On the one hand, let
$\operatorname{res}_{D}:\mathrm{H}^{2}_{{\mathrm{dR}},{\mathrm{c}}}(U)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}$
be induced by

$\sum_{x\in
D}\operatorname{res}_{x}:\Omega^{1}_{\widehat{D}}(*D)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}.$

On the other hand, we consider the trace morphism

$\operatorname{tr}_{X}:\frac{1}{2\pi\mathsf{i}}\int_{X}\colon\mathrm{H}^{2}(\Gamma(X,\mathscr{E}_{X}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}.$

The following result is standard: the morphisms $\operatorname{tr}_{X}$ and
$\operatorname{res}_{D}$ are isomorphisms and we have

(3.11) $\operatorname{tr}_{X}\circ\operatorname{can}=\operatorname{res}_{D}.$

To see the latter equality, we write, for a union of discs $\Delta_{r}$ of
radius $r>0$ centered at the points of $D$,

$\displaystyle-\frac{1}{2\pi\mathsf{i}}\int_{X}{\mathrm{d}}\mu_{\infty}$
$\displaystyle=-\frac{1}{2\pi\mathsf{i}}\int_{X\smallsetminus\Delta_{r}}{\mathrm{d}}\mu_{\infty}-\frac{1}{2\pi\mathsf{i}}\int_{\Delta_{r}}{\mathrm{d}}\mu_{\infty}$
$\displaystyle=\frac{1}{2\pi\mathsf{i}}\int_{\partial\Delta_{r}}\mu_{\infty}-\frac{1}{2\pi\mathsf{i}}\int_{\Delta_{r}}{\mathrm{d}}\mu_{\infty}.$

We can choose $\mu_{\infty}$ so that, for some $R\\!>\\!0$,
$\mu_{\infty}\\!=\\!\widehat{\mu}^{<0}+\nu_{\infty}$, where
$\widehat{\mu}^{<0}$ is the polar part of $\widehat{\mu}$ and $\nu_{\infty}$
is $C^{\infty}$ on $\Delta_{R}$. Then, since ${\mathrm{d}}\mu_{\infty}$ has
rapid decay, the above integral tends to $\operatorname{res}_{D}\widehat{\mu}$
as $r\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0$.

###  3.c. Pairings

Starting from the tautological pairing

$({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\mkern 1.0mu|\mkern
1.0mu{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}):V\otimes
V^{\scriptscriptstyle\vee}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{O}_{U},$

one defines a pairing

$(\\!({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\mkern 1.0mu|\mkern
1.0mu{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})\\!):\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)\otimes\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V^{\scriptscriptstyle\vee})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}^{2}_{{\mathrm{dR}},{\mathrm{c}}}(U)$

in the following way. For representatives
$(\widehat{m},\omega)\in\mathscr{V}_{\widehat{D}}\oplus\Gamma(U,\Omega^{1}_{U}\otimes
V)$ and $\omega^{\scriptscriptstyle\vee}\in\Gamma(U,\Omega^{1}_{U}\otimes
V^{\scriptscriptstyle\vee})$ (seeCorollary 3.6), we set

$(\\!((\widehat{m},\omega)|\omega^{\scriptscriptstyle\vee})\\!)=(\widehat{m}\mkern
1.0mu|\mkern
1.0mu\iota_{\widehat{D}}\omega^{\scriptscriptstyle\vee})\mod\\{{\mathrm{d}}\widehat{\varphi}-\iota_{\widehat{D}}\eta\\}.$

Indeed, let us check that the formula does not depend on the choice of
representatives of the de Rham cohomology classes.

  * $\scriptscriptstyle\bullet$

If $\omega^{\scriptscriptstyle\vee}=\nabla v^{\scriptscriptstyle\vee}$, we
have

$(\widehat{m}\mkern 1.0mu|\mkern 1.0mu\iota_{\widehat{D}}\nabla
v^{\scriptscriptstyle\vee})={\mathrm{d}}(\widehat{m}\mkern 1.0mu|\mkern
1.0mu\iota_{\widehat{D}}v^{\scriptscriptstyle\vee})-\iota_{\widehat{D}}(\omega\mkern
1.0mu|\mkern 1.0muv^{\scriptscriptstyle\vee}),$

which is of the form ${\mathrm{d}}\widehat{\varphi}-\iota_{\widehat{D}}\eta$.

  * $\scriptscriptstyle\bullet$

Similarly, if $(\widehat{m},\omega)=(\iota_{\widehat{D}}(v),\nabla v)$, then
$(\iota_{\widehat{D}}(v)\mkern 1.0mu|\mkern
1.0mu\iota_{\widehat{D}}\omega^{\scriptscriptstyle\vee})=\iota_{\widehat{D}}(v\mkern
1.0mu|\mkern 1.0mu\omega^{\scriptscriptstyle\vee})$.

The residue pairing $\mathsf{R}^{1}$ is defined as the composition (seeExample
3.10 for the isomorphism $\operatorname{res}_{D}$)

$\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)\otimes\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V^{\scriptscriptstyle\vee})\mathchoice{\xrightarrow{\kern
4.0pt(\\!({\raisebox{1.0pt}{${\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}$}})\\!)}}{{\longrightarrow}}}{}{}\mathrm{H}^{2}_{{\mathrm{dR}},{\mathrm{c}}}(U,\mathbf{C})\xrightarrow[\sim]{~{}\textstyle\operatorname{res}_{D}~{}}\mathbf{C}.$

In other words, on representatives $(\widehat{m},\omega)$ and
$\omega^{\scriptscriptstyle\vee}$ it is expressed by summing up the residues:

$\mathsf{R}^{1}\bigl{(}(\widehat{m},\omega),\omega^{\scriptscriptstyle\vee}\bigr{)}=\operatorname{res}_{D}(\widehat{m}\mkern
1.0mu|\mkern
1.0mu\iota_{\widehat{D}}\omega^{\scriptscriptstyle\vee})=\sum_{x\in
D}\operatorname{res}_{x}(\widehat{m}\mkern 1.0mu|\mkern
1.0mu\iota_{\widehat{x}}\omega^{\scriptscriptstyle\vee}).$

On the other hand, by working with the $C^{\infty}$ realizations of Remark
3.8, we consider the de Rham pairing

(3.12)
$\mathsf{Q}:\operatorname{DR}^{\infty,\mathrm{rd}}(\mathscr{V})\otimes\operatorname{DR}^{\infty,\mathrm{mod}}(\mathscr{V}^{\scriptscriptstyle\vee})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathscr{E}_{X}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}}),$

which gives rise at the cohomology level to the pairing

$\mathsf{S}^{1}:\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)\otimes\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V^{\scriptscriptstyle\vee})\mathchoice{\xrightarrow{\kern
4.0pt\mathsf{Q}^{1}\kern
3.0pt}}{\stackrel{{\scriptstyle\mathsf{Q}^{1}}}{{\longrightarrow}}}{}{}\mathrm{H}^{2}(\Gamma(X,\mathscr{E}_{X}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}))\xrightarrow[\sim]{~{}\textstyle\operatorname{tr}_{U^{\mathrm{an}}}~{}}\mathbf{C}.$

######  Proposition 3.13.

The de Rham pairing $\mathsf{S}^{1}$ and the residue pairing $\mathsf{R}^{1}$
are equal.

###### Proof.

As explained in Remark 3.8, the elements
$(\widehat{m},\omega)\\!\in\\!\mathscr{V}_{\widehat{D}}\oplus\Gamma(U,\Omega^{1}_{U}\otimes
V)$ and
$\omega^{\scriptscriptstyle\vee}\\!\in\\!\Gamma(U,\Omega^{1}_{U}\\!\otimes\\!V^{\scriptscriptstyle\vee})$
are sent in a natural way, by choosing a $C^{\infty}$ lift $m_{\infty}$ of
$\widehat{m}$, respectively to

$(m_{\infty},\omega_{\infty})\in\Gamma(X,\mathscr{V}_{\infty})\oplus\Gamma(X,\mathscr{E}_{X}^{\mathrm{mod},(1,0)}\otimes
V)\quad\text{and}\quad\omega_{\infty}^{\scriptscriptstyle\vee}\in\Gamma(X,\mathscr{E}^{\mathrm{mod},(1,0)}\otimes\mathscr{V}^{\scriptscriptstyle\vee}),$

and $\omega_{\infty},\omega_{\infty}^{\scriptscriptstyle\vee}$ are
$(\nabla+\overline{\partial})$-closed. Set
$\eta_{\infty}=\omega_{\infty}-(\nabla+\overline{\partial})m_{\infty}$ as
before. By reasons of type we have

$\mathsf{Q}(\eta_{\infty},\omega_{\infty}^{\scriptscriptstyle\vee})=\bigl{(}-(\nabla+\overline{\partial})m_{\infty}\mkern
1.0mu|\mkern
1.0mu\omega_{\infty}^{\scriptscriptstyle\vee}\bigr{)}=-{\mathrm{d}}(m_{\infty}\mkern
1.0mu|\mkern 1.0mu\omega_{\infty}^{\scriptscriptstyle\vee}).$

With the notation of Example 3.10, this equality reads

$\mathsf{Q}(\eta_{\infty},\omega_{\infty}^{\scriptscriptstyle\vee})=\operatorname{can}(\widehat{m}\mkern
1.0mu|\mkern
1.0mu\omega^{\scriptscriptstyle\vee})=\operatorname{can}(\\!((\widehat{m},\omega)\mkern
1.0mu|\mkern 1.0mu\omega^{\scriptscriptstyle\vee})\\!).$

The proposition now follows from (3.11). ∎

By working on the real blow-up space we will also show:

######  Theorem 3.14.

The de Rham pairing $\mathsf{S}^{1}$ is a perfect pairing.

In particular, the residue pairing $\mathsf{R}^{1}$ is perfect. Before
starting the proof of the theorem which will be done in Section 3.d, let us
state its “middle” consequence.

######  Corollary 3.15.

The residue pairing $\mathsf{R}^{1}$ vanishes on
$\ker\widehat{\nabla}\otimes\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V^{\scriptscriptstyle\vee})$
and induces a non-degenerate pairing

$\mathsf{R}^{1}_{\mathrm{mid}}:\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)\otimes\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V^{\scriptscriptstyle\vee})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}.$

Moreover, if
$V\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}V^{\scriptscriptstyle\vee}$
is a $\pm$-symmetric isomorphism, it induces a $\mp$-symmetric non-degenerate
pairing

$\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)\otimes\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}.$

###### Proof of Corollary 3.15.

If $\widehat{\nabla}\widehat{m}=0$ and $\omega^{\scriptscriptstyle\vee}$
satisfies
$\iota_{\widehat{D}}\omega^{\scriptscriptstyle\vee}=\widehat{\nabla}\widehat{m}^{\scriptscriptstyle\vee}$
(see Remark 3.7(3.7)), we have

$\operatorname{res}_{D}(\widehat{m}\mkern 1.0mu|\mkern
1.0mu\iota_{\widehat{D}}\omega^{\scriptscriptstyle\vee})=\operatorname{res}_{D}{\mathrm{d}}(\widehat{m}\mkern
1.0mu|\mkern 1.0mu\widehat{m}^{\scriptscriptstyle\vee})=0,$

hence the first assertion. That $\mathsf{R}^{1}_{\mathrm{mid}}$ is non-
degenerate follows from the same property for $\mathsf{R}^{1}$. Assume now
that
$\langle{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\rangle$
is a $\pm$-symmetric pairing on $V$ and let $\omega,\omega^{\prime}$ represent
classes in $\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)$, so that there
exist $\widehat{m},\widehat{m}^{\prime}$ such that
$\nabla\widehat{m}=\iota_{\widehat{D}}\omega$ and
$\nabla\widehat{m}^{\prime}=\iota_{\widehat{D}}\omega^{\prime}$. Then

$\displaystyle\operatorname{res}_{D}\langle\widehat{m},\iota_{\widehat{D}}\omega^{\prime}\rangle$
$\displaystyle=\operatorname{res}_{D}\langle\widehat{m},\nabla\widehat{m}^{\prime}\rangle=-\operatorname{res}_{D}\langle\nabla\widehat{m},\widehat{m}^{\prime}\rangle$
$\displaystyle=\mp\operatorname{res}_{D}\langle\widehat{m}^{\prime},\nabla\widehat{m}\rangle=\mp\operatorname{res}_{D}\langle\widehat{m}^{\prime},\iota_{\widehat{D}}\omega\rangle.\qed$

######  Remark 3.16.

Theorem 3.14 is of course a variant of the theorem asserting compatibility
between duality of $\mathscr{D}$-modules and proper push-forward (see[42]),
but more in the spirit of [45, Cor. 5.2.7]. The presentation given here owes
much to [40, App. 2]. Nevertheless, the present formulation is more precise
and can lead to explicit computations.

In the present form, this theorem has a long history, starting (as far as we
know) with [16] (seealso [17]) in rank one. The computation that we perform
with Čech complexes by using a formal neighbourhood of $D$ is inspired from
loc.cit. The result was used by Deligne for showing compatibility between
duality and the irregular Hodge filtration introduced in loc.cit.

In a series of papers mentioned in the introduction, Matsumoto et al. also
gave duality results of this kind, either in the case of regular singularities
or in rank one cases with irregular singularities (possibly in higher
dimensions). An instance of Proposition 3.13 was already obtained in [41].

###  3.d. Computations on the real blow-up space

In this section, we revisit the results of Section 2.f in the simpler case
where $\dim X=1$ and we give details on the proof that Assumption 2.1(2) holds
for $\widetilde{\mathscr{V}}$ with respect to the tautological pairing
$({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\mkern 1.0mu|\mkern
1.0mu{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$. _In this section, we
use the analytic topology on $U$ and $X$_.

Let
$\varpi:\widetilde{X}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$
denote the real oriented blow-up of $X$ at each point of $D$. Recall that
$\widetilde{X}$ is endowed with holomorphic and $C^{\infty}$ sheaves of
functions with growth conditions. Working on $\widetilde{X}$ makes proofs of
local duality easier since all involved de Rham complexes are sheaves, i.e.,
have cohomology in degree zero only. We denote by
$\widetilde{\jmath}:U\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}\widetilde{X}$
the open inclusion and by
${\widetilde{\imath}}:\varpi^{-1}(D)\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}\widetilde{X}$
the complementary inclusion.

For $\mathscr{V}$ as in Section 3.a, we set

$\widetilde{\mathscr{V}}=\mathscr{A}_{\widetilde{X}}\otimes_{\varpi^{-1}\mathscr{O}_{X}}\varpi^{-1}\mathscr{V}=\mathscr{A}_{\widetilde{X}}(*D)\otimes_{\varpi^{-1}\mathscr{O}_{X}}\varpi^{-1}\mathscr{V}$

(the second equality holds since $\mathscr{V}$ is an
$\mathscr{O}_{X}(*D)$-module), which is a locally free
$\mathscr{A}_{\widetilde{X}}(*D)$-module of finite rank. The $C^{\infty}$ de
Rham complexes with growth conditions
$\operatorname{DR}^{\infty,\mathrm{rd}}(\widetilde{\mathscr{V}})$ and
$\operatorname{DR}^{\infty,\mathrm{mod}}(\widetilde{\mathscr{V}})$ have been
defined in Section 2.f and satisfy Assumptions 2.1(1) (with respect to the
tautological pairing $({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\mkern
1.0mu|\mkern 1.0mu{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$) and
A.18, according to Theorem 2.30. We will check in Proposition 3.18, as
asserted in Theorem 2.30, that they also satisfy Assumption 2.1(2) with
respect to the tautological pairing
$({\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\mkern 1.0mu|\mkern
1.0mu{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$ on $V\otimes
V^{\scriptscriptstyle\vee}$ (as already mentioned, the results of Section 2.b
can easily be adapted to this setting).

The pushforwards by $\varpi$ of these $C^{\infty}$ de Rham complexes are
isomorphic to the corresponding complexes on $X$, allowing for a computation
on $\widetilde{X}$ of the various algebraic de Rham cohomologies, according to
Remark 3.8.

######  Corollary 3.17.

We have natural isomorphisms

∎
$\mathrm{H}^{k}_{{\mathrm{dR}}}(U,V)\simeq\mathrm{H}^{k}(\widetilde{X},\operatorname{DR}^{\infty,\mathrm{mod}}(\widetilde{\mathscr{V}})),\quad\mathrm{H}^{k}_{{\mathrm{dR}},{\mathrm{c}}}(U,V)\simeq\mathrm{H}^{k}(\widetilde{X},\operatorname{DR}^{\infty,\mathrm{rd}}(\widetilde{\mathscr{V}})).$

Recall that we set
$\widetilde{\mathscr{V}}^{\mathrm{rd}}=\mathscr{H}^{0}\operatorname{DR}^{\infty,\mathrm{rd}}(\widetilde{\mathscr{V}})$
and similarly with $\mathrm{mod}$.

######  Proposition 3.18.

The natural pairing

$\widetilde{\mathsf{Q}}_{\mathrm{rd},\mathrm{mod}}^{\infty}:\operatorname{DR}^{\infty,\mathrm{rd}}(\widetilde{\mathscr{V}})\otimes\operatorname{DR}^{\infty,\mathrm{mod}}(\widetilde{\mathscr{V}}^{\scriptscriptstyle\vee})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathscr{E}_{\widetilde{X}}^{\mathrm{rd}},{\mathrm{d}})\simeq\widetilde{\jmath}_{!}\mathbf{C}_{U}$

is perfect, that is, the pairing induced on the $\mathscr{H}^{0}$ is perfect:

$\widetilde{\mathscr{V}}^{\mathrm{rd}}\otimes\widetilde{\mathscr{V}}^{{\scriptscriptstyle\vee}\mathrm{mod}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{\jmath}_{!}\mathbf{C}_{U}$

is perfect.

###### Proof.

As the statement is local on $\widetilde{X}$, we work on an analytic
neighbourhood $\Delta$ of $x\in D$ with coordinate $z$. The Levelt-Turrittin
decomposition (see(1) in Section 2.f) can be lifted locally on
$S^{1}\times\\{0\\}$ with coefficients in $\mathscr{A}_{\widetilde{\Delta}}$,
so that the ramification $\rho$ can be neglected by considering a local
determination of $z^{1/r}$ and $\log z$, where $r$ is the order of
ramification. We are in this way reduced to proving the assertions in the case
of an elementary model entering the Levelt-Turrittin decomposition.

Let $\theta_{o}\in S^{1}$ and let
$\mathrm{nb}(\theta_{o})=(\theta_{o}-\delta,\theta_{o}+\delta)\times[0,\varepsilon)$
be an open neighbourhood of $\theta_{o}$ in $\widetilde{X}$. We set
$\mathrm{nb}(\theta_{o})^{\circ}=\mathrm{nb}(\theta_{o})\cap
U=(\theta_{o}-\delta,\theta_{o}+\delta)\times(0,\varepsilon)$ and consider the
decompositions of $\widetilde{\jmath}$ as the composition of the open
inclusions

$\mathrm{nb}(\theta_{o})^{\circ}\mathrel{\lhook\joinrel\mathchoice{\xrightarrow{\kern
4.0pt\alpha^{+}\kern
3.0pt}}{\stackrel{{\scriptstyle\alpha^{+}}}{{\longrightarrow}}}{}{}}\mathrm{nb}(\theta_{o})^{\circ}\cup\bigl{(}(\theta_{o},\theta_{o}+\delta)\times\\{0\\}\bigr{)}\mathrel{\lhook\joinrel\mathchoice{\xrightarrow{\kern
4.0pt\beta^{+}\kern
3.0pt}}{\stackrel{{\scriptstyle\beta^{+}}}{{\longrightarrow}}}{}{}}\mathrm{nb}(\theta_{o})=\mathrm{nb}(\theta_{o})^{\circ}\cup\bigl{(}(\theta_{o}-\delta,\theta_{o}+\delta)\times\\{0\\}\bigr{)}$

and

$\mathrm{nb}(\theta_{o})^{\circ}\mathrel{\lhook\joinrel\mathchoice{\xrightarrow{\kern
4.0pt\alpha^{-}\kern
3.0pt}}{\stackrel{{\scriptstyle\alpha^{-}}}{{\longrightarrow}}}{}{}}\mathrm{nb}(\theta_{o})^{\circ}\cup\bigl{(}(\theta_{o}-\delta,\theta_{o})\times\\{0\\}\bigr{)}\mathrel{\lhook\joinrel\mathchoice{\xrightarrow{\kern
4.0pt\beta^{-}\kern
3.0pt}}{\stackrel{{\scriptstyle\beta^{-}}}{{\longrightarrow}}}{}{}}\mathrm{nb}(\theta_{o}).$

For an elementary model, three kinds of cases may occur for
$\widetilde{\mathscr{V}}^{\mathrm{rd}}$ and
$\widetilde{\mathscr{V}}^{{\scriptscriptstyle\vee}\mathrm{mod}}$ on such a
sufficiently small neighbourhood:

  * $\scriptscriptstyle\bullet$

$\widetilde{\mathscr{V}}^{\mathrm{rd}}=\widetilde{\jmath}_{*}V^{\nabla}$ and
$\widetilde{\mathscr{V}}^{{\scriptscriptstyle\vee}\mathrm{mod}}=\widetilde{\jmath}_{!}V^{{\scriptscriptstyle\vee}\nabla}$,

  * $\scriptscriptstyle\bullet$

$\widetilde{\mathscr{V}}^{\mathrm{rd}}=\widetilde{\jmath}_{!}V^{\nabla}$ and
$\widetilde{\mathscr{V}}^{{\scriptscriptstyle\vee}\mathrm{mod}}=\widetilde{\jmath}_{*}V^{{\scriptscriptstyle\vee}\nabla}$,

  * $\scriptscriptstyle\bullet$

$\widetilde{\mathscr{V}}^{\mathrm{rd}}=\beta^{+}_{!}\alpha^{+}_{*}V^{\nabla}$
and
$\widetilde{\mathscr{V}}^{{\scriptscriptstyle\vee}\mathrm{mod}}=\beta^{-}_{!}\alpha^{-}_{*}V^{{\scriptscriptstyle\vee}\nabla}$.

Perfectness being clear in the first two cases, let us analyze the third one.
Poincaré-Verdier duality shows perfectness of the natural pairing

$(\beta^{-}_{*}\alpha^{-}_{!}V^{\nabla})\otimes(\beta^{-}_{!}\alpha^{-}_{*}V^{{\scriptscriptstyle\vee}\nabla})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{\jmath}_{!}\mathbf{C}_{\mathrm{nb}(\theta_{o})^{\circ}},$

so one only needs to show the equality of subsheaves of
$\widetilde{\jmath}_{*}V^{\nabla}$:

$\beta^{-}_{*}\alpha^{-}_{!}V^{\nabla}=\beta^{+}_{!}\alpha^{+}_{*}V^{\nabla}.$

These subsheaves clearly coincide away from $\theta_{o}$, and both are zero at
$\theta_{o}$, hence they coincide. ∎

###### Proof of Theorem 3.14.

By applying $\boldsymbol{R}\varpi_{*}$ (which is $\varpi_{*}$ here), the
pairing $\widetilde{\mathsf{Q}}_{\mathrm{rd},\mathrm{mod}}^{\infty}$ induces
the pairing $\mathsf{Q}$ of (3.12). From the compatibility between Verdier
duality and proper pushforward by $\varpi$, together with Proposition C.4, we
deduce that $\mathsf{S}^{1}$ is non-degenerate. ∎

###  3.e. Computation of Betti period pairings

The formula for computing the period pairing

$\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{1}\colon\mathrm{H}_{1}^{\mathrm{rd}}(\widetilde{X},\widetilde{\mathscr{V}})\otimes\mathrm{H}^{1}_{{\mathrm{dR}},\mathrm{mod}}(\widetilde{X},\widetilde{\mathscr{V}}^{\scriptscriptstyle\vee})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}$

(seeSection 2.d) is the natural one described as follows. Let
$\sigma\in\mathrm{H}_{1}^{\mathrm{rd}}(\widetilde{X},\widetilde{\mathscr{V}})$
be represented by a finite sum $\sum_{\ell}c_{\ell}\otimes v_{\ell}$ of
twisted cycles where each
$c_{\ell}\colon[0,1]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{X}$
is a piecewise smooth simplex satisfying $c_{\ell}((0,1))\subset U$ and
$v_{\ell}$ is a horizontal section of
$\mathscr{A}_{\widetilde{X}}^{\mathrm{rd}}\otimes\widetilde{\mathscr{V}}$ on a
neighbourhood of the support of $c_{\ell}$. (The boundary $c_{\ell}(o)$,
$o\in\\{0,1\\}$, may lie in $\widetilde{D}$.) Let
$\omega^{\scriptscriptstyle\vee}\in\Gamma(U,\Omega^{1}_{U}\otimes
V^{\scriptscriptstyle\vee})$ be a representative of a de Rham class in
$\mathrm{H}^{1}_{{\mathrm{dR}},\mathrm{mod}}(\widetilde{X},\widetilde{\mathscr{V}}^{\scriptscriptstyle\vee})=\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V^{\scriptscriptstyle\vee})$.
Then

$\mathsf{P}^{\mathrm{rd},\mathrm{mod}}_{1}(\sigma,\omega^{\scriptscriptstyle\vee})=\sum_{\ell}\int_{0}^{1}c_{\ell}^{*}((v\mkern
1.0mu|\mkern 1.0mu\omega^{\scriptscriptstyle\vee})).$

We will make explicit the formula for computing the period pairing

$\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{1}:\mathrm{H}_{1}^{\mathrm{mod}}(\widetilde{X},\widetilde{\mathscr{V}})\otimes\mathrm{H}^{1}_{{\mathrm{dR}},\mathrm{rd}}(\widetilde{X},\widetilde{\mathscr{V}}^{\scriptscriptstyle\vee})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}$

starting from a representative $(\widehat{m},\omega)$ of a de Rham class in
$\mathrm{H}^{1}_{{\mathrm{dR}},\mathrm{rd}}(\widetilde{X},\widetilde{\mathscr{V}}^{\scriptscriptstyle\vee})=\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V^{\scriptscriptstyle\vee})$.
With respect to the above formula, there is a supplementary regularization
procedure to be performed.

######  Proposition 3.19.

  * $\scriptscriptstyle\bullet$

Let
$\sigma\in\mathrm{H}_{1}^{\mathrm{mod}}(\widetilde{X},\widetilde{\mathscr{V}})$
be represented by a finite sum $\sum_{\ell}c_{\ell}\otimes v_{\ell}$ of
twisted cycles where each
$c_{\ell}\colon[0,1]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{X}$
is a piecewise smooth simplex satisfying $c_{\ell}((0,1))\subset U$ and
$v_{\ell}$ is a horizontal section of
$\mathscr{A}_{\widetilde{X}}^{\mathrm{mod}}\otimes\widetilde{\mathscr{V}}$ on
a neighbourhood of the support of $c_{\ell}$.

  * $\scriptscriptstyle\bullet$

Let
$(\widehat{m},\omega)\in\mathscr{V}_{\widehat{D}}^{\scriptscriptstyle\vee}\oplus\Gamma(U,\Omega^{1}_{U}\otimes
V^{\scriptscriptstyle\vee})$ be a representative of
$[(\widehat{m},\omega)]\in\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V^{\scriptscriptstyle\vee})$.

  * $\scriptscriptstyle\bullet$

For $o\in\\{0,1\\}$, if $c_{\ell}(o)\in\widetilde{D}$, let
$\widetilde{m}_{\ell,o}$ be a germ in
$\widetilde{\mathscr{V}}^{\scriptscriptstyle\vee}_{c_{\ell}(o)}$ having
$\widehat{m}_{\ell,o}$ as asymptotic expansion at $c_{\ell}(o)$. Otherwise let
$\widetilde{m}_{\ell,o}=0$.

Then the following equality holds:

$\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{1}(\sigma,[(\widehat{m},\omega)])=\sum_{\ell}\lim_{\varepsilon\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0}\biggl{[}\biggl{(}\int_{\varepsilon}^{1-\varepsilon}c_{\ell}^{*}((v\mkern
1.0mu|\mkern 1.0mu\omega))\biggr{)}+(v\mkern 1.0mu|\mkern
1.0mu\widetilde{m}_{\ell,0})(c_{\ell}(\varepsilon))-(v\mkern 1.0mu|\mkern
1.0mu\widetilde{m}_{\ell,1})(c_{\ell}(1-\varepsilon))\biggr{]}.$

######  Remark 3.20.

In practice, one only needs to know the first few terms of the asymptotic
expansion of $\widetilde{m}_{\ell,o}$ in order to compute the limit. The
knowledge of $\widehat{m}_{\ell,o}$ up to a finite order is therefore enough.

###### Proof.

We start with the $C^{\infty}$ setting.

######  Lemma 3.21.

Let $\sigma$ be as in the proposition and let
$\eta_{\infty}=\omega_{\infty}-(\nabla+\overline{\partial})m_{\infty}$ be a
representative of a class in
$\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{c}}}(U,V^{\scriptscriptstyle\vee})$ as
in Remark 3.8. Then

$\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{1}(\sigma,[\eta_{\infty}])=\sum_{\ell}\lim_{\varepsilon\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0}\biggl{[}\biggl{(}\int_{\varepsilon}^{1-\varepsilon}c_{\ell}^{*}((v\mkern
1.0mu|\mkern 1.0mu\omega_{\infty}))\biggr{)}+(v\mkern 1.0mu|\mkern
1.0mum_{\infty})(c_{\ell}(\varepsilon))-(v\mkern 1.0mu|\mkern
1.0mum_{\infty})(c_{\ell}(1-\varepsilon))\biggr{]}.$

###### Proof.

Since $\eta_{\infty}$ has rapid decay, we have

$\mathsf{P}^{\mathrm{mod},\mathrm{rd}}_{1}(\sigma,[\eta_{\infty}])=\sum_{\ell}\int_{0}^{1}c_{\ell}^{*}((v\mkern
1.0mu|\mkern
1.0mu\eta_{\infty}))=\sum_{\ell}\lim_{\varepsilon\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0}\int_{\varepsilon}^{1-\varepsilon}c_{\ell}^{*}((v\mkern
1.0mu|\mkern 1.0mu\eta_{\infty})),$

and we write

$\displaystyle\int_{\varepsilon}^{1-\varepsilon}c_{\ell}^{*}((v\mkern
1.0mu|\mkern 1.0mu-(\nabla+\overline{\partial})m_{\infty}))$
$\displaystyle=-\int_{\varepsilon}^{1-\varepsilon}{\mathrm{d}}c_{\ell}^{*}((v\mkern
1.0mu|\mkern 1.0mum_{\infty}))$ $\displaystyle=(v\mkern 1.0mu|\mkern
1.0mum_{\infty})(c_{\ell}(\varepsilon))-(v\mkern 1.0mu|\mkern
1.0mum_{\infty})(c_{\ell}(1-\varepsilon)).\qed$

#### End of the proof of Proposition 3.19

Let $\widetilde{m}_{\ell,o}$ be as in the proposition. Then
$\widetilde{m}_{\ell,o}-m_{\infty}$ has rapid decay at $c_{\ell}(0)$ and
$c_{\ell}(1)$ along the path $c_{\ell}$, so that we can replace $m_{\infty}$
with $\widetilde{m}_{\ell,o}$ in the formula of Lemma 3.21 without changing
the limit. ∎

###  3.f. Quadratic relations in dimension one

We summarize the consequences of the identifications previously obtained in
this section to the form of quadratic relations in dimension one. The setting
is as in Section 3.a. We assume that $(V,\nabla)$ is endowed with a non-
degenerate pairing
$\langle{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\rangle:(V,\nabla)\otimes(V,\nabla)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathscr{O}_{U},{\mathrm{d}})$.
We assume that it is symmetric or skew-symmetric, that we denote by
$\pm$-symmetric. We will make explicit the way to express middle quadratic
relations (Corollaries 2.32, 2.16, and Remark 2.17) in the present setting.

#### Middle de Rham pairing

Let $d$ be the dimension of

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

  * $\scriptscriptstyle\bullet$

We choose $d$ elements $\omega_{1},\dots,\omega_{d}$ in
$\mathrm{H}^{1}_{{\mathrm{dR}}}(U,V)$ and we solve the equation
$\nabla\widehat{m}_{j}=\iota_{\widehat{D}}\omega_{j}$ for each $j$ and at each
point of $D$, with $\widehat{m}_{j}\in V_{\widehat{D}}$. We choose such
solutions $\widehat{m}_{j}$.

  * $\scriptscriptstyle\bullet$

The matrix $\mathsf{S}^{1}_{\mathrm{mid}}$ of the de Rham pairing with respect
to these families has entries $\mathsf{S}^{1}_{{\mathrm{mid}};i,j}=\sum_{x\in
D}\operatorname{res}_{x}\langle\widehat{m}_{i},\iota_{\widehat{D}}\omega_{j}\rangle$.
It is $\mp$-symmetric.

  * $\scriptscriptstyle\bullet$

If $\det\mathsf{S}^{1}_{\mathrm{mid}}\neq 0$, then the family $(\omega_{j})$
is a basis of $\mathrm{H}^{1}_{{\mathrm{dR}},{\mathrm{mid}}}(U,V)$.

#### Middle Betti pairing

The space
$\mathrm{H}_{1}^{\mathrm{mid}}(U,V)=\operatorname{im}\bigl{[}\mathrm{H}_{1}^{\mathrm{rd}}(U,V)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}_{1}^{\mathrm{mod}}(U,V)\bigr{]}$
has also dimension $d$.

  * $\scriptscriptstyle\bullet$

We choose $d$ elements $\beta_{1},\dots,\beta_{d}$ in
$\mathrm{H}_{1}^{\mathrm{mod}}(U,V)$ which are the images of elements
$\alpha_{1},\dots,\alpha_{d}$ in $\mathrm{H}_{1}^{\mathrm{rd}}(U,V)$.

  * $\scriptscriptstyle\bullet$

The matrix $\mathsf{B}_{1}^{\mathrm{mid}}$ of the Betti pairing with respect
to these families is computed for example by means of Proposition 2.25. It is
$\mp$-symmetric.

  * $\scriptscriptstyle\bullet$

If $\det\mathsf{B}_{1}^{\mathrm{mid}}\neq 0$, then the family $(\beta_{i})$ is
a basis of $\mathrm{H}_{1}^{\mathrm{mid}}(U,V)$.

#### Middle period pairing

Let us fix a triangulation of $(X,D)$ that is induced by a triangulation of
$\widetilde{X}$ such that the $\mathbf{R}$-constructible sheaves
$\mathscr{H}^{0}\operatorname{DR}^{\mathrm{rd}}\mathscr{V}$ and
$\mathscr{H}^{0}\operatorname{DR}^{\mathrm{mod}}\mathscr{V}$ are constant on
each open $1$-simplex (seeTheorem 2.30 or the proof of Proposition 3.18). We
assume the simplices are given an orientation. Let us write each $\beta_{i}$
as $\sum_{\ell}c_{\ell}\otimes v_{i,\ell}$, where
$c_{\ell}:[0,1]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$
runs among the oriented $1$-simplices of the chosen triangulation, and
$v_{i,\ell}$ is a (possibly zero) section of $\mathscr{V}^{\mathrm{an}}$ with
moderate growth in the neighbourhood of $c_{\ell}$. The cycle condition reads:

$\text{for each vertex }x\in U^{\mathrm{an}},\quad\sum_{\ell\mid
c_{\ell}(1)=x}v_{i,\ell}(x)-\sum_{\ell\mid c_{\ell}(0)=x}v_{i,\ell}(x)=0.$

The middle period matrix $\mathsf{P}^{1}_{\mathrm{mid}}$ is the $d\times d$
matrix with entries

$\mathrm{Pf}\,\sum_{\ell}\int_{c_{\ell}}\langle v_{i,\ell},\omega_{j}\rangle,$

where the “finite part” $\mathrm{Pf}$ means that, equivalently,

  * $\scriptscriptstyle\bullet$

either we replace $\beta_{i}$ with $\alpha_{i}$ that we realize as
$\sum_{\ell}c_{\ell}\otimes v^{\prime}_{i,\ell}$, where $v^{\prime}_{i,\ell}$
has rapid decay near the boundary points of $c_{\ell}$ that are contained in
$D$ and satisfy the cycle condition, and

$\mathrm{Pf}\int_{c_{\ell}}\langle
v_{i,\ell},\omega_{j}\rangle=\int_{c_{\ell}}\langle
v^{\prime}_{i,\ell},\omega_{j}\rangle,$

  * $\scriptscriptstyle\bullet$

or we replace $\omega_{j}$ with $(\widehat{m}_{j},\omega_{j})$, for each
$\ell$ such that $c_{\ell}(o)\in D$ ($o=0,1$), we choose a germ
$\widetilde{m}_{j,\ell}^{o}$ as in Proposition 3.19, and we set

$\mathrm{Pf}\int_{c_{\ell}}\langle
v_{i,\ell},\omega_{j}\rangle=\lim_{\varepsilon\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0}\biggl{[}\int_{\varepsilon}^{1-\varepsilon}c_{\ell}^{*}(\langle
v_{i,\ell},\omega_{j}\rangle)+\langle
v_{i,\ell},\widetilde{m}_{j,\ell}^{0}\rangle(c_{\ell}(0))-\langle
v_{i,\ell},\widetilde{m}_{j,\ell}^{1}\rangle(c_{\ell}(1))\biggr{]}.$

#### Middle quadratic relations

They now read (seeCorollary 2.32)

(3.22)
$\mp(2\pi\mathsf{i})\,\mathsf{B}_{1}^{\mathrm{mid}}=\mathsf{P}_{1}^{\mathrm{mid}}\cdot(\mathsf{S}^{1}_{\mathrm{mid}})^{-1}\cdot{}^{t}\mathsf{P}_{1}^{\mathrm{mid}}.$

##  Appendix A Twisted singular chains

In this section, we recall classical results from the Cartan Seminars [7, 8].
However, we do not use homology of cosheaves as in [6]. We assume that $X$ is
a compact topological space, so that all locally finite open coverings are
finite. All sheaves are sheaves of vector spaces over some field
${\boldsymbol{k}}$, in order to avoid any problem with torsion.

###  A.a. Presheaves and sheaves

Let $\widetilde{\mathscr{F}}$ be a presheaf on $X$ and let $\mathscr{F}$ be
the associated sheaf. We have $\mathscr{F}_{x}=\varinjlim_{U\ni
x}\widetilde{\mathscr{F}}(U)$. For every open set $U\subset X$, there is a
natural morphism
$\widetilde{\mathscr{F}}(U)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathop{\textstyle\coprod}\displaylimits_{x\in
U}\mathscr{F}_{x}$. We denote by $\widetilde{s}_{x}\in\mathscr{F}_{x}$ the
germ of a section $\widetilde{s}\in\widetilde{\mathscr{F}}(X)$. If
$\widetilde{s}_{x}=0$, there exists $U_{x}$ such that the image of
$\widetilde{s}$ in $\widetilde{\mathscr{F}}(U_{x})$ vanishes. It follows that
the support of $\widetilde{s}$, that is, the subset $\\{x\in
X\mid\widetilde{s}_{x}=0\\}$ is closed.

  * $\scriptscriptstyle\bullet$

We say that $\widetilde{\mathscr{F}}$ has the _surjectivity property_ if

(A.1) $\text{for all }x\in
X,\quad\widetilde{\mathscr{F}}(X)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{F}_{x}\text{
\ is surjective}.$

This property is also called $\Phi$-softness, for $\Phi$ the family of closed
subsets of $X$ consisting of points only.

  * $\scriptscriptstyle\bullet$

We say that $\widetilde{\mathscr{F}}$ has the _injectivity property_ if

(A.2)
$\widetilde{\mathscr{F}}(X)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathop{\textstyle\coprod}\displaylimits_{x\in
X}\mathscr{F}_{x}\text{ \ is injective}.$

The latter condition is equivalent to asking that the natural morphism
$\widetilde{\mathscr{F}}(X)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma(X,\mathscr{F})$
is injective, since (A.2) factorizes through the latter.

  * $\scriptscriptstyle\bullet$

We say that $\widetilde{\mathscr{F}}$ is a _fine presheaf_ if, for any
(locally) finite open covering $\mathscr{U}=(U_{i})$ of $X$, there exist
closed subsets $F_{i}\subset U_{i}$ and endomorphisms $\widetilde{\ell}_{i}$
of $\widetilde{\mathscr{F}}$ (i.e., each $\widetilde{\ell}_{i}$ is a family,
indexed by open sets $U$, of endomorphisms
$\widetilde{\ell}_{i}:\widetilde{\mathscr{F}}(U)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{\mathscr{F}}(U)$
compatible with restrictions $U^{\prime}\subset U$ in an obvious way) such
that

    1. (a)

$\widetilde{\ell}_{i}=0$ on $\widetilde{\mathscr{F}}(U)$ if $U\cap
F_{i}=\varnothing$,

    2. (b)

for any open set $U$ and each $\widetilde{s}\in\widetilde{\mathscr{F}}(U)$,
the sum $\sum_{i}\widetilde{\ell}_{i}(\widetilde{s})$ exists in
$\widetilde{\mathscr{F}}(U)$ and is equal to $\widetilde{s}$. Let
$\ell_{i}:\mathscr{F}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{F}$
be the endomorphism associated with $\widetilde{\ell}_{i}$. If $x\notin
F_{i}$, there exist $U\ni x$ such that $U\cap F_{i}=\varnothing$, and hence
$\ell_{i,x}=0:\mathscr{F}_{x}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{F}_{x}$.
Moreover, for each $U$ and $s\in\Gamma(U,\mathscr{F})$, we have, for any $x\in
U$, the relation $\sum_{i}\ell_{i,x}(s_{x})=s_{x}$ (the sum is finite since
the covering is finite).

######  Proposition A.3.

Assume that $\widetilde{\mathscr{F}}$ is a fine presheaf that satisfies the
surjectivity property (A.1). Then the natural morphism

$\widetilde{\mathscr{F}}(X)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma(X,\mathscr{F})$

is onto. If it moreover satisfies the injectivity property (A.2), then it is
an isomorphism.

###### Proof.

Once the first statement is proved, the second statement is obvious. Let
$s\in\Gamma(X,\mathscr{F})$. By (A.1), for any $y\in X$, there exists a
section $\widetilde{s}^{y}\in\widetilde{\mathscr{F}}(X)$ whose germ at $y$ is
$s_{y}$. Then there exists a neighbourhood $U_{y}$ of $y$ such that
$\widetilde{s}^{y}_{x}=s_{x}$ for any $x\in U_{y}$. Let $\mathscr{U}$ be a
finite open covering of $X$ such that each open set $U_{i}$ is of the form
$U_{y}$ as above. Let $\widetilde{s}^{i}\in\widetilde{\mathscr{F}}(X)$ be such
that $\widetilde{s}^{i}_{x}=s_{x}$ for any $x\in U_{i}$. On the one hand, we
have $\ell_{i,x}(s_{x})=(\widetilde{\ell}_{i}(\widetilde{s}^{i}))_{x}$ for any
$x\in U_{i}$, and on the other hand $s_{x}=\sum_{i\mid U_{i}\ni
x}\ell_{i,x}(s_{x})$, whence

$s_{x}=\sum_{i\mid U_{i}\ni
x}(\widetilde{\ell}_{i}(\widetilde{s}^{i}))_{x}=\sum_{i}(\widetilde{\ell}_{i}(\widetilde{s}^{i}))_{x},$

since $(\widetilde{\ell}_{i}(\widetilde{s}^{i}))_{x}=0$ for $x\notin U_{i}$
according to (a) above. We conclude, since the covering is finite,

$s=\sum_{i}\operatorname{im}(\widetilde{\ell}_{i}(\widetilde{s}^{i}))=\operatorname{im}\Bigl{(}\sum_{i}\widetilde{\ell}_{i}(\widetilde{s}^{i})\Bigr{)},$

where we consider the image of the morphism
$\widetilde{\mathscr{F}}(X)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma(X,\mathscr{F})$.
It follows that the natural morphism
$\widetilde{\mathscr{F}}(X)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma(X,\mathscr{F})$
is surjective. ∎

###  A.b. Homotopically fine sheaves

See [10, §8].

######  Definition A.4.

Let $(\widetilde{\mathscr{F}},{\mathrm{d}})$ be a $\mathbf{Z}$-graded
differential presheaf on $X$. We say that $\widetilde{\mathscr{F}}$ is
homotopically fine if, for any (locally) finite open covering
$\mathscr{U}=(U_{i})$ of $X$, there exist closed subsets $F_{i}$ contained in
$U_{i}$ and endomorphisms $\widetilde{\ell}_{i}$ and $\widetilde{k}$ of
$\widetilde{\mathscr{F}}$ such that

  1. (1)

$\widetilde{\ell}_{i}=0:\widetilde{\mathscr{F}}(U)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{\mathscr{F}}(U)$
if $U\cap F_{i}=\varnothing$,

  2. (2)

$\sum_{i}\widetilde{\ell}_{i}=\operatorname{Id}+\widetilde{k}{\mathrm{d}}+{\mathrm{d}}\widetilde{k}$.

We have a similar definition for sheaves. We note the following properties:

  1. (a)

If a graded differential presheaf is homotopically fine, the associated graded
differential sheaf is also homotopically fine.

  2. (b)

If $(\widetilde{\mathscr{F}},{\mathrm{d}})$ is homotopically fine and if
$\widetilde{\mathscr{G}}$ is any presheaf, then the graded differential
presheaf
$(\widetilde{\mathscr{F}}\otimes\widetilde{\mathscr{G}},{\mathrm{d}}\otimes\operatorname{Id})$
is also homotopically fine. Recall that
$\widetilde{\mathscr{F}}\otimes\widetilde{\mathscr{G}}$ is the presheaf
$U\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\widetilde{\mathscr{F}}(U)\otimes\widetilde{\mathscr{G}}(U)$.
The same property holds for graded differential sheaves.

######  Theorem A.5 ([11, §3, Corollaire]).

Let $(\mathscr{F},{\mathrm{d}})$ be a homotopically fine $\mathbf{Z}$-graded
differential sheaf. For each $p$, the natural morphism

$\mathrm{H}^{p}(\Gamma(X,\mathscr{F}),{\mathrm{d}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{H}^{p}(X,(\mathscr{F},{\mathrm{d}}))$

is an isomorphism.

The proof is obtained by choosing a resolution
$C^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}$ of
${\boldsymbol{k}}_{X}$ by fine sheaves and realizing
$\boldsymbol{H}^{p}(X,(\mathscr{F},{\mathrm{d}}))$ as
$\mathrm{H}^{p}(\Gamma(X,(C\otimes\mathscr{F})^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$,
where
$(C\otimes\mathscr{F})^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}$
stands for the simple complex associated with the double complex
$C^{p}\otimes\mathscr{F}^{q}$. The assertion follows from the degeneration at
$E_{1}$ of the second spectral sequence, since the second filtration is
regular. We can complete this theorem in terms of presheaves as follows.

######  Proposition A.6.

Let $(\widetilde{\mathscr{F}},{\mathrm{d}})$ be a homotopically fine
$\mathbf{Z}$-graded differential presheaf satisfying (A.1) and (A.2). Then the
natural morphism

$\mathrm{H}^{p}(\widetilde{\mathscr{F}}(X),{\mathrm{d}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{H}^{p}(X,(\mathscr{F},{\mathrm{d}}))$

is an isomorphism.

###### Proof.

According to Theorem A.5, it is enough to prove that
$(\widetilde{\mathscr{F}}(X),{\mathrm{d}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\Gamma(X,\mathscr{F}),{\mathrm{d}})$
induces an isomorphism in cohomology.

  * $\scriptscriptstyle\bullet$

Let $s\in\Gamma(X,\mathscr{F})$ be such that ${\mathrm{d}}s=0$, and let
$\widetilde{s}^{i}$ be as in the proof of Proposition A.3. Recall that
$\ell_{i,x}(s_{x})=(\widetilde{\ell}_{i}(\widetilde{s}^{i}))_{x}$ for any
$x\in U_{i}$. Besides, $s_{x}=\sum_{i\mid U_{i}\ni
x}\ell_{i,x}(s_{x})-{\mathrm{d}}ks_{x}$ since $s$ is ${\mathrm{d}}$-closed.
Hence

$s_{x}+{\mathrm{d}}ks_{x}=\sum_{i\mid U_{i}\ni
x}(\widetilde{\ell}_{i}(\widetilde{s}^{i}))_{x}=\sum_{i}(\widetilde{\ell}_{i}(\widetilde{s}^{i}))_{x},$

since $(\widetilde{\ell}_{i}(\widetilde{s}^{i}))_{x}=0$ for $x\notin U_{i}$
according to (a) above. Since the covering is finite, we get

$s+{\mathrm{d}}ks=\sum_{i}\operatorname{im}(\widetilde{\ell}_{i}(\widetilde{s}^{i}))=\operatorname{im}\Bigl{(}\sum_{i}\widetilde{\ell}_{i}(\widetilde{s}^{i})\Bigr{)}.$

Since ${\mathrm{d}}(s-{\mathrm{d}}ks)=0$, (A.2) implies that
$\sum_{i}\widetilde{\ell}_{i}(\widetilde{s}^{i})$ is closed, which gives the
surjectivity of the cohomological map.

  * $\scriptscriptstyle\bullet$

Let $\widetilde{s}$ be a ${\mathrm{d}}$-closed section of
$\widetilde{\mathscr{F}}(X)$ whose image $s\in\Gamma(X,\mathscr{F})$ satisfies
$s={\mathrm{d}}\sigma$ for some $\sigma\in\Gamma(X,\mathscr{F})$. Up to
replacing $\sigma$ with $\sigma+({\mathrm{d}}k+k{\mathrm{d}})\sigma$ (and
$\widetilde{s},s$ with $\widetilde{s}+{\mathrm{d}}\widetilde{k}\widetilde{s}$,
$s+{\mathrm{d}}ks$) we can assume that $\sigma$ is the image of some
$\widetilde{\sigma}\in\widetilde{\mathscr{F}}(X)$ as argued above. The image
of $\widetilde{s}-{\mathrm{d}}\widetilde{\sigma}$ in $\Gamma(X,\mathscr{F})$
is zero, and hence $\widetilde{s}={\mathrm{d}}\widetilde{\sigma}$, according
to (A.2). This proves the injectivity of the cohomological map.∎

###  A.c. Homotopy operator for singular chains

See [9, §3].

Let ($S_{\scriptscriptstyle\bullet}(X),\partial)$ be the complex of singular
chains. Recall that $S_{\scriptscriptstyle\bullet}(X)$ is the infinite-
dimensional vector space having as a basis the set of singular simplices
(continuous maps from a simplex to $X$). The support of a singular simplex is
its image, which is a compact subset of $X$. For a closed subset $Z\subset X$,
we regard $S_{\scriptscriptstyle\bullet}(Z)$ as the subspace of
$S_{\scriptscriptstyle\bullet}(X)$ generated by simplices with support in $Z$
and $S_{\scriptscriptstyle\bullet}(X)/S_{\scriptscriptstyle\bullet}(Z)$ as the
subspace generated by simplices whose support is not contained in $Z$. For any
(locally) finite open covering $\mathscr{U}=(U_{i})$, there exist
(seeloc.cit.) endomorphisms $\ell$ and $k$ of
$S_{\scriptscriptstyle\bullet}(X)$ (where $\ell$ preserves the grading and $k$
has degree one) such that

(A.7) $\begin{cases}{\scriptscriptstyle\bullet}\text{ for any simplex
$\sigma\in S_{\scriptscriptstyle\bullet}(X)$, $\ell(\sigma)$ is a sum of
simplices,}\\\ \hphantom{{\scriptscriptstyle\bullet}}\text{each one being
contained in some~{}$U_{i}\cap|\sigma|$,}\\\ {\scriptscriptstyle\bullet}\text{
if $|\sigma|$ is contained in some $U_{i}$, then $\ell(\sigma)=\sigma$,}\\\
{\scriptscriptstyle\bullet}\text{ $\ell=\operatorname{Id}+k\partial+\partial
k$ (in particular, $\ell\partial=\partial\ell$).}\end{cases}$

For $Z$ closed in $X$, let
$(\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}},\partial)$ be the
chain presheaf

$U\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(U)=S_{\scriptscriptstyle\bullet}(X)/S_{\scriptscriptstyle\bullet}((X\smallsetminus
U)\cup Z)$

and let $(\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}},\partial)$ be the
associated differential sheaf (with the usual convention
$(\eucal{C}_{X,Z}^{-{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}})=(\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}},\partial)$,
we obtain a $(-\mathbf{N})$-graded differential sheaf). For any $x\in X$, we
have

$\eucal{C}_{X,Z,{\scriptscriptstyle\bullet},x}=S_{\scriptscriptstyle\bullet}(X)/S_{\scriptscriptstyle\bullet}((X\smallsetminus\\{x\\})\cup
Z).$

More precisely, $\eucal{C}_{X,Z,{\scriptscriptstyle\bullet},x}$ is identified
with the subspace of $S_{\scriptscriptstyle\bullet}(X)$ with basis consisting
of those singular simplices whose support contains $x$ and is not contained in
$Z$. Since
$\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X)=S_{\scriptscriptstyle\bullet}(X)/S_{\scriptscriptstyle\bullet}(Z)$,
properties (A.1) and (A.2) obviously hold.

######  Proposition A.8 ([10, p. 8]).

Given a finite open covering $\mathscr{U}$ of $X$, there exist a closed
covering $(F_{i})$ with $F_{i}\subset U_{i}$ for all $i$ and endomorphisms
$\widetilde{\ell}_{i}$ and $\widetilde{k}$ of the presheaf
$\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}$ such that

  1. (1)

$\widetilde{\ell}_{i}=0:\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(U)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(U)$
if $U\cap F_{i}=\varnothing$, and in particular $\widetilde{\ell}_{i}$ induces
the zero map on each germ $\mathscr{F}_{x}$ with $x\notin F_{i}$,

  2. (2)

for any singular simplex $\sigma$, each simplex component $\tau$ of
$\widetilde{\ell}_{i}(\sigma)$ satisfies $|\tau|\subset F_{i}\cap|\sigma|$,

  3. (3)

$\sum_{i}\widetilde{\ell}_{i}=\operatorname{Id}+\widetilde{k}{\mathrm{d}}+{\mathrm{d}}\widetilde{k}$.

######  Corollary A.9.

The presheaf
$(\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}},\partial)$ is
homotopically fine.∎

It follows that the chain complex
$(\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}},\partial)$ is homotopically
fine.

###### Proof of Proposition A.8.

Let $(U_{i})_{i\in I}$ be a finite open covering of $X$. One can find closed
subsets $F_{i}\subset U_{i}$ such that $(F_{i})_{i\in I}$ is a closed covering
of $X$. We fix a total order on $I$ and we write $I=\\{1,\dots,n\\}$. We also
fix $\ell$ and $k$ as in (A.7).

Let $\sigma$ be a simplex in $S_{\scriptscriptstyle\bullet}(X)$. Then we
define $\widetilde{\ell}_{i}(\sigma)$ inductively on $i=1,\dots,n$ as the sum
of components of the chain
$\ell(\sigma)-\sum_{j<i}\widetilde{\ell}_{j}(\sigma)$ whose underlying
simplices have support in $F_{i}$. The support is also contained in
$|\sigma|$. We can then extend the definition to any finite chain $\sigma\in
S_{\scriptscriptstyle\bullet}(X)$. Then
$\ell(\sigma)=\sum_{i}\widetilde{\ell}_{i}(\sigma)$ in
$S_{\scriptscriptstyle\bullet}(X)$ and
$\sum_{i}\widetilde{\ell}_{i}(\sigma)=\sigma+k\partial\sigma+\partial
k\sigma$.

Let $U$ be an open subset of $X$. Since $\ell,k,$ and $\partial$ preserve the
support, the above construction also preserves $S((X\smallsetminus U)\cup Z)$,
and thus defines endomorphisms $\widetilde{\ell}_{i}$ and $\widetilde{k}$ of
$\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(U)$ satisfying the
desired properties. ∎

###  A.d. Singular chains with coefficients in a sheaf

We make a statement of [10, p. 8] precise. Let $\mathscr{F}$ be any sheaf. We
denote by
$\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(\mathscr{F})$ the
presheaf

$U\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(U)\otimes\Gamma(U,\mathscr{F})$

and by $\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}}(\mathscr{F})$ the
associated sheaf
$\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}}\otimes\mathscr{F}$. By Corollary
A.9 and (a) and (b) after Definition A.4, both
$\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(\mathscr{F})$ and
$\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}}(\mathscr{F})$ are homotopically
fine. These are chain complexes indexed by $\mathbf{N}$, when equipped with
the boundary $\partial\otimes\operatorname{Id}$, that we simply denote by
$\partial$. As usual, we regard them as differential sheaves with grading
indexed by $-\mathbf{N}$ by setting
e.g.$\widetilde{\eucal}{C}_{X,Z}^{-p}(\mathscr{F})=\widetilde{\eucal}{C}_{X,Z,p}(\mathscr{F})$
and with degree-one differential ${\mathrm{d}}$ identified with $\partial$.
Let us set, by definition,

$\mathrm{H}_{p}(X,Z;\mathscr{F})=\boldsymbol{H}^{-p}(X,(\eucal{C}_{X,Z}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}(\mathscr{F}),{\mathrm{d}})).$

Then, since each term of the complex
$\eucal{C}_{X,Z}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}(\mathscr{F})$
is homotopically fine, Theorem A.5 implies

$\mathrm{H}_{p}(X,Z;\mathscr{F})\simeq\mathrm{H}_{p}(\Gamma(X,(\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}}(\mathscr{F}),\partial)).$

In particular, given an exact sequence
$0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{F}^{\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{F}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{F}^{\prime\prime}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0$
of sheaves, there is a long exact sequence

(A.10)
$\cdots\rightarrow\mathrm{H}_{p}(X,Z;\mathscr{F}^{\prime})\rightarrow\mathrm{H}_{p}(X,Z;\mathscr{F})\rightarrow\mathrm{H}_{p}(X,Z;\mathscr{F}^{\prime\prime})\rightarrow\mathrm{H}_{p-1}(X,Z;\mathscr{F}^{\prime})\rightarrow\cdots$

Besides, let us consider the vector space333It should not be confused with
$\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(\mathscr{F})(X)$,
that is,
$\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X)\otimes\mathscr{F}(X)$.

(A.11)
$\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F})=\mathop{\textstyle\bigoplus}\displaylimits_{\sigma}\bigl{[}{\boldsymbol{k}}\sigma\otimes\Gamma(|\sigma|,\mathscr{F})\bigr{]},$

where the sum runs over singular simplices $\sigma$ whose support is not
contained in $Z$, together with the boundary map induced by
$\partial\otimes\operatorname{Id}$. More precisely, if
$\partial\sigma=\sum_{k}\tau_{k}$, then $\partial(\sigma\otimes
f)=\sum_{j}\tau_{j}\otimes f|_{|\tau_{j}|}$, where the sum is taken over $j$
such that the support of $\tau_{j}$ is not contained in $Z$. A $p$-cycle is a
sum $\sum_{a}(\sigma_{a}\otimes f_{a})$ such that, for each $(p-1)$-simplex
$\tau$ with support not contained in $Z$,

(A.12) $\sum_{\begin{subarray}{c}a,k\\\
\tau_{a,k}=\tau\end{subarray}}f_{a}|_{|\tau_{a,k}|}=0\in\Gamma(|\tau|,\mathscr{F}).$

Such an element $\sigma\otimes f$, with $f\in\Gamma(|\sigma|,\mathscr{F})$,
defines a global section in
$\Gamma(X,\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}}(\mathscr{F}))$. Indeed,
let $U$ be an open neighbourhood of $|\sigma|$ on which $f$ is defined. Then
$\sigma\otimes f$ defines an element of
$\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(U)\otimes\Gamma(U,\mathscr{F})$,
and hence of
$\Gamma(U,\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}}(\mathscr{F}))$.
Moreover, since the image of $\sigma$ is zero in
$\Gamma(U,\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}})$ if
$U\cap|\sigma|=\varnothing$, this element is supported on $|\sigma|$, and
hence extends in a unique way to a global section of
$\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}}(\mathscr{F})$ on $X$.

The natural morphism

(A.13)
$\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma(X,\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}}(\mathscr{F}))$

is clearly compatible with $\partial$. Moreover, it is injective. Indeed, let
$(\sigma_{k})_{k}$ be distinct singular simplices and, for each $k$, let
$f_{k}\in\Gamma(|\sigma_{k}|,\mathscr{F})$ be such that
$\sum_{k}\sigma_{k,x}\otimes\nobreak f_{k,x}=0$ for all $x$, where
$\sigma_{k,x}$ denotes the image of $\sigma_{k}$ in
$\eucal{C}_{X,Z,{\scriptscriptstyle\bullet},x}$. Let $K_{x}$ be the set of
indices $k$ such that $x\in|\sigma_{k}|$. Then $(\sigma_{k,x})_{k\in K_{x}}$
is part of a basis of $\eucal{C}_{X,Z,{\scriptscriptstyle\bullet},x}$, and
$\sum_{k\in K_{x}}\sigma_{k,x}\otimes f_{k,x}=0$ implies that $f_{k,x}=0$ for
any $k\in K_{x}$. Since $\mathscr{F}$ is a sheaf, this implies the vanishing
$f_{k}=0$ in $\Gamma(|\sigma_{k}|,\mathscr{F})$ for all $k$.

######  Proposition A.14.

The chain map
$(\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\Gamma(X,\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}}(\mathscr{F})),\partial)$
induces an isomorphism

$\mathrm{H}_{p}(\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial)\simeq\mathrm{H}_{p}(X,Z;\mathscr{F}).$

###### Proof.

We argue as for Proposition A.6.

  * $\scriptscriptstyle\bullet$

Let
$s\in\Gamma(X,\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}}\otimes\mathscr{F})$.
There exists an open covering $\mathscr{U}$ and
$\widetilde{s}^{i}\in\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(U_{i})\otimes\mathscr{F}(U_{i})$
such that $s_{x}=\widetilde{s}^{i}_{x}$ for any $x\in\nobreak U_{i}$. We
decompose $\widetilde{s}^{i}=\sum_{j}\sigma_{ij}\otimes f_{ij}$, where
$\sigma_{ij}$ are singular simplices with support not contained in
$(X\smallsetminus U_{i})\cup Z$, and $f_{ij}\in\mathscr{F}(U_{i})$. Let
$(F_{i})$ be a closed covering such that $F_{i}\subset U_{i}$ for each $i$ and
let $(\widetilde{\ell}_{i},\widetilde{k})$ be as in Proposition A.8, tensored
with $\operatorname{Id}$ so that they act on the presheaf
$\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}\otimes\mathscr{F}$.
We denote by $\ell_{i}$ and $k$ the induced sheaf morphisms. We have
$s=\sum_{i}\ell_{i}(s)-(\partial k+k\partial)s$. For any $x$,
$\ell_{i}(s)_{x}=0$ if $x\notin F_{i}$, according to Proposition A.8(1), and
$\ell_{i}(s)_{x}=\widetilde{\ell}_{i}(\widetilde{s}^{i})_{x}$ otherwise, and
therefore

$\displaystyle s_{x}=\sum_{i\mid U_{i}\ni
x}\widetilde{\ell}_{i}(\widetilde{s}^{i})_{x}-((k\partial+\partial k)s)_{x}$
$\displaystyle=\sum_{i\mid U_{i}\ni
x}\sum_{j}\widetilde{\ell}_{i}(\sigma_{ij})_{x}\otimes
f_{ij,x}-((k\partial+\partial k)s)_{x}$
$\displaystyle=\sum_{i}\sum_{j}\widetilde{\ell}_{i}(\sigma_{ij})_{x}\otimes
f_{ij,x}-((k\partial+\partial k)s)_{x},$

since $\widetilde{\ell}_{i}(\sigma_{ij})_{x}=0$ if $x\notin U_{i}$. Moreover,
if we write $\widetilde{\ell}_{i}(\sigma_{ij})\otimes
f_{ij}=\sum_{a}\sigma_{a}\otimes f_{a}$, where $\sigma_{a}$ are singular
simplices satisfying $|\sigma_{a}|\subset|\sigma_{ij}|\subset F_{i}$, and
$f_{a}\in\mathscr{F}(U_{i})$, we have $(\sigma_{a}\otimes f_{a})_{x}=0$ if
$x\notin|\sigma_{a}|$, and hence we can replace $f_{a}$ with its restriction
in $\Gamma(|\sigma_{a}|,\mathscr{F})$. Therefore,

$s=\operatorname{im}\Bigl{[}\sum_{i}\sum_{j}\widetilde{\ell}_{i}(\sigma_{ij})\otimes
f_{ij}\Bigr{]}-((k\partial+\partial k)s),$

where the term between brackets belongs to
$\widetilde{\eucal}{C}_{X,Z,{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}(X,\mathscr{F})$.
If $s$ is closed, then $s+\partial ks$ is the image of
$\sum_{i}\sum_{j}\widetilde{\ell}_{i}(\sigma_{ij})\otimes f_{ij}$ and the
latter is closed since $\partial(s+\partial ks)=0$ and (A.13) is injective.
This implies surjectivity of the homology map.

  * $\scriptscriptstyle\bullet$

Let $\widetilde{s}=\sum_{a}\sigma_{a}\otimes f_{a}$, where $\sigma_{a}$ are
singular simplices and $f_{a}\in\Gamma(|\sigma_{a}|,\mathscr{F})$, be closed
and such that its image $s$ is equal to $\partial s^{\prime}$. The above
formula shows that, up to replacing $s^{\prime}$ with
$s^{\prime}+(k\partial+\partial k)s^{\prime}$ (and $s$ with $s+\partial ks$),
we can assume that $s^{\prime}$ lies in the image of
$\widetilde{s}^{\prime}\in\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F})$.
Then $\widetilde{s}-\partial\widetilde{s}^{\prime}$ has image zero, and hence
is zero according to the injectivity of (A.13), so the homology map is
injective.∎

######  Proposition A.15 (Excision).

Let $Y$ be a closed subset contained in the interior $Z^{\circ}$ of $Z$. Then
the natural morphism

$\mathrm{H}_{p}(X\smallsetminus Y,Z\smallsetminus
Y;\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}_{p}(X,Z;\mathscr{F})$

is an isomorphism for all $p$.

###### Proof.

We consider the open covering $\mathscr{U}=(X\smallsetminus Y,Z^{\circ})$. Let
$(\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}^{\mathscr{U}}(X,\mathscr{F}),\partial)$
be the subcomplex of
$(\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial)$
for which the spaces
${\boldsymbol{k}}\sigma\otimes\Gamma(|\sigma|,\mathscr{F})$ occurring as
components are those for which $|\sigma|$ is contained in one of the open sets
of $\mathscr{U}$. By using $(\ell,k)$ adapted to this covering (see(A.7)), one
obtains that any closed $s\in\widetilde{\eucal}{C}_{X,Z,p}(X,\mathscr{F})$ can
be written as $\ell(s)+\partial ks$, thus showing the surjectivity of the
homology map. On the other hand, if
$s^{\prime}\in\widetilde{\eucal}{C}_{X,Z,p}^{\mathscr{U}}(X,\mathscr{F})$
satisfies $s^{\prime}=\partial s$ with
$s\in\widetilde{\eucal}{C}_{X,Z,p+1}(X,\mathscr{F})$, we have
$s^{\prime}=\ell(s^{\prime})=\ell(\partial s)=\partial\ell(s)$, hence the
injectivity. ∎

######  Remark A.16.

The long exact sequence (A.10) is not easily seen in the model
$\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F})$,
while the excision property is better understood in that model, as well as the
long exact sequence, for the closed inclusions $Z\subset Y\subset X$:

$\cdots\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}_{p}(Y,Z;\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}_{p}(X,Z;\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}_{p}(X,Y;\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}_{p-1}(Y,Z;\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots$

On the other hand, if $\mathscr{F}$ is the constant sheaf with fiber $F$
(assume that $X$ is connected), then for each singular simplex $\sigma$, the
support $|\sigma|$ is connected, so $\Gamma(|\sigma|,\mathscr{F})$ is
canonically identified with $F$ by the restriction morphism

$F=\Gamma(X,\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma(|\sigma|,\mathscr{F}),$

so

$(\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial)\simeq(\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X)\otimes
F,\partial)$

and $\mathrm{H}_{p}(X,Z;\mathscr{F})$ is the usual singular homology with
coefficients in $F$.

###  A.e. Piecewise smooth and simplicial chains

Assume that $X$ is a manifold (possibly with boundary). We denote by
$(\widetilde{\eucal}{C}^{\mathrm{sm}}_{X,Z,{\scriptscriptstyle\bullet}},\partial)$
the subcomplex of
$(\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}},\partial)$
consisting of piecewise smooth singular chains (i.e., having a basis formed of
piecewise smooth singular simplices), and by
$(\eucal{C}^{\mathrm{sm}}_{X,Z,{\scriptscriptstyle\bullet}},\partial)$ the
associated complex of sheaves. On the other hand, for a sheaf $\mathscr{F}$,
we define
$\widetilde{\eucal}{C}^{\mathrm{sm}}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F})$
in a way similar to (A.11).

######  Proposition A.17.

The inclusions of chain complexes

$\displaystyle(\eucal{C}^{\mathrm{sm}}_{X,Z,{\scriptscriptstyle\bullet}},\partial)$
$\displaystyle\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}(\eucal{C}_{X,Z,{\scriptscriptstyle\bullet}},\partial),$
$\displaystyle(\widetilde{\eucal}{C}^{\mathrm{sm}}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial)$
$\displaystyle\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}(\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial)$

induce isomorphisms in homology.

###### Proof.

We will prove the statement for the second inclusion, the proof for the first
one being similar.

Let $s=\sum_{a}\sigma_{a}\otimes f_{a}$, with
$f_{a}\in\Gamma(|\sigma_{a}|,\mathscr{F})$, be a $p$-cycle in
$\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F})$. We
can assume that $f_{a}$ is the germ along $|\sigma_{a}|$ of
$f_{a}\in\Gamma(U_{a},\mathscr{F})$ for some open neighbourhood $U_{a}$ of
$|\sigma_{a}|$. The cycle condition is as in (A.12) and for each $\tau$ there
exists $U_{\tau}$ such that the corresponding sum $\sum f_{a}$ is zero on
$U_{\tau}$. We can approximate each $\sigma_{a}$ by piecewise smooth simplices
with image contained in $U_{a}$ and more precisely construct a family of
approximations $\Sigma_{a}(t)=\sigma_{a}^{t}$ ($t\in[0,1]$) with
$\sigma_{a}^{0}=\sigma_{a}$ and $\sigma_{a}^{t}$ piecewise smooth for each
$t\in(0,1]$ such that $\tau\times[0,1]$ has image in $U_{\tau}$ for each face
$\tau$ of $\sigma_{a}$. We can then subdivide $\Delta_{p}\times[0,1]$ to
regard each $\Sigma_{a}$ as a $(p+1)$-chain and the total family
$S=\sum_{a}\Sigma_{a}\otimes f_{a}$ as an element of
$\widetilde{\eucal}{C}_{X,Z,p+1}(X,\mathscr{F})$ (we restrict $f_{a}$ to
$|\Sigma_{a}|\subset U_{a}$). The boundary chain $\partial S$ reads

$\sum_{a,k}T_{a,k}\otimes f_{a}|_{|T_{a,k}|}+\sigma_{a}^{1}\otimes
f_{a}-\sigma_{a}^{0}\otimes f_{a},$

where $T_{a,k}$ is a map
$\Delta_{p-1}\times[0,1]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}U_{a}$
satisfying $T_{a,k}^{0}=\tau_{a,k}$ and $|T_{a,k}|\subset U_{\tau_{a,k}}$. The
cycle condition on $s$ implies that, for each
$\tau:\Delta_{p-1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$,
we have

$\sum_{a,k\mid\tau_{a,k}=\tau}f_{a}|_{U_{\tau}}=0,$

so that $\partial S=\sigma_{a}^{1}\otimes f_{a}-\sigma_{a}^{0}\otimes f_{a}$.
This shows the surjectivity of the homology map, and its injectivity follows.
∎

On the other hand, assume that $X$ is endowed with a simplicial structure
$\mathscr{T}$ compatible with $Z$. By this, we mean that $X$ is the support of
a simplicial complex $\mathscr{T}$ and $Z$ is the support of a subcomplex
$\mathscr{T}_{Z}$ of $\mathscr{T}$. We denote by
$(\widetilde{\eucal}{C}^{\scriptscriptstyle\triangle}_{\mathscr{T},\mathscr{T}_{Z},{\scriptscriptstyle\bullet}},\partial)$
the subcomplex of
$(\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}},\partial)$
consisting of simplicial chains of the simplicial structure. We define
correspondingly the simplicial chain complex
$(\widetilde{\eucal}{C}^{\scriptscriptstyle\triangle}_{\mathscr{T},\mathscr{T}_{Z},{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial)$.
We make the following assumption on $\mathscr{F}$ and the simplicial
structure.

######  Assumption A.18.

For any simplex $\sigma$ of the simplicial complex $\mathscr{T}$, there exists
an open neighbourhood $U_{\sigma}$ of $\sigma$ that retracts onto $\sigma$ and
such that the sheaf $\mathscr{F}_{|U_{\sigma}}$ can be decomposed as a direct
sum of sheaves $\mathscr{F}_{\sigma,i}$, each of which is constant of finite
rank on $U_{\sigma}\smallsetminus T_{\sigma,i}$ and zero on $T_{\sigma,i}$,
for some closed subset $T_{\sigma,i}$ of $U_{\sigma}$ that intersects $\sigma$
along a (possibly empty or full) closed face $\tau_{i}$ of $\sigma$.

We note that, if $\mathscr{T}^{\prime}$ is a subdivision of $\mathscr{T}$ and
$\mathscr{T}^{\prime}_{Z}$ is the corresponding subdivision of
$\mathscr{T}^{\prime}_{Z}$, then
$(\mathscr{F},\mathscr{T}^{\prime},\mathscr{T}^{\prime}_{Z})$ satisfies
Assumption A.18 if $(\mathscr{F},\mathscr{T},\mathscr{T}_{Z})$ does so.

######  Proposition A.19.

Under Assumption A.18, the inclusion of chain complexes

$(\widetilde{\eucal}{C}^{\scriptscriptstyle\triangle}_{\mathscr{T},\mathscr{T}_{Z},{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial)\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}(\widetilde{\eucal}{C}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial)$

induces an isomorphism in homology.

We immediately note the following consequence, which justifies that we now
denote by
$(\widetilde{\eucal}{C}^{\scriptscriptstyle\triangle}_{X,Z,{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial)$
the complex
$(\widetilde{\eucal}{C}^{\scriptscriptstyle\triangle}_{\mathscr{T},\mathscr{T}_{Z},{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial)$
when Assumption A.18 holds.

######  Corollary A.20.

The homology of
$(\widetilde{\eucal}{C}^{\scriptscriptstyle\triangle}_{\mathscr{T},\mathscr{T}_{Z},{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial)$
is independent of $\mathscr{T}$ provided Assumption A.18 is satisfied. In
particular, it is equal to the homology of
$(\widetilde{\eucal}{C}^{\scriptscriptstyle\triangle}_{\mathscr{T}^{\prime},\mathscr{T}^{\prime}_{Z},{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial)$
for any subdivision $\mathscr{T}^{\prime}$ of $\mathscr{T}$.∎

###### Proof of Proposition A.19.

We can argue separately with $X$ and $Z$ and obtain the result for the pair
$(X,Z)$ due to compatibility between long exact sequences of pairs. So we
argue with $X$, as in [46, §34], by induction on the (finite) number of
simplices in the simplicial decomposition $X$. Let $\sigma_{o}$ be a simplex
of maximal dimension (that we can assume $\geqslant 1$) and let $X^{\prime}$
be the simplicial set $X$ with $\sigma_{o}$ deleted (but the boundary
simplices of $\sigma_{o}$ kept in $X^{\prime}$). The underlying topological
space $X^{\prime}$ is $X$ with the interior of $\sigma_{o}$ deleted. Then
$(X^{\prime},\mathscr{F})$ also satisfies Assumption A.18. We consider, in
both simplicial and singular homology, the long exact sequences of pairs

$\cdots\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}_{p}(\sigma_{o};\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}_{p}(X;\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}_{p}(X,\sigma_{o};\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{H}_{p-1}(\sigma_{o};\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots$

and the natural morphism between them. The assertion follows from Lemma A.21
below. ∎

######  Lemma A.21.

With the notation and assumption from Proposition A.19,

  1. (1)

the natural morphism

$(\widetilde{\eucal}{C}^{\scriptscriptstyle\triangle}_{\sigma_{o},{\scriptscriptstyle\bullet}}(\sigma_{o},\mathscr{F}),\partial)\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}(\widetilde{\eucal}{C}_{\sigma_{o},{\scriptscriptstyle\bullet}}(\sigma_{o},\mathscr{F}),\partial)$

induces an isomorphism in homology;

  2. (2)

both natural inclusions

$\displaystyle(\widetilde{\eucal}{C}^{\scriptscriptstyle\triangle}_{X^{\prime},\partial\sigma_{o},{\scriptscriptstyle\bullet}}(X^{\prime},\mathscr{F}),\partial)$
$\displaystyle\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}(\widetilde{\eucal}{C}_{X,\sigma_{o},{\scriptscriptstyle\bullet}}^{\scriptscriptstyle\triangle}(X,\mathscr{F}),\partial),$
$\displaystyle(\widetilde{\eucal}{C}_{X^{\prime},\partial\sigma_{o},{\scriptscriptstyle\bullet}}(X^{\prime},\mathscr{F}),\partial)$
$\displaystyle\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}(\widetilde{\eucal}{C}_{X,\sigma_{o},{\scriptscriptstyle\bullet}}(X,\mathscr{F}),\partial),$

induce an isomorphism in homology.

###### Proof.

(1) Let us decompose $\mathscr{F}_{|\sigma_{o}}$ as in Assumption A.18, let us
fix a corresponding component $\mathscr{F}_{\sigma_{o},i}$ of $\mathscr{F}$
that we still call $\mathscr{F}$, and let $\tau_{o}$ be the face of
$\sigma_{o}$ on which it is zero. The assertion is trivial if
$\tau_{o}=\sigma_{o}$. If $\tau_{o}=\varnothing$, then $\mathscr{F}$ is
constant and the proof is done in [46, §34].

Assume now that $\tau_{o}$ is non-empty and different from $\sigma_{o}$. For
any face $\tau$ of $\sigma_{o}$ intersecting $\tau_{o}$, $\mathscr{F}_{|\tau}$
is constant on $\tau\smallsetminus(\tau\cap\tau_{o})$ and zero on
$\tau\cap\tau_{o}$, so that $\Gamma(\tau,\mathscr{F})=0$. Let
$\tau^{\prime}_{o}$ be the union of faces of $\sigma_{o}$ not intersecting
$\tau_{o}$. It is a single face of $\sigma_{o}$, and hence a simplex. Now,
$\mathscr{F}$ is constant on $\tau^{\prime}_{o}$ and, obviously,
$\mathrm{H}_{p}^{\scriptscriptstyle\triangle}(\sigma_{o};\mathscr{F})=\mathrm{H}_{p}^{\scriptscriptstyle\triangle}(\tau^{\prime}_{o};\mathscr{F})$.

Let us now compute $\mathrm{H}_{p}(\sigma_{o};\mathscr{F})$. Let
$\sigma:\Delta_{p}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\sigma_{o}$
be a singular $p$-simplex of $\sigma_{o}$, with image $|\sigma|$. Note that
the natural morphism
$\Gamma(|\sigma|,\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma(\Delta_{p},\sigma^{-1}\mathscr{F})$
is injective (since it preserves germs). If
$|\sigma|\cap\tau_{o}\neq\varnothing$, the argument above shows that
$\Gamma(\Delta_{p},\sigma^{-1}\mathscr{F})=0$, and hence so is
$\Gamma(|\sigma|,\mathscr{F})$. Thus,
$\widetilde{\eucal}{C}_{\sigma_{o},p}(\sigma_{o},\mathscr{F})$ only involves
singular simplices $\sigma$ not intersecting $\tau_{o}$. Let $F$ be the
constant value of $\mathscr{F}$ on $\sigma_{o}\smallsetminus\tau_{o}$. For
such a simplex $\sigma$, we thus have
$\Gamma(|\sigma|,\mathscr{F})\simeq\Gamma(\Delta_{p},\sigma^{-1}\mathscr{F})=F$,
and then
$\widetilde{\eucal}{C}_{\sigma_{o},p}(\sigma_{o},\mathscr{F})=\widetilde{\eucal}{C}_{\sigma_{o}\smallsetminus\tau_{o},p}(\sigma_{o}\smallsetminus\tau_{o})\otimes
F$. In other words,
$\mathrm{H}_{p}(\sigma_{o};\mathscr{F})=\mathrm{H}_{p}(\sigma_{o}\smallsetminus\tau_{o})\otimes
F$. Since $(\sigma_{o}\smallsetminus\tau_{o})$ retracts to
$\tau^{\prime}_{o}$, this group is also equal to
$\mathrm{H}_{p}(\tau^{\prime}_{o};\mathscr{F})$, and we conclude with [46,
§34].

(2) The assertion for the simplicial complexes is obvious since it is already
an equality. We thus consider the singular chain complexes.

Let us prove surjectivity of the homology map. Let $s$ be a closed singular
chain in $\widetilde{\eucal}{C}_{X,\sigma_{o},p}(X,\mathscr{F})$. We can cover
$X$ by $U_{o}$ and $U^{\prime}$, where $U_{o}=U_{\sigma_{o}}$ is given by
Assumption A.18 and $U^{\prime}\subset X^{\prime}$. Choosing $(\ell,k)$ as in
(A.7), we write $s+\partial ks=s_{o}+s^{\prime}$, with $s_{o},s^{\prime}$
closed in $\widetilde{\eucal}{C}_{U_{o},\sigma_{o},p}(X,\mathscr{F})$ and in
$\widetilde{\eucal}{C}_{U^{\prime},p}(X,\mathscr{F})\subset\widetilde{\eucal}{C}_{X^{\prime},\sigma_{o},p}(X^{\prime},\mathscr{F})$
respectively. We decompose $s_{o}$ according to Assumption A.18. For the
component corresponding to “$\mathscr{F}$ constant on $\sigma_{o}$”, it is
proved in [46, §34] that it is homologous to a chain in
$\widetilde{\eucal}{C}_{U_{o}\cap
X^{\prime},\partial\sigma_{o},p}(X,\mathscr{F})$. Assume thus that
$\mathscr{F}$ is zero on $T_{o}$ with
$\tau_{o}=T_{o}\cap\sigma_{o}\neq\varnothing$, and constant on
$U_{o}\smallsetminus T_{o}\neq\varnothing$. Arguing as in Case (1), we show
that singular simplices occurring with a non-zero coefficient in $s_{o}$ do
not intersect $T_{o}$. There is a projection
$\sigma_{o}\smallsetminus\tau_{o}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\tau^{\prime}_{o}$
(with $\tau^{\prime}_{o}$ as above), so that if we compose $s_{o}$ with the
corresponding retraction, we obtain a homologous closed $p$-chain in
$\widetilde{\eucal}{C}_{U_{o}\cap
X^{\prime},\partial\sigma_{o},p}(X,\mathscr{F})$.

Assume now that
$\widetilde{s}\\!\in\\!\widetilde{\eucal}{C}_{X^{\prime},\partial\sigma_{o},p}(X,\mathscr{F})$
is equal to $\partial s$ with $s$ in
$\widetilde{\eucal}{C}_{X,\sigma_{o},p+1}(X,\mathscr{F})$. It follows that
$\partial\widetilde{s}=0$ in
$\widetilde{\eucal}{C}_{X,\sigma_{o},p-1}(X,\mathscr{F})$, and hence in
$\widetilde{\eucal}{C}_{X^{\prime},\partial\sigma_{o},p-1}(X,\mathscr{F})$.
With $U_{o},U^{\prime}$, and $(\ell,k)$ as above, we have

$\widetilde{s}+\partial
k\widetilde{s}=\ell(\widetilde{s})=\partial\ell(s)=\partial s_{o}+\partial
s^{\prime}.$

The above argument shows that $s_{o}$ is homologous to a $(p+1)$-chain in

$\widetilde{\eucal}{C}_{U_{o}\cap
X^{\prime},\partial\sigma_{o},p+1}(X,\mathscr{F}),$

and hence $\widetilde{s}$ is the boundary of a chain in
$\widetilde{\eucal}{C}_{X^{\prime},\partial\sigma_{o},p+1}(X,\mathscr{F})$,
which shows injectivity. ∎

###  A.f. The dual chain complex with coefficient in a sheaf

In this section, we set $(X,Z)=(M,{\partial M})$, where $M$ is a manifold with
corners, and we consider a simplicial decomposition $\mathscr{T}$ of
$(M,{\partial M})$. Let $\mathscr{T}^{\prime}$ denote the first barycentric
subdivision of $\mathscr{T}$. For any simplex $\sigma$ in $\mathscr{T}$, let
$\widehat{\sigma}$ denote its barycenter and let $D(\sigma)$ be the open cell
dual to $\sigma$, which is the sum of open simplices of $\mathscr{T}^{\prime}$
having $\widehat{\sigma}$ as their final vertex (see[46, §64] for definitions
and details). If $\sigma$ is not contained in ${\partial M}$, then the closure
$\overline{D}(\sigma)$ does not intersect ${\partial M}$. If $\sigma$ is
contained in ${\partial M}$, then $D(\sigma)$ is contained in $M^{\circ}$. In
any case, if we regard $\overline{D}(\sigma)$ as a chain of
$\mathscr{T}^{\prime}$ relative to ${\partial M}$, then
$\partial\overline{D}(\sigma)$ is a sum of terms $\overline{D}(\sigma_{i})$
for some $\sigma_{i}$ in $\mathscr{T}$. The following is clear.

######  Lemma A.22.

Let $\sigma$ be a simplex of $\mathscr{T}$ of dimension $p$. Then $\sigma$ is
the only simplex of dimension $p$ of $\mathscr{T}$ intersected by
$\overline{D}(\sigma)$, and the intersection is transversal.

Figure 1. Various examples of dual cells

We consider a sheaf $\mathscr{F}$ on $M$ satisfying Assumption A.18 with
respect to $\mathscr{T}$, with the supplementary condition that $\mathscr{F}$
is locally constant on $M^{\circ}=M\smallsetminus{\partial M}$, that is, the
closed subsets $T_{\sigma,i}$ of Assumption A.18 are contained in ${\partial
M}$. We note that $\mathscr{F}$ also satisfies the previous properties with
respect to $\mathscr{T}^{\prime}$. The chain complex
$(\widetilde{\eucal}{C}_{\mathscr{T}^{\scriptscriptstyle\vee},{\partial
M},{\scriptscriptstyle\bullet}}^{\scriptscriptstyle\square}(M,\mathscr{F}),\partial)$
whose term of index $p$ consists of the direct sum of the terms
$\Gamma(\overline{D}(\sigma),\mathscr{F})$ with $\sigma$ of codimension $p$,
is thus a subcomplex of
$(\widetilde{\eucal}{C}_{\mathscr{T}^{\prime},{\partial
M},{\scriptscriptstyle\bullet}}^{\scriptscriptstyle\triangle}(M,\mathscr{F}),\partial)$.

######  Lemma A.23.

Under the previous assumptions, the inclusion of chain complexes

$(\widetilde{\eucal}{C}_{\mathscr{T}^{\scriptscriptstyle\vee},{\partial
M},{\scriptscriptstyle\bullet}}^{\scriptscriptstyle\square}(M,\mathscr{F}),\partial)\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}(\widetilde{\eucal}{C}_{\mathscr{T}^{\prime},{\partial
M},{\scriptscriptstyle\bullet}}^{\scriptscriptstyle\triangle}(M,\mathscr{F}),\partial)$

induces an isomorphism in homology.

###### Proof.

Similar to that of [46, Th. 64.2]. ∎

It follows from Proposition A.19 that the chain complex
$(\widetilde{\eucal}{C}_{\mathscr{T}^{\scriptscriptstyle\vee},{\partial
M},{\scriptscriptstyle\bullet}}^{\scriptscriptstyle\square}(M,\mathscr{F}),\partial)$
also computes the relative homology
$\mathrm{H}_{\scriptscriptstyle\bullet}(M,{\partial M};\mathscr{F})$.

##  Appendix B Poincaré lemma for currents

In this section, we work in a local setting. We let $U$ be a convex open
subset of $\mathbf{R}^{m}$ (with coordinates $x_{1},\dots,x_{m}$) containing
the origin, we fix $r$ such that $1\leqslant r\leqslant m$, we set
$g(x)=\textstyle\prod_{i=1}^{r}x_{i}$ and

$\displaystyle M$ $\displaystyle=\\{x_{i}\geqslant 0\mid i=1,\dots,r\\}\cap
U,$ $\displaystyle\partial M$ $\displaystyle=\\{g(x)=0\\}\cap M,$
$\displaystyle M^{\circ}$ $\displaystyle=\\{g(x)>0\\}\cap M$ $\displaystyle
U^{\circ}$ $\displaystyle=\\{g(x)\neq 0\\}.$

A $C^{\infty}$ function on $M$ can be defined either as the restriction to $M$
of a $C^{\infty}$ function defined on some neighbourhood of $M$ in $U$, or
such that the derivatives to any order computed in $M$ exist and are
continuous in $M$. It is known that both definitions define the same sheaf of
functions (e.g.by mirroring a function on $M$ with respect to the hyperplanes
$x_{i}=0$, $i=1,\dots,r$). The first definition means that the sheaf
$\mathscr{C}^{\infty}_{M}$ is the sheaf-theoretic restriction of
$\mathscr{C}^{\infty}_{U}$.

###  B.a. Functions with moderate growth and rapid decay along ${\partial M}$

######  Definition B.1.

  1. (1)

A $C^{\infty}$ function $f$ on $U^{\circ}$, resp.on $M^{\circ}$, has moderate
growth along $g^{-1}(0)$, resp.along ${\partial M}$, if, for any relatively
compact open subset $W\subset U$, resp.$W\subset M$, and any multi-index
$\alpha$, there exists $N\geqslant 0$ and $C>0$ such that
$\partial^{\alpha}f_{|W^{\circ}}$ is bounded by $C|g|^{-N}$.

  2. (2)

A $C^{\infty}$ function $f$ on $U$, resp.$M$, has rapid decay along
$g^{-1}(0)$, resp.along ${\partial M}$, if all its derivatives vanish at any
point of $g^{-1}(0)$, resp.${\partial M}$. Equivalently, the function
$f/g^{N}$ is $C^{\infty}$ for every $N$. A $C^{\infty}$ function $f$ on
$U^{\circ}$, resp.$M^{\circ}$ is said to have rapid decay along $g^{-1}(0)$,
resp.along ${\partial M}$, if its extension by $0$ to $U$, resp.to $M$, is
$C^{\infty}$ with rapid decay.

The corresponding sheaves are denoted respectively by
$\mathscr{C}_{U}^{\infty,\mathrm{mod}}$,
$\mathscr{C}_{U}^{\infty,\mathrm{rd}}$ and
$\mathscr{C}_{M}^{\infty,\mathrm{mod}}$,
$\mathscr{C}_{M}^{\infty,\mathrm{rd}}$. Let $x_{o}$ be a point in the
intersection of exactly $\ell$ smooth components of $g^{-1}(0)$. Then, in the
neighbourhood of $x_{o}$, $U^{\circ}$ has $2^{\ell}$ connected components and
$M^{\circ}$ is one of them. A $C^{\infty}$ function with moderate growth,
resp.rapid decay, on such a neighbourhood is nothing but the family of its
restrictions to the connected components, since these restrictions trivially
glue together along $g^{-1}(0)$.

The corresponding de Rham complexes on $M$ are denoted by
$(\mathscr{E}_{M}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}})$
and
$(\mathscr{E}_{M}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}})$
(we omit the boundary ${\partial M}$ in the notation). On the other hand, we
will also consider the sheaf of $C^{\infty}$ functions on $M$ with poles along
$\partial M$, that we simply denote by $\mathscr{C}_{M}^{\infty}(*)$. It is a
subsheaf of $\mathscr{C}_{M}^{\infty,\mathrm{mod}}$.

######  Proposition B.2 (Poincaré lemma).

The complexes
$(\mathscr{E}_{M}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}})$
and
$(\mathscr{E}_{M}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}})$
have cohomology in degree zero only, and are a resolution of
$j_{*}\mathbf{C}_{M^{\circ}}=\mathbf{C}_{M}$ and $j_{!}\mathbf{C}_{M^{\circ}}$
respectively.

###### Proof.

We prove the statements on global sections. Let
$\eta=\sum_{|I|=p}\eta_{I}{\mathrm{d}}x_{I}\in\mathscr{E}^{\mathrm{rd},p}(M)$
(resp.$\eta\in\mathscr{E}^{\mathrm{mod},p}(M^{\circ})$) be such that
${\mathrm{d}}\eta=0$ and $p\geqslant 1$. There is an explicit formula (see for
example [20, (1.23)]) computing $\psi\in\mathscr{E}^{p-1}(M)$ such that
${\mathrm{d}}\psi=\eta$:

$\psi(x)=\sum_{\begin{subarray}{c}|I|=p\\\
k\in[1,p]\end{subarray}}\biggl{(}\int_{0}^{1}t^{p-1}\eta_{I}(tx){\mathrm{d}}t\biggr{)}(-1)^{k-1}x_{i_{k}}{\mathrm{d}}x_{i_{1}}\wedge\cdots\wedge\widehat{{\mathrm{d}}x_{i_{k}}}\wedge\cdots\wedge{\mathrm{d}}x_{i_{p}}.$

This formula shows that $\psi$ belongs to $\mathscr{E}^{\mathrm{rd},p-1}(M)$
(resp.to $\mathscr{E}^{\mathrm{mod},p-1}(M^{\circ})$), hence the first
assertion. The second assertion in both cases is clear. ∎

###  B.b. Poincaré lemma for currents

The sheaf of distributions $\mathfrak{Db}_{M}$ on $M$ is defined as the sheaf-
theoretic restriction to $M$ of $\mathfrak{Db}_{U}$. As in [13, 14], we define
the chain complex of currents
$(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}},\partial)$. By definition, it
is the sheaf-theoretic restriction to $M$ of the complex
$\mathfrak{C}_{U,{\scriptscriptstyle\bullet}}$ that we consider now. A current
$T$ of dimension $q$ on $U$ can be paired with a test $q$-form $\varphi$ on
$U$ ($C^{\infty}$ with compact support) and the Stokes formula holds:
$\langle\partial T,\varphi\rangle=\langle T,{\mathrm{d}}\varphi\rangle$. If
$\omega$ is a $C^{\infty}$ $(m-q)$-form, it defines a $q$-dimensional current
by the formula $\langle
T_{\omega},\varphi\rangle=\int_{U}\omega\wedge\varphi$. We can consider a
current as a differential form with distributional coefficients, i.e., we can
write $T=\sum_{\\#J=m-q}\theta_{J}{\mathrm{d}}x_{J}$ with
$\theta_{J}\in\mathfrak{Db}(U)$. When considered as a form, the differential
${\mathrm{d}}$ on currents extends that on $C^{\infty}$ forms. The relation
between ${\mathrm{d}}$ and $\partial$ is given, for a current of dimension $q$
(i.e., degree $m-q$) by

$\partial T_{q}=(-1)^{m-q+1}{\mathrm{d}}T_{q}.$

For a tensor product taken over $\mathbf{C}$, like
$(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}},\partial)\otimes(\mathscr{E}_{M}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}})$,
the boundary is given by the formula, for a current $T_{q}\otimes\omega^{p}$
(of dimension $q-p$):

(B.3) $\partial(T_{q}\otimes\omega^{p})=(-1)^{p}(\partial
T_{q}\otimes\omega^{p}-T_{q}\otimes{\mathrm{d}}\omega^{p})=(-1)^{p}\partial
T_{q}\otimes\omega^{p}+T_{q}\otimes\partial\omega^{p}.$

######  Proposition B.4 (Poincaré lemma for
$(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}},\partial)$).

The complex of currents on $M$ satisfies

$\mathscr{H}_{q}(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}},\partial)=\begin{cases}0&\text{if
}q\neq m,\\\ \mathbf{C}_{M}&\text{if }q=m.\end{cases}$

More precisely, the inclusion
$(\mathscr{E}_{M}^{m-{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}{\mathrm{d}})\mathrel{\lhook\joinrel\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}},\partial)$
is a quasi-isomorphism.

###### Proof.

We prove the result on an open subset $U$ of $\mathbf{R}^{m}$, and we obtain
the proposition by sheaf-theoretic restriction to $M\cap U$. We refer to [20,
§2.D.4] for the proof of the lemma below, originally in [13, 14]. Let
$T\in\mathfrak{C}_{q}(U)$.

######  Lemma B.5.

For any $\varepsilon\in(0,1)$, there exist $\mathbf{C}$-linear morphisms

$R_{\varepsilon}:\mathfrak{C}_{q}(U)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathfrak{C}_{q}(U)\quad\text{and}\quad
S_{\varepsilon}:\mathfrak{C}_{q}(U)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathfrak{C}_{q+1}(U)$

such that

  1. (1)

$R_{\varepsilon}$ takes values in $\mathscr{E}^{m-q}(U)$,

  2. (2)

$R_{\varepsilon}(T)-T=\partial S_{\varepsilon}(T)+S_{\varepsilon}(\partial
T)$,

  3. (3)

$R_{\varepsilon}(\partial T)=\partial R_{\varepsilon}(T)$ and
$\lim_{\varepsilon\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0}R_{\varepsilon}(T)=T$
weakly.

In particular, $R_{\varepsilon}$ is a morphism of complexes
$(\mathfrak{C}_{\scriptscriptstyle\bullet}(U),\partial)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathscr{E}^{m-{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}(U),{\mathrm{d}})$,
and (2) implies that it is a quasi-isomorphism. Since
$(\mathscr{E}^{m-{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}(U),{\mathrm{d}})$
has homology concentrated in degree
${\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}=m$, it follows that the
same holds for $(\mathfrak{C}_{\scriptscriptstyle\bullet}(U),\partial)$. Then
we can sheafify the construction. ∎

######  Remark B.6.

Arguing as in [14, §15, Th. 12], we can globalize the construction on a
manifold with corners. More precisely, given a covering $(U_{i})_{i\in I}$ of
$M$ by charts, and $\varepsilon_{i}>0$ small enough ($i\in I$), one can
construct morphisms

$R_{\boldsymbol{\varepsilon}}:\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{E}_{M}^{m-{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\quad\text{and}\quad
S_{\boldsymbol{\varepsilon}}:\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathfrak{C}_{M,{\scriptscriptstyle\bullet}+1}$

such that the conclusion of Lemma B.5 holds for any
$T\in\Gamma(M,\mathfrak{C}_{M,{\scriptscriptstyle\bullet}})$, up to replacing
$\varepsilon$ with ${\boldsymbol{\varepsilon}}=(\varepsilon_{i})_{i\in I}$. In
particular,
$R_{\boldsymbol{\varepsilon}}:(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}},\partial)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathscr{E}_{M}^{m-{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}})$
is a quasi-isomorphism.

Let us end by recalling the computation of the Kronecker index made by de
Rham. Let $T_{p}$ and $T^{\scriptscriptstyle\vee}_{m-p}$ be two currents of
complementary dimension on $U$. If both currents are closed, let us choose a
decomposition $T_{p}=\Theta^{m-p}+\partial S_{p+1}$ and
$T^{\scriptscriptstyle\vee}_{m-p}=\Theta^{{\scriptscriptstyle\vee},p}+\partial
S^{\scriptscriptstyle\vee}_{m-p+1}$, where $\Theta^{m-p}$ and
$\Theta^{{\scriptscriptstyle\vee},p}$ are $C^{\infty}$ closed forms. Then
$\mathsf{B}_{\mathrm{dR}}(T_{p},T^{\scriptscriptstyle\vee}_{m-p})$ is by
definition the integral
$\mathsf{Q}(\Theta^{m-p},\Theta^{{\scriptscriptstyle\vee},p})$ of the wedge
product and is independent of the choices. Without the closedness assumption,
but if one of the supports is compact, the intersection
$\mathsf{B}_{\mathrm{dR}}(T_{p},T^{\scriptscriptstyle\vee}_{m-p})$ is defined
by the limit, when it exists

$\lim_{\varepsilon^{\prime},\varepsilon\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0}\mathsf{Q}(R_{\varepsilon^{\prime}}(T_{p}),R_{\varepsilon}(T^{\scriptscriptstyle\vee}_{m-p}))=\lim_{\varepsilon\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0}\langle
T_{p},R_{\varepsilon}(T^{\scriptscriptstyle\vee}_{m-p})\rangle.$

We now consider the setting of the end of Section A. Let us choose a simplex
$\sigma_{p}$ in $M$ not contained in $\partial M$ and let
$\overline{D}(\sigma_{p})$ denote the dual cell in $M^{\circ}$, which
intersects $\sigma_{p}$ at its barycenter $\widehat{\sigma}_{p}$. Let us set
$T_{p}=\int_{\sigma_{p}}$,
$T_{m-p}^{\scriptscriptstyle\vee}=\int_{\overline{D}(\sigma_{p})}$ and
$\mathsf{B}_{\mathrm{dR}}(\sigma_{p},\overline{D}(\sigma_{p}))=\mathsf{B}_{\mathrm{dR}}(T_{p},T^{\scriptscriptstyle\vee}_{m-p})$.

######  Proposition B.7 ([14, p. 85–86]).

Under the previous assumptions, the intersection
$\mathsf{B}_{\mathrm{dR}}(\sigma_{p},\overline{D}(\sigma_{p}))$ exists and is
given by the formula

$\mathsf{B}_{\mathrm{dR}}(\sigma_{p},\overline{D}(\sigma_{p}))=\pm 1,$

where $\pm$ is the orientation change between
$\sigma_{p}\times\overline{D}(\sigma_{p})$ and $M^{\circ}$.

###### Proof.

Let us choose coordinates
$(x^{\prime},x^{\prime\prime})=(x_{1},\dots,x_{p},x_{p+1},\dots,x_{m})$ so
that $\sigma_{p}$ (with orientation) (resp.$\overline{D}(\sigma_{p})$) is
contained in the coordinate plane $(x_{1},\dots,x_{p})$
(resp.$(x_{p+1},\dots,x_{m})$). The formula for $R_{\varepsilon}$ in Lemma B.5
is:

$\displaystyle R_{\varepsilon}(T_{m-p}^{\scriptscriptstyle\vee})$
$\displaystyle=\theta_{m-p,\varepsilon}(x){\mathrm{d}}x_{1}\wedge\cdots\wedge{\mathrm{d}}x_{p},$
with $\displaystyle\theta_{m-p,\varepsilon}(x)$
$\displaystyle=\int_{\overline{D}(\sigma_{p})}\frac{\chi_{\varepsilon}((x-y)/\psi(y))}{\psi(y)^{m}}\,{\mathrm{d}}y_{p+1}\wedge\cdots\wedge{\mathrm{d}}y_{m},$

where
$\chi:B(0,1)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{R}_{+}$
is any $C^{\infty}$-function with compact support and integral equal to $1$
and, for $\varepsilon>0$, let
$\chi_{\varepsilon}(v)=\varepsilon^{-m}\chi(v/\varepsilon)$ with support in
$B(0,\varepsilon)$. Hence, if we assume that $\chi$ is even with respect to
the variables $x^{\prime\prime}$,

$\displaystyle\mathsf{B}_{\mathrm{dR}}(\sigma_{p},\overline{D}(\sigma_{p}))$
$\displaystyle=\lim_{\varepsilon\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0}\int_{\sigma_{p}}{\mathrm{d}}x_{1}\wedge\cdots\wedge{\mathrm{d}}x_{p}\cdot\theta_{m-p,\varepsilon}(x^{\prime})$
$\displaystyle=\lim_{\varepsilon\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0}\int_{\sigma_{p}\times\overline{D}(\sigma_{p})}\frac{\chi_{\varepsilon}((x^{\prime},-x^{\prime\prime})/\psi(x^{\prime\prime}))}{\psi(x^{\prime\prime})^{m}}\,{\mathrm{d}}x_{1}\wedge\cdots\wedge{\mathrm{d}}x_{m}=\pm
1$

by the properties of $\chi$. ∎

###  B.c. Poincaré lemma for current with moderate growth and rapid decay

Let us first consider the sheaf $\mathfrak{Db}^{\mathrm{mod}}_{U}$ on $U$,
defined as the subsheaf of $j_{*}\mathfrak{Db}_{U^{\circ}}$ consisting of
distributions extendable to $U$. Its space of sections on an open set
$W\subset U$ is the topological dual of the space of $C^{\infty}$ functions on
$W$ with compact support and with rapid decay along $g^{-1}(0)$, endowed with
its usual family of semi-norms. It follows from [39, Chap. VII] that
$\mathfrak{Db}^{\mathrm{mod}}_{U}=\mathscr{C}_{U}^{\infty}(*)\otimes_{\mathscr{C}_{U}^{\infty}}\mathfrak{Db}_{U}$,
where we have set $\mathscr{C}_{U}^{\infty}(*)=\mathscr{C}_{U}^{\infty}[1/g]$.
We also have an exact sequence

(B.8)
$0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathfrak{Db}_{[g^{-1}(0)]}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathfrak{Db}_{U}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathfrak{Db}_{U}^{\mathrm{mod}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0,$

where $\mathfrak{Db}_{[g^{-1}(0)]}$ denotes the subsheaf of distributions
supported on $g^{-1}(0)$.

The sheaf $\mathfrak{Db}^{\mathrm{mod}}_{M}$ is defined as the subsheaf of
$j_{*}\mathfrak{Db}_{M^{\circ}}$ consisting of distributions extendable to
$M$, and its space of section on an open set $V\subset M$ is the topological
dual of the space of $C^{\infty}$ functions on $V$ with compact support and
with rapid decay along ${\partial M}$, endowed with its usual family of semi-
norms.

Near a point $x_{o}\in g^{-1}(0)$ where $g^{-1}(0)$ has exactly $\ell$
components, the space of rapid decay functions decomposes as the direct sum of
the spaces of rapid decay functions in each local connected component of
$U^{\circ}$, and by duality so does the space of moderate distributions in
this neighbourhood. If $x_{o}\in{\partial M}$, then the space of moderate
distribution on the intersection of this neighbourhood with $M$ is one of the
summands.

######  Definition B.9 (Currents with moderate growth).

We set

$\mathfrak{C}_{U,p}^{\mathrm{mod}}=\mathscr{E}_{U}^{m-p}\otimes_{\mathscr{C}_{U}^{\infty}}\mathfrak{Db}_{U}^{\mathrm{mod}}\quad\text{and}\quad\mathfrak{C}_{M,p}^{\mathrm{mod}}=\mathscr{E}_{M}^{m-p}\otimes_{\mathscr{C}_{M}^{\infty}}\mathfrak{Db}_{M}^{\mathrm{mod}}.$

In a way similar to currents, a current $T$ of dimension $q$ with moderate
growth on $M$ can be paired with a test $q$-form $\varphi$ on $M$ with rapid
decay and Stokes formula holds.

######  Proposition B.10 (Poincaré lemma for
$(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{mod}},\partial)$).

The chain complex
$(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{mod}},\partial)$
satisfies Poincaré lemma as in Proposition B.4.

###### Proof.

We will first work on $U$ and then restrict to $M$, with the cohomological
version
$(\mathfrak{C}_{M}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}})$,
that we wish to prove to be a resolution of its $\mathscr{H}^{0}$, that is,
$\mathbf{C}_{M}$. In other words, we wish to prove that the natural morphism
$(\mathfrak{C}_{M}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\mathrm{d}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathfrak{C}_{M}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}})$
is a quasi-isomorphism.

######  Lemma B.11.

The complex
$(\mathfrak{C}_{[g^{-1}(0)]}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\mathrm{d}})$
has cohomology in degree one only and, for $x_{o}$ in the intersection of
$\ell$ smooth components of $g^{-1}(0)$, we have
$\dim\mathscr{H}^{1}(\mathfrak{C}_{[g^{-1}(0)]}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\mathrm{d}})_{x_{o}}=2^{\ell}-1$.

###### Proof.

Let $Z\subset U$ denote the $\ell$-th intersection defined by
$x_{1}=\cdots=x_{\ell}=0$, so that
$\mathfrak{Db}_{[Z]}\simeq\mathfrak{Db}_{Z}[\partial_{x_{1}},\dots,\partial_{x_{\ell}}]$.
We write
$(\mathfrak{C}_{[Z]}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\mathrm{d}})$
as the simple complex associated to the $\ell$-cube with $i$-th edges all
equal to

$\mathfrak{C}_{Z}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}[\partial_{x_{1}},\dots,\partial_{x_{\ell}}]\mathchoice{\xrightarrow{\kern
4.0pt\partial_{x_{i}}\kern
3.0pt}}{\stackrel{{\scriptstyle\partial_{x_{i}}}}{{\longrightarrow}}}{}{}\mathfrak{C}_{Z}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}[\partial_{x_{1}},\dots,\partial_{x_{\ell}}],$

which is quasi-isomorphic to
$\mathfrak{C}_{Z}^{{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}-\ell}$,
so $\mathfrak{C}_{[Z]}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}$ has
cohomology in degree $\ell$ only, equal to the constant sheaf $\mathbf{C}_{Z}$
(by the Poincaré lemma for $Z$).

Let us set $Z_{i}=\\{x_{i}=0\\}$ and
$Z^{(\ell)}=\mathop{\textstyle\bigcup}\displaylimits_{i=1}^{\ell}Z_{i}$. We
prove by induction on $\ell$ that the lemma hods for $Z^{(\ell)}$, the case
$\ell=1$ begin proved in the first point. According to [39, Prop. VII.1.4],
the natural sequence of complexes

$0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathfrak{C}_{[Z_{\ell}^{(\ell-1)}]}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\mathrm{d}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathfrak{C}_{[Z_{\ell}]}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\mathrm{d}})\oplus(\mathfrak{C}_{[Z^{(\ell-1)}]}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\mathrm{d}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathfrak{C}_{[Z^{(\ell)}]}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},{\mathrm{d}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0$

is exact. Let us restrict the complexes on
$Z=\mathop{\textstyle\bigcap}\displaylimits_{i=1}^{\ell}Z_{i}$. By induction,
the middle complex has cohomology in degree one only, of dimension
$1+(2^{\ell-1}-1)=2^{\ell-1}$. Also by induction, the left complex has
cohomology in degree $2$ only, of dimension $2^{\ell-1}-1$. This completes the
proof. ∎

The exact sequence (B.8) gives rise to an exact sequence of complexes

$0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathfrak{C}_{[g^{-1}(0)]}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathfrak{C}_{U}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathfrak{C}_{U}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}0$

and the lemma above reduces it to an exact sequence

$0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{C}_{U}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{H}^{0}(\mathfrak{C}_{U}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{H}^{1}(\mathfrak{C}_{[g^{-1}(0)]}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\nobreak
0.$

At a point $x_{o}\in g^{-1}(0)$ at the intersection of exactly $\ell$ smooth
components,
$\mathscr{H}^{0}(\mathfrak{C}_{U}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}})_{x_{o}}$
has thus dimension $2^{\ell}$. If $U^{\circ}_{x_{o},a}$ are the $2^{\ell}$
local connected components of $U^{\circ}_{x_{o}}$, then

$\mathscr{H}^{0}(\mathfrak{C}_{U}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}})_{x_{o}}=\mathrm{H}^{0}(\mathfrak{C}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}(U^{\circ}_{x_{o}}))=\mathop{\textstyle\bigoplus}\displaylimits_{a}\mathrm{H}^{0}(\mathfrak{C}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}(U^{\circ}_{x_{o},a})),$

so that each term in the sum is isomorphic to $\mathbf{C}$. In particular, for
$x_{o}\in{\partial M}$, considering the component $M^{\circ}_{x_{o}}$, the
natural composed morphism (after sheaf-theoretically restricting to $M$)
$\mathfrak{C}_{U|M}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathfrak{C}_{U|M}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathfrak{C}_{M}^{\mathrm{mod},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}}$
induces an isomorphism on the $\mathscr{H}^{0}$ at each $x_{o}$, and hence is
an isomorphism. ∎

We define the complex of currents with rapid decay as
$(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{mod}},\partial)\otimes(\mathscr{E}_{M}^{\mathrm{rd},{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},{\mathrm{d}})$,
with boundary operator given by (B.4). From Propositions B.10 and B.2, we thus
obtain:

######  Proposition B.12 (Poincaré lemma for
$(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{rd}},\partial)$).

The chain complex
$(\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{rd}},\partial)$ has
homology in degree $m$ only, which is equal to $j_{!}\mathbf{C}_{M^{\circ}}$.

######  Remark B.13.

If $M$ is compact, the homology of the complex
$(\Gamma(M,\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{rd}}),\partial)$
is isomorphic to
$\mathrm{H}_{\scriptscriptstyle\bullet}(M^{\circ},\mathbf{C})$, while that of
$(\Gamma(M,\mathfrak{C}_{M,{\scriptscriptstyle\bullet}}^{\mathrm{mod}}),\partial)$
is isomorphic to the Borel-Moore homology
$\mathrm{H}^{\scriptscriptstyle\mathrm{BM}}_{\scriptscriptstyle\bullet}(M^{\circ},\mathbf{C})$.

###  B.d. Dolbeault lemmas

We recall here the various Dolbeault lemmas that we have used in the main
text. Let $X$ be a complex manifold, let $D$ be a normal crossing divisor and
let
$\varpi:\widetilde{X}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$
be the real oriented blow-up of the components of $D$ (local coordinates on
$\widetilde{X}$ are local polar coordinates on $X$ with respect to a
coordinate system adapted to $D$). Then $\widetilde{X}$ is a $C^{\infty}$
manifold with corners as in the beginning of this section, so is endowed with
the sheaves $\mathscr{C}^{\infty,\mathrm{mod}}_{\widetilde{X}}$ and
$\mathscr{C}^{\infty,\mathrm{rd}}_{\widetilde{X}}$ as in Definition B.1.
Moreover, one checks that the $\overline{\partial}$ operator defined on
$\mathscr{C}^{\infty}_{X}$ can be lifted to these sheaves of functions, and we
can define the corresponding Dolbeault complexes
$(\mathscr{E}_{\widetilde{X}}^{\mathrm{mod},0,{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\overline{\partial})$
and
$(\mathscr{E}_{\widetilde{X}}^{\mathrm{rd},0,{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\overline{\partial})$
whose $\mathscr{H}^{0}$ are respectively denoted by
$\mathscr{A}_{\widetilde{X}}^{\mathrm{mod}}$ and
$\mathscr{A}_{\widetilde{X}}^{\mathrm{rd}}$.

######  Lemma B.14.

These Dolbeault complexes are a resolution of their $\mathscr{H}^{0}$.

###### Proof.

For the moderate case, one can use the proof of Dolbeault’s lemma on bounded
domains of $\mathbf{C}^{n}$ with piecewise $C^{1}$ boundary using the Bochner-
Martinelli kernel (seee.g.[34, §III.2]). The rapid decay case is treated in
[37] (seealso [50, Lem. II.1.1.18] and [45, Prop. 4.2.2]). ∎

One can similarly define the sheaves on $X$ of $C^{\infty}$ functions with
moderate growth or rapid decay along $D$, together with their corresponding
Dolbeault complexes, which do not enter the frame considered at the beginning
of this section. Nevertheless, one has obviously

$(\mathscr{E}_{X}^{\mathrm{mod},0,{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\overline{\partial})\simeq(\varpi_{*}\mathscr{E}_{\widetilde{X}}^{\mathrm{mod},0,{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\overline{\partial})\quad\text{and}\quad(\mathscr{E}_{X}^{\mathrm{rd},0,{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\overline{\partial})\simeq(\varpi_{*}\mathscr{E}_{\widetilde{X}}^{\mathrm{rd},0,{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\overline{\partial}).$

Furthermore,
$\varpi_{*}\mathscr{A}_{\widetilde{X}}^{\mathrm{mod}}\simeq\mathscr{O}_{X}(*D)$.

######  Lemma B.15.

The Dolbeault complex
$(\mathscr{E}_{X}^{\mathrm{mod},0,{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\overline{\partial})$
is a resolution of $\mathscr{O}_{X}(*D)$, and the Dolbeault complex
$(\mathscr{E}_{X}^{\mathrm{rd},0,{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}},\overline{\partial})$
is quasi-isomorphic to the two-term complex
$\mathscr{O}_{X}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{O}_{\widehat{D}}$.

###### Proof.

The moderate case is treated as in that on $\widetilde{X}$, while the rapid
decay case is proved more generally in [4]. ∎

##  Appendix C Remarks on Verdier duality

To pass from local results on pairings to global ones, we use compatibility of
Verdier duality with proper pushforward. In this section is made precise an
“obvious result” (Corollary C.6) which we could not find in the literature. We
refer to [28, Chap. 2 & 3] for standard results of sheaf theory.

We fix a field ${\boldsymbol{k}}$ and we work in the category of sheaves of
${\boldsymbol{k}}$-vector spaces. All topological spaces we consider are
assumed to be locally compact, and all maps are assumed to have finite local
cohomological dimension. Let $\mathbf{D}_{X}$ be the dualizing complex. If
$a_{X}:X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{pt}$
denotes the constant map, one has $\mathbf{D}_{X}=a_{X}^{!}{\boldsymbol{k}}$.

###  C.a. The duality isomorphism

Let $f:X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y$
be a continuous map, let $\mathscr{F}$ be an object of
$\mathsf{D}^{\mathrm{b}}({\boldsymbol{k}}_{X})$ and $\mathscr{G}$ an object of
$\mathsf{D}^{\mathrm{b}}({\boldsymbol{k}}_{Y})$. There is a bifunctorial
isomorphism

(C.1)
$\boldsymbol{R}f_{*}\boldsymbol{R}\\!\mathscr{H}\\!om(\mathscr{F},f^{!}\mathscr{G})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\boldsymbol{R}\\!\mathscr{H}\\!om(\boldsymbol{R}f_{!}\mathscr{F},\mathscr{G})$

in $\mathsf{D}^{\mathrm{b}}({\boldsymbol{k}}_{Y})$ (see[28, Prop. 3.1.10]). By
applying
$\boldsymbol{R}\Gamma(Y,{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$,
one obtains

$\boldsymbol{R}\operatorname{Hom}(\mathscr{F},f^{!}\mathscr{G})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\boldsymbol{R}\operatorname{Hom}(\boldsymbol{R}f_{!}\mathscr{F},\mathscr{G}),$

and taking cohomology in degree $0$,

$\operatorname{Hom}_{\mathsf{D}^{\mathrm{b}}(X)}(\mathscr{F},f^{!}\mathscr{G})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Hom}_{\mathsf{D}^{\mathrm{b}}(Y)}(\boldsymbol{R}f_{!}\mathscr{F},\mathscr{G}).$

There exists a natural morphism of functors
$\boldsymbol{R}f_{!}f^{!}\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\operatorname{Id}$.
Indeed, taking $\mathscr{F}=f^{!}\mathscr{G}$ above, one finds

$\operatorname{Hom}_{\mathsf{D}^{\mathrm{b}}(X)}(f^{!}\mathscr{G},f^{!}\mathscr{G})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Hom}_{\mathsf{D}^{\mathrm{b}}(Y)}(\boldsymbol{R}f_{!}f^{!}\mathscr{G},\mathscr{G})$

and the desired morphism is the image of the identity.

One can thus write the duality isomorphism (C.1) as the composition of two
bifunctorial morphisms (see [28, (2.6.25)] for the first one)

$\boldsymbol{R}f_{*}\boldsymbol{R}\\!\mathscr{H}\\!om(\mathscr{F},f^{!}\mathscr{G})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}\\!\mathscr{H}\\!om(\boldsymbol{R}f_{!}\mathscr{F},\boldsymbol{R}f_{!}f^{!}\mathscr{G})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}\\!\mathscr{H}\\!om(\boldsymbol{R}f_{!}\mathscr{F},\mathscr{G}).$

###  C.b. Poincaré-Verdier duality

We set
$\boldsymbol{D}\mathscr{F}=\boldsymbol{R}\\!\mathscr{H}\\!om(\mathscr{F},\mathbf{D}_{X})$.
One then has an isomorphism

$\boldsymbol{R}\Gamma(X,\boldsymbol{D}\mathscr{F})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Hom}(\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F}),{\boldsymbol{k}}),$

and taking hypercohomology,

$\boldsymbol{H}^{\ell}(X,\boldsymbol{D}\mathscr{F})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\boldsymbol{H}^{-\ell}_{\mathrm{c}}(X,\mathscr{F})^{\scriptscriptstyle\vee}.$

Via this isomorphism and by using the natural duality pairing, we deduce a
perfect pairing

(C.2)
$\boldsymbol{H}^{-\ell}_{\mathrm{c}}(X,\mathscr{F})\otimes\boldsymbol{H}^{\ell}(X,\boldsymbol{D}\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\boldsymbol{k}}.$

On the other hand, one has a natural morphism
$\mathscr{F}\otimes\boldsymbol{D}\mathscr{F}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{D}_{X}$
and, in addition, according to [28, (2.6.23)] for $f=a_{X}$, a morphism

$\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F})\otimes\boldsymbol{R}\Gamma(X,\boldsymbol{D}\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F}\otimes\boldsymbol{D}\mathscr{F}).$

By composing, one obtains a morphism

$\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F})\otimes\boldsymbol{R}\Gamma(X,\boldsymbol{D}\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F}\otimes\boldsymbol{D}\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathbf{D}_{X})=\boldsymbol{R}a_{X!}a_{X}^{!}{\boldsymbol{k}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\boldsymbol{k}}.$

By taking hypercohomology, one obtains for each $\ell\in\mathbf{Z}$ a pairing

(C.3)
$\boldsymbol{H}^{-\ell}_{\mathrm{c}}(X,\mathscr{F})\otimes\boldsymbol{H}^{\ell}(X,\boldsymbol{D}\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\boldsymbol{k}}.$

######  Proposition C.4.

The pairings (C.2) and (C.3) coincide. In particular, (C.3) is a perfect
pairing.

For the sake of completeness, we will prove the following lemma at the end of
this section.

######  Lemma C.5.

Let $\mathscr{F}$ and $\mathscr{G}$ be objects of
$\mathsf{D}^{\mathrm{b}}({\boldsymbol{k}}_{X})$ and let
$f:X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}Y$ be a
morphism. The two following composed natural morphisms coincide:

$\displaystyle\boldsymbol{R}f_{!}\mathscr{F}\otimes\boldsymbol{R}f_{*}\boldsymbol{R}\\!\mathscr{H}\\!om(\mathscr{F},\mathscr{G})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}f_{!}\bigl{(}\mathscr{F}\otimes\boldsymbol{R}\\!\mathscr{H}\\!om(\mathscr{F},\mathscr{G})\bigr{)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}f_{!}\mathscr{G},$
$\displaystyle\boldsymbol{R}f_{!}\mathscr{F}\otimes\boldsymbol{R}f_{*}\boldsymbol{R}\\!\mathscr{H}\\!om(\mathscr{F},\mathscr{G})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}f_{!}\mathscr{F}\otimes\boldsymbol{R}\\!\mathscr{H}\\!om(\boldsymbol{R}f_{!}\mathscr{F},\boldsymbol{R}f_{!}\mathscr{G})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}f_{!}\mathscr{G}.$

###### Proof of Proposition C.4.

Taking $f=a_{X}$ and $\mathscr{G}=\mathbf{D}_{X}$, we have equality of the
morphisms

$\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F})\otimes\boldsymbol{R}\Gamma(X,\boldsymbol{D}\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F}\otimes\boldsymbol{D}\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathbf{D}_{X})$

and

$\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F})\otimes\boldsymbol{R}\Gamma(X,\boldsymbol{D}\mathscr{F})\\\
\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F})\otimes\operatorname{Hom}(\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F}),\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathbf{D}_{X}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathbf{D}_{X}).$

Besides, we have a commutative diagram, by considering the natural morphism
$\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathbf{D}_{X})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\boldsymbol{k}}$,

$\textstyle{\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F})\otimes\operatorname{Hom}(\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F}),\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathbf{D}_{X}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathbf{D}_{X}).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F})\otimes\operatorname{Hom}(\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F}),{\boldsymbol{k}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\boldsymbol{k}}$

Hence, both composed natural morphisms

$\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F})\otimes\boldsymbol{R}\Gamma(X,\boldsymbol{D}\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\boldsymbol{D}\mathscr{F}\otimes\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\boldsymbol{k}}$

and

$\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F})\otimes\boldsymbol{R}\Gamma(X,\boldsymbol{D}\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F})\otimes\operatorname{Hom}(\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F}),{\boldsymbol{k}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\boldsymbol{k}}$

coincide, and we deduce the proposition for every $\ell$. ∎

Let now $\mathscr{F},\mathscr{G}$ in
$\mathsf{D}^{\mathrm{b}}({\boldsymbol{k}}_{X})$. We have a natural isomorphism
(see[28, (2.6.8)])

$\operatorname{Hom}(\mathscr{F}\otimes\mathscr{G},\mathbf{D}_{X})\simeq\operatorname{Hom}(\mathscr{F},\boldsymbol{D}\mathscr{G})\simeq\operatorname{Hom}(\mathscr{G},\boldsymbol{D}\mathscr{F}).$

Giving a pairing
$\varphi:\mathscr{F}\otimes\mathscr{G}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbf{D}_{X}$
amounts thus to giving a morphism
$\mathscr{F}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{D}\mathscr{G}$,
or as well a morphism
$\mathscr{G}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{D}\mathscr{F}$.
We say that the pairing $\varphi$ is _perfect_ if the corresponding morphism
$\mathscr{F}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{D}\mathscr{G}$
(or as well
$\mathscr{G}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{D}\mathscr{F}$)
is an isomorphism.

A pairing $\varphi$ induces in a natural way a pairing

$\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{G})\otimes\boldsymbol{R}\Gamma(X,\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathscr{F}\otimes\mathscr{G})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{R}\Gamma_{\mathrm{c}}(X,\mathbf{D}_{X})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\boldsymbol{k}}$

(and a similar pairing by permuting the roles of $\mathscr{F}$ and
$\mathscr{G}$), and thus, for every $\ell\in\mathbf{Z}$, pairings

$\varphi^{\ell}:\boldsymbol{H}^{-\ell}_{\mathrm{c}}(X,\mathscr{G})\otimes\boldsymbol{H}^{\ell}(X,\mathscr{F})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{H}^{0}_{\mathrm{c}}(X,\mathscr{F}\otimes\mathscr{G})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\boldsymbol{H}^{0}_{\mathrm{c}}(X,\mathbf{D}_{X})={\boldsymbol{k}}.$

######  Corollary C.6.

If $\varphi$ is a perfect pairing, the pairings $\varphi^{\ell}$ are also
perfect.

###### Proof.

By construction, the composition of $\varphi^{\ell}$ and the isomorphism

$\boldsymbol{H}^{\ell}(X,\mathscr{F})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\boldsymbol{H}^{\ell}(X,\boldsymbol{D}\mathscr{G})$

induced by $\varphi$ is the pairing (C.3) for $\mathscr{G}$. ∎

###### Proof of Lemma C.5.

We first show that, for sheaves $\mathscr{F}$ and $\mathscr{G}$ of
${\boldsymbol{k}}$-vector spaces, the natural morphisms

(C.7) $\displaystyle f_{*}\mathscr{F}\otimes
f_{*}\mathscr{H}\\!om(\mathscr{F},\mathscr{G})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}f_{*}\bigl{(}\mathscr{F}\otimes\mathscr{H}\\!om(\mathscr{F},\mathscr{G})\bigr{)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}f_{*}\mathscr{G},$
(C.8) $\displaystyle f_{*}\mathscr{F}\otimes
f_{*}\mathscr{H}\\!om(\mathscr{F},\mathscr{G})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}f_{*}\mathscr{F}\otimes\mathscr{H}\\!om(f_{*}\mathscr{F},f_{*}\mathscr{G})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}f_{*}\mathscr{G}$

coincide. Let us first recall the construction of the natural morphism

$\mathscr{F}\otimes\mathscr{H}\\!om(\mathscr{F},\mathscr{G})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{G}.$

We denote by
$[\mathscr{F}\otimes\mathscr{H}\\!om(\mathscr{F},\mathscr{G})]^{\sim}$ the
presheaf

$U\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}\Gamma(U,\mathscr{F})\otimes\Gamma(U,\mathscr{H}\\!om(\mathscr{F},\mathscr{G}))$

whose associated sheaf is
$\mathscr{F}\otimes\mathscr{H}\\!om(\mathscr{F},\mathscr{G})$ by definition.
Recall that

$\Gamma(U,\mathscr{H}\\!om(\mathscr{F},\mathscr{G}))=\operatorname{Hom}(\mathscr{F}_{|U},\mathscr{G}_{|U})$

(compatible families of morphisms
$\operatorname{Hom}(\Gamma(U^{\prime},\mathscr{F}),\Gamma(U^{\prime},\mathscr{G}))$
for $U^{\prime}$ open in $U$), so that there is a forgetful morphism

$\Gamma(U,\mathscr{H}\\!om(\mathscr{F},\mathscr{G}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{Hom}(\Gamma(U,\mathscr{F}),\Gamma(U,\mathscr{G})).$

By composition, we obtain a morphism

$[\mathscr{F}\otimes\mathscr{H}\\!om(\mathscr{F},\mathscr{G})]^{\sim}(U)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma(U,\mathscr{F})\otimes\operatorname{Hom}(\Gamma(U,\mathscr{F}),\Gamma(U,\mathscr{G}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma(U,\mathscr{G}).$

Since $\mathscr{G}$ is a sheaf, it induces a sheaf morphism
$\mathscr{F}\otimes\mathscr{H}\\!om(\mathscr{F},\mathscr{G})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathscr{G}$.
Considering only open sets of the form $f^{-1}(V)$ with $V$ open in $Y$ gives
rise to the natural morphism
$f_{*}(\mathscr{F}\otimes\mathscr{H}\\!om(\mathscr{F},\mathscr{G}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}f_{*}\mathscr{G}$.
But the presheaf

$V\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}[\mathscr{F}\otimes\mathscr{H}\\!om(\mathscr{F},\mathscr{G})]^{\sim}(f^{-1}(V))$

is identified with the presheaf giving rise to $f_{*}\mathscr{F}\otimes
f_{*}\mathscr{H}\\!om(\mathscr{F},\mathscr{G})$, so

$f_{*}(\mathscr{F}\otimes\mathscr{H}\\!om(\mathscr{F},\mathscr{G}))=f_{*}\mathscr{F}\otimes
f_{*}\mathscr{H}\\!om(\mathscr{F},\mathscr{G}).$

We thus have obtained (C.7). Finally, the morphism above reads

$[f_{*}\mathscr{F}\otimes
f_{*}\mathscr{H}\\!om(\mathscr{F},\mathscr{G})]^{\sim}(V)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Gamma(V,f_{*}\mathscr{G}),$

and its sheafification is nothing but (C.8).

By suitably restricting to $f$-proper supports, we obtain the non-derived
version of the identification of morphisms in the lemma. In order to get the
general case, one replaces $\mathscr{G}$ with an injective resolution
$\mathscr{G}^{\scriptscriptstyle\bullet}\in\mathsf{D}^{+}({\boldsymbol{k}}_{X})$.
It follows that, for
$\mathscr{F},\mathscr{G}\in\mathsf{D}^{\mathrm{b}}({\boldsymbol{k}}_{X})$,
$\boldsymbol{R}\\!\mathscr{H}\\!om(\mathscr{F},\mathscr{G})=\mathscr{H}\\!om(\mathscr{F},\mathscr{G}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$
is flabby (see[28, Prop. 2.4.6(vii)]) and, by considering a flabby resolution
$\mathscr{F}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\in\mathsf{D}^{+}({\boldsymbol{k}}_{X})$
of $\mathscr{F}$, we replace
e.g.$\mathscr{F}\otimes\boldsymbol{R}\\!\mathscr{H}\\!om(\mathscr{F},\mathscr{G})$
with
$\mathscr{F}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}}\otimes\mathscr{H}\\!om(\mathscr{F},\mathscr{G}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$,
so that we can apply the result for sheaves (we use flabbiness of
$\mathscr{H}\\!om(\mathscr{F},\mathscr{G}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$
since we cannot consider
$\mathscr{H}\\!om(\mathscr{F}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}},\mathscr{G}^{\raisebox{1.0pt}{${\scriptscriptstyle\bullet}$}})$,
as $\boldsymbol{R}\\!\mathscr{H}\\!om$ is not defined on
$\mathsf{D}^{+}({\boldsymbol{k}}_{X})\times\mathsf{D}^{+}({\boldsymbol{k}}_{X})$).
∎

######  Example C.9.

In the case of a differentiable manifold $M$ with corners as in Section 2.a,
the dualizing complex $\mathbf{D}_{M}$ can be represented by
$j_{!}\mathbf{C}_{M^{\circ}}[m]$. Proposition B.2 implies that the shifted
complex of rapid decay forms
$(\mathscr{E}_{M}^{\mathrm{rd},{\scriptscriptstyle\bullet}},\partial)[m]$ is a
representative of the dualizing complex. Perfectness of
$\mathscr{V}^{\mathrm{rd}}\otimes\mathscr{V}^{\mathrm{mod}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}j_{!}\mathbf{C}_{M^{\circ}}$
as stated in Assumption 2.1 means perfectness, in the above sense, of the
shifted morphism
$\varphi:\mathscr{V}^{\mathrm{rd}}[m]\otimes\mathscr{V}^{\mathrm{mod}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}j_{!}\mathbf{C}_{M^{\circ}}[m]\simeq\mathbf{D}_{M}$.

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