text:
\begin{document}
\begin{abstract}
We give methods to compute $l^2-$cohomology groups of a covering manifold obtained by removing pullback of a (normal crossing) divisor to a covering of a compact K\"ahler manifold.

We prove that in suitable quotient categories, these groups admits natural mixed Hodge structure whose graded pieces are given by expected Gysin maps.
\end{abstract}

\maketitle

\bibliographystyle{plain}

\section{introduction.}

\subsection{}
Let $p: \tilde X\to (X,\omega)$ be a Galois cover of a compact K\"{a}hler manifolds with covering group $G$. Let $N(G)$ be the (left) group Von Neumann algebra of $G$. Let $(S^{k-q,q}(\tilde X),d)$ and $(S^{k-q,(p,q)}(\tilde X),\dbar)$  be the complexes  built on differential $q-$forms (resp. (p,q)-forms) of Sobolev class $k-p$ ($k\in \NN$ big enough). We have the following morphism of (left) $N(G)-$modules:
\begin{eqnarray*}
\text{ Kodaira decomposition: } H^{r}_{d(2)}(\tilde X)=\H^{r}_{d(2)}(\tilde X)\oplus\frac{\Adh{Im d}}{Imd}  &\text{ and }& H^{(p,q)}_{\dbar(2)}(\tilde X)=\H^{(p,q)}_{\dbar(2)}\oplus\frac{\Adh{Im \dbar}}{Im\dbar} \\

 \text{Hodge to De Rham spectral sequence:}  \,\,\,H^{p,q}_{(2)\dbar}(\tilde X) &\abupt & H^{p+q}_{(2)d}(\tilde X)\\

 \text{K\"{a}hler condition:}\,\,\, \H^{r}_{d(2)}(\tilde X) &= &\oplus_{p+q=r} \H^{(p,q)}_{\dbar(2)}(\tilde X)
 \end{eqnarray*}

with $\H^{r}_{d(2)}(\tilde X)$ and  $\H^{(p,q)}_{\dbar(2)}(\tilde X)$ respective harmonic spaces associated to Laplace operators $\Delta_{d}$ and $\Delta_{\dbar}$.

A torsion theory $\tau=(\T,\F)$ on $\Mod(N(G))$ (category of $N(G)-$modules,
 $\T$ is class of torsion modules, $\F$ class of free modules; see \ref{torsion theory} and \ref{example of torsion modules}) is given by a Serre class $\T$ of $N(G)-$modules: $\T$ is a stable by extension, quotient and submodule.
Then in quotient category $\Mod(N(G))_{/\tau}$, a module $A\in \T$ is isomorphic to $\{0\}$, a morphism $\alpha\in Hom_{N(G)}(.,.)$ defines an inversible $[\alpha]\in Hom_{N(G)_{/\T}}(.,.)$   iff its kernel and cokernel are in $\T$. Moreover reduction functor $\Mod(N(G))\to \Mod(N(G))_{/\tau}$ is exact.

Then Theorem \ref{structurepure} shows that interpretation of K\"ahler condition gives:

\begin{theorem-introduction}Let  $\tau=(\T,\F)$ be a real torsion theory such that $\frac{\Adh{Im \dbar}}{Im\dbar}\in \T$. Then Hodge to de Rham spectral sequence $H^{p,q}_{(2)\dbar}(\tilde X) \abupt  H^{p+q}_{(2)d}(\tilde X)$ degenerates at $E_{1}$ in quotient category $Mod(N(G))_{/\tau}$ and isomorphism $\oplus_{p+q=r} \H^{(p,q)}_{\dbar(2)}(\tilde X)\simeq H^{r}_{d(2)}(\tilde X)$  in $Mod(N(G))_{/\tau}$ defines a $\tau-$Hodge structure on $H^{r}_{d(2)}(\tilde X)$.
 \end{theorem-introduction}
\subsection{}

Recall that a mixed Hodge structure is built as successive extension of pure Hodge structures.

The purpose of this work is to put a mixed Hodge structure modulo a torsion theory (localisation at a full thick subcategory) on $l^2-$cohomology groups $H^{.}_{d(2)}(\widetilde{ X\setminus D})$ when $D\subset X$ is a normal crossing divisor.

Let $\p\RR\to X,\, \p\CC\to X$ be the sheaves of $L^2-$direct image of locally constant function (locally isomorphic to the constant sheaf $l^{2}(G,\RR)$ and $l^2(G,\CC)$).
 Let $(\p\dlog{.}{D},W,d)\to X$ be the sheaves of $L^2-$direct image  (\cite{CamDem}) of the complex of logarithmic forms with pole on $D$ with its weight filtration $W$. Theorem \ref{theoreme de structure mixte} implies

 \begin{theorem-introduction}Let $p:\tilde X\to X$ be a Galois covering of a compact K\"{a}hler manifold. Let $G$ be its group of deck transforms and let $N(G)$ be its von Neumann algebra.

Let $D=D_{1}\cup\ldots\cup D_{k}\to X$ be a normal crossing divisor in $X$.
Let $\tau=(\T,\F)$ be a real torsion theory on $\Mod(N(G))$ and let $\Mod(N(G))_{/\tau}$ be the quotient category.
\item[A)]   Isomorphism
\begin{eqnarray}
H^{.}(X\setminus D,\p\CC)\simeq_{N(G,\CC)} \HH^{.}(X,(\p\dlog{.}{D},d))
\end{eqnarray}
(valid for any complex manifold $X$) induces a weight filtration $W$ and a Hodge filtration $F_{ind}$.

\item[B)] Assume the Hodge to de Rham spectral sequences of each $\tilde D_{I}:=p^{-1}(\cap_{i\in I}D_{i})$ ($\tilde D_{\emptyset}:=\tilde X$) degenerates in $\Mod(N(G))_{/\tau}$ :
\begin{eqnarray}
\forall I\in \P(\{1,\ldots, k\}),\forall r\in \NN,\, H^{r}_{d(2)}(\tilde D_{I})\simeq \oplus \H^{p,q}_{\dbar(2)}(\tilde D_{I}) \text{ in } \Mod(N(G))_{/\tau}\,.\label{un}
\end{eqnarray}
Then
\item[i)]  weight spectral sequence
\begin{eqnarray}
E_{1}^{-p,p+n}(W)=\HH^{n}(X,(\p Gr^{W}_{p}\dlog{.}{D})\simeq H^{n-p}_{d (2)}(\sqcup_{|I|=p}\widetilde {D_{I}})\abupt H^{n}(X\setminus D,\p\CC)\label{deux}
\end{eqnarray}
 degenerates  at $E_{2}$ in $\Mod(N(G))_{/\tau}$.
\item[ii)] Hodge spectral sequence $E_{1}^{p,q}=H^{q}(X,\p\dlog{p}{D})\abupt H^{p+q}_{d(2)}(\widetilde{X\setminus D})$ degenerates at $E_{1}$ in $\Mod(N(G))_{/\tau}$:
induced filtration $F_{ind}$ and recurrente filtration $F_{rec}$ are equal in $\Mod(N(G))_{/\tau}$.
\end{theorem-introduction}

\subsection{ }
 Above spectral sequences are valid on general complex manifold $X$ and may be realised through various Sobolev spaces.

From  \Del, one deduce quickly that category of mixed Hodge structure mod $\tau$ (see def. \ref{definition of MHS} ) is abelian.

We will give some details of the following basic principle in theory of mixed Hodge structures.

Degenerescence (mod some torsion theory $\tau$) of Hodge to de Rham spectral sequence of each covering $\tilde D_{I}\to D_{I}$
 implies degenerescence of the weight spectral sequences at $E_{2}$

(in simple words: It is enough to control first constituant within a torsion theory to obtain control of the whole spectral sequence).

\subsection{} It is a priori not clear wich object becomes isomorphic to $\{0\}$ in a quotient category .
If $X$ is a compact K\"{a}hler manifold,  there exists a  scale of torsion theories wich gives non trivial mixed Hodge structure:

Let $\tau_{dim}$ be the torsion theory such that module of  zero $G-$dimension are torsion.
 From \Ati, reduced cohomology spaces are of finite $G-$dimension and $\dim_{N(G)}\frac{\Adh{Im d}}{Im d}=0$, \\ $\dim_{N(G)}\frac{\Adh{Im \dbar}}{Im \dbar}=0$.
 Reduction of above  spectral sequence to $\Mod(N(G))_{\tau_{dim}}$ always  defines a mixed Hodge structure and isomorphism $E_{2}\simeq E_{\infty}$ in $\Mod(N(G))_{/\tau_{dim}}$ gives expected realisation in term of harmonic spaces and Gysin maps:
 \begin{eqnarray*} E_{1}^{p,q}(W)\simeq \H^{q-2p}_{(2)}(\tilde D_{p})\\
  E_{1}^{p,q}(F)\simeq \H^{q}_{\dbar(2)}(\dlog{p}{\tilde D})
  \end{eqnarray*}
  (after a choice of hermitian metric on vector bundles $\dlog{.}{D}$). Count of dimensions prove for example that $Gr^{W}_{n}H^{n}_{(2)}(\widetilde{X\setminus D})$ is isomorphic (in $\Mod(N(G))_{/\tau_{dim}}$) to $\H_{1}(T,l^2(G))$ the first (reduced) homology group of the dual CW-complex associated to $\tilde D\to D$.

\subsection{ }

Finer torsion theories are available: We prove in lemma \ref{example of torsion modules}, that for any  $\alpha \in \Adh{Im \dbar}$, there exists $r\in N(G)$ an almost isomorphism (injective with dense range) such that $r\alpha\in Im{\dbar}$.

This is a particular case of the $T-$lemma \ref{T-lemma} wich is well known to operator algebraists.

  Let $\U(G)$ be the ring of affiliated operator of $N(G)$: it is the quotient ring of $N(G)$ by the multiplicative set of almost isomorphism (see \Luc Chap. 8). Above lemma reads $$\U(G)\otimes_{N(G)}\frac{\Adh{Im \dbar}}{Im\dbar}=0\,.$$ Therefore working in $\Mod(N(G))_{/\tau_{\U(G)}}$ allows easy computation with Hilbert Modules (one could also work directly in category of $\U(G)-$modules and associated sheaves).

\subsection{}Let $\N(G)$ be the local constant sheaf associated to $N(G)$. There is a natural exact functor $N$ from sheaves on $X$ to sheaves of $\N(G)-$modules given by $\F\to \N(G)\otimes_{\CC[G]}p_{!}p^*\F $. ($p_{!}$ is direct image with compact support).
Let $\K$ be a cohomological Hodge complex of sheaves on $X$. Then $N(\K)$ view as a complexes of $\N(G)-$sheaves modulo $\tau_{dim}$ should be a  minimal $(\tau_{dim},\tau_{dim})-$CHMC (see \ref{definition of CHMC}). In this article we obtain a $(0,\gamma^{*}\tau_{\dbar})-$CHMC, wich might be view as a maximal object (\ref{definition of CHMC}).

\subsection{} In last section, we treat simple examples through various choice of torsion theories.

\subsection{ }
In this article, we focus on the technical part of mixed Hodge structure modulo some torsion theory. In a subsequent paper, functoriality, geometrical and analytical applications will be given.

\subsection{}
We give a short list of references:
\begin{trivlist}
\item[-]
We tried to follow articles of \Del, \cite{Deltrois}. The  reference books of \Elz ,\Voi , and  \PetSte~  contains steps to mixed Hodge structures and we extracted from \cite{PetSte} presentation of this theory.

\item[-] Article of \Far~  defines an abelian category (objects are morphisms $(f:A_{1}\to A_{2})$ of finitely generated $N(G)-$module,   morphisms are the natural one) in order  to deal with non closed range. According to \Fre, this is the smallest abelian category wich contains category of finitely generated hilbertian modules. Farber already develops homological algebra for theses modules.

However calculus in this category are not so easy (Objects are $2-$term complexes) and  localisation seems more appropriate in order to use sheaf theory.
\item[-] Article of \Ati, \CheGrodeux, \Shu~ give foundational results. In \cite{CheGrodeux}, \cite{Shu} exact sequences of Fredholm complexes are strudied.
\item[-] Article of \Shu   contains already results on existence of distributional sections with prescribed singularity. It covers (in a short format) needed analytical results on Von Neumann algebra, $G-$Fredholm operators and Sobolev spaces (see also \MaMar).

\item[-] Article of \Eys~ propose, following  method of Farber, a construction of $L^2-$direct image  for coherent sheaves and compute its derived category  (It is also suggested in the introduction to developpe $L^2-$mixed Hodge structure and may be $L^2-$intersection cohomology).
\item[-] Article of \CamDem~  defines $L^2-$direct image of coherent sheaf and gives functorial properties that will be used  in this article.

\item[-] In Chapter 6  of \Luc, an algebraic treatment of $l^2-$cohomology is given. A dimension for any (abstract) $N(G)-$module is defined, and it is proved that the ring $N(G)$ is semi hereditary (finitely generated submodule of a projective module is projective). Then subsequent results are essentially stated using dimension torsion theory.

In chapter 8
and article of \SauTho,  torsion theory and homological algebra associated with dimension function or affiliated operators is used.
 \end{trivlist}

\subsection{ } My hearty thanks go to S. Vassout and G. Skandalis for illuminating discussions on Von Neumann algebra.

\section{preliminaries.}
\subsection{real structures.}
\subsubsection{Godement resolution}\label{Godement-resolution}(See
\God or \Bre):
Let $X$ be a topological space.
Let $R$ be a ring and  $\R$ be the sheaf of rings it defines. Let $\A$ be a sheaf of
(left) $\R-$modules, let $\C^*(X,\A):=(\Cg^{.}(X,\A),d)$ be Godement resolution (\cite{God} p. 168)

 by  sheaves of $\R-$module with differential $\R-$linear. Then
 \begin{eqnarray*}
 \A\to \C^*(X,\A) \,,& & \A\to C^*(X,\A)=(\Gamma(\Cg^*(X,\A)),d)
 \end{eqnarray*}
    are
covariant additive exact functors with value in the category of differential $\R-$sheaf,
resp. cochain $R-$complexes. If $X$ is clear from the context, we write $\C^*(\A),\ldots$

If $(\F^{.},d)$ is a differential sheaf, let $\C^{.}(\F^{.})$ be the total complex associated to the double complex $\Cg^{q}(X,\F^{p})$. Let $j:Y\to X$ be a continuous map and $(\F^.,d)$ be a diferential sheaf of $\R-$modules on $Y$. One sets  $R^{.}j_
{*}\F^{.}:=j_{*}\C^{.}(\F^{.})$

\subsubsection{real structure}
Assume that $R$ is a $\RR-$algebra, then any sheaf of $\R-$modules is
also a sheaf of $\RR-$vector spaces so that $\A\otimes_{\RR}\CC$ is a sheaf of $\R
\otimes_{\RR}\CC-$left modules. Let $i:\A\to \A\otimes_{\RR}\CC$.

Then
\begin{eqnarray}\label{real structure}
  j_{*}(\A)\otimes_{\RR}\CC&\to& j_{*}(\A\otimes_{\RR}\CC)\\

\C^*(X,\A)\otimes_{\RR}\CC\overset{\C^{.}(i)\otimes 1_{\CC}}{\longrightarrow}\C^*(X,\A\otimes_{\RR}\CC)& &

C^*(X,\A)\otimes_{\RR}\CC\overset{C^{.}(i)\otimes 1_{\CC}}{\longrightarrow}C^*(X,\A\otimes_{\RR}\CC)

  \end{eqnarray}
    are $\R\otimes \CC-$isomorphisms (resp. $R\otimes\CC-$isomorphisms).

\subsubsection{ }

A real structure on a $\R\otimes_{\RR}\CC$ sheaf $\B$ is a $\R-$subsheaf
$\A\overset{i}{\to}\B$ such that $\A\otimes_{\RR}\CC\overset{i\otimes 1_{\CC}}{\to}\B$ is an isomorphism. It induces
a real structure on Godement resolution.

Then $H^{.}(X,\,\A)\otimes_{\RR}\CC\to H^{.}(X, \,\A\otimes_{\RR}\CC)$ is a $R\otimes_{\RR}\CC$ isomorphism.

\subsubsection{ }

Let $G$ be a discret group with left action of $\ZZ[G]$. Let $N(G)=N_{l}(G)$ be the left von Neumann algebra of $G$.

Then $\RR[G]-$isomorphism
 $l^2(G,\CC)\ni a\to Re(a)+iIm(a)\in l^2(G,\RR)\otimes\CC$   induces  decompositions
 \begin{eqnarray}
  End_{\CC}( l^2(G,\RR)\otimes\CC) &\simeq & End_{\RR}( l^2(G,
\RR))\otimes\CC \\
  End_{\CC[G]}( l^2(G,\RR)\otimes\CC) &\simeq & End_{\RR[G]}( l^2(G,
\RR))\otimes\CC \\
 N_{r}(G,\CC)&\simeq &N_{r}(G,\RR)\otimes \CC
\end{eqnarray}

Left action of $G$ on  $l^2(G,\RR)$ defines a left action of $G$ on $End_{\RR}( l^2(G,\RR))$
and $End_{\CC} l^2(G,\CC)$ so that $g(m(a))=(gm)(ga)$.  Continuous invariant morphism for this action are
$N_{r}(G,\RR)$ and $N_{r}(G,\CC)$.
 An element of $ \rho(f)\in N(G,\CC)$ is represented by right convolution  with a function $f\in l^{2}(G,\CC)$ wich is moderate (\Dix 13.8.3): $\exists C\geq 0:\, \forall g\in \CC[G],\, ||g\conv f||_{2}\leq C||g||_{2}$. Then $\rho(f)\in N_{r}(G,\RR)$ is equivalent to $f$ real valued.

 Above isomorphism induces isomorphism $N(G,\CC)\simeq
N(G,\RR)\otimes \CC$. Note that $\CC[G]$ is identified to a subring of $N(G,\CC)$.
\subsection{A $T-$lemma on von Neumann algebra.}
Survey on von Neumann algebra is given in \Luc chap.9. See also \Ped, \Dix.
\begin{definition}
Let $\A$ be a Von Neumann algebra on a separable Hilbert space. Let $\A_{+}$ be the cone of positive operator in $\A$.
\item[1)]A  trace is a function $t: \A_{+}\to [0,+\infty]$  such that if $\lambda>0$ and $x,y\in \A_{+}$ then $t(\lambda x)=\lambda t(x)$, $t(x+y)=t(x)+t(y)$ and for all unitary $u\in A$, $t(u^*xu)=t(x)$.
\item[2)] A state  is a positive functional of norm one: $\varphi\in \A^{'}$ such that $\varphi(\A_{+})\subset \RR^+$ and $\varphi(1)=1$.
\item[3)] A trace or a state is normal if it is continuous on increasing limit of net.
\item[3)] Defines the left kernel of a trace or a state $\varphi$ by $L_{\varphi}=\{x\in\A:\,\varphi(x^*x)=0\}$. One says that $\varphi$ is faithful if $L_{\varphi}=0$.
\item[4)] A faithfull trace is called finite if $t(1)<+\infty$ (and $\A$ is then called finite von Neumann algebra). It is called semi finite if $\A_{+}^{t}=\{ y\in \A_{+}:\, t(y)<+\infty\}$ is weakly dense in $\A_{+}$. Then for all $x\in\A_{+}$, $t(x)=\sup_{y\leq x,\ y\in \A_{+}^{t}}t(y)$.
\end{definition}
\begin{example}
\item[1)]
Let $e\in l^2(G)$ be the function $g\to \delta_{e}(g)$ and $e$ the unit element. Then trace of $n\in N_{l}(G)$ or $N_{r}(G)$ is $\tr n:=<n(e),e>$.
\item[2)]
Let $Tr_{H}$ be the Hilbert Schmidt trace on $\B(H)$: $Tr_{H}\alpha=\sum_{i}||\alpha(e_{i})||^2$ with $(e_{i})_{i\in\NN}$ an orthonormal basis.
Then $Tr_{H}\otimes \tr$ defines a semi-finite trace on $\B(H)\otimes N(G)$. If $t\in \B(H)\otimes N(G)$ is a positive element represented (in a orthonormal basis) by an infinite matrix $(n_{ij})_{i,j}$ of element in $N(G)$ then $Tr T=\sum_{i}\tr n_{ii}$.

\item[3)] If $\pi:\A\to \B(H)$ is a representation, let $\zeta\in H$ of unit norm. Then $x\mapsto <\pi(x)\zeta,\zeta>$ is a positive functional of norm $1$, called the vector state associated to $(\pi,H,\zeta)$.
\end{example}

Traces and states satisfies Cauchy-Schwartz inequality:
\begin{eqnarray}
 |\varphi(y^*x)|^2\leq \varphi(x^*x)\varphi(y^*y) &\text{ on } & A_{2}^{\varphi}=\{x\in\A:\, \varphi(x^*x)<+\infty\}\,.
\end{eqnarray}
Therefore, left kernel $L_{\varphi}$ of $\varphi$ is a linear space.
Define a scalar product on $A_{2}^{\varphi}/L_{\varphi}$ (wich is equal to $\A/Ker\varphi$ if $\varphi$ is a state) by $(\zeta_{x},\zeta_{y}):=\varphi(y^*x)$ with $x\mapsto \zeta_{x}$ be the quotient map.

 The GNS (Gelfand-Naimark-Segal) construction is  obtain by completion of this pre-Hilbert space (see \Ped 3.3):
 \begin{lemma}\label{GNS}
 \item[1)] Let $\varphi$ be a positive functional on $\A$, then there exists a  cyclic representation $(\pi_{\varphi}, H_{\varphi},\zeta_{\varphi})$ such that
 $<\pi_{\varphi}(x)\zeta_{\varphi},\zeta_{\varphi}>=\varphi(x)$.
 \item[2)] Let $\varphi'$ be a positive functional such that $\varphi'\leq \varphi $, then there exists a unique $a\in \pi_{\varphi}(\A)'$, $0\leq a\leq 1$, such that $\varphi'(x)=(\pi_{\varphi(x)}a\zeta_{\varphi},\zeta_{\varphi})$.
\item[3)] Hence $(\pi_{\varphi'},H_{\varphi'}, \zeta_{\varphi'})$ is a subrepresentation of $(\pi_{\varphi}, H_{\varphi}, \zeta_{\varphi})$.
\item[4)] Let $\varphi$ be the vector state associated to $(\pi, H, \zeta)$, then $\A/Ker\varphi\ni x\mapsto x\zeta\in H$ defines a $\A-$linear isometry between $(\pi_{\varphi},H_{\varphi},\zeta_{\varphi})$ and $(\pi, \Adh{\pi(A)\zeta}^{H}, \zeta)$.
\item[5)] Assume that $\varphi$ is a faithfull state, then for any state $\varphi'$ on $\A$, $(\pi_{\varphi'},\, H_{\varphi'},\, \zeta_{\varphi'})$ is a sub- representation of $(\pi_{\varphi}, H_{\varphi},\zeta_{\varphi})$.
\end{lemma}
Recall that a representation $(\pi,H,\zeta)$ of $\A$ is said to be cyclic if $\pi(\A)\zeta$ is dense in $H$.
\begin{proof} \item[1)] Let $H_{\varphi}$ be the Hilbert space completion of $(A/(Ker\varphi),(.,.))$. Then $A/(Ker\varphi)\to H_{\varphi}$ is dense. Defines $\pi_{\varphi}(x)\zeta_{y}=\zeta_{xy}$. One checks that this defines a continuous operator $\pi_{\varphi}(x)$ on $A/Ker\varphi$. It extends as a continuous operator on $H_{\varphi}$. Identity $(\pi_{\varphi}(x)\zeta_{y},\zeta_{z})=\varphi(z^*xy)=(\zeta_{y},\pi_{\varphi}(x^*)\zeta_{z})$ proves that $\pi_{\varphi}$ is a $\star-$representation. Let $e$ be unit of  $\A$. One set $\zeta_{\varphi}=\zeta_{e}$.

 Then orbit of $\zeta_{\varphi}$ is dense in $H_{\varphi}$.
\item[2)] Densely defined positive bilinear form $B(\zeta_{x},\zeta_{y}):=\varphi'(y^*x)$ is bounded by $(\zeta_{x},\zeta_{y})$.  Friedrich extension gives a unique operator $a$ on $H_{\varphi}$ such that $B(\zeta_{x},\zeta_{y})=(a\zeta_{x},\zeta_{y})$. One checks that $a\in \pi_{\varphi}(\A)'$ and $0\leq a\leq 1$.
\item[3)]  Let $a^{\frac{1}{2}}$ be some positive square roots in $\pi_{\varphi}(\A)'$. Then

$(\pi_{\varphi'}(z)\zeta'_{x},\zeta'_{y})_{\varphi'}=\varphi'(y^*zx)=(\pi_{\varphi}(z)a^{\frac{1}{2}}\zeta_{x},a^{\frac{1}{2}}\zeta_{y})_{\varphi}$. Hence densely defined morphism

 $a^{\frac{1}{2}}: A/(Ker\varphi')\to A/(Ker\varphi)$ extends as an isometric morphism $U:H_{\varphi'}\to H_{\varphi}$ wich interwines $\pi_{\varphi'}$ and $\pi_{\varphi}$ and such that $U(\zeta_{\varphi'})=a^{\frac{1}{2}}\zeta_{\varphi}$.
\item[4)] is clear.
\item[5)] From $(2)$, $(3)$, one gets a subrepresentation $(\pi_{\varphi}, H_{\varphi},\zeta_{\varphi})\to (\pi_{\varphi+\varphi'}, H_{\varphi+\varphi'},\zeta_{\varphi+\varphi'})$ with dense image (for $\varphi$ is faithfull). Hence representation $\pi_{\varphi}$ and $\pi_{\varphi+\varphi'}$ are isomorphic. But $\pi_{\varphi'}$ is a sub representation of  $\pi_{\varphi+\varphi'}$.
\end{proof}
Note that Hilbert space completion is solution of a universal problem. This implies that natural bounded morphism $(A/(Ker\varphi),(.,.)_{\varphi})\to (A/(Ker\varphi'),(.,.)_{\varphi'})$ induces a morphism $H_{\varphi}\to H_{\varphi'}$.
\subsubsection{Standard form }\label{standarde}
A similar statements holds for the GNS construction associated to a trace. Note however that unitary invariance of  a trace  $t$ implies that $t(xy)=t(yx)$ on $\A_{2}^{t}$. This gives further property to associated GNS space:
Let $(\A,t)$ be a von Neumann algebra with a normal faithfull tracial state (a continuous positive linear form$\varphi$ such that $\varphi(1)=1$ and $\varphi$ is a normal trace). Then $(x,y)\mapsto t(y^*x)=(x,y)_{t}$ is a scalar product.

Let $(\pi_{t}, l^{2}(\A,t), \zeta_{t})$  be the Hilbert space obtains through GNS construction  and called it  the standard form associated to $(\A,t)$.

A vector $\zeta\in l^{2}(\A,t)$ defines two closed, densely defined, unbounded operators:
\begin{eqnarray}\lambda(\zeta): D(\lambda(\zeta))=\zeta_{t}\A' &\ni & \zeta_{t}x'\mapsto \zeta x'\in l^2(\A,t)\\
\rho(\zeta): D(\rho(\zeta))=\A\zeta_{t} &\ni & x\zeta_{t}\mapsto x\zeta\in l^2(\A,t)
\end{eqnarray}
The isometric densely defined operator $\A\ni x\to x^{*}\in\A$ extends to   $J:L^2(\A,t)\to L^{2}(\A,t)$ wich is conjugate linear, isometric and involutive.

\begin{lemma}\label{standard}
\item[1)]
$\U(\A)=\lambda(\A)$ and $\V(\A)=\rho(\A)$ are von Neumann subalgebra on $l^2(\A,t)$ such that $J\U(\A)J=\V(\A)$ and $\U(\A)'=\V(\A)$.
\item[2)] Let $\zeta\in l^2(\A,t)$. Then $\rho(\zeta)$   is bounded iff $\lambda(\zeta)$   is bounded.
\end{lemma}
\begin{proof}\item[1)] see \Dix I.6.2 theorem 2 and I.5.2.
\item[2)] see \Dix I.5.3.
\end{proof}

Therefore we assume that $\A$ is a von Neumann sub-algebra of $\B(L^{2}(\A,t))$ and let $\A'$ acts on $L^2(\A,t)$ on the right.

 \begin{example}
Let $G$ be a discret group, let $N_{l}(G)$, $N_{r}(G)$ left (resp. right) von Neumann algebra generated by left (resp. right) translation.

Then $N_{l}(G)'=N_{r}(G)$ and $N_{r}(G)'= N_{l}(G)$ (\Dix I.5.2).

If $L\in N_{l}(G)$, there exists a unique $l\in l^2(G)$ such that $L(f)=l\conv f$ (convolution product).
Let $h\in l^2(G)$ then
$$\lambda(h)(f):=h\conv f\in l^2(G) \text{ and } \rho(h)(f):=f\conv h\in l^2(G)$$
on respective domains.
Then $\rho(h)$ is bounded iff $\lambda(h)$ is bounded and in this case, $\lambda(h)\in N_{l}(G)$, $\rho(h)\in N(G)_{d}$ (\Dix III 7.6). It is  apparent on $x\to h\conv x\conv h'$ that right and left representation commute.
Note that  standard form of $(N_{l}(G),tr)$ is $(\lambda, l^2(G), e)$.
\end{example}
Closed densely defined unbounded operator $\rho(x)$ with $x\in l^2(G)$ are  particular cases of operators affiliated to $N_{r}(G)$:

\begin{definition}(\MurVon  chap.XVI)
Let $\A$ be a finite von Neumann algebra on $H$.
 A closed densely defined operator $h: D(h)\to H$ is said to be affiliated to $\A$ if it commutes with $\A'$: for all unitary $u\in \A'$, $uD(h)=D(h)$ and $uh=hu$.
\end{definition}
Then (see \Ped 5.3.10)
from the bicommutant theorem, it follows that $h$ is affiliated to $\A$ iff $f(h)\in \A$ for every bounded Borel function on $Spec (h)$. In particular, if $h\geq 0$ then $h$ is affiliated to $\A$ iff $(1+\epsilon h)^{-1}h\in \A$ for some $\epsilon>0$.

An essential property of affiliated operator to $(\A,t)$, a finite von Neumann algebra, is that they form an algebra (a property valid for more general semi-finite von Neumann algebra, see \Luc Chap.8, \MurVon Chap. XVI, \Shu ):

The following lemma is a variation on the description of $\Adh{N_{l}(G)f}$ in term of affiliated operator as in \MurVon  9.2,  developped as Radon-Nykodim theorems in \Dye, \Ped 5.3 :

\begin{lemma}[The $T-$lemma]\label{T-lemma}Let $(\A,t)$ be a finite von Neumann algebra. Let $\pi:\A\to\B(H)$ be a faithfull  represention of $\A$ as a von Neumann subalgebra of $\B(H)$. Let $T$ be an operator in the commutant  $\pi(\A)'$ of $\pi(\A)$.
For any $\zeta\in \Adh{Im T}$, there exists $r\in \A$ such that
 $\pi_{t}(r)$(\ref{standarde}) is injective with dense range and $\pi(r)\zeta\in Im(T)$.
\end{lemma}
\begin{proof}\item[1)] Assume that $H=l^{2}(\A,t)$ and that $\rho(\zeta),\,T\in \A'$ have dense ranges. Then $T=\rho(r)$ with $r\in \A$ (\ref{standard}). Let $T^{-1}\circ \rho(\zeta)=up$ be the polar decomposition of operator $T^{-1}\rho(\zeta)$ affiliated to $\A'$. Then $u$ is an isometry, $p$ is positive, $u$, $p$ are affiliated to $\A'$. But $up=[u(1+p)^{-1}p](1+p)$. Note that $1+p\geq 1$ hence $(1+p)^{-1},\,(1+p)^{-1}p\in \A'$. From (\ref{standard}), there exists $a,y\in \A$, such that $(1+p)^{-1}=\rho(a)$ and $\rho(y)=u(1+p)^{-1}p$. Then $\rho(\zeta)\circ\rho(a)=  T\circ \rho(y)$. But $\rho(a)\circ \rho(\zeta)=\rho(a.\zeta)$ and $T\circ \rho(y)=\rho(T(y))$ (identity is valid on the dense subset $\A$). Then $Im(\rho(a))=D(1+p)$ is dense.
\item[2)]

 Let $\zeta\in \Adh{Im T}$. Then $\Adh{\pi(A)\zeta}\subset \Adh{Im T}$. Let $T_{\perp}:Ker T^{\perp}\to \Adh{Im T}$.
Then $$T'=T_{\perp}\oplus id_{l^2(\A,t)}: T_{\perp}^{-1}[\pi(\A)\zeta]\oplus l^2(\A,t)\to [\pi(\A)\zeta]\oplus l^2(\A,t)$$ is a $\A-$linear injective morphism with dense range. Moreover vector state associated to $(\zeta,1)$ ($x\to <\pi(x)\zeta,\zeta>+t(x)$) dominates $t$.
Using GNS construction \ref{GNS} (5), one deduces that
there exists $\A-$isomorphism (not isometric) $U_{1}:T_{\perp}^{-1}[\pi(\A)\zeta]\oplus l^2(\A,t)\to l^2(\A,t)$ and  $U_{2}:[\pi(\A)\zeta]\oplus l^2(\A,t)\to l^2(\A,t)$. Setting $\tilde\zeta=U_{1}(\zeta,1)$ and $\tilde T=U_{2} \circ T'\circ U_{1}^{-1}$, we apply $(1)$ to conclude.

\end{proof}
In particular, if $x\in l^2(G)$,  conductor of $\rho(x)\in \U(G)$ to $N_{r}(G)$ is non trivial.

\subsection{$N(G)-$Hilbert Modules and von Neumann dimension.}
A (left) $N(G)-$Hilbert module is a Hilbert space $V$ with an unitary action $U(.)$ of $G$ such that $V$ is $G-$isometric to a $G$ invariant subspace of the free Hilbert $N(G)-$module $H\hat\otimes l^2(G)$. Then the von Neumann generated by $\{U(.)\}$ is isomorphic to $N_{l}(G)$. If $\dim_{\CC}H<+\infty$, then $V$ is said to be finitely generated.
\begin{example}(see \Shu~ section 3) If $\tilde E\to \tilde X$ is a pullback of  smooth vector bundle $E\to X$ under a $G-$covering map $\tilde X\to X$, then space $W^s(\tilde X,\tilde E)$ of section of $\tilde E$ with coefficient in Sobolev space are $N(G)-$Hilbert modules.
\end{example}

If $V$ is embedded as a closed $G-$invariant subset of $H\otimes l^2(G)$, let $P\in N_{r}(G)\otimes\B(H)$ be its projector. Defines $dim_{N(G)}V:=Tr P$.

A left $N(G)-$Hilbert submodule $V$ of $l^2(G)^n$ is defined by a $G-$equivariant orthogonal projection $p$ onto $V$. Then $p$ is identified with a matrix $(p_{ij})\in M_{n}(N_{r}(G))$ and $\dimG V=\tr p:=\sum_{i=1}^{n}\tr p_{ii}$.

A finitely generated projective $N(G)-$module is represented by an idempotent matrix $A\in M_{n}(N(G))$. Then $\dimG P:= \tr A$. Let $M\in \Mod(N(G))$. In \Luc Chap. 6 a dimension function on $\Mod(N(G))$ is defined:\label{definition of dimension}
\begin{eqnarray*}
 \dimG M:=\sup\{\dimG P\,:\, P\subset M\text{ a finitely generated projective submodule}\}\,.
 \end{eqnarray*}

 Then if $V$ is a Hilbert $N(G)-$module the two dimension functions agree:
both of them satisfies that dimension is supremum over finite dimensional subspaces (\Luc p.21 th.1.12 and th. 6.24).

\subsection{Localisation at a Torsion theory}\label{torsion theory}

\begin{definition}A Serre subcategory $\T$ of an abelian category $\A$  is a full subcategory $\T$ of $\A$ such that  for any exact sequence $0\to A\to B\to C\to 0$ in $\A$,
then $$B\in \T\iff A\in\T \text{ and } C\in \T\,.$$
\item[2)] In category  $\Mod(R)$ of $R-$modules over a ring $R$, a Serre class $\T$ defines a hereditary torsion theory  $\tau=(\T,\F)$ on $R$ (\Vas, \Gol ):
Module in $\T$ are torsion modules. Defines class of free modules $\F=\{F\in \Mod(R)\,: \forall T\in\T,\, Hom_{R}(T,F)=0\}$.
\end{definition}

\begin{lemma} Let $\tau=(\T,\F)$ be a hereditary torsion theory, then
\item[1)] $\F$ is closed under submodules, direct products and extension.
\item[2)] Any $M\in\Mod(R)$ has a unique maximal $\tau-$torsion submodule, denoted $T_{\tau}(M)$.
\item[3)]\label{cogenerated by injective} A hereditary torsion theory is cogenerated by an injective module $E$: $\tau=(\T,\F)$ with $\T\ni S\iff Hom_{R}(C,E)=0$.
\item[4)]\label{torsion functor} Functor $T_{\tau}(.): \Mod(R)\to \T$ is a left exact functor ($N\subset M$ implies that $T_{\tau}(M)\cap N=T_{\tau}(N))$.
\end{lemma}

\begin{proof}\item[2)] Let $\{N_{i},\, i\in\Lambda\}$ be the set of all torsion submodule of $M$. Trivial module $\{0\}$ belongs to this set. Then $T_{\tau}(M):=\sum_{i\in\Lambda}N_{i}$ satisfies required properties for it is a homomorphic images of $\oplus_{i\in\Lambda}N_{i}$ wich is torsion.
\item[3)\,, 4)] See \Gol~ p.5 and p. 24.
\end{proof}

If $\tau_{1}$ and $\tau_{2}$ are torsion theories, then $\tau_{1}$ is smaller than $\tau_{2}$ ($\tau_{1}\leq \tau_{2}) \text{ if  }\T_{1}\subset \T_{2}$ iff $ \F_{1}\supset \F_{2}\,.$

Then if $\C$ is a class in $\Mod(R)$, the hereditary torsion theory generated by $\C$ is the smallest hereditary torsion theory $\tau$ such that $\C\subset\T$.

\subsubsection{Quotient category. }

According to \Grot (1.11) and \Gab Chap.3, if $\T$ is a Serre subcategory of an Abelian category $\A$, there exists a quotient category $\A/\T$ with same object than $\A$ and obtained by extending class of isomorphism. Moreover functor $\A\to \A/\T$ is exact.

\begin{definition}Let $\tau=(\T,\F)$ be a hereditary torsion theory on $\Mod(R)$. Defines $\Mod(R)_{/\tau}$ to be the quotient category of $\Mod(R)$ by the Serre class $\T$:
\item[i)] Object of $\Mod(R)_{/\tau}$ are identical with object of $\Mod(R)$; \item[ii)]Morphisms are elements of the following inductive limite $$\varinjlim\{Hom_{R}(M',N/N')\,:M'\subset M,\, N'\subset N \text{ and } M/M',\, N'\in\T\}\,.$$
\end{definition}

Let $\alpha\in Hom_{N(G)}(M,N)$ and let $[\alpha]$ be its image in  quotient category. Then $[\alpha]$ is a monomorphism (resp. epimorphism, resp. isomorphism) iff $Ker\alpha$ (resp. $Coker\alpha$, resp. $Ker\alpha$ and $Coker\alpha$) is a torsion object.

\subsubsection{Torsion theory and complexification}

Let $R$ be a ring wich is also an $\RR-$algebra then $R\otimes_{\RR}\CC$ is a $\CC-$algebra.
If $A\in \Mod(R)$ then
$\CC\otimes_{\RR}A$  has a structure of
$R\otimes_{\RR}\CC-$module through $(R,\CC)-$isomorphism $A \otimes_{R}(R\otimes_{\RR}\CC)\to A\otimes_{\RR}\CC$.

Forgetful functor $\phi:\Mod(R\otimes\CC)\ni B\to B_{R}\in \Mod(R)$ is faithfull and exact and has a left adjoint
\begin{eqnarray}
(\CC\otimes_{\RR}R)\otimes \, . \; : Hom_{R}(A,E_{R}) &\simeq& Hom_{\CC\otimes_{\RR} R}((\CC\otimes_{\RR} R)\otimes_{R} A, E)
\end{eqnarray}

Let $J=\phi(iId_{B})$ be automorphism in $B_{R}$ defined by multiplication by $i$ in $B$ and let $J_{\CC}=J\otimes1_{\CC}:B_{R}\otimes_{\RR}\CC\to B_{R}\otimes\CC$ be its complex linear extension. Let $\bar B$ be the complex conjugate $R\otimes_{\RR}\CC-$module associated to $B$: Its underliying abelian group is $B$, and module structure is given by the following representation $\rho$ of $R\otimes_{\RR}\CC$ in  $End(B)$: $\rho(r\otimes c)(b)=(r\otimes\bar c).b$.
Then there is a natural antilinear $R-$isomorphism $B\to \bar B$. This defines a functor on $\Mod(S)$.  Moreover antilinear $R-$linear map from $B$ to $B'$ are in bijection with $R\otimes\CC-$linear map from $B$ to $\Adh{ B'}$ (or from $\Adh{B}$ to $B'$). A $R\otimes\CC-$isomorphism from $B$ to $\bar B$ is equivalent with a real structure $B\simeq_{R\otimes\CC} A\otimes_{\RR}\CC$ on $B$ ($A\in \Mod(R)$). This defines antilinear $\RR-$isomorphism $Hom_{R\otimes\CC}(A\otimes\CC,\bar E)\simeq Hom_{R\otimes\CC}(A\otimes\CC,E)$.

\begin{lemma}Let $B\in \Mod(R\otimes\CC)$, then $B_{R}\otimes_{\RR}\CC\simeq_{R\otimes\CC} B\oplus\bar B$.
\end{lemma}
\begin{proof} Let $P_{\pm}=\frac{Id\mp iJ_{\CC}}{2}: B_{R}\otimes\CC\to B_{R}\otimes\CC$ be projectors onto eigenspaces $\{J_{\CC}=\pm i Id\}$.

One checks that  $B\ni b\to P_{+}(b\otimes 1)\in \{J_{\CC}= i Id\} \text{ and } \bar B\ni b\to P_{-}(b\otimes 1) \in \{J_{\CC}= i Id\}\, $ are $R\otimes\CC-$isomorphisms.

\end{proof}

Note that $B\ni b\to P_{+}(b\otimes 1)\in B_{R}\otimes_{\RR}\CC$ is a splitting of natural surjection $B_{R}\otimes_{\RR}\CC\to B$.

\begin{definition}(see \cite{Gol}.)\label{ring extension} Let $\gamma:R\to R\otimes_{\RR}\CC$ be above ring extension.
\item[i)] Let $\tau=(\T,\F)$ be a torsion theory on $\Mod(R)$. Then $\gamma_{*}\tau$ is the torsion theory on $\Mod(R\otimes_{\RR}\CC)$ such that $B\in \Mod(S)$ is $\gamma_{*}\tau$ torsion iff $B_{R}$ is $\tau-$torsion.
\item[ii)] Let $\tau_{\CC}=(\T_{\CC},\F_{\CC})$ be a torsion theory on $\Mod(R\otimes_{\RR}\CC)$. Then $\gamma^{*}\tau_{\CC}$ is the torsion theory on $\Mod(R)$ such that $A\in \Mod(R)$ is $\gamma^{*}\tau_{\CC}$ torsion iff $A\otimes_{\RR}\CC$ is $\tau_{\CC}-$torsion.
\end{definition}

One checks that if $\tau$ is cogenerated by injective $I$ (\ref{cogenerated by injective}), then $\gamma_{*}\tau$ is cogenerated by injective $I\otimes_{\RR} \CC$. Hence $\gamma^{*}\gamma_{*}\tau=\tau$ for $I\otimes_{\RR}\CC\simeq_{R}I^2$.
However if $\tau_{\CC}$ is a torsion theory on $\Mod(R\otimes_{\RR}\CC)$, $\gamma_{*}\gamma^{*}\tau_{\CC}$ is in general strictly smaller than  $\tau_{\CC}$: $B$ torsion does not implies $B\oplus\bar B$ torsion.

\begin{definition} Let $\tau_{\CC}=(\T_{\tau_{\CC}},\F_{\tau_{\CC}})$ be a torsion theory on $\Mod(R\otimes_{\RR}\CC)$ with torsion functor $T_{\tau_{\CC}}$ (\ref{torsion functor}). The following properties are equivalent:
\item[i)] $\gamma_{*}\gamma^{*}\tau_{\CC}=\tau_{\CC}$.
\item[ii)] There exists a torsion theory $\tau$ on $\Mod(R)$ such that $\tau_{\CC}=\gamma_{*}\tau$.
\item[iii)] $\T_{\tau_{\CC}}$ is stable by conjugation.
\item[iv)] $\forall B\in \Mod(R\otimes_{\RR}\CC)$, $T_{\tau_{\CC}}(\bar B)=\Adh{T_{\tau_{\CC}}(B)}$.

\item[ ]
A (hereditary) torsion theory on $\Mod(R\otimes_{\RR}\CC)$ is real if it satisfies one of the above properties.
\end{definition}
\begin{example}Let $\tau_{\CC}$ be any torsion theory then $\gamma_{*}\gamma^{*}\tau_{\CC}$ is a real torsion theory.
\end{example}

\newpage
\begin{corollary}\item[1)]Let $(\tau,\tau_{\CC},\tau')$ be a torsion theories on $(R,R\otimes\CC,\,R)$ such that $  \tau\leq\gamma^{*}\tau_{\CC}$ and $\tau_{\CC}\leq \gamma_{*}\tau'$ (iff $ \T_{\tau}\otimes\CC\subset\T_{\tau_{\CC}}$ and $[\T_{\tau_{\CC}}]_{R}\subset \T_{\tau'}$). There exists exact functors
\begin{eqnarray*}
.\otimes\CC:\Mod(R)_{/\tau}\to \Mod(R\otimes\CC)_{/\tau_{\CC}}& &
(.)_{R}:\Mod(R\otimes\CC)_{/\tau_{\CC}}\to \Mod(R)_{\tau'}
\end{eqnarray*}
such that the following diagrams

$$
\begin{diagram}\node{\Mod(R)}\arrow{e,t}{.\otimes\CC}\arrow{s,l}{}\node{\Mod(R\otimes\CC)}\arrow{s,l}{}    \node{\Mod(R\otimes\CC)}\arrow{e,t}{(.)_{R}}\arrow{s,l}{}\node{\Mod(R)}\arrow{s,l}{}\\
\node{\Mod(R)_{/ \tau}}\arrow{e,t}{.\otimes\CC}\node{\Mod(R\otimes\CC) _{/ \tau_{\CC}}}  \node{\Mod(R\otimes\CC)_{/ \tau_{\CC}}}\arrow{e,t}{(.)_{R}}\node{\Mod(R) _{/ \tau'} }
\end{diagram}
$$
commute up to an equivalence of functors.
\item[2)] In particular if $\tau=\tau'$ and $\tau_{\CC}=\gamma_{*}\tau$, then $(.\otimes\CC,\, (.)_{R})$ are adjoint functors.
\item[3)] Let $(\tau',\gamma_{*}\tau')$ be a torsion theories on $(R,R\otimes\CC)$ such that $\tau'\leq\tau$. Then there exists an exact functor from
$(\Mod(R)_{/\tau'},\Mod(\CC\otimes R)_{/\gamma_{*}\tau'})$ to  $(\Mod(R)_{/\tau},\Mod(\CC\otimes R)_{/\gamma_{*}\tau})$ wich commute to tensor product up to an equivalence.
\end{corollary}
\begin{proof}Kernel of $\Mod(R)\overset{\CC\otimes .}{\to} \Mod(\CC\otimes_{\RR}R) \to\Mod(\CC\otimes R)_{/\tau_{\CC}}$ is spanned by module $M$ such that $\CC\otimes M$ is $\tau_{\CC}$ torsion wich follows from $M$ being $\tau$ torsion. One conclude from  \Gab~ p. 368 Cor. 2 and Cor. 3 or \Fai 15.9.
Other assertion are proved in the same way.

\end{proof}
\begin{notations}Let $\tau$ be a (hereditary) torsion theory on $\Mod(R)$. Then $\tau\otimes\CC:=\gamma_{*}\tau$ is the real torsion theory on $\Mod(R\otimes_{\RR}\CC)$ associated to $\tau$.
\end{notations}

\begin{definition}Defines a category $\R$ of modules with real structure mod$\tau$:
\item[]Objects are pair $(B,\alpha)$, $B\in \Mod(\CC\otimes R)_{\tau\otimes\CC}$ with $\alpha:B\to \CC\otimes A$ an isomorphism in $\Mod(\CC\otimes R)_{\tau\otimes\CC}$.
\item[]  A morphism $f:(B,\alpha)\to (B',\alpha')$ is a map $f:B\to B'$ such that there exists $g:A\to A'$ with $1_{\CC}\otimes g\circ \alpha=\alpha' \circ f$.
\end{definition}

\begin{lemma}  Then $\R$ is abelian.
\end{lemma}

\subsection{Examples of torsion theories}

\subsubsection{}
From \Dic section 3, if $\C$ is a class of modules defines
\begin{eqnarray}L(\C):=\{B\in\Mod(R):\, \forall C\in\C,\,\, Hom_{R}(B,C)=0\},\\
R(\C):=\{B\in\Mod(R):\, \forall C\in\C,\,\, Hom_{R}(C,B)=0\}.
\end{eqnarray}
Then torsion theory generated by $\tau_{\C}$ is given by $\T_{\C}=LR(\C)$.
\subsubsection{}
A multiplicative system $S$ is a subset in $R$ stable by multiplication. Then $\T_{S}=\{M\in \Mod(R):\, \forall m\in M,\, \exists s\in S \text{ s.t.} sm=0\}$ defines a Serre class.
We are interested in torsion theory generated by $A=\oplus_{1\leq i\leq j} \frac{\Adh{Im f_{i}}}{Im f_{i}}$ with $f_{i}$ a bounded morphism between $N(G)-$Hilbert module. If $A\in Mod(R)$, let $$S=\{r\in N(G) \text{with dense range}:\, \exists a\in A\text{ with } ra=0\}\subset\cup_{a\in A}(0:a)\,.$$ Then properties of almost $G-$dense subset (see \Shu or \cite{MurVon} Chap. XVI) imply that $S$ is a multiplicative system.
A $N(G)-$module $M$ is $\tau_{S}=(\T_{S},\F_{S})$ torsion iff $\forall m\in M,\, \exists s\in S$ s.t. $sm=0$.  Note that intersection of multiplicative systems is a multiplicative system.

\subsubsection{}

\label{definition of dim tor}

According to \Luc~Chap. 6, there is a dimension function $\dimG:\Mod(N(G))\to [0,+\infty]$ defined on $N(G)-$modules wich is additive on short exact sequences and wich coincides with Von Neumann dimension for $N(G)-$Hilbert modules.

Therefore $\T_{\dim}=\{M\in\Mod(N(G)):\, \dimG M=0\}$ is a Serre class and defines a torsion theory $\tau_{dim}$ on $N(G)$.
This torsion theory is real for a $N(G)-$module $P$ is projective finitely generated iff $\bar P$ is (see \ref{definition of dimension}).

Standard exemples of $\tau_{dim}$ torsion module  are given in \ref{example of torsion modules}. Modules of the shape $\frac{\bar A}{A}$ with $A$ a $G-$invariant subspace of a {\it finite } $G-$dimensional $G-$Hilbert modules are zero dimensional. This follows from the normality of the dimension function (see a proof in \ref{example of torsion modules}).
\subsubsection{}

Algebra $\U(G)$ of affiliated operator to $N(G)$ is studied in \MurVon, \Luc~ Chap. 8. and  \Rei.

Elements $f\in \U(G)$ is a $G-$equivariant unbounded operator $f: dom(f)\subset l^2(G)\to l^2(G)$. Let $M\in\Mod(N(G))$. Defines $T_{\U}(M):=Ker (M\to \U(G)\otimes_{N(G)}M)$. This defines a torsion class $\T_{\U}$ and torsion theory $\tau_{\U}$.

An element $m\in M$ belongs to $T_{\U}(M)$ iff there exists an $r\in N(G)$ wich is a non right zero divisor (iff $r$ is a weak isomorphism) such that $rm=0$. Indeed $r$ becomes inversible in $\U(G)$.

According to \Vas p.9, a module $F\in \Mod(N(G))$ is $\tau_{\U}-$torsion free iff $F$ is flat.

This torsion theory is real for $F$ is flat iff $\bar F$ is.

Exemples of $\tau_{\U}$ torsion modules  are given in \ref{example of torsion modules}.

\subsection{Direct $L^2$ image}

Let $p: \tilde X\to X$ be a covering map between complex manifolds. Let $G$ be the group  of deck transformations.

Let $N(G):=N(G,\CC)$ be the its (left) von Neumann algebra, and let $\N(G)$ be the sheaf of rings it defines.
According to \CamDem, if $\F$ is a coherent analytic sheaf on $X$, there exists a sheaf
$\p\F$ called direct $L^2$ image such that $\p(.)$ is an exact functor on the
category of coherent analytic sheaves to the category of sheaves (\CamDem
prop2.6). We change from notation of \CamDem: our $\p\F$ is written there $\p
\tilde\F$.

\CamDem Cor 2.7 proves
\begin{lemma}\label{local isomorphism}For any analytic coherent sheaf $\F$,  morphism  $\p\O\otimes\F\to \p\F$ is an isomorphism.
\end{lemma}

This isomorphism defines on  $\p{\F}$  a structure of $\N(G)-$ sheaf compatible
with natural structure of $\ZZ[G]-$sheaf.
\begin{definition} Let $K$ be a subring of $\CC$. Let $\p K$  be the sheaf
defined by the presheaf $$U\to \{ f\in L^{2}(\pi^{-1}(U, K) \text{and locally
constant}\}$$
\end{definition}

Then $\p(\RR)$ is a $\N(G,\RR)-$module and $\p({\CC})$ is a $\N(G,
\CC)-$module.

\begin{lemma}
Let $D:E\to F$ be a differential operator with holomorphic coefficiants between
holomorphic vector bundles. Then there exists an operator $\p D:\p E\to \p F$
\end{lemma}

\begin{proof}In local trivialisation $\O^n\simeq E$, $\O^m\simeq F$, $D$ is given
has $\sum_{|I|\leq r}a_{I}\partial_{I}$ with $a_{I}$ holomorphic function. $\p E$, and
$\p F$ are $\O$ modules, hence  It suffice to study $D=\partial_{I}$. But the
claim is then a consequence of Cauchy inequalities.
\end{proof}

\begin{lemma}\label{squareholomorphicpoincare}$$0\to\p\un{\CC}\to\p\O\overset
{d}{\to}\p\Omega^1\ldots\overset{d}{\to}\p\Omega^n\to 0$$ is well defined and
exact.
\end{lemma}

Question:
It seems natural to extends functor direct $L^2-$images to an exact functor from category of $D_{X}$coherent modules to category of $\N(G)-$sheaves. So that above lemma and proof of prop. \ref{filteredquasiisomorphism} will be simple consequences.

\section{Hodge to de rham spectral sequence.}

Let $p:\tilde{X}\to X$ be a $G-$cover.
Fix a hermitian (later K\"{a}hler) metric on $X$, takes pullback metric on $\tilde X$.
Let $\Delta$ be the maximal extension of the Hodge laplacian $dd^*+d^*d$ on
complex forms. Then $A=(1+\Delta)^{\frac{1}{2}}$ is a positive injective operator
with dense domain.
Let ${\A'}^{p,q}$ be the sheaf of currents of bidegree $(p,q)$, and let $\A'^{k}$ be the sheaf of current of degree $k$.
Let $s\in\RR$. Define Sobolev space $S^s(\tilde X)$ to be space of current $\alpha
$ such that $A^s\alpha\in L^2(\tilde X)$
 Let $U$ an open subset in $X$.
Define uniform (with respect to $p)$ local Sobolev space  $\p S^{s}_{loc}(U)=\{\alpha\in{\A'}^{.}(p^{-1}(U)),\,\forall \theta\in\C^
{\infty}_{c}(U),\,  (\theta\circ p) \alpha\in S^s(\tilde X)\}$

Let $\p\S^{j}\A^{p,q}$  be the sheaf associated to the presheaf
$U\to \p S^{j}_{loc}(\pi^{-1}(U))\cap {\A'}^{p,q}(\pi^{-1}(U))$ and set $\p\S^{j}\A^{n}=\oplus_{p+q=n}\p\S^{j}\A^{p,q}$, $\p\S^{+\infty}\A^{.}=\cap_{n\in\NN}\p\S^{n}\A^{.}$.

Let $D:\C^{\infty}(X,E)\to \C^{\infty}(X,E')$ be a differential operator on $X$, acting
on hermitian vector bundles.
Let $\E_{(2)}\cap Dom D$ be  the sheaf generated by the presheaf $U\to L^{2}(U,E)
\cap Dom D$ of square summable section $\alpha$ such that $D\alpha$ is square
summable.

Also $\p(\E \cap Dom d)$ is  the sheaf generated by the presheaf $U\to [ \E_
{(2)}\cap Dom D](p^{-1}(U))$.

Note that these are sheaves of $\N(G,\CC)-$modules.

\begin{lemma}\item[1)] $(\p\Omega^{*},d)$ is a $\N(G,\CC)-$resolution of $\p\CC$.
\item[2)]Let $k\in\NN\cup\{+\infty\}$, then $ (\p\S^{k-*}\A^{*}_{\RR},d)$ is a $\N(G,\RR)-$resolution of $\p\RR$ and $ (\p\S^{k-*}\A^{*}_{\CC},d)$ is a $\N(G,\CC)-$resolution of $\p\CC$.
\item[3)]  $\p\S^{j}\A^{p,q}\simeq \p\S^j\A^{0}\otimes_{\A}\A^{p,q}$ is a
fine $\N(G)$ sheaf.
\end{lemma}

\begin{lemma}\label{holomorphic to de rham} Let $k\in\NN$, $k\geq n$. Let $F$ be the Hodge filtration.

\item[1)] Morphism $(\p\Omega^{*},\partial,F)\to (\p\S^{k-*}\A^{*},d,F)$ is a filtered quasi-isomorphism.
 \item[2)]  Following complex is exact in $\N(G,\CC)$: $$0\to\p\Omega^{p}\overset{i}{\to}\p\S^k\A^{p,0}\overset{\dbar}{\to}
\p\S^{k-1}\A^{p,1}\overset{\dbar}{\to}\ldots \p\S^{k-n}\A^{p,n}\to 0\,.$$
\end{lemma}
\begin{proof}

One filters complex $(\p\A^{*}\cap Dom d,d)$ with the Hodge filtration. Then \\ $(\p\S^{k-*}\A^{*},d,F)\to (\p\A^{*}\cap Dom d,F,d)$ is a filtered
morphism.

 $E_{1}-$term of the spectral sequence of this complex is  $\frac{\p\A^{p,q}
\cap Dom d\cap\ker \dbar}{\dbar(\p\A^{p,q-1}\cap Dom d)}\,.$

  Let $f$ be a  $(p,q)-$form ($q\geq 1$) on a strictly pseudoconvex domain $U_{1}
\subset\subset \CC^n$ wich is in  $Dom d\cap Ker \dbar$. Then $f\in Dom_
{\dbar}$.  From \Tak,  $f=\dbar\dbar^*Nf+\dbar^*\dbar Nf$ the last term is vanishing
for $\dbar N=N\dbar$ on $Dom_{\dbar}$. Hence $f=\dbar(\dbar^*Nf)$ and  $
(\dbar^*Nf)$ is in $H^{s+1}_{loc}(U_{1})$ (usual Sobolev space in euclidian
space) if $f$ is in $H^s_{loc}(U_{1})$ .

 Let $U$ be an open charts of $X$ biholomorphic to some ball $B(0,2)$ in $\CC^n
$.
 Then $\pi^{-1}(U)\simeq U\times G$, let $B(0,1)\simeq U_{1}\subset\subset U$.
Then any $\alpha\in (\p\A^{p,q}
\cap Dom d\cap\ker \dbar)(U)$ ($q\geq1$)  defines   $(\dbar^*N\alpha_{| U_{1}
\times\{g\}})_{g\in G} \in [L^2(\pi^{-1}(U_{1})\cap Dom\dbar]\cap \p\S^{s+1}\A^{p,q-1}(U_{1})$ if $\alpha\in \p\S^{s}A^{p,q}_{2}(U)$.
This proves that   $$0\to\p\Omega^{p}\overset{i}{\to}\p(\A^{p,0}\cap Dom d)\overset
{\dbar}{\to}\p(\A^{p,1}\cap Dom d)\overset{\dbar}{\to}\ldots \p(\A^{p,n}\cap Dom d)\to 0$$
and
  $$0\to\p\Omega^{p}\overset{i}{\to}\p\S^k\A^{p,0}\overset{\dbar}{\to}\p\S^{k-1}\A^{p,
1}\overset{\dbar}{\to}\ldots \p\S^{k-n}\A^{p,n}\to 0$$ are exacts. Hence $(\p\S^{k-*}\A^{*},d,F)\to (\p\A^{*}\cap Dom d,F,d)$ is a filtered quasi isomorphism.
\end{proof}

\subsubsection{}

Defines $d_{p}: S^{k-p}(\tilde X)\to  S^{k-p-1,p+1}(\tilde X)$ where $S^{j,p}(\tilde X)=\Gamma(\tilde X,\Lambda^{p})\cap S^{j}(\tilde X)$ with Sobolev norm $||\alpha||_{j}=||A^j\alpha||_{L^2}$.

Defines $\dbar_{q}: S^{k-q,(p,q)}(\tilde X)\to  S^{k-q-1,(p,q+1)}(\tilde X)$ where $S^{j,(p,q)}(\tilde X)=\Gamma(\tilde X,\Lambda^{p,q})\cap S^
{j}(\tilde X)$ with Sobolev norm $||\alpha||_{j}=||A^j\alpha||_{L^2}$.

 Then $A^k$ is an
isometric isomorphism from $S^{j,.}(\tilde X)\to S^{j-k,.}(\tilde X)$ and
operator $\dbar$ and $A$ commute if the metric is K\"{a}hler .

\begin{definition}[\Shu prop 1.13, \Luc chap. 1] \item[i)]
Recall that a $G-$equivariant bounded operator $f:H_{1}\to H_{2}$ is $G-$Fredholm if $\dimG Ker f^*<+\infty$ and if $f^*f=\int \lambda dE_{\lambda}$ is the spectral decomposition of $f^*f$, then there exists $\lambda>0$ such that $Tr_{G}E_{\lambda}=\dimG Im E_{\lambda}<+\infty$.
\item[ii)] A bounded complex $(L^.,d^.)$ of Hilbert $G-$modules is $G-$Fredholm if $\oplus_{i}d_{i}:\oplus_{i}L_{i}\to \oplus_{i}L_{i}$ is $G-$Fredholm.
 \end{definition}

\begin{lemma}[\Ati, see also \Shu, \Luc]

The complexes of Hilbert modules
\begin{eqnarray}
(\Gamma(\tilde X,\Lambda^{p,q})\cap S^{k-q}
(\tilde X),\dbar) &\text{ and }&(\Gamma(\tilde X,\Lambda^{p})\cap S^{k-p}(\tilde X),d)
\end{eqnarray}
are $G-$Fredholm.
\end{lemma}

\begin{corollary}\label{example of torsion modules}

\item[1)] $\frac{ \Adh{ Im \dbar_{ q} }^{ S^{k-q-1,(p,q+1)}(\tilde X)}  }{Im \dbar_
{ q} }$ is  a $\dimG $ torsion module.
\item[2)] $\forall x\in \Adh{Im\dbar_{q}}$, $\exists r\in N(G)$ such that $ker (r)=0$ and $rx\in Im\dbar_{q}$, so that $\U(G)\otimes_{N(G)} \frac{\Adh{Im\dbar_{q}}}{Im\dbar_{q}}=0$.
\end{corollary}

\begin{proof}  \item[1)] Lemma 2.12 of \Shu implies that $T=\dbar_{q}: H_{1}= \Adh{Im\dbar^{*}_{q
+1}}\to H_{2}=\Adh{Im \dbar_{q}}$ is $G-$Fredholm, hence  lemma 1.15 of \Shu implies
that $Im T$ is $G-$dense in its closure: $\forall\epsilon >0$, there exists $L_{\epsilon}\subset Im T$, a closed $G-$invariant subspace, such that $\\dimG \Adh{Im T}^{H_{2}}\ominus L_{\epsilon}\leq \epsilon$. In this example, one may take $L_{\epsilon}:= Im(\dbar\circ 1_{[\eta_{\epsilon},+\infty[}(\Delta_{\dbar}))$ (fonctional calculus) with $\eta_{\epsilon}>0$ small enough.

 This is equivalent to
$\dimG ImT=\dimG \Adh{ImT}$ hence $\dimG \frac{\Adh{Im
T}}{Im T}=0$.
\item[2)] Applies the $T-$lemma \ref{T-lemma}.
\end{proof}

Analogue proof holds for elliptic  morphism between vector bundles:
\begin{theorem} Let $(E^{.},d)$ be an elliptic complex ( $d$ a differential operator of order one) between vector bundles. Let $(S^{j-.}(\tilde X,p^{*}(E^.),d)$ be the associated Sobolev complex on $p: \tilde X\to X$.
\item[1)]  The complex $(S^{j-.}(\tilde X,p^{*}(E^.),d)$ is $G-$Fredholm
\item[2)] Module $\frac{\Adh{Im d}}{Im d}$ is of $G-$dimension zero and $\U(G)\otimes_{N(G)} \frac{\Adh{Im d}}{Im d}=0$.
\end{theorem}

This implies that if $d:L^2(\tilde X,E)\to L^2(\tilde X, F)$ acts as unbounded elliptic operator, then $\U(G)\otimes\frac{\Adh{Im d}}{Im d}=0$: One uses conjugation by $(1+d^*d)^{-1}$ and $(1+dd^{*})^{-1}$. Example of  current in negative Sobolev scale (e.g. Dirac mesure on a point) is also instructive.

\begin{corollary}[\Dod] Combinatorial reduced $l^2-$cohomology and analytical reduced \\ $L^2-$cohomology are isomorphic in $N(G)_{\tau_{dim}}$.
\end{corollary}

\begin{proof} Let $(C_{.},\delta)$ be a simplicial structure on $X$: $C_{k}$ is a collection of closed differentiable \\ $k-$dimensional simplices and $\delta$ is  simplicial boundary.

 If $\A$ is a sheaf on $X$, let $(C^{.}(\A),\delta)$ be the differential sheaf defines by $U\to \Pi_{S\in C_{.}}\A(S\cap U)$ (\God II.5.2).
  Then  $(C^{.}(\p\CC),\delta)$ is a resolution of $\p\CC$ and \\
  $(C^{.}(\p\CC),\delta)\to (C^{.}(\p\S^{k-.}\A^{.}_{\CC}),\delta+d)$ is a \qi of $\N(G)-$sheaves.  Each term in theses complexes are $\Gamma(X,.)-$acyclic, sheaves  $\S^{k-.}\A^{.}_{\CC}$ are fine. Hence (\God II.5.2):
  $$H^{k}(X,\p\CC)\simeq H^{k}_{\delta}(\Gamma(X,C^{.}(\p\CC)))\simeq H^{k}_{d(2)}(\tilde X)\,.$$
   Let $(\tilde C_{.},\delta)$ be the pullback simplicial structure. Then $$ H^{k}_{\delta}(\Gamma(X,C^{.}(\p\CC)))\simeq_{N(G)} H^{k}(Hom_{\CC[G]}(\tilde C_{.},l^{2}(G)))\,.$$ Reduction with respect to torsion dimension $\tau_{dim}$ gives result for combinatorial (see \ref{definition of dim tor}) or analytical (see above) modules $\frac{\Adh{Im d}}{Im d}$ are of vanishing dimension.
\end{proof}

Let $\H_{d(2)}^{.}$ be space of square integrable $\Delta_{d}-$harmonic forms and $ \H^{p,q}_{\dbar(2)}$ be  space of square integrable $\Delta_{\dbar}-$ harmonic $(p,q)-$forms. Let $(\H^{.}_{d(2)},F)$ be the complex with trivial differential, and Hodge filtration.
Then $E_{0}^{p,q}=\H^{p,q}_{ \dbar (2)}=E_{r}^{p,q}$, $\forall r\geq 0$, for the metric is K\"{a}hler.

\begin{lemma}

Assume metric is K\"{a}hler. Let $ \H^{p,q}_{\dbar(2)}$ be the space of square integrable harmonic $(p,q)-$forms.
\item[1)]Let $\tau$ be a torsion theory such that $\H^{p,q}_{\dbar (2)}\to H^{p,q}_{\dbar (2)}$ is an isomorphism in $\Mod(N(G,\CC))_{/\tau}$.
\item[] Then:
\item[i)]
Spectral sequence for $(\Gamma(\tilde X,\Lambda^{*})\cap S^
{k-*}(\tilde X),d,F)$ in the category  $\Mod(N(G,\CC))_{/\tau}$  degenerates. Differential $d$ is strictly compatible with $F$ in $\Mod(N(G,\CC))_{/\tau}$.
\item[ii)] $E^
{p,q}_{1}\simeq E^{p,q}_{\infty}\simeq \H^{p,q}_{\dbar (2)}$ in $N(G,\CC)_{/\tau}$.
\item[2)] Dimension torsion theory fullfilled assumption of $1)$.
\end{lemma}

\begin{proof} $F^{p}\Gamma(\tilde X,\Lambda^{r})\cap S^{k-r}(\tilde X)=\oplus_
{p'+q=r,\, p'\geq p}\Gamma(\tilde X,\Lambda^{p',q})\cap S^{k-(p'+q)}(\tilde X)$

Hence, $$(E^{p,q}_{0},d_{0})\simeq_{N(G)} (\Gamma(\tilde X,\Lambda^{p,q})\cap S^
{k-p-q}(\tilde X),\dbar)$$ and $$(E_{1}^{p,q},d_{1})\simeq_{N(G,\CC)} \frac{ Ker \dbar_
{q}}{Im \dbar_{q}}\simeq_{N(G,\CC)}\frac{\H^{p,q}_{\dbar}\oplus \Adh{ Im\dbar_{q}} }{Im \dbar_
{q}}\simeq_{N(G,\CC)} \H^{p,q}_{\dbar (2)}\oplus \frac{\Adh{ Im\dbar_{q}} }{Im \dbar_
{q}}\,,$$

from the Kodaira decomposition.

Then $i:(\H^{.}_{d(2)},F)\to  (\Gamma
(\tilde X,\Lambda^{*})\cap S^{k-*}(\tilde X),d,F)$ is a morphism of filtered $N(G)-
$module.
Let $\tau$ be a torsion theory on $N(G,\CC)$ such that
 $E_{1}(i)$ is an
isomorphism in $N(G)_{/\tau}$. Then $E_{r}(i)$ are isomorphisms for any $r\geq 1$. But spectral sequence of $(\H^{.}_{d(2)},F)$ degenerates so that spectral sequence for $(\Gamma(\tilde X,\Lambda^{*})\cap S^
{k-*}(\tilde X),d,F)$ degenerates at $E_{1}$  in $N(G,\CC)_{/\tau}$.

 \end{proof}
 \begin{definition}\label{Dolbeault torsion theory} Torsion theory  generated by $\C=\{Coker(\H^{p,q}_{\dbar (2)}\to H^{p,q}_{\dbar (2)}),\,\, p,q\geq 0\,\}$ is the smallest torsion theory on $\Mod(N(G))$ wich satisfies above lemma. Let $\tau_{\dbar}$ (or $\tau_{\dbar,\tilde X}$) be the torsion theory it defines on $\Mod(N(G))$.
\end{definition}

\begin{theorem}\label{structurepure}Let $X$ be a K\"{a}hlerian manifold and $p:\tilde X\to X$ be a $G-$covering.
Let $k\geq n$.
\item[1)\,(i)]$((\p\Omega^{.},d),\, F)\to ((\p\S^{k-.}\A^{.},d),\,F)$ is a filtered \qi of $\N(G)-$sheaves such that $Gr_{F}\p\S^{k-.}\A^{.}$ is $\Gamma-$acyclic.
\item[ii)] $(\p\Omega^{.},d) \to (\p\S^{k-.}\A^{.},d)$ is a \qi
of $\N(G)-$sheaves.

This last quasi isomorphism defines a real structure on $\HH_{\N(G)}(X,(\p\Omega^{.},d))$ compatible with real structure given by Godement resolution and
pseudo isomorphism $\C^{.}\p\CC\dto (\p\Omega^{.},d) $.

\item[2)] Let $\tau\geq \tau_{\dbar}$ be a torsion theory on $\Mod(N(G))$.
 Fr\"{o}licher spectral sequence $$H^q(X,\p\Omega^p)\abupt \HH^{p+q}(X,\p\Omega^*)\simeq_{N(G,\CC)} H^{p+q}(X,\p\CC)\,,$$ wich is isomorphic to $$H^{p,q}_{\dbar(2)}(\tilde X)\abupt H^{p+q}_{d(2)}(\tilde X)\,,$$ degenerates in $\Mod(N(G))_{/\tau}$ so that
 $d$ is strict for $F$ in $\Mod(N(G))_{/\tau}$: $$Gr_{F}^{p}H^{p+q}(X,\p\CC)\simeq_{\Mod(N(G,\CC))_{/\tau}}H^{q}(X,\p\Omega^{p})$$

\item[3)] Assume that $\tau$ is real. Then Hodge filtration on hypercohomology\\  $F^{.}\HH(\p\Omega^*):=\im(\HH(F^.\p\Omega^*)\to \HH(\p\Omega^*))$  and its complex conjugate $\bar F$ is
$k-$opposed on $\HH^{k}(\p\Omega^{.})$ in $\Mod(N(G,\CC))_{/\tau}$.
 It defines a pure Hodge structure of
weight $k$ on  $\HH^{k}(\p\RR)$ in $\Mod(N(G))_{/\tau}$.
(see definition \ref{definition of MHS}).

\end{theorem}

\begin{proof}\item[1)\, i)] Filtered \qi was proved in lemma \ref{holomorphic to de
rham}.
\item[ii)]
Second quasi-isomorphism is a consequence of $\p\CC\to (\p\Omega^{.},d)$ and $\p\CC\to (\p\S^{k-.}\A^{.},d)$ are resolutions.
But $(\A^{.}_{\CC},d)=(\A^{.}_{\RR},d)\otimes_{\ZZ}\CC$ and the following diagram is commutative: $$
\begin{diagram}\node{\p\RR}\arrow{e,t}{}\arrow{s,l}{}\node{(\p\Omega^{.},d)}\arrow{s,l}{}\\
\node{(\p\S^{k-.}\cap\A^{.}_{\RR},d)}\arrow{e,t}{}\node{(\p\S^{k-.}\A^{.}_{\CC},d)}
\end{diagram}
$$
 \item[(2)] These sheaves are $\Gamma(X,.)-$acyclic, and $\Gamma(X,\p\S^j\A^{.})= S^{j,.}(\tilde X)$
 for $X$ is compact.

Hence $\HH^{.}(X,\p\Omega^{.})\simeq_{\N(G)} H^{.}(S^{k-.,.}(\tilde X),d)$.

Note that $\p\S^j\A^{.}$ is such that $\p\S^j\A^{p,q}\tilde{\to}Gr_{F}^p(\p\S^j\A^{p
+q})$ is $\Gamma(X,.)-$acyclic.
Hence spectral sequence for $(\HH^{.}(\p\Omega^{.}),\sigma)$ is (isomorphic) given
by spectral sequence of $N(G)-$modules
$$H^{p+q}(S^{k-.,.}(\tilde X),d)\Leftarrow E_{1}^
{p,q}=H^{q}(S^{k-.-p,(p,.)},\dbar)\,.$$

It degenerates in $\Mod(N(G,\CC))_{/\tau}$.

\end{proof}

\begin{definition}\label{Hodgecomplex}Let $R$ be a ring wich is also an $\RR-$algebra. Let $\tau'$ and $\tau$ be torsion theory on $\Mod(R)$ such that $\tau'$ is smaller than $\tau$.
\item[A] A $(\tau',\tau)$ Hodge complex  $(K_{R},(K_{R\otimes \CC},F))$ of weight $m$
 consists in
\item[1)] A bounded below complex  of modules $K_{R}$ in $\Mod( R)_{\tau'}$
\item[2)] A bounded below filtered complex of module $(K_{R
\otimes\CC},F)$ in  $\Mod(R\otimes\CC)_{\tau'\otimes\CC}$.
\item[3)] A pseudomorphism of bdd below complex $\alpha:K_{R}
\dto K_{R\otimes\CC}$ (first comparison morphism)  in $\Mod(R)_{\tau'}$ such that $\alpha\otimes Id:K_
{R}\otimes\CC\dto K_{R\otimes \CC}$ is a pseudo-isomorphism.

Isomorphism $H(K_{R})
\otimes\CC\tilde\to H(K_{R\otimes\CC})$ (in $\Mod(R\otimes\CC)_{\tau'\otimes\CC'}$) defines a real structure on $H(K_{R\otimes
\CC})$.
\item[]
One requires that
\item[1)] $d$ is strictly compatible with $F$ in $\Mod(\R\otimes\CC)_{/\tau\otimes\CC}$.
\item[2)]  $F$ and $\bar F$ are $m+k-$opposed on $H^k(K_{R\otimes\CC})\simeq H^
{k}(K_{R})\otimes\CC$  in $\Mod(R\otimes\CC)_{\tau\otimes\CC}$.

\end{definition}
Note that category $\Mod(R)_{/\tau}$ is abelian and admits inductive limits (\Gab~ prop. 9 p. 378). Hence, one may consider the category $\M(R)_{/\tau'}$ of sheaf with value in $\Mod(R)_{/\tau'}$. Then functor localisation $\Mod(R)\to \Mod(R)_{/\tau}$ commute to inductive limits.

\begin{definition}\label{definition of CHC}
\item[B] On a topological space $X$, a $(\tau',\tau)-$cohomological Hodge complex $(\K_{R},(\K_{R\otimes \CC},F))$ consists in
\item[1)] A bounded
below complex  $\K_{R}$ in $\M(R)_{/\tau'}$
\item[2)] A bounded below filtered complex  $(\K_{R\otimes\CC},F)$ in $\M(R\otimes\CC)_{/\tau'\otimes\CC}$.
\item[3)] A pseudo-morphism of bdd below complexes $
\alpha:\K_{R}\dto \K_{R\otimes\CC}$ (first comparison morphism) in $\M(R)_{/\tau'} $
such that $\alpha\otimes Id:\K_{R}\otimes\CC\dto \K_{R\otimes \CC}$ is a pseudo-isomorphism in $\M(R\otimes\CC)_{\tau'\otimes\CC}$ and
\item[] $R\Gamma(\K)$ is a $(\tau',\tau)-$Hodge complex.
\end{definition}
\begin{notations}\item[C] If $\tau'=(0,\Mod(R))$  is the trivial torsion theory, one says that
 $(\K_{R},(\K_{R\otimes \CC},F))$ is a \\ $\tau-$cohomological Hodge complex
of sheaves.
\end{notations}
In the following, we will mostly deals with $\tau'=(0,\Mod(R))$ the trivial torsion theory. Hence we will mostly used complexes of sheaves of $N(G)-$modules.

\newpage
\begin{example}\item[1)] \label{structurepure2}We have seen that $(\p\RR,(\p\Omega^{.}),\alpha)$ is a
CHC mod $dim_{\N(G)}$ torsion, or $\gamma^{*}\tau_{\dbar}$ (\ref{ring extension}), with $\alpha:\p\RR\to (\p\Omega^{.})$ natural map
such that $\alpha\otimes 1_{\CC}$ is a \qi. In diagram
$$
\begin{diagram}
\node{R\Gamma(\p\RR)}\arrow{e,t}{R\Gamma(\alpha)}\node{R\Gamma(\p\Omega^
{.},F)}\arrow{e,t}{R\Gamma(i)}\node{R\Gamma(\p \S^{k-*}\A^*,d,F)}\\

\node{}\node{}\node{\Gamma(\p \S^{k-*}\A^*,d,F)}\arrow{n,l}{m}

\end{diagram}
$$
maps $R\Gamma(i)$ (Hodge to de Rham) and $m$ ($\Gamma$acyclic sheaves) are
filtered quasi isomorphism. Therefore if $r\geq 1$, $$E_{r}(R\Gamma(i),F) \circ E_{r}(m,F)^
{-1}:E_{r}(\Gamma(\S^{k-*}\A^*,d,F))\to E_{r}(R\Gamma(\p\Omega^{.},F)$$ defines
isomorphism of spectral sequences. First degenerates mod $dim_{\N(G)}$.
\item[2)] Assume that $\tilde X\to X$ is such that each connected componants $
\tilde X_{i}$ of $\tilde X$ is compact (and $G$ is transitive on fiber). Let $G_{i}$ be
the stabilisator of $\tilde X_{i}$ and let $p_{i}$ be restriction of $p$ to $X_{i}$.

 Then $d,\, \dbar$ have closed range. Indeed $$S^
{k,p}(\tilde X)\ni\alpha\mapsto (g\mapsto g_{*}\alpha_{|\tilde X_{i}})\in l^{2}(G,  S^{k,p}(\tilde X_{i}))^{G_{i}}$$ is a $G-$equivariant isometric isomorphism wich commutes
with $d,\, \dbar$. But taking invariant with respect to $G_{i}$ is exact on $\QQ[G_{i}]-$module, hence $H^{k}_{(2)}(\tilde X)\simeq l^2(G,H^{k}(\tilde X_{i}))^{G_{i}}$. Then cohomology is reduced, and Hodge structure is isomorphic to that of  $\tilde X_{i}$(twisted by that of $l^{2}(G,\CC)$).

In general, if $\F$ is a sheaf for wich $\p\F$ is defined then $L^{2}-$cohomology is separated and
\begin{eqnarray*}
H^{k}_{(2)}(X,\p\F) &\simeq_{N(G)} &l^2(G,H^{k}(X_{i},p_{i}^{*}\F))^{G_{i}}\\ \text{ and }\,\,\, \dimG   & l^2(G,H^{k}(X_{i},p_{i}^{*}\F))^{G_{i}} = &\frac{dim_{\CC}H^{k}(X_{i},p_{i}^{*}\F)}{|G_{i}|}\,.
\end{eqnarray*}

\end{example}
\subsection{Addendum: Smoothing of cohomology}
 It is well known that cohomology of current is isomorphic to cohomology of smooth form. Here we insist on uniformity with respect to action of $G$.

\begin{lemma}\label{smoothingofcohomology}

Let $U$ be some open set in $X$. Then $H^{.}(\Gamma(U,\S^{\infty}\A^{.}))\to H^{.}(\Gamma(U,\S^{k-.}\A^{.})$ is an isomorphism. In particular, for any closed form $\alpha$ in $\Gamma(\tilde X,\A^{p})\cap Dom(1+\Delta)^{r}$, there exists $(\beta,\gamma)\in  \Gamma(\tilde X,\S^{r-1}\A^{p-1})\times \Gamma(\tilde X,\S^{\infty}\A^{p})$ such that $\alpha=d\beta+\gamma$
 \end{lemma}
 \begin{proof} Note that $(\p\S^{\infty-.}\A^.,d)\to (\p\S^{k-.}\A^.,d)$ is a quasi isomorphism. Result follows for theses sheaves are flabby.
\end{proof}

\section{mixed Hodge structures.}

In this section, one follows notations of \PetSte Chap. 4:

Let $X$ be a  complex manifold and $D\subset X$ be a normal crossing
divisor. Let $j:U=X\setminus D\to X$ be the injection.
Then $\dlog{.}{D}$ is the $\O_{X}$ subsheaf of $j_{*}\Omega^{.}_{U}$ of
meromorphic form with logarithmic poles on $D$: $\alpha\in \dlog{p}{D}$ if $\alpha
$ and $d\alpha:=j_{*}dj^{*}\alpha$ have pole of order at most one on $D$. If $p\in X
$ is such that in a chart $(z):V\to D(0,1)^n$ one has $D\cap V=\{z_{1}.
\ldots . z_{k}=0\}$  then $\frac{dz_{1}}{z_{1}},\ldots \frac{dz_{k}}{z_{k}},dz_{k+1},
\ldots,dz_{n}$ is a free basis of $\dlog{1}{D}$. Then $\dlog{p}{D}=\wedge^{p}\dlog
{1}{D}$ is a locally free $\O_{X}$ sheaf.

Call $(V,(z))$ a standard chart and let $(\Lambda^*\{ \frac{dz_{1}}{z_{1}},\ldots \frac{dz_{k}}{z_{k}}\})$ be the free $\CC-$antisymetric algebra built on $ \frac{dz_{1}}{z_{1}},\ldots \frac{dz_{k}}{z_{k}}$.

Weight filtration $W$ is defined by
$$W_{k}\dlog{p}{D}=\left\lbrace
\begin{array}{ll}
0 & \text{for } m<0\\
\Omega^{p-m}\wedge \dlog{m}{D} &  \text{for } 0\leq m\leq p\\
\dlog{p}{D} &  \text{for }  p\leq m\\
\end{array}\right. $$

 \subsubsection{Residues}
Let $D=\cup_{i\in T}D_{i}$ be the decomposition of $D$ into irreducible
smooth componants. If $I$ is a subset of $T$, set  $a_{I}:D_{I}=\cap_{i\in I}D_{i}\to X$ , $D(m)=\sqcup_{|I|=l}
D_{I}$ and $a_{m}=\sqcup a_{I}$ ($D(\emptyset)=X$,  $a_{\emptyset}=Id$ ).

Let $ i\in T$. Defines

$$Res_{i}=Res_{D_{i}}:\dlog{.}{D}\to a_{i*}\Omega_{D_
{i}}^{.-1}(Log D_{i}\cap(\sum_{j\not= i}D_{j}))\, .$$ In a coordinate neighborhood $(U,(z))$,  $Res_{D_{i}}(\eta\wedge
\frac{dz_{k_{i}}}{z_{k_{i}}}+\eta')=\eta_{|D_{i}}$ if $\eta$ and $\eta'$ do not contain $\frac
{dz_{k}}{z_{k}}$.
 Then  $Res_{D_{k}}$ is $\O_{X}$ linear, commute with $d$.

Let $I=(i_{1},\ldots,i_{k})$ be an ordered $k$ uplet.
Defines $$Res_{I}=Res_{i_{1}}\circ \ldots \circ Res_{i_{k}}:\dlog{.}{D}\to a_{I*}
\Omega_{D_{I}}^{.-m}(Log D_{I}\cap(\sum_{j\not\in I}D_{j}))\,.$$ Then $Res_{I}$ is $\O_
{X}-$linear, commutes with $d$.
A permutation $\sigma$ of $1,\ldots,k$ defines another $k$uplet $\sigma(I)$. Then
$Res_{\sigma(I)}=(-1)^{sign(\sigma)}Res_{I}$.
Let $\Lambda \{e_{i},\, i\in T \}$ be the free antisymetric algebra on $T$. Let $I\in T^k$, set $\Lambda^{I}e:=e_{i_{1}}\wedge\ldots\wedge e_{i_{k}}$. Let $I,\, J$ be $k-$upplet such that the unordered sets are equals then $$\wedge e^{I}\otimes Res_{I}=\wedge e^{J}\otimes Res_{J}\,.$$ Therefore if $\{I\}$ is a $k-$subset of $\{1,\ldots,N\}$, there is a well defined map $Res_{\{I\}}:=\wedge e^{I}\otimes Res_{I}$.
One sets $Res_{m}=\oplus_{|\{I\}|=m}Res_{\{I\}}$.
 Choice of an ordering on $T$

  gives a trivialisation

$$Res_{m}\simeq \oplus_{i_{1}<\ldots<i_{m}}Res_{(i_{1},\ldots,i_{m})}\,.$$
such that
\begin{eqnarray}
Res_{m}:(Gr^{W}_{m}\dlog{.}{D},d)\to a_{m*}(\Omega^{.-m}_{D(m)},d)
\end{eqnarray}
 is an isomorphism of complexes (\Del 3.1.5.2).

Let $p: \tilde X\to X$ be a covering (not necessarly connected) with group of
deck transformation $G$. If $f:Y\to X$ is a continuous map, let $f^{*}p:\tilde Y\to
Y$ be induced covering.

From lemma \ref{squareholomorphicpoincare}, the following complexes are well
defined.
\begin{proposition}\label{filteredquasiisomorphism}
\item[1)] Maps of filtered complexes

\begin{eqnarray}(\p\dlog{.}{D},W,d)\overset{\alpha}{\leftarrow} (\p\dlog{.}{D},
\tau,d)\overset{\beta}{\to} (j_{*}(j^{*}p)_{*(2)}\Omega^{.}_{U},\tau,d)
\end{eqnarray}
 are filtered quasi isomorphism.
 \item[2)]
 This defines an isomorphism between the Leray spectral sequence for $j_{*}(j^{*}p)_{*(2)}\CC$ and the spectral sequence for the hypercohomology of the filtered complex $(\p\dlog{.}{D},W,d)$
 \item[3)]One deduces $N(G,\CC)-$isomorphisms:  $$\HH(X,\p\dlog{.}{D})\simeq \HH^{.}(X, j_{*}\p\Omega_{X}^{.})\simeq \HH^{.}(X\setminus D,\p\Omega_{X^*}^{.})\simeq H^{.}(X\setminus D,\p\CC)\,.$$
\end{proposition}
\begin{proof}
\item[0)] Corresponding statement without $\p$ is proposition 3.1.8 of \Del. Moreover let $(V,(z))$ be a standard chart for $D$ in $X$ ( so that $D\cap V=\{z_{1}=\ldots z_{k}=0\}$).

Define a residu map
$R_{m}:\Gamma(V\setminus D,\Omega^m)\cap Ker d\to    \CC^{c(m,k)}$ through integration on  $m-$cycles $\{|z_{i_{1}}|=\epsilon_{1},\ldots, |z_{i_{m}}|=\epsilon_{m}\}$, $1\leq i_{1}<\ldots <i_{m}\leq k$.

It is known (see\GriHar) that  $d \Gamma(V\setminus D,\Omega^{m-1})=\Ker (R_{m})$. An explicit (continuous) left inverse $S_{m}$ is construct in \GriHar. It maps logarithmic forms to logarithmics forms.

Fix a trivialisation $p^{-1}(V)\simeq \cup_{g\in G}\, gV_{1}$.

This defines $R_{m(2)}: \Gamma(p^{-1}(V\setminus D),\p\Omega^m)\cap Ker d\to l^{2}(G)^{c(m,k)}$ and $S_{m(2)}:\Ker R_{m(2)}\to  \Gamma(p^{-1}(V\setminus D),\p\Omega^{m-1})$ a left inverse to $d$ wich maps logarithmic forms to logarithmic forms.

Then $l^2(G,\CC)\otimes (\Lambda^m\{ \frac{dz_{1}}{z_{1}},\ldots \frac{dz_{k}}{z_{k}}\})\to H^{m}(\Gamma(V,\p\dlog{.}{D}),d)\to H^{m}(\Gamma(V\setminus D,\p\Omega^{.}),d)$ provides required isomorphism.

\item[1)]
Maps $\alpha$ and $\beta$ are filtered morphisms.
We proved that $p_{2*}\dlog{.}{D},d)\to (j_{*}(j^{*}p)_{*(2)}\Omega^{*}_{U},d)$ is a \qi hence a filtered \qi for the canonical filtration.

 One has $$\p Res_{p}:Gr^{W}_{p}p_{2*}\dlog{.}{D}\simeq a_
{*}\Omega^{.-p}_{\tilde D_{p}}\otimes_{\O}p_{2*}\tilde\O\simeq a_{*}p_{2*}\Omega^{.-
p}_{\tilde D_{p}}$$ for $\p\O$ is a flat $\O-$sheaf, later isomorphism is proved in \CamDem prop. 2.9 .

 These isomorphisms commute with $d$. But $\tilde D_{p}\to D_{p}$ is a covering of a
manifold. Hence  $\H^{i}p_{2*}\Omega^{.-p}_{\tilde D_{p}}=0$ if $i\not=p$  and is
isomorphic (say as $\N(G)-$sheaf) to $\p\CC_{\tilde D_{p}}$ if $i=p$ ( Lemma \ref
{squareholomorphicpoincare}).

 Hence
 Spectral sequence degenerates, one deduce that  $\H^{i}(\p\dlog{}
{D},d))\simeq a_{*}\p\CC_{\tilde D_{p}}$ as $\N(G)-$sheaf and that $\alpha$ is a \qi.

\item[2)]

Note that $(j^{*}(p)_{*(2)}\Omega^{*}_{U},d)$ is a $\N(V)-$resolution of $(j^{*}p)_{*(2)}\CC$
 by $j_{*}$ acyclic sheaves: This follows from \CamDem theorem 3.6

 and $j$ is a Stein morphism. Then $R^{p}j_{*}(j^{*}p)_{*(2)}\CC\simeq_{\N(V)}\H^p(j_{*}(j^{*}p)_
{*(2)}\Omega^{*}_{U},\partial)\,.$

This defines a pseudo- isomorphism $(Rj_{*}(j^{*}p)_{*(2)}\CC,\tau)=(j_{*}\C^{*}(j^{*}p)_{*(2)}\CC,\tau)\dto (p_
{(2)*}\Omega^{*}_{X^*},\partial)$
 wich composed with $E_{r}(\alpha)\circ E_{r}(\beta)^{-1}$ induces an isomorphism of spectral sequences.
 \item[3)] Quasi-isomorphism $(\p\dlog{.}{D},d)\to (j_{*}\p\Omega_{X^*},d)$ gives  $$\HH^{.}(X,\p\dlog{.}{D})\simeq \HH^{.}(X, j_{*}\p\Omega_{X^*}^{.})\simeq \HH^{.}(X\setminus D,\p\Omega_{X^*}^{.})\simeq H^{.}(X\setminus D,\p\CC)$$ as $N(G,\CC)-$modules.

\end{proof}

\begin{definition}\label{normal crossing Dolbeault torsion theory}Let $\tau_{\dbar,\tilde D_{.}}$ be the Serre category generated by $\cup_{I\subset T}\tau_{\dbar,\tilde D_{I}}$ (see \ref{Dolbeault torsion theory})

\end{definition}

\begin{lemma}
\item[1)]
 In the following diagram, $f_{1}$ and $f_{2}$ are quasi-isomorphisms:

$$
\begin{diagram}

\node{(Rj_{*}(j^{*}p)_{*(2)}\RR,\tau)} \arrow{e,t}{i}\node{(Rj_{*}(j^{*}p)_{*(2)}\CC,\tau)} \arrow{e,t}{f_{1}}
\node{(Rj_{*}(j^{*}p)_{*(2)}\Omega^{.}_{U},\tau)} \node{}\\

\node{(\p\dlog{.}{D},W)}
 \node{(\p\dlog{.}{D},\tau)}\arrow{w,t}{\alpha}\arrow{e,t}{\beta}\node{(j_{*}(j^{*}p)_{*(2)}\Omega^
{.}_{U},\tau)}\arrow{n,t}{f_{2} }
\end{diagram}
$$

\item[2)]This defines Second comparison morphism $\tilde\beta :(Rj_{*}\jp\RR,\tau)\dto (\p\dlog
{.}{D},W)$  such that $\tilde\beta\otimes1_{\CC}:(Rj_{*}\jp\CC,\tau)\dto (\p\dlog{.}
{D},W)$ is a pseudo- isomorphism.
\item[3)] Let $\tau\geq \gamma^{*}\tau_{\dbar,\tilde D_{.}}$ be a real torsion theory.

Then $\p\K:=[(Rj_{*}\jp\RR,\tau=W);(\p\dlog{.}{D},W,F),\beta]$ is a mixed Hodge
complex of $\N(G,\RR)-$sheaves: $R\Gamma(Gr^{W}_{p}\K)$ is a Hodge complex of
weight $p$ in $\Mod(N(G,\CC))_{/\tau}$ (see \ref{Hodgecomplex}).
\end{lemma}

\begin{proof}
\item[]  According to \ref{Godement-resolution},
   $$Rj_{*}(\jp\RR)\otimes\CC\overset{i\otimes1_{\CC}}{\to} Rj_{*}(\jp\CC)$$ is a $
\N(G)-$isomorphism. This defines a filtered isomorphism for the canonical
filtration.

From Prop. \ref{filteredquasiisomorphism} and exactness of Godement resolution, $f_{1}$ is a \qi.
Also $f_{2}$ is \qi.
Hence $f_{1},\,f_{2}$ are  filtered \qi. Then $\tilde\beta$ is defined through $$E_{1}(\tilde\beta)=E_{1}(\alpha)E_{1}(f_{2}
\circ \beta)^{-1}E_{1}(f_{1})E_{1}(i) $$ so that $$E_{1}^{-p,q}(\tilde\beta):\H^{p}(j_{*}
\jp\RR)\delta_{p,-p+q}\Rightarrow \H^{-p+q}((Gr^{W}_{p}\dlog{.}{D},d))\, .$$
 Then $\tilde\beta\otimes 1_{\CC}$ is an isomorphism ($E_{r}(\tilde\beta\otimes 1_{\CC})\, r\geq 1$ is an isomorphism).

 \item[3,a)]

 \begin{lemma}The residue morphism $$Res_{p}: Gr^{W}_{p}\p\dlog{.}{D}\to a_
{\tilde D_{p},*}\Omega^{.-p}$$ maps $$R^p{j_{*}\p\RR}[-p]\overset{qi}{\to}Gr^{\tau}_
{p}Rj_{*}\p\RR\overset{ Res_{p}}{\to} a_{\tilde D_{p},*}(2i\pi)^{-p}\p\RR[-p]$$
\end{lemma}
\begin{proof}
See \Del prop.3.1.9.
\end{proof}

\item[3,b)] According to example \ref{structurepure2}  $$\K_{\tilde D_{p}}:=(\p\RR_{\tilde D_{p}}[-p](-p),(\p\Omega_{\tilde D_{p}}[-p],d),\alpha_{\tilde D_{p}})$$ is a $\N(G,\RR)-$Hodge complex of sheaves mod $\tau$. So does its $-p-$th Tate twist (see \PetSte p. 51).

 Hence $(R^p{j_{*}\p\RR}[-p],  Gr^{W}_{p}\p\dlog{.}{D},\alpha_{p}:=(2i\pi)^p\alpha)$ is a cohomological Hodge complex of sheaves mod $\tau$ :
 $R\Gamma(Gr^{W}_{p}\K)\simeq R\Gamma(\K_{\tilde D_{p}}) $ is a Hodge complex.

\end{proof}

\subsection{Mixed Hodge structures}
\begin{definition}\label{definition of MHS}Let $R$ be a ring wich is also an $\RR-$algebra. Let $\tau$ be a (hereditary) torsion theory on $\Mod(R)$.
\item[1)]A mixed Hodge structure $H=(H_{R},W,F)$ in $\Mod(R)_{/\tau}$
is given by

\item[i)] a left $R-$module $H_{R}$
\item[ii)] a filtration $W$ of $H_{R}$ in $\Mod(R)_{/\tau}$
\item[iii)] a filtration $F$ of $\CC\otimes H_{R}$ in $\Mod((\CC\otimes R))_{/\tau\otimes\CC}$ such that $W_{\CC},F,\bar F$ are opposed (see \Del 1.2) in $\Mod((R\otimes\CC))_{/ \tau\otimes\CC}$.

\item[2)] A morphism of mixed Hodge stucture $f:H\to H'$ in $\Mod(R)_{/\tau}$  is a morphism \\$f\in Hom_{\Mod(R)_{/\tau}}(H_{R}, H'_{R})$ such that $f$ is compatible with $W$ and $ f_{\CC}$ is compatible with $F$.
\end{definition}
From \Del theorem 1.2.10, one deduces:
\begin{theorem}\label{MHSabelian} Category of mixed Hodge structure in $\Mod(R)_{/\tau}$ is abelian. A morphism $f:H\to H'$ between MHS in $\Mod(R)_{/\tau}$ is strict for filtrations.
\end{theorem}

In this article, we use only mixed Hodge complexe over $\RR-$algebra. This reflects the use of $l^2(G,\RR)$. Therefore one does not use first comparison morphism.
\begin{definition}(Following \cite{PetSte})
 \item[] Let $R$ be a ring wich is also an $\RR-$algebra. Let $\tau',\tau$ be  torsion theories on $R$ such that $\tau'$ is smaller than $\tau$. \\
MHC mod $(\tau',\, \tau)$:
\item[] A mixed Hodge complex mod$(\tau',\, \tau)$  $((K_{R},W), (K_{R\otimes \CC},W,F))$ consits in

\item[1)] A bounded below filtered complex  $(K_{R},W)$ in $\Mod(R)_{/\tau'}$
\item[2)] A bdd below bifiltered complex $(K_{R\otimes \CC},W,F))$ in $\Mod({R\otimes\CC})_{/\tau'\otimes\CC}$  and a pseudo-morphism $\beta:(K_{R},W)\dto (K_{R\otimes \CC},W)$ (second comparison morphism) in the category of bounded below filtered complex  in $\Mod(R)_{/\tau'}$ inducing a pseudo- isomorphism  $\beta\otimes Id_{\CC}:(K_{R}\otimes\CC,W)\dto (K_{R\otimes \CC},W)$.
\item[3)] such that for each $n$, $(Gr^{W}_{n}(K_{R}),(Gr^{W}_{n}(K_{R\otimes \CC},F))$ with pseudo- morphism
 $$Gr^{W}_{n}(\beta):Gr^{W}_{n}(K_{R})\dto Gr^{W}_{n}(K_{R\otimes\CC})$$ is a Hodge complex mod$(\tau',\,\tau)$ of weight $n$ .

\end{definition}

\begin{example}
 If $\tau'=(0,\Mod(R))$ is the trivial torsion theory (no torsion submodule), then a $(\tau',\tau)-$mixed Hodge complex will be refered as a $\tau-$mixed Hodge complex. Therefore complexes and pseudo-morphisms are data in $\Mod(R)$ and properties of degenerescence are assumed in $\Mod(R)_{/\tau}$.

\end{example}

The following is the  ''lemme des deux filtrations'' (\Del 1.3 and \Deltrois 7.2):
\begin{theorem} Let $((K_{R},W), (K_{R\otimes \CC},W,F))$ be a mixed Hodge complex mod$(\tau',\, \tau)$. Then recurrente filtration and direct filtration induced by $F$ on $E_{r}^{p,q}(K_{R\otimes \CC},W)$ are equal in $\Mod(R\otimes\CC)_{/\tau\otimes\CC}$.

\end{theorem}
\begin{proof}
 We have following data within a torsion theory $\tau'$ smaller than $\tau$:
There exists a left fraction
$$
\begin{diagram}

\node{} \node{'K_{R\otimes\CC}}\node{}\\

\node{(K_{R},W)} \arrow{ne,t}{\beta_{1}}\node{} \node{(K_{R\otimes\CC},W,F)}\arrow{nw,tb}{\beta_{2}}{\tilde{qis}}

\end{diagram}
$$ with $\beta_{1}$ a morphism  in $\Mod(R)_{\tau'}$ and $\beta_{2}$ a morphism in $\Mod(R\otimes_{\RR}\CC)_{\tau'\otimes\CC}$ such that $\beta_{1}\otimes 1_{\CC}$ is a filtered quasi isomorphism.
If, $r\geq 1$, morphisms
\begin{eqnarray}
 E_{r}(\beta_{1}):=E_{r}(K_{R},W)\to E_{r}(K'_{R\otimes\CC},W)\\
E_{r}(\beta_{1}\otimes 1_{\CC}):=E_{r}(K_{R}\otimes \CC,W)\to E_{r}(K'_{R\otimes\CC},W)\\
E_{r}(\beta_{2}):=E_{r}(K_{R\otimes\CC},W)\tilde\to E_{r}(K'_{R\otimes\CC},W)
\end{eqnarray}
define morphisms of spectral sequences
\begin{eqnarray}
E_{r}(\beta)=  E_{r}(\beta_{2})^{-1} \circ E_{r}(\beta_{1})\\
E_{r}(\beta_{\CC})=  E_{r}(\beta_{2})^{-1} \circ E_{r}(\beta_{1}\otimes 1_{\CC})
\end{eqnarray}
Then $E_{r}(\beta_{1})\otimes1_{\CC}\simeq E_{r}(\beta_{1}\otimes 1_{\CC})$ is an isomorphism. Moreover morphism of spectral sequences
$E_{r}(\beta)$ defines a real structure $\alpha_{r}$ on $E_{r}(K_{R\otimes\CC},W)$ such that  $d_{r}$ is real, for one has $$d_{r}\simeq (E_{r}(\beta)\otimes 1_{\CC}) (d_{r}\otimes 1_{\CC})$$
Note that real structure induced on $E_{r+1}\simeq H(E_{r})$ by $E_{r}$ is the same than its real structure.
$d_{r}$ is compatible with direct filtrations and their conjugates.

One reduces mod $\tau$:
\begin{trivlist}
\item[a)]
One obtains a real structure mod $\tau$ on each term of the spectral sequence, with $d_{r}$ real and compatible with direct filtrations.
\item[b)]  From \Del $(1.2.10)$ generalised in \ref{MHSabelian},  category of mixed Hodge module $mod\tau$ is abelian.
\end{trivlist}
\item[1)] By hypothesis, $F_{d}=F_{d^*}=F_{rec}$ and their conjugates define a Hodge structure mod $\tau$ of weight $-p+(p+q)=q$ on $E_{1}^{p,q}=H^{p+q}(Gr_{-p}^{W})$ . But $d_{1}$ is real and compatible with $F$ so that it is a morphism of Hodge structure in $\Mod(R)_{/\tau}$. It is therefore strict for $F$  in $\Mod(R\otimes\CC)_{/\tau\otimes\CC}$.
\item[2)]Proposition $(7.2.5)$ in \Deltrois~ implies that on $E_{2}$, $F_{d}=F_{d^*}=F_{rec}$ mod $\tau\otimes\CC$. From $b$ above,
 $(E_{2}^{p,q},\alpha_{2}^{p,q},F_{rec})$ is a  Hodge structure of weight $q$ mod $\tau$.
But $d_{2}$ is real and compatible with $F_{rec}$ and its conjugate: $d_{2}$ is a morphism of Hodge sructure in $\Mod(R)_{/\tau}$. This implies that $d_{2}$ is strict mod $\tau\otimes\CC$.
But $d_{2}: E_{2}^{p,q}\to E_{2}^{p+2,q-1}$ is a morphism of Hodge structures of different weights. It must vanish.

An induction argument implies that $d_{r}=0$ if $r\geq2$.
\item[3)]  Therefore the following sequence is exact for any $1\leq r\leq +\infty$ and for any $p$: $$0\to E_{r}(F^pK,W)\to E_{r}(K,W)\to E_{r}(K/F^pK)\to 0$$ and one conclude from section $(7.2)$ of \Deltrois.
\end{proof}

One gets our first theorem on mixed Hodge structure on $l^2-$cohomology\\ (compare with theorem 3.18 of \cite{PetSte}):

\begin{theorem}\label{theoreme de structure mixte}
Let $\Gamma$ be  functor of global section over $X$, from the category of $\N(G,\RR)-$sheaves of modules (resp. $\N(G)-$sheaves of modules) to the
category of  $N(G,\RR)-$modules (resp. \\ $N(G)-$modules).
Let $R\Gamma$ be its derived functor realized through Godement resolution.

\item[] Assume that a torsion theory $\tau$ is chosen so that for each $p\in\ZZ$, $$R
\Gamma(Gr^{W}_{p}\K):=[(\Gamma R^pj_{*}\p\RR),\, (R\Gamma Gr^{W}_{p}\p\dlog{}
{D},F),\, R\Gamma(Gr^{W}_{p}\tilde\beta)]$$ is a Hodge complex in $N(G,\RR)_{/\tau}$ , then
\item[]$$R\Gamma(\K):=[(\Gamma Rj_{*}\p\RR, (\Gamma Rj_{*}\p\RR,\tau), (R\Gamma
\p\dlog{.}{D},W,F), R\Gamma(\tilde\beta)]$$ is a mixed Hodge complex in $N(G,\RR)_{/\tau}$. Therefore:

\item[i)] spectral sequence in $N(G,\CC)_{/\tau\otimes\CC}$ $$E_{1}^{-p,q}(R\Gamma\p\dlog{}{D},W)\simeq\HH^{-p
+q}(Gr^{W}_{p}\p\dlog{}{D})\abupt Gr^{W}_{p}\HH^{-p+q}(\p\dlog{}{D})$$
degenerates at $E_{2}-$terms.
\item[ii)] Differential $d_{1}$ on $E_{1}^{-p,q}(R\Gamma\p\dlog{.}{D},W)\simeq \HH^
{-p+q}(Gr^{W}_{p}\p\dlog{.}{D})$ is real, and is a morphism of Hodge structures in $\Mod(N(G))_{/\tau}$. In particular $d_{1}$ is strictly compatible with $F$.
\item[iii)]Through isomorphism $Gr^{W}_{p}\HH^{-p+q}(\p\RR)\otimes\CC\to Gr^{W}
_{p}\HH^{-p+q}(\p\dlog{.}{D})$, the Hodge filtration induces a Hodge structure (mod
$\tau$). It is the same than the Hodge structure induced by  isomorphism $E_{2}(R
\Gamma\p\dlog{.}{D},W)\simeq E_{\infty}(R\Gamma\p\dlog{.}{D},W)$.
\item[iv)] Spectral sequence in $N(G,\CC)_{/\tau\otimes\CC}$
$$E_{1}^{p,q}(R\Gamma\p\dlog{.}{D},F)=\HH^{p+q}(X,\p
\dlog{p}{D})\abupt \HH^{p+q}(X,\p\dlog{.}{D})$$ degenerates at $E_{1}$

\end{theorem}
\begin{proof} \item[1)]Recal that $\C^{.}(.)$ is an exact functor to the category of
Flabby Sheaves.

Let
$$
\begin{diagram}

\node{(R\Gamma Rj_{*}\p\RR,\tau)} \arrow{e,t}{R\Gamma (i)} \arrow{s,l,..}{R\Gamma
(\tilde\beta)}\node{(R\Gamma Rj_{*}\p\CC,\tau)} \arrow{e,tb}{R\Gamma(f_{1})}{g_{1,
\CC}} \node{(R\Gamma Rj_{*}(\p\Omega^{.}_{U},\tau)} \\

\node{(R\Gamma \p\dlog{.}{D},W)}\arrow{e,tb}{g_{3}}{R\Gamma(\beta)}\node{(R
\Gamma(\p\dlog{.}{D}),\tau)}\arrow{e,t}{R\Gamma(\alpha)}\arrow{ne,t}{g_{2}}  \node{(R\Gamma (j_{*}\p\Omega^{.}),\tau)}\arrow{n,l}{R\Gamma(f_{2})}

\end{diagram}
$$

This defines  pseudo- morphisms

\begin{eqnarray}
\beta_{2}=R\Gamma(\tilde\beta):(R\Gamma Rj_{*}\p\RR,\tau)\dto (R\Gamma \p\dlog{.}{D},W)\\
\beta_{2,\CC}: (R\Gamma Rj_{*}\p\CC,\tau)\tilde\dto (R\Gamma \p\dlog{.}{D},W)
\end{eqnarray}

Set $g_{1}=R\Gamma(f_{1})\circ R\Gamma(i)$. Then

 $g_{1,\CC}=g_{1}\otimes 1_{\CC}$.

 With notation of previous theorem defines $\beta_{2}=g_{2}\circ g_{3}$ and $\beta_{1}=g_{1}$.
 Then $$E_{1}^{p,q}(R\Gamma \p\dlog{.}{D},W_{-})=H^{p+q}(Gr^{p}_{W_{-}}R\Gamma
\p\dlog{.}{D})\simeq H^{p+q}(R\Gamma Gr^{p}_{W_{-}} \dlog{.}{D})$$

 has a Hodge structure of weight $q$ mod $\tau$. One conclude from above theorem and section $7.2$ of \Deltrois.

\end{proof}

\begin{example}\label{ }Let $\tau_{\dbar,\tilde D_{.}}$ be the Serre category generated by $$\{\frac{\Adh{Im \dbar}}{Im \dbar},\, \dbar:\oplus_{p,q,I}S^{k-q,(p,q)}(\tilde D_{I})\to \oplus_{p,q,I}S^{k-q-1,(p,q+1)}(\tilde D_{I})\,\}$$
($k$ big enough, $D_{\emptyset}=\tilde X$). Then $\gamma^{*}\tau_{\dbar,\tilde D_{.}}$ (\ref{ring extension}) is the smallest torsion theory wich fullfils above assumption.
\end{example}

Let  $\Sigma$ be a (finite) simplicial structure on $X$ such that $D$ is a subcomplex. Let $F=\{F_{i}\}_{i\in P}$ be the collection of closed $2n-$dimensional simplex. Set $F\setminus D=\{F_{i}\setminus D\}_{i\in I}$.  Then $H^{k}(\cap_{i\in I}F_{i},K)=0$ and $H^{k}(\cap_{i\in I}(F_{i}-D),K)=0$ for any $k\geq 1$ and any locally constant sheaf $K$. Therefore $H^{k}(X\setminus D,\jp\CC)\simeq_{N(G,\CC)}\check{H}^{k}(F\setminus D,\jp\CC)$ for each closed (in $X\setminus D$) set $(F\setminus D)_{I}:=\cap_{i\in I}(F_{i}-D)$ is simply connected (see \God p.207). Natural complex $\ZZ[F\setminus D]$ lifts to a $\ZZ[G]-$complex  $\ZZ[\widetilde{F\setminus D}]$. Then $$\check{H}^{k}(F\setminus D,\jp\CC)\simeq_{N(G,\CC)}H^{k}(Hom_{\CC[G]}(\ZZ[\widetilde{F\setminus D}],\, l^{2}(G)))\,.$$

\begin{corollary}\label{MHS on simplicial} There exists mixed Hodge structure on $H^{k}(Hom_{\CC[G]}(\ZZ[\widetilde{F\setminus D}],\, l^{2}(G)))$ \\
in $\Mod(N(G))_{\gamma^{*}\tau_{\dbar,\,\tilde D_{.}}}$. In particular in $\Mod(N(G))_{\tau_{dim}}$, there exists a mixed Hodge structure on  $\bar H^{.}_{(2)}(\widetilde{X\setminus D})$.
\end{corollary}

\subsection{CHMC $\mod(\tau',\tau)$}\label{definition of CHMC}

A $(\tau',\tau)-$cohomological mixed Hodge complex of sheaves \\$\K=((\K_{\RR},W);\,(\K_{\CC},F,W),\beta) $ could be define by complexes of sheaves of $N(G)_{/\tau'}-$modules and quasi isomorphism $\beta$ in that category, such that for all $m\in \ZZ$, $Gr_{W}^{m}\K$ is a $(\tau',\tau)-$cohomological  Hodge complex of sheaves (see \ref{definition of CHC}) .

\begin{example}
\item[1)] $\K:=[(Rj_{*}\p\RR, (Rj_{*}\p\RR,\tau), (\p\dlog{}{D},W,F), (\tilde\beta)]$ is a $(0,\gamma^{*}\tau_{\dbar,\tilde D_{.}})-$CHMC.
\item[2)]
Recal (\Ive p.99 and p. 315) that if $p:\tilde X\to X$ is a covering map with group $G$ of deck transforms  between locally compact space, then functor $p_{!}p^*$ from category of sheaves on $X$ to category of $\ZZ[G]-$sheaves on $X$ is exact  (moreover, one has isomorphism $ p_{!}p^*(F)\simeq (p_{!}\ZZ)\otimes F$ has $\ZZ[G]-$sheaves).

Then tensoring over $\CC[G]$ by $N(G)$ gives a functor from category of $\RR-$mixed Hodge complexes of sheaves on $X$ to category of $N(G)-$mixed Hodge complexes of sheaves modulo dimension function:

 functor $N:\text{Sheaf over X}\to\N(G)-\text{sheaf over X}$ given by $\F\to \N(G)\otimes_{\CC[G]}p_{!}p^*\F$ is exact for a local model is induced trivial module tensorised with $\CC[G]$.
 There are  natural maps
 \begin{eqnarray}

 \N(G)\otimes_{\CC[G]}p_{!}p^*\F\to l^2(G)\otimes_{\CC[G]}p_{!}p^*\F& \text{ s.t. } &  n\otimes f\longrightarrow n(\delta_{e})\otimes f
\end{eqnarray}

    In case of holomorphic coherent sheaf (or locally constant sheaf)  this map falls into $\p\F$. But is far from being surjective.

However, if one reduce to the category of sheaves in $\Mod(N(G))_{\tau_{\dim}}$, then one gets from usual CHMC $\K$ on $X$ a CHMC $mod(\tau_{\dim},\tau_{\dim})$ $\N(G)\otimes p_{!}p^{*}\K$.
\end{example}

\subsection{Interpretation of $(E_{1},d_{1})$}

In order to interpret $d_{1}$ through Gysin morphism, one uses the smooth logarithmic complex (see \GriSch p.73):
\begin{definition} Let $\p\Slog{.}{D}$ be subsheaf of $j_{*}[\jp\S^{\infty}\A^{.}]$ such that a germ\\$\alpha$ belongs to $[\p\Slog{.}{D}]_{x}$ iff
\begin{eqnarray}
h\alpha\in j_{*}[\jp\S^{\infty}\A^{.}]_{x} &\text{ and } & h d\alpha\in j_{*}[\jp\S^{\infty}\A^{.}]_{x}
\end{eqnarray}
with $h$ a defining equation for $D$ at $x$.
\end{definition}
From its definition, $(\p\Slog{.}{D},d)$ is a complex. Moreover
\begin{lemma} $\p\Slog{.}{D}=\p\S^{\infty}\A^{.}\otimes_{\O} \dlog{.}{D}$.
\end{lemma}
\begin{proof}
\item[1)]
 One first recall a proof in a coordinate chart  $(V,(z))$: Hypothesis implies that $h\alpha$ and $(h\alpha)\wedge \frac{dh}{h}$extends as smooth forms. Assume that $h=z_{1}\ldots z_{k}$ in $(V,(z))$ and that $h\alpha=\sum_{I}\beta_{I}dz^{I}$ with $\beta^{I}$  smooth forms on $V$ without any $dz_{1},\ldots, dz_{k}$. Then  $\gamma=(h\alpha)\wedge \frac{dh}{h}=\sum_{i,I}\frac{\beta_{I}}{z_{i}}dz^{I}\wedge dz^{i}$
 is a smooth form on $V\setminus \{z_{1}\ldots z_{k}=0\}$ such that any of its partial derivative $\frac{\partial^{|A|+|B|}}{\partial z^{A}\partial \bar z^{B}}\gamma$
admits a limite on $V\cap D$. Let $p\in (z_{1}=0)\setminus (z_{2}.\ldots .z_{k}=0)$. Then

\begin{eqnarray}
0=\lim_{V\setminus D\ni p'\to p} z_{1} \frac{\partial^{|B|}}{ \partial\bar z^{B}}\gamma&=&
\lim_{V\setminus D\ni p'\to p} \sum_{i,I}\frac{ z_{1}}{z_{i}} \frac{\partial^{|B|}}{\partial \bar z^{B}}\beta_{I}dz^{I}\wedge dz^{i}\\
&= &\sum_{I\not\ni 1}  \frac{\partial^{|B|}}{ \partial\bar z^{B}}\beta_{I}(p)dz^{I}\wedge dz^{1}
 \end{eqnarray}

 Hence $1\not\in I\Rightarrow \frac{\partial^{|B|}}{\partial \bar z^{B}}\beta_{I}(p)=0$.
 By continuity, this holds for any $p\in \{z_{1}=0\}$. This implies (\Mal, \Sch th.2) that $\alpha_{I}\in z_{1}\A^{.}(V)$.
 Renaming of coordinates proves therefore that $i\not\in I\Rightarrow \alpha_{I}\in z_{i}\A^{.}(V)$, hence $\beta_{I}\in z^{K-I}\A^{.}(V)$ (\cite{Sch} th.3) and
  $\alpha=\sum_{I}\frac{\beta_{I}}{z^{K-I}}\frac{dz^{I}}{z^{I}}$ is in $\A^{.}\otimes_{\O} \dlog{.}{D}(V)$.

 \item[2)]
But $z^{K-I}\A^{.}(V)$ is a closed ideal, therefore surjective maps
$ \alpha_{I}\to z^{K-I}\alpha_{I}$ are open:
For all compact $K_{1}\subset V$, all $m_{1}\in \NN$, all $I\subset \{1,\ldots, K\}$; $\exists K_{2}\subset V$, $m_{2}\in \NN$ and a constant $C>0$ such that
$||\alpha_{I}||_{\C^{m_{1}}(K_{1})}\leq C ||z^{K-I}\alpha_{I}||_{\C^{m_{2}}(K_{2})}$. This implies that $\p\S^{\infty}\dlog{.}{D}=\p\S^{\infty}\A^{.}\otimes_{\O} \dlog{.}{D}$.
\end{proof}

\begin{lemma}Sheaf $\p\S^{\infty}\A^{0}$ is a flat $\O_{X}$ module.
\end{lemma}
\begin{proof}From \Mal, this is true without growth: For any finitely generated ideal $\I_{x}\subset \O_{x}$, let $V$ a neighborhood of $x$ such that one has a finite presentation $$\O^{k}(V)\overset{r}{\to}\O^{p}(V)\overset{g}{\to} \I(V)\to 0$$ (map $r$ gives module of relations and $g:h\to\sum_{i=1}^{p}h_{i}g_{i}$ is given by generator of $\I(V)$).

Flatness of $\C^{\infty}(V)$ on $\O(V)$ implies that $(\C^{\infty}(V))^k\to (\C^{\infty}(V))^p\to \I(V).\C^{\infty}(V)$ is exact. Hence $Im(r)$ is closed and above exact sequence splits. If $V$ is small enough, pullback of this splitting defines a splitting of $\p\S^{\infty}\A^{0}(V)^k\to \p\S^{\infty}\A^{0}(V)^p\to \I(V).\p\S^{\infty}\A^{0}(V)$. So that $Tor_{1}^{\O_{x}}(\O_{x}/I_{x},\p\S^{\infty}\A^{0}_{x})=0$ (see \Tou I.4).
\end{proof}

\begin{corollary}
\item[1)] $(\p\dlog{}{D},d)\overset{ }{\to} (\p\Slog{.}{D},d)$ is a quasi-isomorphism.  Moreover $$(\p\dlog{}{D},d, W, F)\to (\p\Slog{.}{D},d, W, F)$$ is a bifiltered $\Gamma$ acyclic resolution.
\item[2)] Let $I$ be a $p-$upplet from $\{1,\ldots,\, N\}$ then  there is a  map $$Res_{I} : \p\Slog{t}{D}\to \p\Slog{t-p}{D_{I}\cap\sum_{j\not\in I}D_{j}}\,.$$
\item[4)]
 In the following diagram, maps are filtered quasi isomorphisms.
\begin{equation*}
\begin{diagram}
\node{(Gr^{W}_{l}\p\dlog{.}{D},d, F)} \arrow{e,t}{Res_{l}}\arrow{s,l}{}\node{ ( a_{l*}\p\Omega^{.-l}_{\tilde D_{l}},d,F^{-l})} \arrow{s,l}{}\\

\node{(Gr^{W}_{l}\p\Slog{.}{D}, d, F)} \arrow{e,t}{Res_{l}} \node{( a_{l*}(a\circ p)_{*(2)}\S^{\infty}\A^{.-l}_{\tilde D_{l}},d, F^{-l}) }
\end{diagram}
\end{equation*}
\end{corollary}

\subsubsection{Gysin morphism.}

 Let $I$ be a $p-$subset of $\{1,\ldots,,N\}$.

 Fix (hermitian) metric on $D_{I}$ and choose an orthogonal splitting $T(X)_{|D_{I}}\simeq T_{D_{I}}\oplus N_{D_{I}/X}$.  For any $\epsilon$ small enough, exponential map induces a diffeomorphism
 $$\begin{diagram}
 \node{N_{D_{I}/X}(\epsilon)}\arrow[2]{e,t}{e_{I}}\node{}\node{U_{I}}\\
 \node{}\node{D_{I}}\arrow{nw,l}{i_{0}}\arrow{ne,l}{i}
 \end{diagram}
 $$
 This defines a retraction $r_{I}$ by composition of $e^{-1}_{I}$ with bundle projection $N_{D_{I}/X}\to D_{I}$. In the following, one sets  $r_{I}:=U_{D_{I}}(\epsilon_{p})\to D_{I}$ with $\epsilon_{1},\ldots,\epsilon_{n}$ ($n=dim X$) choosen as follows: If $I$ is a $p-$subset, let $p>q$ and considers those $q-$subset $J$  contained in $I$. Then one required $\cap_{ J\subset I}U_{J}(\epsilon_{q})\subset U_{I}(\epsilon_{p})$.
 (e.g. intersection of tubular neighborhoods of $D_{1}$, and $D_{2}$ is included in the tubular neighborhood of $D_{1}\cap D_{2}$).
Then $r_{I}$ is smooth submersion hence $r_{I}^{*}$ maps local Sobolev space of order $s$ to local Sobolev space of order $s$ (tensor product of separated variables commutes to Fourier transform).

Let $s_{k}$ be a  section of $[D_{k}]$ vanishing on $D_{k}$. Fix hermitian metrique $h_{k}$ on $[D_{k}]$, such that $|s_{k}(x)|=1$ if $x\not \in U_{k}(\epsilon_{1}')$ with $\epsilon_{1}'<\epsilon_{1}$. Let  $\eta_{k}=\frac{1}{2i\pi}\partial log|s_{k}|$. Poincar\'{e}-Lelong formula reads $d[\eta_{k}]=[D_{k}]-c_{1}(h_{k})$ and these currents are supported in  compact neighborhood of $D_{k}$. Let $I\in \{1,\ldots, N\}^p$, defines $\eta_{I}=\eta_{i_{1}}\wedge\ldots\wedge \eta_{i_{p}}$. Then
\begin{eqnarray}\label{support} supp( \omega_{i}\wedge \eta_{J})\subset\subset U_{iJ}(\epsilon_{|J|+1})\,.
\end{eqnarray}

Lift of this construction by the covering map gives same diagram with bounded geometry.
Note that  a connected componant of $\pi^{-1}(U_{I}(\epsilon_{p}))$ contains only one connected componants of $\pi^{-1}(D_{I})$.
Let $I\in\{1,\ldots,N\}^k$. Let $r_{I}, \eta_{I}, \ldots$ denotes restriction of  $p^{*}(r_{p(I)}), p^{*}\eta_{p(I)},\ldots$ to $p^{-1}(U_{I})$.

Let $[\alpha]\in H^{q-2p}( D_{I}, \p\S^{k-.}\A^{.})$. From \ref{smoothingofcohomology}, one may assume  $\alpha\in \Gamma(p^{-1} (D_{I}),\A^{q-2p})\cap_{n\in\NN}Dom(1+\Delta_{\tilde D_{I}})^n$.
Then $r_{I}^{*}(\alpha)\eta_{I}\in \Gamma( U_{I}(\epsilon'_{1}),\p\Slog{q-p}{D})$. Denote by the same symbole its extension by zero to $\tilde X$. Then $r_{I}^{*}(\alpha)\eta_{I}\in \Gamma(\tilde X, \Slog{q-p}{\tilde D})$ and $Res_{I}(r_{I}^{*}\alpha\eta_{I})=\alpha$. Any class in $\HH^{-p+q}(X,\p (Gr^{W}_{p}\dlog{.}{ D},d))$ has a representative
$\frac{1}{p!}\sum_{I}r_{I}^{*}\alpha_{I}\eta_{I}\in \Gamma(\tilde X,\Slog{.}{D})$,

with the convention that $I\to \alpha_{I}$ is antisymetrical in $I$. So that $\alpha_{I}=(-1)^{i}\alpha_{(i,\Rm{i}{I})}$.

Then,
\begin{eqnarray*}

 d(r_{I}^{*}\alpha_{I}\eta_{I})=0+(-1)^{q-2p}r_{I}^*(\alpha_{I})\sum_{i\in I}(-1)^{i}\omega_{i}\Rm{i}{\eta_{I}}=(-1)^{q-2p}\sum_{i\in I}r_{I}^*(\alpha_{(i,\Rm{i}{I})})\omega_{i}\Rm{i}{\eta_{I}}\\

d \sum_{I}r_{I}^{*}\alpha_{I}\eta_{I}=(-1)^{q-2p}\sum_{J}\sum_{i}r_{i,J}^*(\alpha_{(i,J)})\omega_{i}\eta_{J}\\

Res_{\{J\}}d\sum_{I}r_{I}^{*}\alpha_{I}\eta_{I}=(-1)^{q-2p}a_{J}^{*}(\sum_{i}\omega_{i}r_{i,J}^{*}\alpha_{i,J})\otimes \Lambda^{J}e
\end{eqnarray*}
Hence
\begin{eqnarray} \label{formulepourd1} d_{1}[\sum_{I\in T^p}\alpha_{I}\otimes\Lambda^{I}e]=[\sum_{J\in T^{p-1}}(-1)^{q-2p}a_{J}^{*}(\sum_{i\in T}\omega_{i}r_{i,J}^{*}\alpha_{i,J})\otimes \Lambda^{J}e]
\end{eqnarray}

\subsection{$L^{2}-$characteristics}

Let $\tau_{dim}$ be torsion theory such that $\T_{dim}=\{M\in\Mod(N(G)\,:\dim_{N(G)}M=0\}$ (see \Luc). Corollary \ref{MHS on simplicial} implies that $H^{.}(X\setminus D,\,\jp\CC)$ is computable  (in $\Mod(N(G(R))_{\tau_{dim}})$ through simplicial structure. Moreover  in $\Mod(N(G))_{/\tau_{dim}}$,  one has \\$H^{k}(X,\p\dlog{p}{D})\simeq \bar H^{k}_{\dbar (2)}(\tilde X,\dlog{p}{\tilde D})$ (reduced cohomology).

Then weight spectral sequence degenerates at $E_{2}$ and Hodge spectral sequence degenerates at $E_{1}$. Hence  in  $\Mod(N(G))_{/\tau_{\dim}}$

$$Gr_{F}^{p} H^{n}(X\setminus D,\jp\CC)=\bar H^{n-p}_{(2),\dbar}(\tilde X,\dlog{p}{\tilde D}) \text{ (reduced cohomology)}$$ is finite $N(G)-$ dimensional and

\begin{eqnarray}

\sum_{i=0}^{n}(-1)^{i}dim_{N(G)}Gr_{F^p}H^{i}(X\setminus D,\jp\CC)&=&\sum_{i=0}^{n}(-1)^{i}dim_{N(G)}Gr_{F^p}H^{i}(X\setminus D,\CC)\\
\chi_{(2)}(\widetilde{X\setminus D})=\chi(X\setminus D)

\end{eqnarray}

Note in particular that $F^{n}H^{k}=Gr_{F}^{n}H^{k}\simeq H^{k-n}(K\otimes [D])$ in $\Mod(N(G))_{/\tau_{dim}}$.

\subsubsection{MHS on normal crossing divisor}

MHS on  normal crossing divisors  as in \Elz section 3.5 is clearly transposable to $L^2-$cohomology as soon as a torsion theory is choosen so that $E_{1}$ term of weight spectral sequence is a Hodge structure in $\Mod(N(G))_{/\tau}$.

\section{Examples.}

\subsection{}
First terms of  unreduced $(E_{1},d_{1})$ complex of the weight spectral sequence

reads:

$$
\begin{diagram}
\node{6}    \node{H^{0}_{(2)}(\tilde D_{3}) } \arrow{se,..}  \arrow{e,t}{0}   \node{ H^{2}_{(2)}(\tilde D_{2}) }   \arrow{se,..}\arrow{e,t}{d_{1}}   \node{ H^{4}_{(2)}(\tilde D_{1}) } \arrow{se,..}  \arrow{e,t}{d_{1}}
\node{ H^{6}_{(2)}(\tilde X) }  \\

\node{5}    \node{0 }    \node{ H^{1}_{(2)}(\tilde D_{2}) }   \arrow{se,..}\arrow{e,t}{d_{1}}   \node{ H^{3}_{(2)}(\tilde D_{1}) } \arrow{se,..}  \arrow{e,t}{d_{1}}
\node{ H^{5}_{(2)}(\tilde X) }  \\

\node{4}    \node{ 0}    \node{ H^{0}_{(2)}(\tilde D_{2}) }   \arrow{se,..}\arrow{e,t}{d_{1}}   \node{ H^{2}_{(2)}(\tilde D_{1}) } \arrow{se,..}  \arrow{e,t}{d_{1}}      \node{ H^{4}_{(2)}(\tilde X) }  \\

\node{3}       \node{}\node{0}  \node{ H^{1}_{(2)}(\tilde D_{1}) }  \arrow{e,t}{d_{1}}  \arrow{se,..}   \node{ H^{3}_{(2)}(\tilde X) }   \\

\node{2}     \node{}   \node{0}      \node{ H^{0}_{(2)}(\tilde D_{1}) }  \arrow{e,t}{d_{1}}  \arrow{se,..}   \node{ H^{2}_{(2)}(\tilde X) }  \\

\node{1}    \node{ }   \node{}    \node{0  }   \node{ H^{1}_{(2)}(X) }  \\

\node{0}       \node{}  \node{}      \node{ 0 }     \node{ H^{0}_{(2)}(\tilde X) }    \\

\node{}      \node{Gr_{W_{-}}^{-3}}  \node{Gr_{W_{-}}^{-2}} \node{Gr_{W_{-}}^{-1}}   \node{Gr_{W_{-}}^{0}}

\end{diagram}
$$

\subsubsection{}
Homology at column $Gr^{0}_{W_{-}}$ gives $Gr_{0}^{W}\HH^{.}(U)=Im(H^{.}_{(2)}(X)\to H_{(2)}(X\setminus D))$. Hence kernel of restriction mapping is isomorphic modulo some torsion theory to image of the Gysin homomorphism.

\subsubsection{}
 First unreduced $E_{1}-$term of the Hodge filtration is (surface case):
$$
\begin{diagram}

\node{2}     \node{ H^{2}_{(2)}(\O) }  \arrow{e,t}{d_{1}} \node{H^2(\p\dlog{1}{D})}  \arrow{e,t}{d_{1}=\partial}     \node{\overset{ H^{2}(\p\dlog{2}{D})}{\simeq H^{2}_{(2)}(K\otimes [D])} } \\% \arrow{e,t}{d_{1}}   \node{ 0 }  \\

\node{1}    \node{H^{1}_{(2)}(\O)}  \arrow{e,t}{d_{1}} \node{H^1(\p\dlog{1}{D})}  \arrow{e,t}{d_{1}}    \node{ H^{1}_{(2)}(K\otimes[D]) }  \\%\arrow{e,t}{d_{1}} \node{0}   \\

  \node{0}       \node{H^{0}_{(2)}(\O) }  \arrow{e,t}{d_{1}} \node{H^0(\p\dlog{1}{D})}  \arrow{e,t}{d_{1}}   \node{ H^{0}_{(2)}(K\otimes D) }\\

   \node{}      \node{Gr_{F}^{0}}  \node{Gr_{F}^{1}} \node{Gr_{F}^{2}}
\end{diagram}$$

\subsubsection{Simplicial structure}

Top line of $(E_{1}(W),d_{1})$ is interpreted through simplicial complex associated to divisors:
  In general $D_{I}$ is not connected. Let $T_{k}$ be a set wich parametrized connected componant of $\sqcup_{|I|=k}D_{I}$.
Let $\tilde T_{k}\ni I\to \tilde D_{I}$ be a set wich parametrize connected componant of $\pi^{-1}(\sqcup_{|I|=k}D_{I})$.
Then $G$ acts on $\tilde T_{k}$ and $p$ induces a projection  $p: \tilde T_{k} \to T_{k}$.

If $I\in \tilde T^k$ or $T_{k}$, then $ r_{I},  \ldots$ denotes restriction of  $p^{*}(r_{p(I)}), \ldots$ to $U_{I}$ that connected componant of $p^{-1}(U_{p(I)})$ wich contains $\tilde D_{I}$.

 If $dim_{\CC}X=n$,
then $\bar H^{2n-2k}_{(2)}(\tilde D_{I})$ (top reduced cohomology) is either vanishing ($\tilde D_{I}$ non compact) or one dimensional spanned by a fundamental class $C_{I}$.

In general closedness of $Im\dbar:L^{2}(\tilde D_{I},\Lambda^{2n-2k-1})\to L^{2}(\tilde D_{I},\Lambda^{2n-2k})$ is controlled through amenability of $G/G_{I}$ with $G_{I}$ stabilisator of one irreducible componant of $\tilde D_{I}$.

A $l-$simplexe is an element $(I,o)\in  \tilde T_{l+1}\times \{1,\ldots,\,N\}^{l+1}$ such that $\tilde D_{I}$ is a compact (non empty) submanifold of $p^{-1}(\cap_{i\in o}D_{i})$ (hence one remember order of intersection). $(J,o)$ is a face of $(I,o)$ if $\tilde D_{I}\subset \tilde D_{J}$.

Choice of support (\ref{support}) implies that $a^{*}_{J}\omega_{i} r^{*}_{I}C_{I}$ (see (\ref{formulepourd1})) is closed in $\tilde D_{J}$.

Then $$\int_{\tilde D_{J}}\omega_{i}r^{*}_{I}C_{I}=\int_{\tilde D_{i}\cap \tilde D_{J}}r^{*}_{I}C_{I}=1$$ for now there exists only one $i\in \tilde T_{1}$ such that $\tilde D_{i}\cap \tilde D_{J}=\tilde D_{I}$ .

Identifications
\begin{eqnarray}\bar H^{2n-2p}_{(2)}(\tilde{ D_{p}})\ni a= (a_{I}C_{I}) &\to & (a_{I}) \in l^{2}(\tilde T_{p})
\end{eqnarray}
and $H^{-p+q}(Gr^{W}_{p}\S^{k-.}\Alog{.}{\tilde D})\ni  \alpha\to \frac{1}{p!}\sum_{(I,o)}r_{I}^{*}\alpha_{(I,o)}\eta_{(I,o)}\in \Gamma(\tilde X,\p\Slog{.}{ D})$

(with  $ \alpha_{(I,o)}$ is antisymetrical in $o$),
 gives  $$(d_{1}\alpha)_{J,o}=\sum_{I:\,\, \tilde D_{J}\supset \tilde D_{I} }\alpha_{(I,o)}\,.$$
This is combinatorial boundary map.
\subsection{}
 Hence  weight $Gr_{q}^{W}H^{n-q}_{(2)}(\widetilde{X\setminus D})$ occur (in $\Mod(N(G))_{/\tau_{dim}}$) only through compact connected componant of codimension $q$.

\subsubsection{Nori string} Assume  connected componant of pullback of smooth divisors $D_{1},\ldots, D_{n}$ are compact submanifolds. Then only the first column of the $E_{1}$ term are non reduced. But if one reduced modulo $\gamma^{*}\tau_{\dbar, \tilde X}$ (the real torsion theory generated by $Coker\{\H^{p,q}_{\dbar(2)}(\tilde X)\to H^{p,q}_{\dbar(2)}(\tilde X)\}$), then any term has a Hodge structure. Therefore cohomology of $\widetilde {X\setminus D}$ has a mixed Hodge structure in $\Mod(N(G))_{\gamma^{*}\tau_{\dbar,\tilde X}}$.

If $dim X=2$,
 homology of
$H^{0}(\widetilde D_{2})\to H^{2}(\widetilde D_{1})\to 0$
is  $l^2-$homology of graph $T$  whose edge $E$ are points in $\widetilde D_{2}$ and  vertices $S$ are connected componant of $\widetilde D_{1}$. Hence one  has multiple edges between two vertices.
Then
\begin{enumerate}
\item[1i)] $Gr_{2}^{W}H^{2}$ is $N(G)_{/\gamma^{*}\tau_{\dbar,\tilde X}}-$isomorphic to $H_{1(2)}(T)=Ker(\partial: l^2(E)\to l^{2}(S))$ (after a choice of $G-$orientation).

\item[1ii)] $Gr^{W}_{1}H^{3}$ is $N(G)_{/\gamma^{*}\tau_{\dbar,\tilde X}}-$isomorphic to $Coker(\partial : l^2(E)\to l^{2}(S))$.

Assume furthemore that group $G$  is non amenable, then both boundary map $\partial$ and coboundary map $\partial^*$ have close range for  graph $T$ is quasi isometric to a Cayley graph of $G$. There is no harmonic square integrable function on this graph, hence $Coker(\partial :\, l^2{E}\to l^{2}(S))=0$.
Therefore $Gr^{W}_{1}H^3=0$ in $N(G)_{/\gamma^{*}\tau_{\dbar,\tilde X}}$.

\item[1iii)] $H^{4}_{(2)}(\tilde X)=0$ for non amenability implies that $d_{3\to 4}$ has closed range.

\item[2)] $Gr_{1}^{W}H^{2}$ is $N(G)_{/\gamma^{*}\tau_{\dbar,\tilde X}}-$isomorphic to kernel of $$q\circ d_{1}: H^{1}_{2}(\tilde D_{1})\ni (\alpha_{i}) \to \sum_{i} r^{*}(\alpha_{i})\wedge\omega_{i}\overset{q}{\to } \H^3_{(2)}(\tilde X)\,\text{(orthogonal projection)}.$$

This map vanishes if group $G=\pi_{1}(X)$ is non fibered and this is the case if there exists a Nori string in $\tilde X$ the universal covering space of $X$.

\item[3)] Weight $1$ part of $H^{1}_{(2)}(\widetilde {X\setminus D})$ is $N(G)_{/\gamma^{*}\tau_{\dbar,\tilde X}}-$isomorphic to kernel of $$(\lambda_{i})\to \sum_{i}\lambda_{i}\omega_{i}\to \H^{2}_{(2)}(\tilde X)\,\text{ (orthogonal projection) }.$$ Hence described by forms $\sum_{i}\lambda_{i}\omega_{i}$ orthogonal to $\H^{2}_{(2)}(\tilde X)$. Let $\omega_{i}=h_{i}+\theta_{i}$ with $\theta_{i}\in \Adh{Im\dbar}$ and $h^{i}$ harmonic part of $\omega_{i}$. Then $\int_{\tilde X}h_{i}\wedge\omega_{j}=\int\omega_{i}\wedge\omega_{j}$ for $\int(\lim_{k}\dbar\alpha_{ki})\omega_{j}=\lim_{k}\int\dbar\alpha_{ki}\omega_{j}=0$ by compactness of $supp\omega_{j}$.
Therefore $\int_{\tilde X}\omega_{i}\wedge \omega_{j}=\int_{\tilde X}h_{i}\wedge h_{j}$.
Therefore an element $(\lambda_{i})$ in $Ker (q\circ d_{1})$ must satisfies $$l^2(T)\ni\lambda \mapsto Q(\lambda):=\int_{\tilde X}\sum_{i}\lambda_{i}\omega_{i}\wedge  \sum_{i}\bar\lambda_{i}\omega_{i}\,.$$ is vanishing. But

\begin{lemma}Intersection form on $\tilde D_{1}$ is negative definite.
\end{lemma}
\begin{proof} It is well known that compact submanifold in non compact  covering of compact manifold $X$ cannot be of strictly positive self intersection (see e.g. \Cam). But if $\tilde D_{i}^2=0$ then any divisor $\tilde D_{j}$ different from $\tilde D_{i}$ wich intersects $\tilde D_{i}$ would satisfies $(\tilde D_{j}+k\tilde D_{i})^2>0$ if $k$ is big enough.
\end{proof}

\begin{corollary} There exists a constant $c>0$ such that $-Q(\lambda)\geq c||\lambda||_{(2)}$ on function with compact support. If $G$ is non amenable, then this hold on $l^2(T)$.
\end{corollary}

In particular, $Gr_{W}^1H^{1}_{(2)}(\widetilde{X\setminus D})=0$ in $\Mod(N(G))_{\gamma^{*}\tau_{\dbar,\tilde X}}$.

\end{enumerate}