.82373pt\raise-36.46385pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\delta}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
86.37927pt\raise-41.89441pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
86.37927pt\raise-41.89441pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega^{1}_{\widehat{D}}\otimes\mathscr{V}_{\widehat{D}}.}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{array}$

We claim that the morphism
$\nabla:\mathscr{V}_{\widehat{D},-1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{1}_{\widehat{D}}(D)\otimes\mathscr{V}_{\widehat{D},-1}$
is bijective. Indeed, recall that $\mathscr{V}_{\widehat{D}}$ has a canonical
decomposition (see[40, Th. (2.3), p. 51])

$\mathscr{V}_{\widehat{D}}=\mathscr{V}_{\widehat{D}}^{\mathrm{reg}}\oplus\mathscr{V}_{\widehat{D}}^{\mathrm{irr}}$

into its regular and irregular components and that the Kashiwara-Malgrange
filtration $\mathscr{V}_{\widehat{D},{\scriptscriptstyle\bullet}}$ decomposes
accordingly. Moreover, the filtration
$\mathscr{V}_{\widehat{D},{\scriptscriptstyle\bullet}}^{\mathrm{irr}}$ is
constant and equal to $\mathscr{V}_{\widehat{D}}^{\mathrm{irr}}$, and we know
that

$\widehat{\nabla}:\mathscr{V}_{\widehat{D}}^{\mathrm{irr}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{1}_{\widehat{D}}\otimes\mathscr{V}_{\widehat{D}}^{\mathrm{irr}}$

is bijective (use for example ii) in loc.cit. with
$M^{\prime}\\!=\\!\mathscr{O}_{\widehat{D}}$ and
$M^{\prime\prime}\\!=\\!\mathscr{V}_{\widehat{D}}^{\mathrm{irr}}$). On the
other hand, for the regular holonomic part,

$\widehat{\nabla}:\mathscr{V}_{\widehat{D},-1}^{\mathrm{reg}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{1}_{\widehat{D}}(D)\otimes\mathscr{V}_{\widehat{D},-1}^{\mathrm{reg}}$

is bijective: indeed, by the very definition of the Kashiwara-Malgrange
filtration, this property holds true for the morphism induced by
$\widehat{\nabla}$ on
$\mathscr{V}_{\widehat{D},-1}/\mathscr{V}_{\widehat{D},-1-k}$ for each
$k\geqslant 1$, hence on the projective limit; for
$\mathscr{V}_{\widehat{D},-1}^{\mathrm{reg}}$, this projective limit is
isomorphic to $\mathscr{V}_{\widehat{D},-1}^{\mathrm{reg}}$ itself, as follows
e.g.from [49, Chap. I, Lem. 6.2.6].

It follows from this claim that $\operatorname{DR}\mathscr{V}(!D)$ is quasi-
isomorphic to the complex given in the proposition, and the statement for
$\boldsymbol{R}\Gamma(X,\operatorname{DR}\mathscr{V}(!D))$ results
immediately. ∎

######  Corollary 3.5.

The formal de Rham complex $\operatorname{DR}_{\widehat{D}}\mathscr{V}(!D)$ is
quasi-isomorphic to zero.

###### Proof.

Indeed, $\operatorname{DR}_{\widehat{D}}\mathscr{V}(!D)$ is the simple complex
associated with the double complex
$\operatorname{DR}_{\widehat{D}}\mathscr{V}\mathchoice{\xrightarrow{\kern
4.0pt\operatorname{Id}\kern
3.0pt}}{\stackrel{{\scriptstyle\operatorname{Id}}}{{\longrightarrow}}}{}{}\operatorname{DR}_{\widehat{D}}\mathscr{V}$.
∎

######  Corollary 3.6.

The de Rham cohomology space with compact support
$\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)$ consists of pairs

$(\widehat{m},\omega)\in\mathscr{V}_{\widehat{D}}\oplus\Gamma(U,\Omega^{1}_{U}\otimes
V)$

satisfying $\widehat{\nabla}\widehat{m}=\iota_{\widehat{D}}\omega$, modulo
pairs of the form $(\iota_{\widehat{D}}(v),\nabla v)$ for $v\in\Gamma(U,V)$.∎

######  Remark 3.7.

The natural morphism
$\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V)$
is described in terms of the previous representatives as
$(\widehat{m},\omega)\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\omega$.

(1) The image $\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)$ of this
morphism (see(3.1)) consists of classes of sections $\omega\in\Gamma(U,V)$
such that
$\iota_{\widehat{D}}\omega\in\operatorname{im}(\widehat{\nabla}:\mathscr{V}_{\widehat{D}}\rightarrow\mathscr{V}_{\widehat{D}})$.
Therefore, any family consisting of
$\dim\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)$ linearly independent
classes $[\omega]$ in $\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V)$ for which there
exists $\widehat{m}\in\mathscr{V}_{\widehat{D}}$ satisfying
$\widehat{\nabla}\widehat{m}=\iota_{\widehat{D}}\omega$ is a basis of
$\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)$.

(2) The kernel of this morphism consists of pairs of the form
$(\widehat{m},\nabla w)$ for some $w\in\Gamma(U,V)$, modulo pairs
$(\iota_{\widehat{D}}(v),\nabla v)$. Each class has thus a representative of
the form $(\widehat{m},0)$ with $\widehat{\nabla}\widehat{m}=0$, and there is
a surjective morphism

$\ker\bigl{[}\widehat{\nabla}:\mathscr{V}_{\widehat{D}}\rightarrow\mathscr{V}_{\widehat{D}}\bigr{]}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\ker\Bigl{[}\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)\rightarrow\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V)\Bigr{]}.$

(3) If $V$ has no constant subbundle with connection (e.g.if $V$ is
irreducible and non-constant), then a representative $(\widehat{m},0)$ with
$\widehat{\nabla}\widehat{m}=0$ is unique. We thus conclude in such a case
that

$\ker\Bigl{[}\widehat{\nabla}:\mathscr{V}_{\widehat{D}}\rightarrow\mathscr{V}_{\widehat{D}}\Bigr{]}\simeq\ker\Bigl{[}\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)\rightarrow\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V)\Bigr{]}.$

(4) With the assumption in (3.7), a basis of
$\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)$ can thus be obtained as the
union of the following sets:

  * $\scriptscriptstyle\bullet$

a basis of $\ker\widehat{\nabla}$ in $\mathscr{V}_{\widehat{D}}$,

  * $\scriptscriptstyle\bullet$

a family of $\dim\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)$
representatives $(\widehat{m},\omega)$ for which the classes of $\omega$ in
$\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V)$ are linearly independent (and thus form
a basis of $\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)$).

######  Remark 3.8 ($C^{\infty}$ computation).

We consider the sheaves

$\mathscr{V}_{\infty}=\mathscr{C}_{X}^{\infty}\otimes_{\mathscr{O}_{X}}\mathscr{V}\quad\text{and}\quad\mathscr{V}(!D)_{\infty}=\mathscr{C}_{X}^{\infty}\otimes_{\mathscr{O}_{X}}\mathscr{V}(!D)$

and the corresponding $C^{\infty}$ de Rham complexes
$\operatorname{DR}^{\infty}(\mathscr{V})$ and
$\operatorname{DR}^{\infty}(\mathscr{V}(!D))$ with differential
$\nabla+\overline{\partial}$. These complexes are fine resolutions of
$\operatorname{DR}^{\mathrm{an}}(\mathscr{V})$ and
$\operatorname{DR}^{\mathrm{an}}(\mathscr{V}(!D))$. We also consider the
sheaves of $C^{\infty}$ functions on $U$ having respectively moderate growth
and rapid decay at the points of $D$, defined in a way similar to that of
Definition B.1. The Dolbeault complex
$(\mathscr{E}_{X}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\overline{\partial})$
(seeAppendix B) is a resolution of $\mathscr{O}_{X}(*D)$. Hence, the complex

$\operatorname{DR}^{\infty,\mathrm{mod}}(\mathscr{V})=(\mathscr{E}_{X}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\otimes\mathscr{V},\nabla+\overline{\partial})$

is a fine resolution of $\operatorname{DR}^{\mathrm{an}}\mathscr{V}$.
Therefore, there is an isomorphism

$\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V)\simeq\mathrm{H}^{1}(\Gamma(X,\operatorname{DR}^{\infty,\mathrm{mod}}(\mathscr{V})))$

induced by

$\Gamma(U,\Omega^{1}_{U}\otimes
V)=\Gamma(X,\Omega^{1}_{X}(*D)\otimes\mathscr{V})\ni\omega\\\
\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\omega_{\infty}=\omega\in\Gamma(X,\mathscr{E}_{X}^{\mathrm{mod},(1,0)}\otimes\mathscr{V})\subset\Gamma(X,\mathscr{E}_{X}^{\mathrm{mod},1}\otimes\mathscr{V}).$

On the other hand, $\operatorname{DR}^{\infty}(\mathscr{V}(!D))$ is quasi-
isomorphic to the double complex
$\operatorname{DR}^{\infty}(\mathscr{V})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{DR}^{\infty}_{\widehat{D}}(\mathscr{V})$.
By Borel’s lemma (applied to the coefficients in a basis of
$\mathscr{E}_{X}^{k}\otimes\mathscr{V}$), this morphism is termwise
surjective, hence the double complex is isomorphic to the $C^{\infty}$ rapid
decay complex
$\operatorname{DR}^{\infty,\mathrm{rd}}(\mathscr{V})=(\mathscr{E}_{X}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\otimes\mathscr{V},\nabla+\overline{\partial})$.
Therefore, there is an isomorphism

(3.9)
$\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)\simeq\mathrm{H}^{1}(\Gamma(X,\operatorname{DR}^{\infty,\mathrm{rd}}(\mathscr{V})))$

described as follows. For a class $(\widehat{m},\omega)$ with
$\nabla\widehat{m}=\iota_{\widehat{D}}\omega$, let us choose
$m_{\infty}\in\Gamma(X,\mathscr{V}_{\infty})$ whose Taylor series at $D$ is
equal to $\widehat{m}$. As above, we also regard $\omega$ as belonging to
$\Gamma(X,\mathscr{E}_{X}^{\mathrm{mod},1}\otimes\mathscr{V})$ and we denote
it by $\omega_{\infty}$. Then

$\eta_{\infty}=\omega_{\infty}-(\nabla+\overline{\partial})m_{\infty}$

has rapid decay at $D$. Moreover, it is $(\nabla+\overline{\partial})$-closed.
The image of the class of $(\widehat{m},\omega)$ by (3.9) is the class of
$\eta_{\infty}$.

######  Example 3.10 (The case $(V,\nabla)=(\mathscr{O}_{U},{\mathrm{d}})$).

We are interested in $\mathrm{H}^{2}_{{\mathrm{dR}},{\mathrm{c}}}(U)$. This
space is identified with
$\mathrm{H}^{2}(X,\operatorname{DR}\mathscr{O}_{X}(!D))$. By Proposition 3.2
applied to $V=\mathscr{O}_{U}$, we identify
$\mathrm{H}^{2}_{{\mathrm{dR}},{\mathrm{c}}}(U)$ with the quotient of
$\Omega^{1}_{\widehat{D}}(*D)$ by the subspace consisting of elements of the
form ${\mathrm{d}}\widehat{\varphi}-\iota_{\widehat{D}}\eta$, with
$\widehat{\varphi}\in\mathscr{O}_{\widehat{D}}(*D)$ and
$\eta\in\Omega^{1}(U)$. Let us make explicit the canonical isomorphism

$\operatorname{can}:\mathrm{H}^{2}_{{\mathrm{dR}},{\mathrm{c}}}(U)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{H}^{2}(\Gamma(X,\mathscr{E}_{X}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}})).$

For $\widehat{\mu}\in\Omega^{1}_{\widehat{D}}(*D)$, one can choose a lift
$\mu_{\infty}\in\Gamma(X,\mathscr{E}_{X}^{1,0}(*D))$. Since
${\mathrm{d}}\widehat{\mu}=0$, the $2$-form ${\mathrm{d}}\mu_{\infty}$ has
rapid decay. Then, $\operatorname{can}[\widehat{\mu}]$ is the class of
$-{\mathrm{d}}\mu_{\infty}$.