text:
\section{Introduction}

\noindent Let $p:\mathcal{X}\to \mathbb D$ be a holomorphic family of smooth, $n$-dimensional compact manifolds whose central fiber -denoted by $X$- is assumed to be K\"ahler. We denote by $\displaystyle K_\mathcal{X}$ the canonical bundle of $\mathcal{X}$ and let $\mathcal{L}\to \mathcal{X}$ be an arbitrary line bundle.
\smallskip

\noindent We are interested in the extension properties of global sections of the adjoint bundle $K_\mathcal{X}+ \mathcal{L}$ defined over the $k^{\rm th}$ infinitesimal neighborhood of the central fiber of $p$, where $k\geq 1$ is a positive integer.
In other words, we consider the sheaves
\begin{equation}\label{intr1}
\mathcal{F}_k:= \left(K_\mathcal{X}+ \mathcal{L}\right)\otimes \O_\mathcal{X}/t^{k+1}\O_\mathcal{X},
\end{equation}
and the corresponding spaces of sections $H^0(\mathcal{X}, \mathcal{F}_k)$.
\medskip

\noindent In this article we are establishing sufficient criteria for a given $\displaystyle s\in H^0\left(\mathcal{X}, \mathcal{F}_k\right)$ to be in the image of the map induced by the projection
\begin{equation}\label{intr2}\pi_k: \mathcal{F}_{k+1}\to \mathcal{F}_k.
\end{equation}
We are particularly interested in the case
$\mathcal{L}= (m-1)K_\mathcal{X}$ where $m\geq 1$ is a positive integer,
so we state next our main result in this setting. In order to do so we have to introduce further notations. Let $s$ be a holomorphic section of $\mathcal{F}_k$; we denote by
$h_L$ the metric on the bundle $L:= (m-1)K_X$ induced by the restriction of $s$ to the central fiber $X$.
\medskip

\noindent Then the following holds true.

\begin{thm}\label{MT} Let $s$ be a section of the sheaf \eqref{intr1} with $\mathcal{L}= (m-1)K_\mathcal{X}$, which admits
a $\mathcal C^\infty$ extension $s_k$ so that if we write
$\displaystyle \bar \partial s_k= t^{k+1}\Lambda_k,$
the integral
\begin{equation}\label{intr7}
\int_X\left|\frac{\Lambda_k}{dt}\right|^2e^{-(1-\varepsilon)\varphi_L}dV< \infty
\end{equation}
converges for any positive $\varepsilon> 0$.
Then there exists a section $\widehat s$ of $\mathcal{F}_{k+1}$ such that $s= \pi_k(\widehat s)$.
\end{thm}
\noindent The notations in \eqref{intr7} are as follows: we measure the $(n,1)$-form $\displaystyle \frac{\Lambda_k}{dt}\Big|_X$ with respect to an arbitrary Kähler metric on $X$ and the metric $e^{-(1-\varepsilon)\varphi_L- \varepsilon \phi_L}$ where $e^{-\phi_L}$ is a smooth metric on $L$ and $e^{-\varphi_L}=h_L$ is the singular metric induced by $s$.

\medskip

\noindent Coming back to the general setting, let $\mathcal{E}_1,\dots, \mathcal{E}_K$ be a set of line bundles
 on $\mathcal{X}$ such that
\begin{equation}\label{intr5}
(1- r_0)c_1(\mathcal{L})= r_0c_1(K_\mathcal{X})+ \sum_{i=1}^K r_ic_1(\mathcal{E}_i),
\end{equation}
where $0\leq r_0< 1$ and  $r_j\geq 0$ for $j=2,\dots, K$ are rational numbers.
In \eqref{intr5} we denote by $c_1(\mathcal{E})$ the first Chern class of $\mathcal{E}$.
Let $s$ be a section of $\mathcal{F}_k$ and and for each $i= 1,\dots K$ let $\sigma_i$ be a section of the sheaf $\displaystyle \mathcal{E}_i\otimes \O_\mathcal{X}/t^{k+2}\O_\mathcal{X}$ (notice that this is one order higher than $s$). We denote by $h_L$ the metric induced on the bundle $L:= \mathcal{L}|_X$ by the set of holomorphic sections $(s, \sigma_i)$ restricted to the central fiber
--notice that this makes sense thanks to \eqref{intr5}.
\medskip

\noindent Then we have the following version of Theorem \ref{MT}.
\begin{thm}\label{007} Let $\displaystyle (s, \sigma_i)_{i=1,\dots, K}$ be a family of holomorphic sections of $\displaystyle (\mathcal{F}_k, \mathcal{E}_i)_{i=1,\dots, K}$ respectively as above. We assume that $s$ admits a $\mathcal C^\infty$ extension $s_k$ such that if we write
$\displaystyle \bar \partial s_k= t^{k+1}\Lambda_k$
then \begin{equation}\label{intr6}\int_X\left|\frac{\Lambda_k}{dt}\right|^2e^{-(1-\varepsilon)\varphi_L}dV< \infty\end{equation} for any positive $\varepsilon> 0$. Then there exists a section $\widehat s$ of $\mathcal{F}_{k+1}$ such that $s= \pi_k(\widehat s)$.
\end{thm}
\medskip

\noindent A very interesting result in connection to Theorem \ref{007} was established by Cao, Demailly and Matsumura in \cite{CDM}. They show that if $\mathcal{L}$ admits a global, semi-positively curved metric
$h_\mathcal{L}$ such that the analogue of \eqref{intr6} holds on $\mathcal{X}$, then $s\in H^0(\mathcal{X}, \mathcal{F}_k)$ admits an extension to $\mathcal{X}$.
In our case, the curvature hypothesis is weaker and as a consequence the conclusion is that we can only extend $s$ "one step further".
\smallskip

\noindent Theorem \ref{007} is motivated by a conjecture in K\"ahler geometry that we next recall. After Y.-T. Siu's invariance of plurigenera articles \cite{Siu98, Siu00} (see also \cite{Pau07}) concerning the extension of $s$ in case of a \emph{projective family} $\mathcal{X}$,
the following very important problem is still open (despite of the \emph{avalanche} of articles and crucial achievements that followed Siu's work).

\begin{conjecture}\label{Siuconj}\cite{Siu00}
  Let $p:\mathcal{X}\to \mathbb D$ be a family of smooth, $n$-dimensional compact manifolds whose central fiber is denoted by $X$. We assume that the total space $\mathcal{X}$ admits a K\"ahler metric. Then any holomorphic pluricanonical section defined on $X$ extends holomorphically to $\mathcal{X}$.
\end{conjecture}
\smallskip

\noindent  Among the articles dedicated to Conjecture \ref{Siuconj} we first mention here \cite{Lev83} and \cite{Lev85}, due to M. Levine. A direct consequence of Theorem \ref{MT} is the following statement, which represents a more general version of the results in \cite{Lev83, Lev85}.
\begin{cor} \label{corlev}
Let $p:\mathcal{X}\to \mathbb D$ be a smooth holomorphic family whose central fiber $X$ is K\"ahler, and let $\displaystyle s\in H^0(X, mK_{X})$
be a pluricanonical section defined on $X$. We assume that the set of zeros of the
ideal $$\mathfrak I:= \lim_{\varepsilon\to 0}\mathcal I\left((1-\varepsilon)\frac{m-1}{m}\Sigma\right)$$ is discrete,
where $\Sigma$ is the divisor corresponding to $s$.
 Then $s$ admits a holomorphic extension to $\mathcal{X}$.
\end{cor}
\smallskip

\noindent  Furthermore we obtain a new proof of the next statement, which represents the main result in \cite{Lev83} (cf. Section 6). Our main objective in doing that is to understand better the set-up in which the hypothesis \eqref{intr6} is verified.
\begin{thm}\label{Lev83} \cite{Lev83}
Let $\displaystyle s\in H^0(X, mK_{X})$ be a pluricanonical section defined on the
central fiber. We assume that the set of zeroes $(s=0)\subset X$ of $s$ is non-singular.
Then $s$ admits an extension to $\mathcal{X}$.
\end{thm}
\smallskip

\noindent Many other results in this framework are established in Section 6; since they are a bit more technical to state, we will not reproduce them here. We mention nevertheless Theorem \ref{hypo} in which a sufficient condition for the $L^2$ hypothesis \eqref{intr6} to be satisfied is given.

\medskip

\noindent Finally, we highlight some parts of the proof of Theorem \ref{MT} which seem interesting to us. By hypothesis the section $s$ admits a $\mathcal C^\infty$ extension $s_k$ such that
\begin{equation}\label{int100}
\bar \partial s_k= t^{k+1}\Lambda_k
\end{equation}
for some form $\Lambda_k\in \mathcal C^\infty_{n+1, 1}(\mathcal{X}, (m-1)K_{\mathcal{X}})$ whose restriction to $X$ is $\bar \partial$-closed. The first step is to show that there exist forms $\alpha_k$ and $\beta_k$
which are smooth in the complement of the support of $(s=0)\subset X$ and such that
\begin{equation}\label{int101}
\frac{\Lambda_k}{dt}\Big|_X= \bar \partial \alpha_k+ D'(\beta_k)
\end{equation}
holds pointwise in the complement of $s=0$, where $D'$ is the $(1,0)$-part of the Chern connection on $L:= (m-1)K_X$
induced by the given section $\displaystyle s= s_k|_X$.
\smallskip

\noindent Note that the equality \eqref{int101} is true in general, i.e. without any $L^2$ assumption. But the problem is that the $L$-valued forms $\alpha_k$ and $\beta_k$ have singularities of type $\displaystyle \frac{\Phi}{s^{k+1}}$, where $\Phi$ is smooth. We therefore find ourselves in a rather strange situation, in which the LHS of \eqref{int101} is non-singular but the primitives appearing in
the RHS are meromorphic.
\medskip

\noindent Nevertheless, after the log-resolution of $\big(X, \mathrm{Div}(s)\big)$ combined with a few other reductions, the singularities of $\alpha$ and $\beta$ can be improved significantly and we show that it would be enough to prove the following statement (of independent interest).
\begin{thm}\label{007'} Let $\lambda$ be a smooth $(n, 1)$-form on a compact K\"ahler manifold $X$ with values in the bundle $E+ L$
 such that the following hold.
\begin{enumerate}

\item[\rm (a)] The bundle $L$ is endowed with a metric $h_L$ such that $\displaystyle
  \varphi_L\simeq \sum \delta_i\log|f_i|^2$ modulo smooth functions, where $0<\delta_i< 1$ are rational numbers.

\item[\rm (b)] The curvature
  $\Theta(L, h_L)= 0$ in the complement of $F:= (\varphi_L= -\infty)$.

\item[\rm (c)] We have $\displaystyle E:= \sum_{i\in I}E_i$ and the divisor $E+ F$ is snc.

\item[\rm (d)] There exists a $(n-1,1)$-form $\beta$ with log poles on $E$ and a $(n, 0)$-form $\alpha$ with values in $L$ and log poles along $E$ such that
\begin{equation}\nonumber
  \frac{\lambda}{s_E}= \bar \partial\alpha+ D'(\beta)
\end{equation}
on $X\setminus (E\cup F)$, where $D'$ is induced by the singular metric $h_L$ on $L$.
\end{enumerate}
Then $\lambda$ is $\bar \partial$-exact.
\end{thm}

\noindent It is for this reduction step that the hypothesis \eqref{intr6} is needed. Theorem \ref{007'} is proved by induction on the number of components of the divisor $E$, combined with the Hodge decomposition results we establish in Section
\ref{hodge}, cf. Theorem \ref{Hodge},Theorem \ref{deprim}. The results we obtain in Section \ref{currents}
concerning the Hodge decomposition analogue for bundle-valued currents
are equally crucial for our proof.

\medskip

\noindent Other than the aforementioned references, the articles \cite {Nog95},
\cite{LRW} by J. Noguchi, K. Liu, S. Rao and X.~Wan contain important ideas in connection with our work here. Recent and exciting contributions in the direction of Conjecture 1.1 are due to
J.-P. Demailly in \cite{Dem20} as well as to S.~Rao and I.-H. Tsai in \cite{RT}.

\medskip

\noindent{\bf Acknowledgments.} During part of the preparation of this article we have enjoyed the hospitality and excellent working conditions offered by the \emph{Freiburg Institut for Advanced Study}. At FRIAS we had the opportunity to meet and discuss with many visitors, but the math exchanges we have had with S. Kebekus and C. Schnell have been exceptionally stimulating. We thank A. Höring for valuable discussions. Also, J. Lott, R. Mazzeo and Y. Rubinstein answered patiently and professionally to our "Hodge theory" questions. Many thanks to all of them!

\medskip

\noindent This paper is organised as follows.
\tableofcontents

\section{First order differential operators}\label{comm}

In this subsection we will recall a few facts from differential geometry of line
bundles. We take this opportunity to fix some notations as well.

\subsection{Connection induced by a smooth section}

\begin{notation}\label{notation:1}
  In the setting of our article, let $L\to \mathcal{X}$ be a holomorphic line
  bundle endowed with a connection whose $(0,1)$-component is given by the
  $\bar \partial$ operator,
  \begin{equation}\label{eq73}
    \nabla= D^\prime_{\mathcal{X}}+ \bar \partial.
  \end{equation}
  Since no confusion is likely, we will use the same symbol to denote the
  induced operator on the space of smooth, $L$-valued $(p,q)$-forms,
  \begin{equation}\label{eq74}
    D^\prime_{\mathcal{X}}: \mathcal{C}^\infty\left(\mathcal{X}, \Omega_{\mathcal{X}}^{p, q}\otimes L\right)\to \mathcal{C}^\infty\left(\mathcal{X}, \Omega_{\mathcal{X}}^{p+1, q}\otimes L\right).
  \end{equation}
 \end{notation}

\begin{construction}\label{cons:1}
 Let $L\to \mathcal{X}$ be a holomorphic line bundle, and let
  $\widetilde s$ be a smooth section of $L$, with vanishing locus $Z \subseteq \mathcal{X}$.
  Assume that the open cover $(U_i)_i$ trivialises the bundle $L$ and choose
  trivialisations, $L|_{U_i} \cong \mathcal{O}_{U_i}$.  The section $\widetilde s$ will
  therefore give rise to smooth functions $\widetilde s_i$ on $U_i$.  Set
  $U^\circ_i := U_i \setminus Z$.

  Given any index $i$ we can now define a differential operator
  $D'_i : \mathcal{O}_{U^\circ_i} \to \Omega^1_{U^\circ_i}$ on $U^\circ_i$ as
  follows,
  \begin{equation}\label{eq76}
    D'_i : \mathcal{O}_{U^\circ_i} \to \Omega^1_{U^\circ_i}, \qquad %
    \sigma \mapsto  D^\prime_t(\sigma_i)dt+ \sum_\alpha D^\prime_\alpha(\sigma_i)dz_i^\alpha
  \end{equation}
  where
  \begin{align*}\label{eq77}
    D^\prime_t(\sigma_i) & := \partial_t \sigma_i- \frac{\partial_t \widetilde s_i}{\widetilde s_i}\sigma_i, & D_\alpha^{\prime}(\sigma_i) & :=  \partial_\alpha \sigma_i- \frac{\partial_\alpha \widetilde s_i}{\widetilde s_i}\sigma_i. \\
    \partial_t & := \partial/\partial t & \partial_\alpha & := \partial/\partial z_i^\alpha
  \end{align*}
  Using the trivialisations chosen above, we can view $D'_i$ as differential
  operators
  $D'_i : L|_{U^\circ_i} \to L|_{U^\circ_i} \otimes \Omega^1_{U^\circ_i}$ on
  $U^\circ_i$.  One computes without much pain that these differential operators
  glue to give a globally defined operator
  $$
  D'_{\mathcal{X}} : L \to L \otimes \Omega^1_{\mathcal{X}}
  $$
  on $\mathcal{X} \setminus Z$.  In particular, we obtain a connection
  $\nabla := D'_{\mathcal{X}} + \bar \partial$ on $L|_{\mathcal{X} \setminus Z}$.
\end{construction}

\begin{lemme}
  Setting as in Construction~\ref{cons:1}.  Then, we have the following graded
  commutator identity
  \begin{equation}\label{eq78}
    [D^\prime_\mathcal{X}, \bar \partial]= -\bar \partial\left(\frac{\partial \widetilde s}{\widetilde s}\right)\wedge
  \end{equation}
 \end{lemme}
\begin{proof}
  Direct computation that we skip. We note that the symbol $\displaystyle
  \bar \partial\left(\frac{\partial \widetilde s}{\widetilde s}\right)$ which appears in \eqref{eq78}
  is a global $(1,1)$ form on the complement of the set $\widetilde s= 0$, as we now explain. The global section $\widetilde s$ corresponds to local smooth functions denoted by $\widetilde s_i$ such that we have
  \begin{equation}\label{exp_1}
\widetilde s_i= g_{ij}\widetilde s_j
  \end{equation}
  on overlapping subsets of $\mathcal{X}$. Then we have
\begin{equation}\label{exp_2}
\frac{\partial \widetilde s_i}{s_i}= \frac{\partial g_{ij}}{g_{ij}}+ \frac{\partial \widetilde s_j}{s_j}
  \end{equation}
and since $\displaystyle \bar \partial\left(\frac{\partial g_{ij}}{g_{ij}}\right)= 0$, we obtain
\begin{equation}\label{exp_3}
\bar \partial \left(\frac{\partial \widetilde s_i}{s_i}\right)= \bar \partial\left(\frac{\partial \widetilde s_j}{s_j}\right)\end{equation}
which proves our claim. We note that if $\widetilde s$ is holomorphic, then
the $(1,1)$-from is simply zero on $\mathcal{X} \setminus Z$.
\end{proof}
\smallskip

\begin{remark} We note that in general, the differential operator $\nabla$ defined in \eqref{eq73}
does not coincides with the Chern connection induced by the metric corresponding to
$\widetilde s$. This is of couse the case if $\widetilde s$ is holomorphic.
\end{remark}
\smallskip

\begin{remark}
As in the usual case, a smooth section $\widetilde s$ of the bundle $mL$ induces a connection on $L$ itself by a slight modification of the construction above that is to say, by multiplication with a constant, cf. next subsection.
\end{remark}

\subsection{Lie derivative and commutation relations}
Let
\begin{equation}\label{eq16}
  s\in H^0(X, m K_X)
\end{equation}
be a holomorphic section of the pluricanonical bundle, where $m\geq 1$ is a positive integer.

Let $\widetilde s$ be arbitrary smooth extension of the section $s$. As already hinted, we can use the section $\widetilde s$ in order to
define a connection on the bundle
$L:= (m-1) K_{\mathcal{X}/\mathbb D}$ restricted to the complement of the set of zeros of
$\widetilde s$. The local differential operators corresponding to $D^\prime_\mathcal{X}$
are given by
\begin{equation}\label{eq200}
  D^\prime_t(\sigma):= \partial_t \sigma- \frac{m-1}{m}
    \frac{\partial_t \widetilde s_i}{\widetilde s_i}\sigma, \quad D_\alpha^{\prime}(\sigma):=
    \partial_\alpha \sigma- \frac{m-1}{m}
  \frac{\partial_\alpha \widetilde s_i}{\widetilde s_i}\sigma.
\end{equation}
where $\partial_t:= \partial/\partial t$ and
$\partial_\alpha:= \partial/\partial z_i^\alpha$
for each $\alpha=1, \dots, n$. In \eqref{eq200} the symbol $\sigma$ represents a local section of the bundle $\displaystyle (m-1)K_{\mathcal{X}/D}$. The sum
$$D^\prime_t(\sigma)dt+ \sum_\alpha D_\alpha^{\prime}(\sigma)dz_i^\alpha$$ corresponds to a global connection of $(1, 0)$ type on $L:= (m-1)K_{\mathcal{X}}$. If $\widetilde s$ would have been holomorphic, then this is nothing but the Chern connection; in any case, this is well (i.e. globally) defined. It is clear that we have
\begin{equation}
D' _\mathcal{X} \circ D' _\mathcal{X} =0 .
\end{equation}

We consider next a vector field $\displaystyle \Xi$ on the total space $\mathcal{X}$
which projects into $\frac{\partial}{\partial t}$ on $\mathbb D$.
It can be written in local coordinates as follows
\begin{equation}\label{eq18}
  \Xi|_{U_i}= \frac{\partial}{\partial t}+ \sum_{\alpha=1}^n
  v_i^\alpha\frac{\partial}{\partial z_i^\alpha}
\end{equation}
where $v_i^\alpha$ for $i=1,\dots, n$ are smooth functions.
\medskip

\noindent Our vector field induces a Lie derivative operator $\mathcal{L}_\Xi$ as follows
\begin{equation}\label{eq19}
\mathcal{L}_\Xi(\sigma):= D'_\mathcal{X}(\Xi\rfloor \sigma)
\end{equation}
where $\sigma$ is any $(n+1, q)$-form with values in $L$. The result $\mathcal{L}_\Xi(\sigma)$
is a form of the same type as $\sigma$, and it will play an important role in what follows.
\medskip

\noindent Let $i_\Xi$ be the operator of degree (-1, 0) given by the contraction with the vector field $\Xi$. Since we have $\mathcal{L}_{\Xi}= [D' _{\mathcal{X}} , i_{\Xi}]$ on $(n+1, q)$-differential forms, the following Jacobi identity
$$
[\bar \partial, \mathcal{L}_{\Xi}] +[D' _{\mathcal{X}}, [i_{\Xi}, \bar \partial]] + [i_\Xi, [\bar \partial, D' _{\mathcal{X}}]]= 0$$
holds true, and therefore we obtain the next formula over $(n+1, q)$-forms
\begin{equation}\label{eq20}
[\bar \partial, \mathcal{L}_{\Xi}]= -D' _{\mathcal{X}} \circ  (\bar \partial \Xi)\rfloor  -  [i_\Xi, [\bar \partial, D' _{\mathcal{X}}]].
\end{equation}
\medskip

\noindent Our next statement is playing an important role in the ``algebra" part of the proof of our main results.

\begin{lemme}\label{Komm} Let $\rho$ be a $(n-1, 1)$-form with values in $L$ on $\mathcal{X}$. Then we have the equality
\begin{equation}\label{eq21}
\mathcal{L}_\Xi \big(D' _\mathcal{X} (dt\wedge \rho)\big) = D' _\mathcal{X} \left( dt\wedge \Big( \Xi  \rfloor D' _\mathcal{X} \big(\Xi \rfloor (dt \wedge \rho)\big) \Big)\right)
\end{equation}
on $\mathcal{X}$.
\end{lemme}
\begin{proof}
The argument is quite clear; we start with the left hand side and we remark that we have
\begin{equation}\label{eq22}
D' _\mathcal{X} (dt\wedge \rho)= -dt\wedge D' _\mathcal{X} (\rho)
\end{equation}
and so
\begin{equation}\label{eq23}
\Xi\rfloor \big(D' _\mathcal{X} (dt\wedge \rho)\big)= - D' _\mathcal{X} (\rho)+ dt\wedge\big(\Xi\rfloor D' _\mathcal{X} (\rho)\big)
\end{equation}
since the contraction with the vector field $\Xi$ is a derivation. The LHS of \eqref{eq21} is therefore equal
to
\begin{equation}\label{eq24}
-dt\wedge D' _\mathcal{X}\big(\Xi\rfloor D' _\mathcal{X} (\rho)\big).
\end{equation}

\noindent For the RHS, we have
\begin{equation}\label{eq25}
\Xi \rfloor (dt \wedge \rho)= \rho- dt \wedge (\Xi \rfloor \rho)
\end{equation}
so
\begin{equation}\label{eq26}
D' _\mathcal{X} \big(\Xi \rfloor (dt \wedge \rho)\big)= D' _\mathcal{X}\rho+ dt \wedge D' _\mathcal{X}(\Xi \rfloor \rho).
\end{equation}
A contraction with $\Xi$ gives
\begin{equation}\label{eq27}
\Xi\rfloor \Big(D' _\mathcal{X} \big(\Xi \rfloor (dt \wedge \rho)\big)\Big)\equiv \Xi\rfloor \big(D' _\mathcal{X}\rho\big)+ D' _\mathcal{X}(\Xi \rfloor \rho)
\end{equation}
modulo a term in $dt\wedge \cdot$, so we have
\begin{equation}\label{eq28}
dt\wedge \left(\Xi\rfloor \Big(D' _\mathcal{X} \big(\Xi \rfloor (dt \wedge \rho)\big)\Big)\right)= dt\wedge \Big(\Xi\rfloor \big(D' _\mathcal{X}\rho\big)\Big)+ dt\wedge D' _\mathcal{X}(\Xi \rfloor \rho)
\end{equation}
and a further derivative of \eqref{eq28} shows that the LHS of \eqref{eq21} equals
\begin{equation}\label{eq29}
-dt\wedge D' _\mathcal{X}\big(\Xi\rfloor D' _\mathcal{X} (\rho)\big),
\end{equation}
so our statement is proved. In the argument just finished, we have used many times the fact that $D' _\mathcal{X}\circ D' _\mathcal{X}= 0$.
\end{proof}

\section{A few results from $L^2$ Hodge theory}\label{hodge}

\subsection{Hodge decomposition for metrics with Poincaré singularities}
Let $X$ be a $n$-dimensional compact Kähler manifold, and let $(L, h_L)$ be a
line bundle  endowed with a (singular) metric $h_L= e^{-\varphi_L}$ satisfying the following hypothesis:
\begin{enumerate}
\smallskip

\item[(a)] $h_L$ has log poles, i.e. its local weights can be written as follows
\begin{equation}\nonumber
\varphi_L\equiv \sum a_i\log |f_i|^2
\end{equation}
modulo a smooth function, where $a_i$ are positive real numbers and $f_i$ are holomorphic functions.
\smallskip

\item[(b)] The curvature current $\displaystyle \Theta_{h_L}(L)$
corresponding to the metric $h_L$ is equal to zero in the complement of the set
$(h_L= \infty)$.
\end{enumerate}

\noindent We consider a modification $\pi: \widehat X\to X$ of $X$ such that the support of the singularities of
$\varphi_L\circ \pi$ is a simple normal crossing divisor $E$. As usual, we can construct $\pi$ such that its restriction to $\widehat X\setminus E$ is an biholomorphism . We write
\begin{equation}\label{eq30}
\varphi_L\circ \pi|_{\Omega}\equiv \sum_{\alpha=1}^p e_\alpha\log |z_\alpha|^2
\end{equation}
modulo a smooth function. Here $\Omega\subset \widehat X$ is a coordinate subset, and
$(z_\alpha)_{\alpha=1,\dots, n}$ are coordinates such that
$E\cap \Omega= (z_1\dots z_p= 0)$.

\noindent Let $\widehat \omega_E$ be a complete K\"ahler metric on $\widehat X\setminus E$,
with Poincar\'e singularities along $E$, and let
\begin{equation}\label{eq31}
\omega_E:= \pi_\star(\widehat \omega_E)
\end{equation}
be the direct image metric. We note that in this way $(X_0, \omega_E)$ becomes a
complete K\"ahler manifold, where $X_0:= X\setminus (h_L= \infty)$.
\smallskip

\noindent The following statement is well-known, but we recall the precise version we need here.
We denote by $s$ the section of a line bundle for which the support of its zero divisor coincides with the support of $(h_L= \infty)$.
\begin{lemme}\label{cutoff}
There exist a family of smooth functions $(\mu_\varepsilon)_{\varepsilon> 0}$ with the following properties.
\begin{enumerate}

\item[\rm (a)] For each $\varepsilon> 0$, the function $\mu_\varepsilon$ has compact support in $X_0$, and
$0\leq \mu_\varepsilon\leq 1$.

\item[\rm (b)] The sets $(\mu_\varepsilon= 1)$ are providing an exhaustion of $X_0$.

\item[\rm (c)] There exists a positive constant $C> 0$ such that
we have $$\displaystyle \sup_{X_0}\left(|\partial \mu_\varepsilon|_{\omega_E}^2+ |\partial\bar{\partial} \mu_\varepsilon|^2_{\omega_E}\right)\leq C.$$
\end{enumerate}
\end{lemme}

\begin{proof}
It is enough to obtain the corresponding statement on $\widehat X$, so that the divisor $E$ is snc. Then we take
\begin{equation}\label{cut1}
\mu_\varepsilon= \rho_\varepsilon\Big(\log\log\frac{1}{|\widehat s|^2}\Big)
\end{equation}
where $\rho_\varepsilon$ is equal to 1 on the interval $[1, \varepsilon^{-1}]$ and is equal to zero on
$[1+ \varepsilon^{-1}, \infty]$. Also, we denote by $|\widehat s|$ the inverse image of the norm
of the section $s$ with respect to an arbitrary, smooth metric.
Actually we can also impose the condition that
\begin{equation}\label{cut2}
\max_{j= 1,\dots N}\sup_{\varepsilon> 0}\sup_{\mathbb{R}_+}|\rho^{(j)}_\varepsilon|< C_N<\infty,
\end{equation}
for any positive integer $N$.
The properties (a)-(c) are then verified by a simple computation. In what follows we will use this statement for
$$\displaystyle L:= (m-1)K_X, \qquad \varphi_L:= \frac{m-1}{m}\log|f|^2$$
and $s$ the pluricanonical section we have to extend (the holomorphic function $f$ above corresponds locally to $s$).
\end{proof}

\medskip

\noindent In this context we have the following statement, which is a slight
generalisation of the usual result in $L^2$ Hodge theory.

\begin{thm}\label{Hodge}
Consider a line bundle $(L, h_L)\to X$ endowed
with a metric $h_L$ such that the requirements {\rm (a)} and {\rm (b)} above are satisfied, as well as
the corresponding complete K\"ahler manifold $(X_0, \omega_E)$,
cf. \eqref{eq31}.

\noindent Then the following assertions are true.
  \begin{enumerate}
  \item[\rm (i)] We have the following Hodge decompositions
  $$
  L^2_{n, 1}(X_0, L)= {\mathcal H}_{n,1}(X_0, L)\oplus
  \mathrm{Im} \, \bar \partial\oplus \mathrm{Im} \, \bar \partial^\star   ,$$
  and
  $$
  L^2_{n, 0}(X_0, L)= {\mathcal H}_{n,0}(X_0, L) \oplus \mathrm{Im} \, \bar \partial^\star   .
  $$

  \item[\rm (ii)] Let $u$ be a $\bar \partial$-closed $L^2$-form of type $(n,1)$ on $X_0$, and
    assume that we
    have $u= D^\prime w$, where $w\in L^2_{n-1, 1}(X_0, L)$.  Then, there exists $v\in L^2_{n, 0}(X_0, L)$ such
    that $u= \bar \partial v$.
  \end{enumerate}
\end{thm}

\medskip

\noindent The proof of Theorem~\ref{Hodge}, which we give below, makes use of the
following statement which is the $\bar \partial$-version of the Poincar\'e inequality established in \cite{Auv17}.

\begin{thm}\label{prop:2}
  Let $p\leq n$ be an integer.  There exists a positive constant $C> 0$ such
  that the following inequality holds
  \begin{equation}\label{eq32}
    \int_{X_0}|u|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}\leq C \int_{X_0}|\bar \partial u|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}
  \end{equation}
  for any $L$-valued form $u$ of type $(p, 0)$ which belongs to the domain of $\bar \partial$ and which is orthogonal to the
  space of $L^2$ harmonic $(p,0)$--forms.
\end{thm}
\begin{proof}[Proof of Theorem~\ref{prop:2}]
The first observation is that we can assume from the very beginning that the polar set
$h_L= \infty$ has snc support. Indeed, via the map $\pi$ the hypothesis and the conclusion of
our Poincar\'e inequality transform as follows.
\smallskip

\noindent $\bullet$ The form
\begin{equation}
\widehat u= \pi^\star u
\end{equation}
on $\widehat X$ is $\pi^\star L$-valued and $L^2$ with respect to $\widehat \omega_E$ and $\pi^\star h_L$.
\smallskip

\noindent $\bullet$ The form $\widehat u$ is orthogonal on the space of $L^2$--harmonic $(p,0)$ forms with values in $\pi^\star L$. Indeed, given any $\pi^\star L$-valued form of
$(p, 0)$-type $\widehat \gamma$ on $\widehat X\setminus E$ there exists a form $\gamma$ on
$X\setminus Z$ such that
\begin{equation}
\widehat \gamma= p^\star \gamma
\end{equation}
because $p:\widehat X\setminus E\to X\setminus Z$ is a biholomorphism. Also, $p$ establish an
isometry between $(\widehat X\setminus E, \widehat \omega_E)$
and $(X\setminus Z, \omega_E)$ respectively. The same is true for the pairs
$(p^\star L, p^\star h_L)$ and $(L, h_L)$, so our claim is clear.
\medskip

\noindent This means that it is enough to assume that $\widehat X= X$, i.e. the metric $\omega_E$ has
Poincar\'e singularities along an snc divisor $E$ which coincides with the support of the analytic set $h_L= \infty$.
\smallskip

\noindent The following statement represents an important step towards the proof of Theorem \ref{prop:2}.

\begin{proposition}\label{poinc} Let $\displaystyle (\Omega_j)_{j=1,\dots, N}$ be a finite union of coordinate sets of $X$ covering $E$, and let $U$ be any open subset contained in their union.
Let $\tau$ be a $(p, 0)$-form with compact support
in a set $U\setminus E\subset X$ and values in $(L, h_L)$. Then we have
\begin{equation}\label{eq57}
\frac{1}{C}\int_U|\tau|^2e^{-\varphi_L}dV_{\omega_E}\leq \int_U|\bar \partial\tau|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}
\end{equation}
where $C$ is a positive numerical constant.
\end{proposition}
\medskip

\noindent The arguments in \cite{Auv17}, Lemma 1.10 correspond to the case $p=0$ and $\varphi_L=0$. For the general case we need the
following auxiliary statement.

\begin{lemme}\label{fourier} Let $\delta> 0$ be a positive real; then there exists $x_\delta< 1$ such that the following holds.
For any smooth function $f$ defined on the closure of the unit disk $\mathbb D$ in $\mathbb C$, such that
there exists $\varepsilon> 0$ with the property that $f(z)= 0$ for all $|z|\leq \varepsilon$ or $|z|\geq 1- x_\delta$ we have
\begin{equation}\label{d1}\int_{\mathbb D}|f|^2 \frac{d\lambda(z)}{|z|^{2\delta}} \leq C \int_{\mathbb D}\left| \frac{\partial f }{\partial \overline z}\right|^2 |z|^{2- 2\delta} \log^2 |z| d\lambda(z)
\end{equation}
as well as
\begin{equation}\label{d2}
\int_{\mathbb D}|f|^2 \frac{d\lambda(z)}{|z|^{2+ 2\delta}\log^2 |z|} \leq C \int_{\mathbb D}\left| \frac{\partial f }{\partial \overline z}\right|^2 \frac{d\lambda(z)}{|z|^{2\delta}}.
\end{equation}
\end{lemme}
\medskip

\noindent We present next the arguments for \ref{fourier}.

\begin{proof} Let $x_\delta< 1$ be a positive real number whose expression will be determined in what follows. Let $v:[0,1]\to \mathbb C$ be a smooth function whose support is contained in the interval $[\varepsilon, x_\delta]$ for some positive, small enough $\varepsilon$.

\noindent We claim that the following inequalities hold
\begin{equation}\label{cc1}
\int_0^1 |v|^2 t^kdt \leq C \int_0^1 |v' |^2 t^{k+2}\log^2t dt
\end{equation}
as well as
\begin{equation}\label{c1bis}
\int_0^1 |v|^2 t^k\frac{dt}{\log^2t} \leq C \int_0^1 |v' |^2 t^{k+2}dt
\end{equation}
where $k$ is any element of the set $\delta+ \mathbb Z$, and $C$ is a numerical constant (in particular independent of $v, \varepsilon, k, ...$).

\noindent We first check that \eqref{cc1} holds true: we have
\begin{equation}\label{e1}
-\int_0^1 |v|^2 t^{k+1}\log^2 d\left(\frac{1}{\log t}\right)= \int_0^1\frac{1}{\log t} \frac{d}{dt}\left(v^2 t^{k+1}\log^2t\right)dt
\end{equation}
and this equals
\begin{equation}\label{e2}
2\int_0^1(\overline v v'+ v\overline{v'}) t^{k+1}\log tdt + \int_0^1\frac{t^kv^2}{\log t}\left((k+1)\log^2t+ 2\log t\right)dt.
\end{equation}
Now two things can happen: either $k+1=0$, and then the second term in \eqref{e2} is equal to twice the quantity we are interested in, or $k+1\neq 0$. Note that in the latter case it is possible to fix $x_\delta$ uniformly with respect to $k$ so that
\begin{equation}\label{e201}|(k+1)\log^2t+ 2\log t | \geq 2 |\log t|\end{equation}
for any $t\in [0, x_\delta]$ precisely because $k\in \delta+ \mathbb Z$.

\noindent As a consequence, we have
\begin{equation}\label{e200}
\left|\int_0^1 (\overline v v'+ v\overline{v'}) t^{k+1}\log tdt\right| \geq  \int_0^1 t^kv^2 dt.
\end{equation}
for any $v$ of compact support in $]0, x_\delta[$.

\noindent Moreover, we have
\begin{equation}\label{e3}
\left|\int_0^1(\overline v v'+ v\overline{v'}) t^{k+1}\log tdt\right|^2 \leq \int_0^1|v'|^2 t^{k+2}\log^2tdt \int_0^1|v|^2 t^{k}dt.
\end{equation}
by Cauchy-Schwarz, so the inequality \eqref{cc1} is settled.
\medskip

\noindent The arguments for \eqref{c1bis} are completely similar: we have
\begin{equation}\label{c11}
\int_0^1 |v|^2 t^k\frac{dt}{\log^2t}=  \int_0^1 |v|^2 \frac{t^{k+1}}{\log^2 t}d\log t
\end{equation}
which equals
\begin{equation}\label{c12}
-\int_0^1 {\log t}\frac{d}{dt}\left(|v|^2 \frac{t^{k+1}}{\log^2 t}\right)dt= \int_0^1
\frac{|v|^2t^{k}}{\log^2t}\left(2+ (k+1)\log \frac{1}{t}\right) dt -
\int_0^1\frac{t^{k+1}}{\log t}(v'\overline v+ v\overline {v'})dt.
\end{equation}
and two things can happen. If $k+1\geq 0$, then we have $\displaystyle 2+ (k+1)\log \frac{1}{t}\geq 2$ for any $t\in [0,1]$ and \eqref{c1bis} follows.
For each element $k\in \delta+ \mathbb Z$ such that $k+1< 0$ we can assume that
\begin{equation}\label{e2011}(k+1)\log \frac{1}{t}+ 2 \leq \frac{1}{2}\end{equation}
for any $t\in [0, x_\delta]$ by imposing a supplementary (but uniform) condition on $x_\delta$ if necessary, and we proceed exactly as we did for \eqref{cc1}.

\smallskip
Therefore, we obtain \eqref{cc1} and \eqref{c1bis} for any
$k\in \delta+ \mathbb{Z}$.
\medskip

\noindent  Lemma \ref{fourier} follows from this: we consider the Fourier series
\begin{equation}\label{c5}
f=\sum_{k\in \mathbb Z}a_k(t)e^{\sqrt{-1}k\theta}
\end{equation}
of $f$, and then we have
\begin{equation}\label{c6}
\frac{\partial v }{\bar \partial z}= \sum_{k\in \mathbb Z}\left(a_k'(t)- \frac{k}{t}a_k(t)\right)e^{\sqrt{-1}k\theta}.
\end{equation}
The identity
\begin{equation}\label{c7}
t^k\frac{d}{dt}\left(\frac{a_k}{t^k}\right)= a_k'(t)- \frac{k}{t}a_k(t)
\end{equation}
reduces the proof of our statement to inequalities of type \eqref{cc1} and \eqref{c1bis} which are already established. In conclusion, Lemma \ref{fourier} is completely established.
\end{proof}
\medskip

\noindent We now return to the proof of Proposition \ref{poinc}. We consider the restriction of the form
$\tau$ to some co-ordinate open set $\Omega$ whose intersection with $E$ is of type $z_1\dots z_r= 0$. It can be written as sum of forms of type
\begin{equation}\label{d111}
\tau_I:= f_Idz_I
\end{equation}
and assume that $I\cap \{1,\dots, r\}= \{1,\dots, p\}$ for some $p$ or $I\cap \{1,\dots, r\}= \emptyset.$

\noindent In the first case we have
\begin{equation}\label{d222}
\frac{|\tau_I|_g^2e^{-\varphi_L}dV_g}{d\lambda}= \frac{|f_I|^2e^{-\varphi_L}}{\prod_{\alpha=p+1}^r|z_\alpha|^2\log^2|z_\alpha|^2}
\end{equation}
where $\displaystyle e^{-\varphi_L}= \frac{1}{\prod_{i=1}^r|z_i|^{2\delta_i}}$
and in the second case this is the same with $p=0$.
In \eqref{d222} we denote by $g$ the model Poincar\'e metric
$$\sum_{i=1}^r\frac{\sqrt{-1}dz_i\wedge d\overline z_i}{|z_i|^2\log^2|z_i|^2}+ \sum_{i=r+1}^n{\sqrt{-1}dz_i\wedge d\overline z_i}.$$
\smallskip

\noindent Now for the $\bar \partial \tau_I$ we have
\begin{align}
\frac{|\bar \partial \tau_I|_g^2e^{-\varphi_L}dV_g}{d\lambda}= & \sum_{i=1}^r\left|\frac{\partial f_I}{\partial \overline z_i}\right|^2\frac{|z_i|^2\log^2|z_i|^2e^{-\varphi_L}}{\prod_{\alpha=p+1}^r|z_\alpha|^2\log^2|z_\alpha|^2}\nonumber \\ + & \label{d3} \sum_{i=r+1}^n\left|\frac{\partial f_I}{\partial \overline z_i}\right|^2\frac{e^{-\varphi_L}}{\prod_{\alpha=p+1}^r|z_\alpha|^2\log^2|z_\alpha|^2} \\
\nonumber
\end{align}
and the observation is that for every $\xi\in \Omega$ there exist some index $j\in \{1,\dots, r\}$ for which
the support condition we require in Lemma \ref{fourier} is satisfied for
\begin{equation}\label{d4}
f(z):= f_I(\xi_1,\dots \xi_{j-1}, z, \xi_{j+1},\dots \xi_n).
\end{equation}
We use \eqref{cc1} or \eqref{c1bis} according to the case where $j\geq p$ in \eqref{d3} or not.
Proposition \ref{poinc} is therefore completely proved.

\bigskip

\noindent The constant $C$ in \eqref{eq57} only depends on the distortion between the model Poincar\'e metric on $\Omega_j$ with singularities on $E$ and
the global metric $\omega_E$ restricted to $\Omega_j$.
Another important observation is that by using the cut-off function $\mu_\varepsilon$ in Lemma \ref{cutoff}, we infer that \eqref{eq57} holds in fact for any $L^2$-bounded form with compact support in $U$.
\smallskip

\noindent The rest of the proof follows the arguments in \cite{Auv17}, proof of Lemma 1.10: if a positive constant as in \eqref{eq32} does not exists, then we obtain a sequence $u_j$ of
$L$-valued forms of type $(p,0)$ orthogonal to the space of holomorphic forms such that
\begin{equation}\label{eq61}
\int_X|u_i|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}= 1, \qquad
\lim_i\int_X|\bar \partial u_i|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}= 0
\end{equation}
It follows that the weak limit $u_\infty$ of $(u_i)$ is holomorphic. On the other hand, each $u_i$
is perpendicular to the space of holomorphic forms, so it follows
that $u_\infty$ is equal to zero. Moreover, by Bochner formula combined with Proposition \ref{poinc} we can assume that $u_i$
converges weakly in the Sobolev space $W^1$, so strongly in $L^2$ to zero. In particular we have
\begin{equation}\label{eq611}
 u_i |_K\to 0
 \end{equation}
in $L^2$ for any compact subset $K\subset X\setminus E$.

The last step in the proof is to notice that the considerations above contradict the fact that the $L^2$ norm of each $u_i$ is equal to one. This is not quite immediate, but is precisely as the end of the proof of Lemma 1.10 in
\cite{Auv17}, so we will not reproduce it here. The idea is however very clear:
we decompose each $u_j= \chi u_j+ (1-\chi)u_j$ where $\chi$ is a cutoff function which is equal to 1 in $U$ and whose norm of the corresponding gradient is small. Then the $L^2$ norm of $\chi u_j$ is small
by \eqref{eq57} and \eqref{eq61}. The $L^2$ norm of $(1-\chi)u_j$ is equally small
by \eqref{eq611}, and this is how we reach a contradiction.
\end{proof}
\medskip

\noindent We have the following direct consequence of Proposition \ref{prop:2}.

\begin{cor}\label{usecor} There exists a positive constant $C> 0$ such that the following holds true.
Let $v$ be a $L$-valued form of type $(n,q)$ for some $q\geq 1$. We assume that $v$ is $L^2$, and orthogonal
to the kernel of the operator $\bar \partial^\star$. Then we have
\begin{equation}\label{eq62}
 \int_{X_0}|v|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}\leq C \int_{X_0}|\bar \partial^\star v|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}.
\end{equation}
\end{cor}
\smallskip

\noindent Indeed, the inequality \eqref{eq62} follows immediately from the observation that
the Hodge star $u:= \star v$ is orthogonal to the space of holomorphic forms (here we use again the fact that the curvature of $(L, h_L)$ equals zero when restricted to $X_0$). This can be seen as follows:
let $F$ be a $L^2$-holomorphic $(n-2)$-form. Then we have
$$D'F= 0$$
and therefore $\displaystyle \bar \partial^\star(\star F)= 0$, in other words the Hodge dual $\star F$ of $F$ belongs to the kernel of $\bar \partial^\star$.
We thus have
$$\int_{X_0} \langle u, F\rangle e^{-\varphi_L} d V_{\omega_E} = \int_{X_0} \langle v, \star F \rangle e^{-\varphi_L} d V_{\omega_E} =0.$$
Note that we have
\begin{equation}\label{eq63}
 \int_{X_0}|\bar \partial u|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}=
 \int_{X_0}|D'^\star v|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}\leq
 \int_{X_0}|\bar \partial^\star v|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}
\end{equation}
where the inequality in \eqref{eq63} is obtained by Bochner formula. This proves the
corollary by applying Proposition \ref{prop:2}.
\medskip

\begin{proof}[Proof of Theorem~\ref{Hodge}]
	This statement is almost contained in \cite[chapter VIII, pages
	367-370]{bookJP}. Indeed, in the context of complete manifolds one has the
	following decomposition
	\begin{equation}\label{eq33}
	L^2_{n, 1}(X_0, L)= {\mathcal H}_{n,1}(X_0, L)\oplus \overline{\mathrm{Im} \, \bar \partial}\oplus \overline{\mathrm{Im} \, \bar \partial^\star} ,
	\end{equation}
	and
	\begin{equation}\label{eq331}
	L^2_{n, 0}(X_0, L)= {\mathcal H}_{n,0}(X_0, L)\oplus  \overline{\mathrm{Im} \, \bar \partial^\star} ,
	\end{equation}
	We also know (see \emph{loc. cit.}) that the adjoints $\bar \partial^\star$ and
	$D^{\prime\star}$ in the sense of von Neumann coincide with the formal
	adjoints of $\bar \partial$ and $D^\prime$ respectively.

	It remains to show that the ranges of the $\bar \partial, \bar \partial^\star$-operators are closed with
	respect to the $L^2$ topology. In our set-up, this is a consequence of the
	particular shape of the metric $\omega_E$ at infinity (i.e. near the support
	of $\pi (E)$).  The Poincar\'e inequality~\eqref{eq32} of Proposition~\ref{prop:2} implies
	that the image of $\bar \partial$ is closed. The inequality \eqref{eq62} implies that  image of $\bar \partial^\star$ is closed.

	The assertion $(ii)$ is an easy consequences of $(i)$, together with standard considerations.
\end{proof}

\medskip

\begin{remark}
Actually the first point of Theorem \ref{Hodge} holds provided that we only impose the condition
$\Theta(L, h_L)\geq 0$. We will come back to this matter in a forthcoming paper.
\end{remark}

\noindent In the setting of the current section, the harmonic $(n,1)$-forms have the following properties.

\begin{lemme}\label{vanishing} We assume that $(X_0, \omega_E)$ and $(L, h_L)$ have the properties {\rm (a)} and {\rm (b)} (at the beginning of this section).
Let $\alpha$ be a $\Delta^{''}$-harmonic $L^2$ form of type $(n,1)$ with values in $L$. We denote by $F$ its Hodge dual. Then the following hold.

	\begin{enumerate}

		\item[\rm (1)] We have
		\begin{equation}\nonumber
		\bar \partial F=0\qquad \text{and } \qquad D'  F=0     .
		\end{equation}

		\item[\rm (2)] Let $\Omega\subset X$ be a coordinate open set, and let $f$ be any holomorphic function defined on $\overline \Omega$. We assume that the support of the set of zeroes of $f$ is contained in
		$(h_L = \infty)$.
		Then the form $\displaystyle F\wedge \frac{df}{f}$ is non-singular on $X$ (but it is not clear whether this form is $L^2$ with respect to $e^{-\varphi_L}$ or not).

		\item[\rm (3)] Let $\Omega\subset X$ be a coordinate open set, and let $g$ be any holomorphic function defined on $\overline \Omega$. Then we have
		\begin{equation}\nonumber
		\int_\Omega |F\wedge dg|^2e^{-\varphi_L}< \infty.
		\end{equation}
	\end{enumerate}
\end{lemme}

\begin{proof}

\noindent Note that given any $\Delta^{''}$-harmonic $(n,1)$-form $\alpha$,
by Bochner formula combined with the fact that the curvature of $(L, h_L)$ is semi-positive,
we have
\begin{equation}\label{snc8-}
D '^{\star} \alpha =0  \qquad\text{on } X \setminus (h_L = \infty) .
\end{equation}
So if we write $\displaystyle \alpha = \omega_E \wedge F$, it follows that
\begin{equation}\label{dps}
F \in H^0 \big(X, \Omega_X ^{n-1} \otimes K_X ^{\otimes (m-1)}\big),
\qquad \int_X|F|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}< \infty
\end{equation}
thanks to the property \eqref{snc8-}.
Moreover, as $\displaystyle \bar \partial^* \alpha= 0$ it turns out that we have $D'  F=0$ as well.
We obtain thus $(1)$ and $(3)$.
Note that \eqref{dps} is valid in a more general setting,
cf. \cite{DPS01}, \cite{Wu19}.
\bigskip

\noindent As for $(2)$, by our choice of the metric $\omega_E$, we show first that we have
\begin{equation}\label{snc10}
|df|^2 _{\omega_E} \sim  |f|^2 \log^2 |f|.
\end{equation}
Indeed, assume that via the bimeromorphism $\pi: \widehat X \rightarrow X$ the function $f$ corresponds locally to
the function $\displaystyle \prod_{i=1}^r z_i^{p_i}$ (notations as in the beginning of this section). Then the metric $\omega_E$
is quasi-isometric to the Poincar\'e metric with singularities along $z_1\dots z_r= 0$. The $\pi$-inverse image of $\displaystyle\frac{df}{f}$ is equal to
$$\sum_{i=1}^rp_i\frac{dz_i}{z_i}
$$
and thus we have
$$\left|\frac{\partial f}{f}\right|^2_{\omega_E}\simeq \sum p_i^2\log^2|z_i|^2$$
and the RHS of this quantity is the same as $\log^2|f|^2$.

\noindent Therefore we infer
\begin{equation}\label{snc11}
\int_{\Omega}  \left|F\wedge \frac{df}{f}\right|^2\frac{e^{-\varphi_L}}{\log^2 |f|} \leq
C\int_X {|F |^2_{\omega_E} }{}e^{-\varphi_L}dV_{\omega_E}.
\end{equation}
Since the RHS of \eqref{snc11} is finite and $\varphi_L$ has non zero Lelong number over any component of $Div (f)$, it follows that $F \wedge df$ vanishes along $Div (f)$.
\end{proof}
\medskip

\noindent Consider the equation
\begin{equation}\label{hodge1}
\bar \partial u= D'w+ \Theta(L, h_L)\wedge \tau
\end{equation}
where $\tau, w$ and $D'w$ are $L^2$, and $D'w+ \Theta(L, h_L)\wedge \tau$ are $\bar \partial$-closed. We show next that
one can solve it in a fair general context, and moreover obtain $L^2$ estimates for the solution with minimal norm. We refer to \cite[Thm A.5]{Wan17} for a similar argument in the non-singular case.

\begin{thm}\label{deprim} Let $(L, h_L)$ be a holomorphic line bundle on $X$ with a possible singular metric $h_L$ with analytic singularities along a subvarities $Z \subset X$ and whose curvature current is semi-positive.
Consider a complete K\"{a}hler metric $(X\setminus Z, \omega_E)$.
Let $w$ be an $L$-valued $(n-1, 1)$--form on $X\setminus Z$ such that $w$ and $D'w$ are in $L^2$. Let also $\tau$ be an $L$-valued $(n-1, 0)$--form on $X\setminus Z$ such that $\tau$
and $\Theta(L, h_L)\wedge \tau$ are $L^2$.
Then there exists a solution $u$ of the equation \eqref{hodge1} such that
\begin{equation}\label{hodge2}
\int_X|u|^2e^{-\varphi_L}\leq \int_X|w|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}
- \int_X\langle[\Theta(L, h_L), \Lambda_{\omega_E}]\tau, \tau \rangle_{\omega_E}e^{-\varphi_L}dV_{\omega_E}
\end{equation}
\end{thm}
\begin{proof}
  Let $\xi$ be a smooth $(n,1)$-form of compact support in $X\setminus Z$. Let $\xi =\xi_1 +\xi_2$ be its decomposition according to $\mathrm{Ker}(\bar \partial)$ and its orthogonal.

\noindent By using the $L^2$-assumptions, we have
$$ \int_{X\setminus Z}\langle D'w+ \Theta(L, h_L)\wedge \tau , \xi \rangle_{\omega_E}e^{-\varphi_L}dV_{\omega_E} =
 \int_{X\setminus Z}\langle D'w+ \Theta(L, h_L)\wedge \tau , \xi_1 \rangle_{\omega_E}e^{-\varphi_L}dV_{\omega_E} .$$
Then the semipositive curvature condition and the Bochner equality imply that
\begin{equation}\label{hodge200} \left|\int_{X\setminus Z}\langle D'w , \xi_1 \rangle_{\omega_E}e^{-\varphi_L}dV_{\omega_E} \right|^2\end{equation}
\noindent is smaller than

\begin{equation}\label{hodge201}\int_X|w|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E} \cdot
  \int_X|\bar \partial^\star \xi|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}.
\end{equation}
 For the term containing the curvature of $(L, h_L)$ we are using Cauchy inequality and we infer that
 \begin{equation}\label{hodge202}\left|\int_{X\setminus Z}\langle  \Theta(L, h_L)\wedge \tau , \xi_1 \rangle_{\omega_E}e^{-\varphi_L}dV_{\omega_E}\right|^2\end{equation}
is smaller than

\begin{equation}\label{hodge203}\int_X\langle -[\Theta(L, h_L), \Lambda_{\omega_E}]\tau, \tau \rangle_{\omega_E}e^{-\varphi_L}dV_{\omega_E} \cdot
  \int_X\langle[\Theta(L, h_L), \Lambda_{\omega_E}]\xi_1, \xi_1 \rangle_{\omega_E}e^{-\varphi_L}dV_{\omega_E}.
\end{equation}
We are using again Bochner equality and we infer that
\begin{equation}\label{hodge204}\left|\int_{X\setminus Z}\langle  \Theta(L, h_L)\wedge \tau , \xi_1 \rangle_{\omega_E}e^{-\varphi_L}dV_{\omega_E}\right|^2
\end{equation}
is smaller than
\begin{equation}\label{hodge205}\int_X\langle -[\Theta(L, h_L), \Lambda_{\omega_E}]\tau, \tau \rangle_{\omega_E}e^{-\varphi_L}dV_{\omega_E} \cdot  \int_X|\bar \partial^\star \xi|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}.
\end{equation}
 Then the theorem is proved by the standard $L^2$-estimate argument.
\end{proof}
\medskip

\subsection{Hodge decomposition for metrics with conic singularities}
\smallskip

\noindent As application of the previous results/techniques, we show here that the analog of Theorem
\ref{Hodge} holds true for metrics with conic singularities. Since most of the arguments in the proof of Theorem \ref{Hodge} will be used here, we will skip a few details.

\noindent Let $X$ be a compact K\"ahler manifold, and let $\displaystyle Y= \sum Y_i$ be a snc divisor. We denote by $\omega_{\mathcal{C}}$ a metric with conic singularities along the $\mathbb{Q}$-divisor
\begin{equation}\label{pave28}
\sum_{i\in I}\left(1-\frac{1}{m}\right)Y_i
\end{equation}
where $m$ is a positive integer. By this we mean that locally we have
\begin{equation}\label{pave99}
\omega_\mathcal{C}\simeq \sum_{i=1}^r\frac{\sqrt{-1}dz_i\wedge d\overline z_i}{|z_i|^{2-\frac{2}{m}}}
+ \sum_{i\geq r+1}\sqrt{-1}dz_i\wedge d\overline z_i
\end{equation}
that is to say, $\omega_\mathcal{C}$ is quasi-isometric with the RHS of \eqref{pave99},
where $z_1\dots z_r=0$ is the local equation of the divisor $Y$.
\smallskip

\noindent Let $(L, h_L)$ be a line bundle endowed with a (singular) metric $h_L$ such that the following holds.
\smallskip

\begin{enumerate}
\smallskip

\item[(i)] The singularities of the metric $h_L$ are contained in the support of $Y$, i.e.
  $\displaystyle \varphi_L\simeq \sum q_i\log|z_i|^2$ modulo a smooth function, where $q_i\in \mathbb{Q}_+$ and as above, $z_i$ represent the local equations of $Y$.
\smallskip

\item[(ii)] For each $q_i\in \mathbb{Q}\setminus \mathbb{Z}$, we assume that $\displaystyle
\left\lfloor q_i- \frac{1}{m}\right\rfloor= \lfloor q_i\rfloor$ holds true.

\smallskip

\item[(iii)] The support of the curvature current of $(L, h_L)$ is contained in $Y$,
  i.e. $\Theta(L, h_L)= \sum q_i[Y_i]$.
\end{enumerate}
\medskip

\noindent Our next result states as follows.
\begin{thm}\label{CoHo}
	Let $(L, h_L)\to X$ be a line bundle endowed
	with a metric $h_L$ such that the requirements {\rm (i)}-{\rm (iii)} above are satisfied. Let $\omega_\mathcal{C}$ be the K\"ahler metric with conic singularities above.
	Then we have the following Hodge decomposition
	\begin{equation}\label{pave29}
	L^2_{n, 1}(X_0, L)= {\mathcal H}_{n,1}(X_0, L)\oplus
	\mathrm{Im} \, \bar \partial\oplus \mathrm{Im} \, \bar \partial^\star    ,
	\end{equation}
	and
	\begin{equation}\label{pave299}
	L^2_{n, 0}(X_0, L)= {\mathcal H}_{n,0}(X_0, L)\oplus \mathrm{Im} \, \bar \partial^\star    ,
	\end{equation}
	where $X_0:= X\setminus Y$.

	Moreover, if $\xi \in {\mathcal H}_{n,1}(X_0, L)$ or ${\mathcal H}_{n,0}(X_0, L)$, we have
	\begin{equation}\label{diffdep}
	\bar \partial \xi =\bar \partial^\star \xi =(D')^{\star} \xi =0 .
	\end{equation}
	In particular, $\star \xi$ is a $L^2$-holomorpic form on $X$.
\end{thm}
\medskip

\noindent The proof of Theorem \ref{CoHo} is based on the corresponding version for metrics with Poincar\'e singularities, together with the following result. Let
\begin{equation}\label{pave35}
\omega_{Y, \varepsilon}:= \omega_\mathcal{C}+ \varepsilon \omega_\mathcal{P}
\end{equation}
be a sequence approximating $\omega_\mathcal{C}$, where $\omega_\mathcal{P}$ is a metric with Poincar\'e singularities on $Y$. We have the following statement.

\begin{thm}\label{Poinc}
  Let $p\leq n$ be an integer.  There exists a positive constant $C> 0$ (independent of $\varepsilon$) such
  that the following inequality holds
\begin{equation}\label{pave30}
    \int_{X_0}|u|^2_{\omega_{Y, \varepsilon}}e^{-\varphi_L}dV_{\omega_{Y, \varepsilon}}\leq C \int_{X_0}|\bar \partial u|^2_{\omega_{Y, \varepsilon}}e^{-\varphi_L}dV_{\omega_{Y, \varepsilon}}
  \end{equation}
  for any $L$-valued form $u$ of type $(p, 0)$ which belongs to the domain of $\bar \partial$ and which is orthogonal to the
  space of $L^2$ harmonic $(p,0)$--forms.
\end{thm}

\noindent We first prove Theorem \ref{Poinc}, and then derive Theorem \ref{CoHo} as consequence.

\begin{proof}[Proof of Theorem \ref{Poinc}] The argument for $\varepsilon= 0$ is quite easy, and even if the general case does not follow from it, we give here a direct proof.
\smallskip

\noindent We argue by contradiction: if \eqref{pave30} does not holds, there exists a sequence
$(u_k)$ of $(p, 0)$-forms orthogonal to $\mathrm{Ker}(\bar \partial)$ such that
\begin{equation}\label{pave31}
\int_X|u_k|^2_{\omega_\mathcal{C}}e^{-\varphi_L}dV_{\omega_\mathcal{C}}= 1, \qquad \lim_{k\to\infty}\int_{X_0}|\bar \partial u_k|^2_{\omega_\mathcal{C}}e^{-\varphi_L}dV_{\omega_\mathcal{C}}= 0.
\end{equation}
\noindent It follows that the weak limit $u_0$ of $(u_k)_{k\geq 1}$ has the following properties
\begin{equation}\label{pave32}
\int_X|u_0|^2_{\omega_\mathcal{C}}e^{-\varphi_L}dV_{\omega_\mathcal{C}}\leq 1, \qquad \bar \partial u_0= 0.
\end{equation}
Since on the other hand $u_0$ is orthogonal to $\mathrm{Ker}(\bar \partial)$, it follows that $u_0= 0$. We show next that this contradicts the first equality of \eqref{pave31} for $k$ large enough.

\noindent Let $\Omega$ be a coordinate ball whose origin belongs to $Y$. The restriction
$\displaystyle \bar \partial u_k|_{\Omega\setminus Y}$ is a closed $(p, 1)$-form, but we can also see it as
$(n, 1)$-form with values in the vector bundle $\displaystyle \Lambda^{n-p}T_X\otimes L|_{\Omega}$.
We equip the tangent bundle $\displaystyle E:= T_X|_{\Omega}$ with the standard metric with conic singularities, and $L$ with $e^{-\phi_L}$ where
\begin{equation}\label{pave33}
\phi_L= \varphi_L+ \Vert z\Vert^2+ C\sum |z_i|^{\frac{2}{m_i}}.
\end{equation}
In \eqref{pave33} we denote by $\displaystyle (z_i)_{i=1,\dots, n}$ a coordinate system adapted to $Y$. Then the curvature of $E$ is greater than $\omega_\mathcal{C}\otimes \mathrm{Id}_E$ and by the $L^2$ estimates we can solve the equation
\begin{equation}\label{pave34}
\bar \partial v_{\Omega, k}= \bar \partial u_k, \qquad \int_\Omega |v_{\Omega, k}|^2e^{-\varphi_L}\leq C \int_{X_0}|\bar \partial u_k|^2_{\omega_\mathcal{C}}e^{-\varphi_L}dV_{\omega_\mathcal{C}}.
\end{equation}
Here we are using the fact that local term added to the global metric of $L$ in \eqref{pave33} is two-sided bounded.
\smallskip

\noindent We cover $X$ with a finite subset of coordinate subsets $\Omega$, and we construct the forms $v_{\Omega, k}$ as above (if the set $\Omega$ do not intersects the support of the divisor $Y$, then we just use the flat metric). The holomorphic forms $\displaystyle (u_k- v_{\Omega, k})$ are converging to zero weakly, hence strongly as well. Moreover, the $L^2$ norm of each $v_{\Omega, k}$ tends to zero as $k\to\infty$, so for $k$ large enough the first equality in \eqref{pave31} cannot hold.
The contradiction we obtained ends the proof of the case $\varepsilon= 0$.
\medskip

\noindent The general case (i.e. for an arbitrary $\varepsilon$) is quite similar, except that the we cannot rely on the "correction" forms $v_{\Omega, k}$ for $\Omega\cap Y\neq \emptyset$. Instead we are using the following two facts.

The first fact is a version of Proposition~\ref{poinc}: let $U$ be a finite union of coordinate subsets of $X$ containing $Y$. There exists a positive constant $C> 0$  (independent of $\varepsilon$) such that for any $L$-valued $(p,0)$ form $u$ with support in $U$ we have the inequality
\begin{equation}\label{pave36}
\frac{1}{C}\int_U|u|^2_{\omega_{Y,\varepsilon}}e^{-\varphi_L}dV_{\omega_{Y,\varepsilon}}\leq
\int_{U}|\bar \partial u|^2_{\omega_{Y,\varepsilon}}e^{-\varphi_L}dV_{\omega_{Y,\varepsilon}}+ \int_{U\setminus \frac{1}{2}U}|u|^2_{\omega_{Y,\varepsilon}}e^{-\varphi_L}dV_{\omega_{Y,\varepsilon}}.
\end{equation}
We will not prove \eqref{pave36} here, because it follows directly from the
arguments invoked for Proposition~\ref{poinc}.

The second fact is that the space of $L^2$ holomorphic
$p$-forms is independent of $\varepsilon\geq 0$.
Let $\Phi$ be a holomorphic $p$--form on $X_0$  with values in $L$.
If $\Phi$ is $L^2$ with respect to $(\omega_{Y, \varepsilon}, h_L)$, it is also $L^2$ integrable with respect to $(\omega_{Y}, h_L)$ because of the inequality
\begin{equation}
\Vert \Phi\Vert_{\omega_{Y, \varepsilon}}^2\geq \Vert \Phi\Vert_{\omega_{Y}}^2.
\end{equation}
We next assume that $\Phi$ is $L^2$ with respect to $(\omega_{Y}, h_L)$. In local coordinates this writes as
\begin{equation}
\int_\Omega\sum|\Phi_I(z)|^2\prod_{j\not\in I} \lambda_j(z) e^{-\varphi_L(z)}d\lambda(z)< \infty
\end{equation}
where $\lambda_j$ are the eigenvalues of $\omega_\mathcal{C}$ with respect to the Euclidean metric on $\big(\Omega, (z_1,\dots, z_n)\big)$. We take the coordinates $z$ so that they are adapted to $(X, Y)$, i.e. $\Omega\cap Y= z_1\dots z_r= 0$.
It follows that the holomorphic functions $\Phi_I$ are divisible by
\begin{equation}\label{pave4411}
\prod_{i\leq r, i\in I} z_i^{\lfloor q_i\rfloor}
\prod_{i\leq r, i\not\in I} z_i^{\lfloor 1+ q_i-\frac{1}{m}\rfloor}
\end{equation}
cf. notations at the beginning.
Recall that we have chosen $m$ so that
\begin{equation}\label{pave422}
\left\lfloor q_i-\frac{1}{m}\right\rfloor= \left\lfloor q_i\right\rfloor,
\end{equation}
for each non-integer coefficient $q_i$ which appears in the expression of the metric $h_L$.

\noindent So summing up we see that by the careful choice of the singularities of the metric $\omega_\mathcal{C}$ cf. $(ii)$,  for any holomorphic form $\Phi$ satisfying $\Vert \Phi\Vert_{\omega_\mathcal{C}} ^2< \infty$, we have
\begin{equation}\label{invarspace}
\Vert \Phi\Vert_{\omega_{Y, \varepsilon}}^2< \infty \qquad\text{and} \qquad
\lim_{\varepsilon \rightarrow 0 } \Vert \Phi\Vert_{\omega_{Y, \varepsilon}}^2 = \Vert \Phi\Vert_{\omega_\mathcal{C}}^2 .
\end{equation}

The rest of the proof is absolutely similar (i.e. by contradiction) to the case
of Theorem \ref{prop:2}: we suppose by contradiction that there exists a sequence $u_\varepsilon$ such that $\|u_\varepsilon\|_{\omega_{Y,\varepsilon}}=1$, $u_\varepsilon$ is orthogonal to the space of $L^2$-holomorphic forms with respect to $\omega_{Y, \varepsilon}$ and
$\lim_{\varepsilon \rightarrow 0} \|\bar \partial u_\varepsilon\|_{\omega_{Y,\varepsilon}} =0$. By passing to some subsequence, $u_\varepsilon$ converges weakly to some $(p,0)$-form $u$ on $X_0$. Then $\|u\|_{\omega_{\mathcal{C}}} \leq 1$ and $\bar \partial u =0$. Therefore $u$ is a holomorphic $L^2$-form on $X$. Since $u_\varepsilon$ is orthogonal to the holomorphic forms, we have $\int_X \langle u, u_\varepsilon\rangle_{\omega_{Y,\varepsilon}} =0$. Together with \eqref{invarspace}, we obtain $u =0$. Then we obtain a contradiction by the same arguments as in Theorem \ref{prop:2} together with \eqref{pave36}.
\end{proof}
\medskip

\noindent We prove next Theorem \ref{CoHo}.

\begin{proof} Let $\xi$ be any $L^2$-form of type $(n,1)$ with values in $(L, h_L)$. Then we also have
$\displaystyle \Vert\xi\Vert_{\omega_{Y, \varepsilon}}< \infty$ and thus we write
\begin{equation}\label{pave37}
\xi= \xi_\varepsilon+ \bar \partial(\eta_\varepsilon)+ \bar \partial^\star(\tau_\varepsilon),
\end{equation}
where $\xi_\varepsilon$ is harmonic, $\eta_\varepsilon$ is orthogonal to $\mathrm{Ker}(\bar \partial)$ and $\tau_\varepsilon$ orthogonal on $\mathrm{Ker}(\bar \partial^\star)$. We have
\begin{equation}\label{pave38}
\Vert \xi_\varepsilon\Vert^2 _{\omega_{Y, \varepsilon}}\leq \Vert \xi\Vert^2 _{\omega_{Y, \varepsilon}}\leq \Vert \xi\Vert^2 _{\omega_{Y}}
\end{equation}
and therefore the limit of $(\xi_\varepsilon)_{\varepsilon> 0}$ will be harmonic with respect to
$\omega_\mathcal{C}, h_L$. Moreover, thanks to Theorem \ref{Poinc}, we have
\begin{equation}\label{pave39}
\Vert \eta_\varepsilon\Vert^2 _{\omega_{Y, \varepsilon}}\leq C \Vert \bar \partial \eta_\varepsilon\Vert^2 _{\omega_{Y, \varepsilon}}\leq
C \Vert \xi\Vert^2 _{\omega_{Y}}
\end{equation}
and thus we can extract a limit of $(\eta_\varepsilon)_{\varepsilon> 0}$ on compacts of $X_0$.

\bigskip

\noindent In order to deal with the family $(\tau_\varepsilon)_{\varepsilon> 0}$, thanks to the uniform estimate of Theorem \ref{Poinc}, the proof of Corollary \ref{usecor} applies. We get thus a constant $C> 0$ so that
\begin{equation}\label{pave43}
\Vert \tau_\varepsilon\Vert^2_{\omega_{Y,\varepsilon}}\leq C\Vert \bar \partial^\star(\tau_\varepsilon)\Vert^2_{\omega_{Y,\varepsilon}}
\end{equation}
and the RHS of \eqref{pave43} is bounded uniformly with respect to $\varepsilon$. We can therefore take the limit $\varepsilon\to 0$ and we get $\xi_0$ harmonic, $\eta_0\in \mathrm {Dom}(\bar \partial)$ and $\tau_0\in \mathrm {Dom}(\bar \partial^\star)$ such that
\begin{equation}\label{pave44}
\xi= \xi_0+ \bar \partial\eta_0+ \bar \partial^\star\!(\tau_0) .
\end{equation}

\medskip

To prove that $\mathcal{H}_{n,1} (X_0, L), \mathrm{Im} \, \bar \partial$ and $\mathrm{Im} \, \bar \partial ^*$ are orthogonal with each other, we use the partition of unity and the local covering. For example, let $\xi_0 \in \mathcal{H}_{n,1} (X_0, L)$ and $\eta_0 \in L^2$. Let $\theta_i$ be a partition of unity. We have
$$\int_{X_0} \langle  \xi_0, \bar \partial \eta_0 \rangle =\sum_i \int_{X_0} \langle  \xi_0,  \bar \partial (\theta_i\eta_0) \rangle .$$
Let $\pi_i : V_i \rightarrow U_i$ be a local ramified cover such that $\pi_i ^* \omega_\mathcal{C}$ is smooth and
$Supp \theta_i \Subset U_i$. Then
$$\int_{X_0} \langle  \xi_0,  \bar \partial (\theta_i\eta_0) \rangle  =
\int_{U_i } \langle  \xi_0,  \bar \partial (\theta_i\eta_0) \rangle = \frac{1}{r} \int_{V_i }
\langle  \pi_i ^*\xi_0,  \bar \partial (\pi_i ^* (\theta_i\eta_0) ) \rangle_{\pi_i ^* \omega_\mathcal{C}} =0 ,$$
where the last equality comes from the fact that $\pi_i ^* \omega_\mathcal{C}$ is smooth on $V_i$ and
$Supp \theta_i \Subset V_i$.

\medskip

To conclude the proof, it remains to check \eqref{diffdep}. Let $\xi \in \mathcal{H}_{n,1} (X_0, L)$ (with respect to the conic metric). Then $\xi$ is still $L^2$ with respect to $\omega_{Y,\varepsilon}$ for every $\varepsilon$. We consider the Hodge decomposition of $\xi$ with respect to $\omega_{Y,\varepsilon}$:
$$\xi = \xi_{\varepsilon} +\bar \partial \eta_\varepsilon + \bar \partial^{\star_\varepsilon}\tau_\varepsilon .$$
Then $\xi_{\varepsilon}$ is harmonic with respect to the complete metric $\omega_{Y,\varepsilon}$.
In particular, by using Bochner, we know that
\begin{equation}\label{diff}
\bar \partial \xi_{\varepsilon} = \bar \partial^{\star_\varepsilon} \xi_{\varepsilon} = (D')^{\star_\varepsilon} \xi_{\varepsilon} =0 .
\end{equation}
By passing to some subsequence, the above argument implies that $\xi_{\varepsilon} , \eta_\varepsilon$ and $\tau_\varepsilon$ converge to some $\xi_0, \eta_0$ and $\tau_0$ which are $L^2$ with respect to $\omega_\mathcal{C}$ and we have
$$\xi = \xi_0 + \bar \partial  \eta_0 + \bar \partial^\star \tau_0 .$$
As we proved that $\mathcal{H}_{n,1} (X_0, L), \mathrm{Im} \, \bar \partial$ and $\mathrm{Im} \, \bar \partial ^*$ are orthogonal with each other, we obtain that $\xi =\xi_0$. Together with \eqref{diff}, \eqref{diffdep} is proved.
The decomposition for the $(n,0)$-forms follows from the same argument.
\end{proof}

$$$$

\section{Towards the proof of the main result: algebra}\label{alg}

\noindent Let $s_i$ be an arbitrary holomorphic extension of
$\displaystyle s|_{U_i\cap X}$. Then we have
\begin{equation}\label{eq34}
  s_idz_i\wedge dt\otimes e_i=
  s_jdz_j\wedge dt\otimes e_j
+ t\Lambda_{ij}dz_j\wedge dt\otimes e_j.
\end{equation}
on overlapping coordinate sets $U_i\cap U_j$. Hence we interpret $s$ as a top form on $X$ with values in $\displaystyle L:= \omega_{\mathcal{X}/\mathbb D}^{\otimes (m-1)}|_{X}$. In \eqref{eq34} the symbol $e_i$ stands for $dz_i^{\otimes (m-1)}$, the local frame of $\displaystyle \omega_{\mathcal{X}/\mathbb D}^{\otimes (m-1)}$.

\noindent We can reformulate this data as follows: \emph{there exists a smooth section $s_{1}$ of the bundle $K_{\mathcal{X}}+ L$ such that}
\begin{equation}\label{eq35}
\bar \partial s_{1}= t\Lambda_{1}
\end{equation}
on $\mathcal{X}$.
\medskip

\noindent Consider next the Lie derivative operator $\mathcal{L}_\Xi$ associated to a vector field $\Xi$ such that $\displaystyle dp(\Xi)= \frac{\partial}{\partial t}$. If we apply $\mathcal{L}_\Xi$ to \eqref{eq35}
on the RHS we obtain
\begin{equation}\label{eq36}
\Lambda_{1}+ t\mathcal{L}_\Xi(\Lambda_{1})
\end{equation}
By the commutation relation \eqref{eq20}, the LHS of \eqref{eq35} becomes
\begin{equation}\label{eq37}
\bar \partial\big(\mathcal{L}_\Xi(s_1)\big)+ D'_\mathcal{X}\big(\bar \partial\Xi\rfloor s_1\big)+ [i_\Xi, [\bar \partial, D' _{\mathcal{X}}]] s_1
\end{equation}
Modulo a factor divisible with $t$, the last term of \eqref{eq37} is equal to
$\displaystyle \frac{m-1}{m}\Lambda_1$ on $X \setminus Z$. Therefore we get
\begin{equation}\label{eq38}
\frac{1}{m}\Lambda_1\equiv \bar \partial\big(\mathcal{L}_\Xi(s_1)\big)+ D'_\mathcal{X}\big(\bar \partial\Xi\rfloor s_1\big)
\qquad\text{on } X \setminus Z
\end{equation}
modulo a forms which is divisible by $t$ and which is non-singular on $X\setminus Z$.
\medskip

\noindent Another interesting observation is that the form $\bar \partial\Xi\rfloor s_1$ can be written as follows
\begin{equation}\label{eq39}
\bar \partial\Xi\rfloor s_1= dt\wedge \rho_1
\end{equation}
given the shape of the vector field $\Xi$, cf. \eqref{eq18}. We have therefore obtained the
first step of the next statement.

\begin{lemme}\label{linalg}
Let $s_k$ be a smooth section of the bundle $K_{\mathcal{X}/\Delta}+ L$, such that
\begin{equation}\label{eq339}
\bar \partial s_k= t^{k+1}\Lambda_k\end{equation}
for some $(n+1, 1)$-form $\Lambda_k$. We assume that the
connection on $L= (m-1)K_\mathcal{X}$ is induced by the section $s_k$. Then we can find the forms $\alpha_k$ and $\beta_k$
such that we have
\begin{equation}\label{eq40}
c_k\Lambda_k\equiv \bar \partial \alpha_k+ D'_\mathcal{X}\big(dt\wedge \beta_k) \qquad\text{on }  X \setminus Z
\end{equation}
modulo the ideal generated by $t$. Moreover, the forms $\alpha_k$ and $\beta_k$ are
smooth in the complement of the set $s_k= 0$ and $c_k$ is a positive constant.
\end{lemme}

\begin{proof}
This is obtained as follows: we take the Lie derivative in \eqref{eq40} and use the commutation relation \eqref{eq20}. The result of this first operation is that we have
\begin{equation}\label{eq41}
\frac{k+1}{m}t^{k}\Lambda_k\equiv \bar \partial \big(\mathcal{L}_\Xi(s_k)\big)+
D'_\mathcal{X}\big(\bar \partial\Xi\rfloor s_k\big) \qquad\text{on }  X \setminus Z
\end{equation}
modulo the ideal generated by $t^{k+1}$. In order to start the inductive process which will prove our statement, we rewrite \eqref{eq41} as follows
\begin{equation}\label{eq50}
\frac{k+1}{m}t^{k}\Lambda_k\equiv \bar \partial u_1+
D'_\mathcal{X}\big(dt\wedge v_1\big) \qquad\text{on }  X \setminus Z,
\end{equation}
modulo $t^{k+1}$. Here we use the notations
\begin{equation}\label{eq51}
u_1:=  \mathcal{L}_\Xi(s_k), \qquad dt\wedge v_1:= \bar \partial\Xi\rfloor s_k.
\end{equation}
We show next that if we apply the operator $\mathcal{L}_\Xi$ to the RHS of \eqref{eq51}
the result is an expression of a similar type. Indeed, we have
\begin{equation}\label{eq52}
\mathcal{L}_\Xi(\bar \partial u_1)= \bar \partial\left(\mathcal{L}_\Xi(u_1)\right)+ D'_\mathcal{X}\big(\bar \partial\Xi\rfloor u_1\big)
\qquad\text{on }  X \setminus Z
\end{equation}
modulo the curvature term which is of order $t^k$: this is one higher than we have to take into account, so we just drop it.

\noindent Also, we have
\begin{equation}\label{eq53}
\mathcal{L}_\Xi \big(D' _\mathcal{X} (dt\wedge v_1)\big) = D' _\mathcal{X} \left( dt\wedge \Big( \Xi  \rfloor D' _\mathcal{X} \big(\Xi \rfloor (dt \wedge v_1)\big) \Big)\right)
\end{equation}
thanks to Lemma \ref{Komm}, so summing up we get
\begin{equation}\label{eq54}
\frac{k(k+1)}{m}t^{k-1}\Lambda_k\equiv \bar \partial u_2+
D'_\mathcal{X}\big(dt\wedge v_2\big),
\end{equation}
modulo $t^{k}$, where we use the notations
\begin{equation}\label{eq55}
u_2:= \mathcal{L}_\Xi(u_1), \qquad dt\wedge v_2:=  \bar \partial\Xi\rfloor u_1+ dt\wedge \Big(\Xi  \rfloor D' _\mathcal{X} \big(\Xi \rfloor (dt \wedge v_1)\big) \Big)
\end{equation}
Our statement follows by induction on $k$ --and moreover we have
\begin{equation}
c_k:= \frac{(k+1)!}{m}
\end{equation}
as we see by taking successive derivatives of \eqref{eq54}.
\end{proof}
\medskip

\begin{remark}
The relations \eqref{eq55} give the explicit process of constructing $\alpha_k$ and $\beta_k$.
They will play an important role in the analysis of the singularities of these forms (this is required by Theorem \ref{Hodge}).
\end{remark}
\vfill\eject

\section{Proof of Theorem \ref{MT}}

\noindent The plan for this section is as follows. After a few reductions, we show that the singularities of the forms $\alpha_k$ and $\beta_k$ in Lemma \ref{linalg} can be improved significantly, cf. section \ref{cos}. Then we state and prove Theorem \ref{log007}, which implies
Theorem \ref{MT}.

\subsection{Preliminaries and some reductions}
To start with, we consider the obstruction form $\Lambda_k$ in
Lemma \ref{linalg}.

\noindent By hypothesis the restriction $\lambda_k$ of $\Lambda_k$ to $X$  verifies
\begin{equation}\label{snc6}
  \int_X\left|\lambda_k\right|^2_{\omega}e^{-(1-\varepsilon)\varphi_L}dV_{\omega}< + \infty
\end{equation}
for each positive $\varepsilon> 0$
and moreover it is $\bar \partial$-closed. Here $L:=  (m-1)K_X$ and
$\varphi_L = \frac{m-1}{m}\log |s|^2$ is the weight of the singular metric $h_L$ on $L$.
Actually this would equally be true for any metric
$\omega_E\geq \omega$.
\medskip

\noindent Let $\pi:\widehat X\to X$ be a log-resolution of $\displaystyle \Big(X, \frac{m-1}{m}\mathrm{Div}(s)\Big).$
Then $\widehat \lambda_k:= \pi^\star \lambda_k$ is a $(n, 1)$-form with values in $\widehat L:= (m-1)\pi^\star(K_X)$, such that
\begin{equation}\label{new1}
\int_X\left|\widehat\lambda_k\right|^2_{\widehat \omega}e^{-(1-\varepsilon)\varphi_{\widehat L}}dV_{\widehat \omega}< + \infty
\end{equation}
holds true for every positive $\varepsilon$, where $\widehat \omega$ is any (non-singular) metric on $\widehat X$.

\noindent We write
\begin{equation}\label{new2}
\frac{m-1}{m}\mathrm{Div}(\pi^\star s)= \sum_{i\in I} m_iE_i+ \sum_{j\in J}q_jF_j
\end{equation}
where the convention is that $m_i\in \mathbb{Z}_+$ and $q_j\in \mathbb{Q}_+\setminus \mathbb{Z}$. Given the integrability
condition \eqref{new1}, we infer that the form
\begin{equation}\label{new3}
\lambda:= \frac{\widehat\lambda_k}{\prod_{i\in I}s_{E_i}^{m_i-1}\prod_{j\in J}s_{F_j}^{\lfloor q_j\rfloor}}
\end{equation}
is smooth -and of course, $\bar \partial$-closed.

\noindent The relation $\lambda_k= \bar \partial \alpha_k+ D\beta_k $ on $X\setminus \mathrm{Div}(s)$ in Lemma \ref{linalg}
 becomes
\begin{equation}\label{new4}
\lambda= \bar \partial \alpha+ D'\beta
\end{equation}
on $\widehat X\setminus (E\cup F)$, where $D'$ is the $(1, 0)$-part of the Chern connection for
singular metric on the line bundle $\O\left(E+ \sum (q_j- \lfloor q_j\rfloor)F_j\right)$.
\smallskip

\noindent Thus, by changing the notations we have the following setup.

\begin{enumerate}
\smallskip

\item[(1)] $\lambda$ is a $\bar \partial$-closed form of type $(n, 1)$ and values in $E+ L$ on a compact K\"ahler manifold $X$.
\smallskip

\item[(2)] $E+ F$ is a snc divisor on $X$ and $L= \O\left(\sum (q_j- \lfloor q_j\rfloor)F_j\right)$
is a line bundle which we endow with the singular metric $h_L$ induced by $F$.
\smallskip

\item[(3)] $\alpha$ and $\beta$ are two $E+ L$-forms of type $(n, 0)$ and $(n-1, 1)$, respectively
with poles at most along $E+F$, i.e.
\begin{equation}\label{twoforms}
\alpha = \frac{1}{s^{N}}\O(1), \qquad \beta= \frac{1}{s^N}\O(1)
\end{equation}
where $s= s_E\cdot s_F$ and such that \eqref{new4} holds in the complement of
the support of $E+F$.
\end{enumerate}

\noindent Our next step would be to improve the "shape" of the two forms in \eqref{twoforms}: we show in the next subsection that we can replace them with forms having logarithmic poles on $E+F$.

\subsection{Improving the pole order of $\alpha$ and $\beta$ (cosmetics)}\label{cos}
Notice that by hypothesis (3) above, the meromorphic form
$\displaystyle \frac{\lambda}{s_E}$ has logarithmic poles along $E$; it can therefore be interpreted as a $(0,1)$-form with values in $\Lambda^n T^\star_X\langle E\rangle\otimes L$.
In this set-up, we formulate the next statement.

\begin{proposition}\label{c1} There exists $\alpha_1$ and $\beta_1$ differential forms with log-poles along $E$, such that
\begin{equation}\label{log1}
\bar \partial \alpha_1+ D'\beta_1= \frac{\lambda}{s_E}
\end{equation}
holds in the complement of $E$, where $D'$ is the $(1,0)$-part of the Chern connection for $(L, h_L)$.
 \end{proposition}

\begin{proof} We start by choosing a finite covering $\displaystyle (\Omega_i)_{i\in I}$ of $X$, together with corresponding coordinates $(z_i^1,\dots, z_i^n)$ such that for every index $i\in I$ we have
\begin{equation}\label{pave90}
E_j\cap \Omega_i= (z_i^j=0), \qquad F_k\cap \Omega_i= (z_i^k=0)
\end{equation}
for each $j= 1,\dots, r$ and $k=r+1,\dots, r+s$.
\smallskip

\noindent Consider next the component $E_1$ of $E$. We then define an associated vector field
\begin{equation}\label{log-20}
V_1 := \sum \theta_i(z)z_i^1\frac{\partial}{\partial z_i^1}
\end{equation}
where $\displaystyle (\theta_i)_{i\in I}$ is a partition of unit and $\Omega_i\cap E_1= (z_i^1= 0).$
Notice that since $E_1$ is a global hypersurface, we have
\begin{equation}\label{log-21}
V_1 |_{\Omega_i}= \big(z_i^1+ \O (z_i^1)^2\big)\frac{\partial}{\partial z_i^1}+
\sum_{k= 2}^{r+s}\O(z_i^1z_i^k)\frac{\partial}{\partial z_i^k}+ \sum_{k> r+s}\O(z_i^1)\frac{\partial}{\partial z_i^k}
\end{equation}
where the notation $\O(z_i^1)$ means a function belonging to the ideal
$\O(-E_1)\otimes\mathcal{C}^\infty(\Omega_i)$. Indeed, given that the transition functions for the coordinate system we have are
\begin{equation}
z_i^\alpha= g_{ij}^{\alpha}(z_j) z_j^\alpha
\end{equation}
the expression \eqref{log-21} obviously corresponds to the restriction of $V_1$ to $\Omega_i$.
\smallskip

\noindent Notice also that the local expression of the covariant derivative $D'$ induced by the singular metric on $E+L$ reads as
\begin{equation}\label{pave91}
D'|_{\Omega_i}= \partial - \sum_{j=1}^r\frac{dz_i}{z_j}- \sum_{k=r+1}^{r+s}q_k\frac{dz_k}{z_k}.\end{equation}

\medskip

\noindent We first modify the form $\alpha$.
Assume that $\alpha$ has a pole of order $N$ along $E_1$ for some $N\geq 1$.
We define	\begin{equation}\label{log4}
	\alpha_N :=  \alpha   +\frac{1}{N} D'\left(V_1 \rfloor\alpha\right)
	\end{equation}
where $D'$ is the $(1,0)$ part of the Chern connection corresponding to the singular metric on $E+ L$,
cf. \eqref{pave91}. By a direct computation, we show now that the pole order of $\alpha_N$ is at most $N-1$. Indeed, we write $\displaystyle \alpha|_{\Omega}= \frac{f}{z_1^N}dz$ (where $\Omega$ is one of the $\Omega_i$; we also drop the index $i$) and then we have
\begin{equation}\label{pave92}
V_1\rfloor \alpha= \left(z_1+ \O (z_1)^2 \right)\frac{f}{z_1^N}\widehat{dz_1}+ \sum_{j\geq 2}(-1)^{j-1}\O (z_1)
\frac{f}{z_1^N}\widehat{dz_j}.
\end{equation}
By the formula \eqref{pave91} we have
\begin{equation}\label{pave93}
D'\left(V_1\rfloor \alpha\right)\equiv -N\frac{f}{z_1^N}dz
\end{equation}
modulo a form whose pole order is smaller than $N-1$, so our claim is proved
\smallskip

\noindent Moreover, we have
	\begin{equation}\label{log5}
	\lambda = \bar \partial\alpha_N+ D' (\beta +\frac{1}{N} \bar \partial \left(V_1 \rfloor\alpha\right) )
\end{equation}
in the complement of $E\cup F$. We repeat this procedure until the exponent $N$ is equal to zero. At this moment we have
	\begin{equation}\label{log6}
	\lambda= \bar \partial \alpha_0 + D'(\beta_0)
	\end{equation}
	where $\alpha_0$ has no poles along $E_1$ and the coefficients of $\beta_0$ are similar to the ones of our initial $\beta$.
\medskip

\noindent We are dealing with the other components of $E$ and $F$ similarly.  For example for $E_2$ we proceed as follows.  We define the vector field
\begin{equation}\label{comp1}
V_2:= \sum \theta_i(z)z_i^2\frac{\partial}{\partial z_i^2}
\end{equation}
constructed by using coordinate system \eqref{pave90}. It then follows that locally the $V_2$ can be written as
\begin{equation}\label{comp3}
V_2 |_{\Omega_i}= \big(z_i^2+ \O(z_i^2)^2\big)\frac{\partial}{\partial z_i^2}+ \O(z_i^1z_i^2)\frac{\partial}{\partial z_i^1}+ \sum_{j\neq 1,2} \O(z_i^2)\frac{\partial}{\partial z_i^k}.
\end{equation}
With this choice of vector field, if $\alpha$ is a smooth form along $E_1$ then the same is true for
\begin{equation}\label{addb}
\alpha + \frac{1}{N}D' \left(V_2\rfloor \alpha\right)
\end{equation}
in other words, we are not loosing what we have gained at the first step.
Furthermore,  the pole order of \eqref{addb} along $E_2$ has dropped by at least one unit if $\alpha$ is of pole order $N\geq 1$ along $E_2$. The same procedure as before applies.
\smallskip

\noindent Finally we get some new $\alpha$ and $\beta$ such that
	\begin{equation}\label{log7}
	\lambda = \bar \partial{\alpha}+ D'{\beta}
      \end{equation}
holds in the complement of $E+F$ and such that the form $\displaystyle {\alpha}$ is smooth on $X$ and the coefficients of $\beta$ are of type $\displaystyle \frac{b}{s^k}$ for some smooth function $b$ and positive integer $k$.

	\medskip

	\noindent We now turn to the form $\beta$, and prove that modulo the image of the $D'$-operator it has no poles along $E+F$.
	Assume that $\beta$ has a pole of order $N\geq 1$ along $E_1$. Then we define
	\begin{equation}\label{log888}\beta_N :=\beta +\frac{1}{N} D' (V_1 \rfloor \beta)\end{equation}
and the same type of calculations as before will show that it has a pole of order at most $N-1$ along $E_1$.

\noindent To see this, we write
	\begin{equation}\label{log8}
	\beta|_{\Omega}= \sum_{j\geq 2, k} \frac{\eta_{j \overline k}}{z_1^{N}}dz_1\wedge \widehat{dz_j}\wedge d\overline z_k
	+ \sum_{k} \frac{\eta_{\overline k}}{z_1^N}\widehat{dz_1}\wedge d\overline z_k
	\end{equation}
in local coordinates.
Since $\lambda$ and $\alpha$ in \eqref{log7} are smooth, $D'\beta$ has no poles along $E_1$. Therefore for each $k$, $\displaystyle \eta_{\overline k}$ is divisible by $z_1$. We can then write
	\begin{equation}\label{log-28}
	\beta |_{\Omega}= \sum_{j\geq 2, k} \frac{\eta_{j \overline k}}{z_1^{N}}dz_1\wedge \widehat{dz_j}\wedge d\overline z_k
	+ \sum_{k} \frac{\eta_{\overline k}}{z_1^{N-1}}\widehat{dz_1}\wedge d\overline z_k.
	\end{equation}
        By using the expression \eqref{log-21}, a direct computation -which we skip- shows that adding $\displaystyle \frac{1}{N}D' (V_1 \rfloor \beta)$ to our form $\beta$ has the effect of removing the first sum in \eqref{log-28}, and so $\beta_N$
has a pole of order at most $N-1$ along $E_1$. We still have
\begin{equation}\label{log71}
\lambda = \bar \partial{\alpha}+ D'{\beta_N} \qquad\text{on } X\setminus (E+F) ,
\end{equation}
where $\alpha$ is unchanged. We repeat this procedure until the exponent $N$ is equal to zero. We get
\begin{equation}\label{log66}
\lambda= \bar \partial \alpha + D'(\beta)
\end{equation}
where $\alpha$ is smooth on $X$ and (the resulting) $\beta$ has no pole along $E_1$.
\smallskip

\noindent A little observation is in order at this point: if we write the form $\beta$ in coordinates
\begin{equation}\label{log-288}
\beta |_{\Omega}= \sum_{j\geq 2, k} {\eta_{j \overline k}}dz_1\wedge \widehat{dz_j}\wedge d\overline z_k
	+ \sum_{k} {\eta_{1\overline k}}\widehat{dz_1}\wedge d\overline z_k,
	\end{equation}
where $\displaystyle \eta_{j\overline k}$ and $\displaystyle \eta_{1\overline k}$ are smooth with respect to $z_1$, the equality \eqref{log66} shows that $D'\beta$ is smooth. By the explicit expression of the operator $D'$ is \eqref{pave91} it follows that $\displaystyle \eta_{1\overline k}$ is divisible by $z_1$, for each $k$.
\smallskip

\noindent In order to treat the remaining components of $E$ and $F$ we are using the argument already employed for $\alpha$,
 namely we consider $\beta + \frac{1}{N} D' (V_2 \rfloor \beta)$. This form has no pole along $E_1$ (this is where the divisibility of the coefficient $\displaystyle \eta_{1\overline k}$ by $z_1$ is playing an important role) and pole order along $E_2$ drops.
\medskip

\noindent Finally we obtain two smooth forms $\alpha$ and $\beta$ such that
\begin{equation}\label{log666}
\lambda= \bar \partial \alpha + D'(\beta)
\end{equation}
holds at each point of $X\setminus (E\cup F)$. If we write
\begin{equation}\label{log6667}
\beta= \sum_{j, k} {\eta_{j \overline k}}\widehat{dz_j}\wedge d\overline z_k
\end{equation}
then we automatically have $\displaystyle \eta_{j\overline k}$ divisible by $z_j$, for all $j=1,\dots, r+s$. In particular \eqref{log666} holds point-wise on $X$ and by division with the section $s_E$ we have
\begin{equation}\label{log6668}
\frac{\lambda}{s_E}= \bar \partial \alpha_1 + D'(\beta_1)
\end{equation}
on $X\setminus E$, where $\alpha_1$ and $\beta_1$ have logarithmic poles along $E$.
The proposition is proved.
\end{proof}

\begin{remark}\label{preciselog}
Assume that $E$ is equal to zero. Then the particular shape of the coefficients of $\beta= \beta_1$
in \eqref{log6667} shows that
\begin{equation}
\int_X|\beta|_{\omega_\mathcal{P}}^2e^{-\varphi_L}< \infty
\end{equation}
in other words, $\beta$ is $L^2$ with respect to the Poincar\'e metric.
\end{remark}
\medskip

\noindent Therefore, in order to complete the proof of Theorem \ref{MT} it would be enough to establish the following statement.

\begin{thm}\label{log007} Let $\lambda$ be a smooth $(n, 1)$-form on a compact K\"ahler manifold $X$ with values in the bundle $E+ L$
 such that the following hold.
\begin{enumerate}

\item[\rm (a)] The bundle $L$ is endowed with a metric $h_L$ such that $\displaystyle
  \varphi_L\simeq \sum \delta_i\log|f_i|^2$ modulo smooth functions, where $0<\delta_i< 1$ are rational numbers.

\item[\rm (b)] The curvature
  $\Theta(L, h_L)= 0$ in the complement of $F:= (\varphi_L= -\infty)$.

\item[\rm (c)] We have $\displaystyle E:= \sum_{i\in I}E_i$ and the divisor $E+ F$ is snc.

\item[\rm (d)] There exists a $(n-1,1)$-form $\beta$ with values in $L$ and log poles on $E$, and a $(n, 0)$-form $\alpha$ with values in $L$ and log poles along $E$ such that
\begin{equation}\label{pave46}
  \frac{\lambda}{s_E}= \bar \partial\alpha+ D'(\beta)
\end{equation}
on $X\setminus E$, where $D'$ is induced by the singular metric on $L$.
\end{enumerate}
\noindent Then $\lambda$ is $\bar \partial$-exact.
\end{thm}

\noindent We will argue by induction on the number of components of the
divisor $E$. In the next subsection, we are dealing with the case $E=0$.

\subsection{The klt case ($E=0$)} \bigskip

\noindent Let $\lambda$ be a $\bar \partial$-closed form of type $(n,1)$ with values in
$L= (m-1)K_X$, which is $L^2$ with respect to $h_L$ and the Poincar\'e metric
$\omega_{\mathcal{P}}$ corresponding to the divisor $F$. Thanks to Theorem \ref{Hodge}, we have a $L^2$- decomposition
\begin{equation}\label{snc7}
\lambda= u+ \bar \partial v
\end{equation}
where $u$ is harmonic. From Lemma \ref{vanishing} it follows that $\lambda$ is $\bar \partial$-exact if and only if
\begin{equation}\label{snc8}
\int_X \lambda \wedge \overline \gamma e^{-\varphi_L}= 0
\end{equation}
for any $L^2$ holomorphic form $\gamma$ of type $(n-1, 0)$ with values in $L$.

\bigskip

\noindent Under the current assumption (i.e. $E=0$) we have
the following equality
\begin{equation}\label{snc4}
\lambda= \bar \partial \alpha+ D'\beta
\end{equation}
on $X$, where $\alpha$ and $\beta$ are smooth. As we have seen in Remark \ref{preciselog}, the form
$\beta$ is $L^2$ with respect to the Poincar\'e metric and $h_L$. It follows that
\begin{equation}\label{snc5}
\int_X\bar \partial\alpha\wedge \overline{\gamma}e^{-\varphi_L}= 0
\end{equation}
as well as
\begin{equation}\label{snc-5-}
\int_XD'\beta\wedge \overline{\gamma}e^{-\varphi_L}= 0
\end{equation}
because $D'\gamma$ and $\bar \partial\gamma$ are equal to zero. The conclusion follows, i.e. we are done with the case $E=0$.

\medskip

\medskip

\subsection{The induction step} We are now ready to complete the proof of Theorem \ref{log007}. In this section we will work with a metric $\omega_\mathcal{C}$ with conic singularities along the divisor $F$.
The forms $\alpha$ and $\beta$ have the properties stated in the point (d) of this result.
\smallskip

\begin{proof} For the remaining part of the proof of Theorem \ref{log007} we proceed in two steps. We first show that it is enough to treat the case where the quotient $\displaystyle \frac{\lambda}{s_E}$ in \eqref{pave46} is a
smooth, $\bar \partial$-closed form on $X$. Then we show that
\eqref{pave46} holds on $X$, modulo a \emph{closed} current whose support is contained in $E$. After this second step is completed it will be easy to conclude.

\subsubsection{Step 1} Even if \eqref{pave46} only holds true in the complement of $E$, we have
\begin{equation}\label{pave24}
\lambda= \bar \partial(s_E\alpha)+ D'(s_E\beta)
\end{equation}
on $X$, given that the forms involved $\lambda, s_E\alpha$ and $s_E\beta$ are smooth.
\smallskip

\noindent On the other hand, thanks to \eqref{log6667}, we can write
\begin{equation}\label{pave25}
\beta= \frac{\partial_1 s_{E_1}}{s_{E_1}}\wedge a+ b
\end{equation}
where $a$ and $b$ are $L$-valued, smooth along $E_1$ and $\partial_1$ is the derivative with respect to a smooth metric on $E_1$. It follows that
\begin{equation}\label{pave26}
s_E\beta= {\partial_1(s_{E_1})}\wedge a_1+ s_E b,
\end{equation}
where $\displaystyle a_1:= \big(\prod_{j\neq 1} s_{E_j}\big) a$ and then
\begin{equation}\label{pave27}
D'(s_E\beta)= -{\partial_1(s_{E_1})}\wedge D'a_1+ s_E D'b
\end{equation}
on $X$, because the term $D'\big(\partial_1(s_{E_1})\big)= 0$ by the computation
$$\partial (dz_1- z_1\partial \phi_1)- \frac{dz_1}{z_1}\wedge (dz_1- z_1\partial \phi_1)= 0,$$
where $z_1=0$ is the local equation of the divisor $E_1$.
We notice that the operator $D'$ on the left and right hand side of \eqref{pave27} respectively is not the same, hopefully this is not too confusing.
\smallskip

\noindent We divide \eqref{pave27} by the section $\displaystyle s_{E_1}$ and then take the $\bar \partial$ in the sense of currents on $X$.
Consider the respective residue on $E_1$: the only contribution we have is $\displaystyle D'a|_{E_1}$.
As a consequence, we infer that the residue of $\displaystyle \frac{\lambda}{s_{E_1}}$ is in the image of
$\bar \partial+ D'$; it is also closed, and hence by the induction hypothesis, it belongs to the image of $\bar \partial$.
\medskip

\noindent In conclusion, the image of $\lambda$ in $\displaystyle H^1(E_1, K_{E_1}+ E'+ L)$ is equal to zero, and thus $\lambda$ belongs to the image of the morphism
\begin{equation}\label{pave47}
H^1(X, K_{X}+ E'+ L)\to H^1(X, K_{X}+ E+ L),
\end{equation}
induced by the multiplication with $\displaystyle s_{E_1}$, where $E':= E_2+\dots + E_N$. We therefore have a smooth $(n,1)$-form $\lambda_1$ with values in $E'+ L$ and a smooth $(n,0)$-form $\rho$ with values in $E+L$ such that
\begin{equation}\label{pave80}
\lambda+ \bar \partial \rho= s_{E_1}\lambda_1,
\end{equation}
and thus
\begin{equation}\label{pave81}
\frac{\lambda_1}{s_{E'}}= \bar \partial \alpha'+ D'(\beta)
\end{equation}
in the complement of the support of $E$, where
\begin{equation}\label{pave82}
\alpha'= \alpha+ \frac{\rho}{s_E}
\end{equation}
has log-poles along $E$.
\smallskip

\noindent By repeating this operation $N$ times it follows that we can assume that the form $\lambda$ in \eqref{pave46}
is divisible by the equation $s_E$ of $E$. Nevertheless, this equality only holds in the complement of the support of $E$.
\subsubsection{Step 2} Our main goal in this subsection is to show that there exist a set $\gamma_i$ of holomorphic $L$-valued forms on $E_i$ such that
\begin{equation}\label{pave3a}
\int_X \left(\frac{\lambda}{s_E}- \sum_i T_{\gamma_i}\right)\wedge \overline \gamma e^{-\varphi_L}= 0
\end{equation}
where $\gamma$ is a $(n-1,0)$-holomorphic $L^2$-form on $X$. In \eqref{pave3a} we denote by
$\displaystyle T_{\gamma_i}$ the current on $X$ defined by the following equality
\begin{equation}\label{pave85}
\left\langle T_{\gamma_i}, \Phi\right\rangle:= \int_{E_i}\gamma_i\wedge \overline\Phi e^{-\varphi_L}
\end{equation}
for any $L$-valued form $\Phi$ of degree $(n-1, 0)$ on $X$. Notice that $\displaystyle T_{\gamma_i}$
is closed, in the sense that $\displaystyle \left\langle T_{\gamma_i}, D'\Psi\right\rangle= 0$ for any
$(n-2, 0)$-form $\Psi$ such that $D'\Psi$ is smooth.

\noindent The results in the previous section show that we can assume that $\displaystyle \frac{\lambda}{s_E}$ is $L^2$ and $\displaystyle \frac{\lambda}{s_E} =\bar \partial \alpha + D' \beta$ on $X\setminus E$ and $\alpha, \beta$ have at most log poles.

\medskip

\noindent By the previous step the quotient $\displaystyle \frac{\lambda}{s_E}$ is smooth, so that we have
\begin{equation}\label{lc3}
\int_X \frac{\lambda}{s_E}\wedge \overline \gamma e^{-\varphi_L}= \lim_{\varepsilon \rightarrow 0}
\int_X \theta_\varepsilon \left( \bar \partial \alpha + D' \beta\right)\wedge \overline \gamma e^{-\varphi_L}
\end{equation}
where $\theta_\varepsilon$ here equals
\begin{equation}\label{lc1}
\theta_\varepsilon(z):=\prod \theta\left(\left|\frac{s_{F_i}}{\varepsilon}\right|^2\right) \prod \theta\left(\left|\frac{s_{E_i}}{\varepsilon}\right|^2\right).
\end{equation}

\noindent We show that
\begin{equation}\label{lc2}
\lim_{\varepsilon \rightarrow 0}
\int_X \theta_\varepsilon D' \beta\wedge \overline \gamma e^{-\varphi_L}= 0,
\end{equation}
despite of the fact that $\beta$ has now log poles along $E$. Locally, we have
\begin{equation}\label{log6669}
\beta|_\Omega= \frac{1}{\prod_{i=1}^rz_i}\sum_{j, k} {\eta_{j \overline k}}\widehat{dz_j}\wedge d\overline z_k
\end{equation}
(this corresponds to the expression in \eqref{log6667} divided by the equation of $E$)
where we recall that $\eta_{j\overline k}$ is divisible by $z_j$ for all $j=1,\dots,r+ s$.

\noindent By using Stokes in \eqref{lc2} we see that we have to evaluate expressions of the following type
\begin{equation}\label{pave95}
\beta\wedge \frac{dz_l}{z_l}
\end{equation}
for $l=1,\dots, r+s$. Given the properties of the coefficients $\eta_{j\overline k}$ we have just mentioned,
the evaluation of \eqref{lc2} is reduced to a sum of integrals
\begin{equation}\label{pave96}
\int_\Omega\theta'\left(\left|\frac{s_{F_k}}{\varepsilon}\right|^2\right) \frac{\Psi  d\lambda}{
\prod_{j=r+1}^{r+s} |z_j|^{2q_j}}, \qquad \int_\Omega\theta'\left(\left|\frac{s_{E_p}}{\varepsilon}\right|^2\right) \frac{\Psi d\lambda}{
\prod_{j=r+1}^{r+s} |z_j|^{2q_j}}
\end{equation}
where $\Psi$ is smooth; clearly, the expressions in \eqref{pave96} are converging to zero.
\medskip

\noindent However, in the actual context there is an important change in the evaluation of the limit
in \eqref{lc3} of the term containing $\alpha$: if we write locally
\begin{equation}\label{lc4}
\alpha|_{\Omega}= g(z)\frac{dz}{z_1\dots z_r}
\end{equation}
where $z_1\dots z_r= 0$ is the local equation of $E\cap \Omega$, then the forms
\begin{equation}\label{lc5}
g(z)\frac{dz_2\wedge\dots \wedge dz_n}{z_2\dots z_r}\Big|_{z_1= 0}
\end{equation}
define a global form on $E_1$ which we denote by $\alpha_1$. We use similar notations $\alpha_i$ for
the residue of $\alpha$ to the other components $E_i$ of $E$.
\smallskip

\noindent We have
\begin{equation}\label{pave87}
\lim_{\varepsilon\to 0}\int_X \theta_\varepsilon\bar \partial \alpha \wedge \overline \gamma e^{-\varphi_L}=
\sum_i \int_{E_i} \alpha_i\wedge \overline \gamma e^{-\varphi_L},
\end{equation}
modulo signs/constants that we disregard.

\noindent On each component $E_i$ we have the -analog of the- Hodge decomposition
\begin{equation}\label{pave86}
\alpha_i = \gamma_i + \Delta''(\tau_i)
\end{equation}
as we shall soon see in the next subsection Remark \ref{currentdecomp}. In \eqref{pave86} we denote
by $\gamma_i:= {\mathcal{H}} (\alpha_{i})$ the harmonic part of $\displaystyle \alpha_i$. Notice that the current $\Delta''(\tau_{i})$ is orthogonal to harmonic (in this case, holomorphic) forms on $E_i$, and $\gamma |_{E_i}$ is still holomorphic $L^2$ (since the divisor $E+F$ is snc).
Then \eqref{pave87} is equal to
\begin{equation}\label{pave88}
\sum_i \int_{E_i} \gamma_i\wedge \overline \gamma e^{-\varphi_L}.
\end{equation}
\smallskip

\noindent In conclusion, we have
\begin{equation}
\int_X \left(\frac{\lambda}{s_E}- \sum_i T_{\gamma_i}\right)\wedge \overline \gamma e^{-\varphi_L}= 0
\end{equation}
and we infer that the current $\displaystyle \frac{\lambda}{s_E}- \sum_i T_{\gamma_i}$ is orthogonal on
the space of harmonic forms. Thanks to the step 1, $\displaystyle \frac{\lambda}{s_E}- \sum_i T_{\gamma_i}$ is closed.
By the results we shall establish in  the next subsection Remark \ref{currentdecomp}, it follows that this current is $\bar \partial$-exact, so that we have
\begin{equation}
\frac{\lambda}{s_E}- \sum_i T_{\gamma_i}= \bar \partial \tau.
\end{equation}
Multiplication with $s_E$ shows that the form $\lambda$ is $\bar \partial$-exact. This completes the proof of Theorem \ref{log007}, modulo the decomposition result for forms with log poles which we discuss in Section \ref{currents}.
\end{proof}
\medskip

\noindent We end this section with the following remark. With a little extra-work, one can obtain the following result: \emph{let $\beta$ be an $L$-valued form of type $(n-1,1)$ and logarithmic poles along $E+F$. Then we have}
\begin{equation}\label{add}
\lim_{\varepsilon\to 0}\int_X\theta_\varepsilon D'\beta\wedge \overline\gamma e^{-\varphi_L}= 0
\end{equation}
\emph{for any holomorphic $L^2$ form $\gamma$.} The proof is using the following elementary statement.

\begin{lemme} Let $\phi$ be a negative, $\mathcal{C}^\infty$ real function defined on the unit disk in $\mathbb{C}$.
	We define
	$${\mathcal D}_\varepsilon:= \{z\in \mathbb D : \varepsilon< |z|e^{-\phi(z)}< 2\varepsilon\}, \qquad {\mathcal D}_{\varepsilon, r}:= \{z\in \mathbb D : \varepsilon< |z|e^{-\phi(z)}< r\}.$$
	The following hold true.
	\begin{enumerate}
		\smallskip

		\item[\rm (a)] For any smooth function $f$ and any $\delta< 1$ we have $\displaystyle \lim_{\varepsilon\to 0}\int_{{\mathcal D}_\varepsilon}
		f(z)\frac{d\lambda}{z|z|^{2\delta}}= 0$.
		\smallskip

		\item[\rm (b)] For any smooth function $f$, any $\delta< 1$ and any fixed $r> 0$ the set of complex numbers $\displaystyle \left(\int_{{\mathcal D}_{\varepsilon, r}}
		f(z)\frac{d\lambda}{z|z|^{2\delta}}\right)_{\varepsilon> 0}$ is bounded.
	\end{enumerate}
\end{lemme}

\noindent We leave the proof of the lemma and of the equality \eqref{add} to the interested readers.

\subsection{Proof of Corollary \ref{corlev} and Theorem \ref{007}}

\begin{proof}[Proof of Corollary \ref{corlev}]
We suppose by induction that there exists a smooth extension $s_k$ such that $\bar \partial s_k = t^{k+1} \Lambda_k$.
To prove the corollary, it is sufficient to prove that $\Lambda_k$ is $\bar \partial$-exact.
Let
\begin{equation}\label{vers1}
\lambda_k:= \frac{\Lambda_k}{dt}\Big|_X
\end{equation}
be the restriction of $\Lambda_k$ to $X$. Then we have $\bar \partial \lambda_k=0$.
Since $\mathfrak J$ is discrete,  the image of $\lambda_k$  via the projection morphism
\begin{equation}\label{vers2}
H^1(X, mK_X)\to H^1\big(X, \O_X (mK_X)\otimes \O_X/\mathfrak J\big)
\end{equation}
is equal to zero. It follows that there exist a $\mathcal C^\infty$ form $a_k$ of type $(n,0)$ with values in $(m-1)K_X$, as well as a $\mathcal C^\infty$ form $b_k$ of type $(n, 1)$ with values the sheaf $K_X^{\otimes(m-1)}\otimes \mathfrak J$ so that
\begin{equation}\label{vers3}
\lambda_k= b_k+ \bar \partial a_k.
\end{equation}
Thanks to Theorem \ref{MT}, we know that $b_k$ is $\bar \partial$-exact. Then $\lambda_k$ is $\bar \partial$-exact. It means that we can find a smooth extension $s_{k+1}$ such that $\bar \partial s_{k+1} = t^{k+1} \Lambda_{k+1}$. By induction, $s$ admits an extension for any order. Therefore $s$ admits a holomorphic extension to the total space $\mathcal{X}$ cf. \cite[Lemma 1.1]{Lev83}.
\end{proof}
\medskip

\noindent The proof of Theorem \ref{007} follows along the same lines as in Theorem \ref{MT}, so we only sketch it very briefly.
\begin{proof}[Theorem \ref{007}]
Notice first that since $\sigma_i$ is  a section of $\mathcal{E}_i\otimes \mathcal{O}_{\mathcal{X}} / t^{k+2} \mathcal{O}_{\mathcal{X}}$,
the curvature term containing $\sigma_i$ vanishes on $X\setminus Div (\sigma_i)$ up to order $k$.
Then the same argument as Lemma \ref{linalg} implies that
\begin{equation}\label{varthm}
\lambda_k = \bar \partial \alpha_k + D' (\beta_k) \qquad\text{on } X \setminus (Div(s) + \sum_i Div (\sigma_i)).
\end{equation}
The scheme of the proof of Theorem \ref{MT} applies \emph{mutatis mutandis} and we infer that $\lambda_k$ is $\bar \partial$-exact. The theorem is proved.
\end{proof}

\subsection{Appendix: currents with values in a line bundle}\label{currents} In this section we show that the results of de Rham-Kodaira in \cite{dRKo} concerning Hodge decomposition for currents on compact manifolds have a complete analogue in our setting, i.e. for $(X, \omega_\mathcal{C})$ and $(L, h_L)$.
\smallskip

\noindent The general setup here is as follows. We consider $(X, E+F)$ a compact K\"ahler manifold, together with an snc divisor
denoted by $E+F$. Here we use the notation $\displaystyle E:= \sum E_i$ as well as
$\displaystyle F:= \sum_i F_i$. We assume that $(L, h_L)$ has the properties set at the beginning of Section 5 and we will use the same notations. Let $\omega_\mathcal{C}$ be a metric with conic singularities on $X$ for which the corresponding multiplicities $m_i$ are all equal to $m$,
an integer large and divisible enough.
\medskip

\noindent
 Let $\Omega \subset X$ be a coordinate open subset,
 on which we have $F\cap \Omega= (z_{r+1}\dots z_{r+s}= 0)$, so as the notations to be consistent with \eqref{pave90}. We denote by $\pi$ the ramified cover
\begin{equation}\label{pave66}
\pi(w)= (w_1,\dots, w_r, w_{r+1}^m,\dots, w_{r+s}^m,\dots, w_n),
\end{equation}
and let $(\Omega_i, \pi_i)_{i\in I}$ be a covering of $X$ together with the
corresponding maps. Notice that for each index $i\in I$ the form $\omega_i:= \pi_i^\star\omega_\mathcal{C}$ is a smooth K\"ahler metric on the domain of $\pi_i$.

\smallskip

\noindent We consider the equation
\begin{equation}\label{pave65}
\Delta''_{\omega_\mathcal{C}}v= u
\end{equation}
where $u$ is an $L^2$-form of type $(p, 0)$ with values in $L$ and orthogonal to the space of
harmonic forms (here $p\leq n$ is a positive integer). The solution $v$ belongs to the domain $\mathrm {Dom}(\bar \partial)$ of the $\bar \partial$ operator as well as to
$\mathrm {Dom}(\bar \partial^\star)$. In general, the existence and regularity properties of $v$ can be quite non-trivial to establish, but in our case this is rather straightforward via the maps $\pi_i$ above.
Alternatively, one can also analyse the equation \eqref{pave65} by a global
reduction to the
smooth case, i.e. one can write
\begin{equation}\label{pave67}
\omega_\mathcal{C}= \lim_{\varepsilon\to 0}\omega_{\mathcal{C}, \varepsilon}
\end{equation}
where $\omega_{\mathcal{C},\varepsilon}$ has the following shape $$\sum_{k=r+1}^{r+s}\sqrt{-1}
\frac{dz_k\wedge d\overline z_k}{(\varepsilon^2+ |z_k|^2)^{1-\frac{1}{m}}}+ \sum_{k\not\in\{r+1,\dots, r+s\}}\sqrt{-1}
{dz_k\wedge d\overline z_k}$$
and solve \eqref{pave65} with a perturbation of $u$ which is orthogonal to the
space of harmonic forms for $(X, \omega_{\mathcal{C}, \varepsilon})$.
\smallskip

\noindent Anyway, if we define the local -smooth!- forms
\begin{equation}\label{pave69}u_i:= \frac{1}{\prod w_j^{q_jm}}\pi^\star_iu,\end{equation}
where $j=r+1,\dots, r+s$
then the equation \eqref{pave65} becomes
\begin{equation}\label{pave68}
\Delta''_{\omega_i}v_i= u_i
\end{equation}
where $v_i$ is defined as in \eqref{pave69}. It follows that $(v_i)_{i\in I}$ is a collection of smooth forms. In what follows we will say that an $L$-valued form $v$ is \emph{smooth in orbifold sense} if the associated set $(v_i)_{i\in I}$ consists of non-singular $(p, 0)$-forms.
\medskip

\noindent Next, let $\alpha$ be an $L$-valued $(n, q)$ form with log poles along the divisor $E$. Given any $L$-valued form $\Phi$ of type $(n-q,0)$ which is smooth in orbifold sense, the integral
\begin{equation}\label{pave70}
\int_X\alpha\wedge \overline \Phi e^{-\varphi_L}
\end{equation}
is convergent, since the set of poles of $\alpha$ intersects completely with the set of singularities of $h_L$. Thus we can interpret $\alpha$ as $L$-valued current on $X$, acting on forms $\overline \Phi$ as in \eqref{pave70}.
\smallskip

\noindent Precisely as in the untwisted case (i.e. without any ``$L$''), we can define the \emph{harmonic part} of $\alpha$ as
\begin{equation}\label{pave71}
\mathcal{H}(\alpha):= \sum_i \langle \alpha, \xi_i\rangle \xi_i
\end{equation}
where $(\xi_i)_i$ is a basis of $L^2$ harmonic $L$-valued forms of type $(n, q)$ and where the notation in \eqref{pave71} is
\begin{equation}\label{pave72}
\langle \alpha, \xi\rangle:= \int_X\alpha\wedge \overline {\gamma_\xi}e^{-\varphi_L},
\end{equation}
and $\gamma_\xi:= \star \xi$.
\smallskip

\noindent We define next by duality the Green operator $\mathcal{G}(\alpha)$.
Let
$\Phi$ be a $(n-q, 0)$-form with values in $L$ which moreover is smooth in orbifold sense. Then we can solve the equation
\begin{equation}\label{pave16}
\Delta''(v)= \Phi- \mathcal{H}(\Phi)
\end{equation}
so that the solution $v$ is orthogonal on the space of harmonic forms.
Such $v$ is unique, and we denote it by $\mathcal{G}(\Phi)$. As discussed above, it is smooth in orbifold sense.
\medskip

\noindent Then we define the current $\mathcal{G}(\alpha)$ as follows
as usual
\begin{equation}\label{pave17}
\langle \mathcal{G}(\alpha), \overline \Phi\rangle:= \int_{X}\alpha\wedge \overline{\mathcal{G}(\Phi)}e^{-\varphi_L}.
\end{equation}
and this induces a decomposition of the current $\alpha$ as in the usual case
\begin{equation}\label{pave18}
\alpha= \mathcal{H}(\alpha)+ \Delta''\big(\mathcal{G}(\alpha)\big).
\end{equation}

\begin{remark}\label{currentdecomp} This discussion applies to any current $T$ of type $(n,1)$ acting on forms
  which are smooth in orbifold sense. Thanks to \eqref{pave18}, there is an associate decomposition
\begin{equation}\label{pave73}
T= \mathcal{H}(T)+ \Delta''\big(\mathcal{G}(T)\big)
\end{equation}
in the sense of currents. If moreover $T$ is closed, then we can write
\begin{equation}\label{pave19}
T= \mathcal{H}(T)+ \bar \partial T_{1}
\end{equation}
where $\displaystyle T_{1}= \bar \partial^\star\big(\mathcal{G}(T)\big)$.

In fact, \eqref{pave19} comes from the following argument:  Thanks to \eqref{diffdep}, we know that $\mathcal{H}(T)$
is $\bar \partial$-closed. Then we have $\Delta^{''} \bar \partial \big(\mathcal{G}(T)\big)=0$. As a consequence $\bar \partial \big(\mathcal{G}(T)\big)$ is smooth in the orbifold sense and we get $\bar \partial^\star \bar \partial \big(\mathcal{G}(T)\big) =0$.
\end{remark}

\medskip

\noindent Coming back to the case of a current induced by a form $\alpha$ with log-poles,
the Green current $\mathcal{G}(\alpha)$ has a few interesting properties that we collect in our next statement.

\begin{proposition}\label{reg2} The following statements hold true.
\begin{enumerate}
\smallskip

\item[\rm (1)] The current $\mathcal{G}(\alpha)$ is a smooth $L$-valued $(n, q)$--form on
$X\setminus (E\cup F)$ which moreover is $L^2$ on $X$.
\smallskip

\item[\rm (2)] For each index $i$ the local form
$\displaystyle  v_i:= \frac{1}{\prod w_i^{q_im}}\pi_i^\star\left(\mathcal{G}(\alpha)\right)$
is a solution of an equation of the following type
\begin{equation}\label{pave8}\nonumber
\Delta''_{\omega_i}(v_i)= u_i
\end{equation}
where $u_i$ is a $(n,q)$-form with logarithmic poles along the local divisor $w_{1}\dots w_{r}= 0$.
\smallskip

\item[\rm (3)] For every $\delta> 0$ form $v_i$ belongs to the Sobolev space $W^{2, 2-\delta}$ (i.e. it has two derivatives in $L^{2-\delta}$).
\smallskip

\item[\rm (4)] The form $\tau_i:= \partial^\star v_i$ is $L^2$ and moreover
``smooth'' with respect to the variables $w_j$ for each $j=r+1,\dots, n$, in the sense that it admits a Taylor expansion
\begin{equation}\label{pave7}\nonumber
  \tau_i(w)= \sum_{|I|+|J|\leq N-1} w^{\prime \alpha}\overline w^{\prime \beta}\tau_{I\overline J}(w'')+
\sum_{|I|+|J|= N} w^{\prime \alpha}\overline w^{\prime \beta}\tau_{I\overline J}(w)
\end{equation}
where $$w'= (w_{r+1},\dots, w_{n}),\qquad w''= (w_{1},\dots, w_r)$$
and $\tau_{I\overline J}$ are $L^2$.
\end{enumerate}
\end{proposition}

\begin{proof} In order to prove the point (1), our first remark is that

\begin{equation}\label{pave10}
\int_X |\mathcal{G}(\Phi)|^2e^{-\varphi}\leq C \int_X|\Phi- \xi_\Phi|^2e^{-\varphi}
\end{equation}
by Poincar\'e inequality. The inequality \eqref{pave10} is valid for $(n-q, 0)$-forms and it implies that
\begin{equation}\label{pave11}
\int_X |\mathcal{G}(\Phi)|^2e^{-\varphi}\leq C \int_X|\Phi|^2e^{-\varphi}
\end{equation}
because the $L^2$ norm of $\xi_\Phi$ is smaller than the $L^2$ norm of $\Phi$. The smoothness of
$\mathcal{G}(\alpha)$ in the complement of the support of the divisor $E+F$ is standard. Then the Sobolev norm of order two of
$\mathcal{G}(\Phi)$ is bounded by the RHS of \eqref{pave11}.
\medskip

\noindent The first part of the point (2) (i.e. the equation satisfied by the form $v_i$) was already discussed in \eqref{pave68}. The fact that the RHS is a form with log-poles is immediate and we skip the details.

\medskip

\noindent The points (3) and (4) are a bit more interesting and we will provide a complete treatment.
\smallskip

\noindent By the definition of the current $\mathcal{G}(\alpha)$ as in \eqref{pave17} it follows that we have
\begin{equation}\label{pave60}
\left|\langle \mathcal{G}(\alpha), \overline \Phi\rangle\right|\leq \Vert\alpha\Vert_{L^{2-\delta}}\Vert \mathcal{G}(\Phi)\Vert_{L^{\frac{2-\delta}{1-\delta}}}.
\end{equation}
Now the $L^{\frac{2-\delta}{1-\delta}}$-norm of $\mathcal{G}(\Phi)$ is certainly controlled by the Sobolev norm above (at least when $\delta$ is small enough) so in conclusion we get
\begin{equation}\label{pave61}
\left|\langle \mathcal{G}(\alpha), \overline \Phi\rangle\right|^2\leq C_\delta \Vert\Phi\Vert^2_{L^2}
\end{equation}
which shows that $\mathcal{G}(\alpha)$ is in $L^2$.
\smallskip

\noindent In particular, this preliminary discussion allow us to assume that the form $v_i$ in \eqref{pave8} is in $L^2$.
Together with the G\aa rding inequality established in \cite{Agm}, Theorem 7.1 we obtain
\begin{equation}\label{pave62}
\Vert v_i\Vert_{W^{2, 2-\delta}}\leq C_p(\Vert v_i\Vert_{L^{2-\delta}}+ \Vert u_i\Vert_{L^{2-\delta}})
\end{equation}
for any $0<\delta<2$.

\noindent Let $\Xi$ be any local vector field with constant coefficients of the following shape
\begin{equation}\label{pave63}
\Xi= \sum_{i=r+1}^n a_i\frac{\partial}{\partial w_i}.
\end{equation}
The equation \eqref{pave8} implies that we have
\begin{equation}\label{pave64}
\Delta''_{\omega_i}(\Xi\cdot v_i)= \Xi\cdot u_i + \mathcal L(v_i)
\end{equation}
where $\mathcal L$ is a second order operator. In \eqref{pave64} we denote by $\Xi\cdot v_i$ the form whose coefficients are the derivative of those of $v_i$ in the direction of $\Xi$.
\smallskip

\noindent
By Sobolev embedding theorem we infer that $\Xi(v)$ is in $L^2$ (provided that $\delta\ll 1$).
It follows from \eqref{pave64} that $\Xi(v)$ has two derivatives in $L^p$. We can iterate this method,
and infer that $\Xi^m(v)\in L^2$ for any $m$.
\medskip

\noindent Finally, the usual argument --by using a Fourier expansion-- implies that $v_i$ admits a Taylor expansion as required in (4). We discuss here only a particular case, for notational reasons.
Let $f\in L^2(\mathbb D)$ be a square integrable function defined on the unit disk in $\mathbb{R}^2$, and such that
$\displaystyle \frac{\partial^m}{\partial x^m}(f)$ is in $L^2$ for any positive integer $m$.

We consider the convolution $f_\varepsilon$ of $f$ with the usual regularisation kernel and as usual we can assume that this functions are 1-periodic. We then write
\begin{equation}\label{pave75}
f_\varepsilon(x, y)= \sum_{k\in \mathbb{Z}}f_{\varepsilon, k}(y)e^{\sqrt{-1}kx}
\end{equation}
and by hypothesis, together with the $L^2$ convergence of convolutions and their derivatives
we infer in particular that for any $m\in \mathbb{Z}_+$ we have
\begin{equation}\label{pave76}
\sum_{k\in \mathbb{Z}}k^{2m}\int_{|y|< 1}|f_{\varepsilon, k}(y)|^2d\lambda(y)< C_m
\end{equation}
where $C_m$ is a constant independent of $\varepsilon$.
\smallskip

\noindent A first consequence of \eqref{pave76} is the existence of $C>0$ such that
\begin{equation}\label{pave77}
\int_{|y|< 1}|f_{\varepsilon}(0, y)|^2d\lambda(y)< C
\end{equation}
which can be seen by observing that $$\displaystyle \left|\sum_{k=-N}^Nf_{\varepsilon, k}(y)\right|^2\leq
2\sum_{k=-N}^Nk^2|f_{\varepsilon, k}(y)|^2\sum_{k= 1}^N\frac{1}{k^2}$$
and integrating.
\smallskip

\noindent We next write $\displaystyle f_\varepsilon(x, y)= f_\varepsilon(0,y)+ x\int_0^1\partial_1f_{\varepsilon}(tx, y)dt$. Remark that we already know that the $L^2$ norm of the first term of the RHS is uniformly bounded,
hence it is weakly convergent towards a function in $L^2$. We will analyse next the other function; notice that
\begin{equation}\label{pave78}
\int_\mathbb D\left|\int_0^1\partial_1f_{\varepsilon}(tx, y)dt\right|^2d\lambda\leq \int_0^1dt\int_\mathbb D\left|\partial_1f_{\varepsilon}(tx, y)\right|^2d\lambda
\end{equation}
which equals
\begin{equation}\label{pave79}
\int_\mathbb D\log\frac{1}{|x|}\left|\partial_1f_{\varepsilon}(x, y)\right|^2d\lambda.
\end{equation}
By H\"older inequality the expression \eqref{pave79} is smaller than
\begin{equation}\label{pave800}
C_\delta\int_{|y|< 1}dy\left(\int_{|x|< 1}\left|\partial_1f_{\varepsilon}(x, y)\right|^{2+ 2\delta}dx\right)^{\frac{1}{1+\delta}},
\end{equation}
and on the other hand, for each fixed $y$ we have
\begin{equation}\label{pave811}
\frac{1}{C}\left(\int_{|x|< 1}\left|\partial_1f_{\varepsilon}(x, y)\right|^{2+ 2\delta}dx\right)^{\frac{1}{1+\delta}}\leq
\int_{|x|< 1}\left|\partial_1f_{\varepsilon}(x, y)\right|^{2}dx+ \int_{|x|< 1}\left|\partial_{11}f_{\varepsilon}(x, y)\right|^{2}dx
\end{equation}
by Sobolev inequality. The integration with respect to $y$ of \eqref{pave811} shows that the LHS of
\eqref{pave78} is uniformly bounded.

\noindent The arguments for the general case are completely similar --modulo the fact that the number of indexes is greater-- and we will not provide any further detail.
\end{proof}

\section{More results and applications}\label{ana}

\noindent The subsections contained in this part of our article are organised as follows.
\smallskip

\noindent $\bullet$  In subsection \ref{smoothdiv},  we provide an alternative argument for Theorem \ref{Lev83}. The main
point is to show that if the divisor $s=0$ is smooth, then for each order $k$ we can construct a particular vector field
$\Xi$, so that the resulting forms $\alpha_k$ and $\beta_k$ are automatically smooth (hence in particular $L^2$, given that the multiplier ideal sheaf corresponding to $\frac{m-1}{m}\mathrm{Div}(s)$ is trivial). Then we can extend $s$ over infinitesimal nbd's of any order and the conclusion follows.
\smallskip

\noindent $\bullet$ In subsection \ref{sufcond},  we provide a sufficient condition for the $L^2$ requirement in Theorem \ref{MT} to be fulfilled. This is formulated as follows: we assume that there exists a divisor $\Sigma$ on the total space $\mathcal{X}$ of our family such that $\Sigma+ X$ is snc and such that the support of the divisor $\mathrm{Div}(s)$ is contained in $\Sigma|_X$. Then for any $k$ we can find a $\mathcal{C}^\infty$ extension $s_k$ such that \eqref{intr7} is satisfied --so in particular, $s$ extends to $\mathcal{X}$.

\smallskip

\noindent $\bullet$ In subsection \ref{2nd}, we discuss the second order extension. The problem extending sections of the sheaf $\mathcal{F}_1$ is already very interesting, since we have to deal with an unreduced space.  The partial result we obtain in this direction is the following.

\begin{thm}\label{order2}
Let $\displaystyle s\in H^0(X, mK_{X})$ be a pluricanonical section defined on the
central fiber $X$ of a family $p:\mathcal{X}\to D$ of compact K\"ahler manifolds. We assume that there exists a non-singular vector field $\Xi$ on the total space $\mathcal{X}$ such that
\begin{equation}\label{stupid_cond1}
  dp(\Xi)= \frac{\partial}{\partial t}, \qquad \sup_X
  \frac{\big|\bar \partial \Xi |_X\big|^2_{\omega_E}}{\log^2\frac{1}{|s|^2}}\leq C,
\end{equation}
where $\omega_E$ is a metric with Poincar\'e singularities along the set
$(s= 0)$.
Then there exists a section $s_2\in \mathcal C^\infty(\mathcal{X}, mK_{\mathcal{X}})$ such that
$\displaystyle s_2|_X= s$ and \begin{equation}\label{int1}
\bar \partial s_2= t^3\Lambda_2,\end{equation} where $\Lambda_2$ belongs to the space $\mathcal C^\infty_{0,1}(\mathcal{X}, mK_\mathcal{X})$.
\end{thm}

\medskip

\noindent We discuss next the main technical result needed in the proof of Theorem \ref{order2}. The general setup is as follows.
Let $(L, h_L)$ be a Hermitian line bundle on $X$, endowed with a metric with analytic singularities. We assume that the curvature current
$\Theta(L, h_L)\geq 0$ is semi-positive.
Let $v$ be an $L$-valued form of type
$(n-1, 1)$, such that $D'v$ is $\bar \partial$-closed. We are interested in the $\bar \partial$-equation
\begin{equation}\label{int2}
\bar \partial u= D'v.
\end{equation}
\noindent We assume further that $X$ is endowed with a Poincar\'e-type metric $\omega_E$,
with poles along the singular locus of $h_L$. The result we obtain is as follows.
\begin{thm}\label{L2I}
  Assume moreover that $v$ is $L^2$ with respect to a Poincar\'e-type metric $\omega_E$ on $X$. Then the equation \eqref{int2} admits a solution $u$ such that
\begin{equation}\label{int2111}
\int_X|u|^2e^{-\varphi_L}\leq \int_X|v|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}.
\end{equation}
\end{thm}
\noindent The new aspect of Theorem \ref{L2I} is that we do not assume
that $D'v$ belongs to $L^2$.
\smallskip

\noindent It is a very interesting question to decide weather the equation \eqref{int2} can be solved with estimates involving an incomplete metric $\omega$ on $X\setminus (h_L= \infty)$. Let $\omega$ be non-singular K\"ahler metric on $X$.
In this direction we obtain the following result.

\begin{thm}\label{L2II} Let $(L, h_L)$ be a Hermitian line bundle, such that $\Theta(L, h_L)\geq 0$ is semi-positive. Let $v$ be an $L$-valued form of type
$(n-1, 1)$, such that $D'v$ is $\bar \partial$-closed. We assume that the following hypothesis are satisfied.
\begin{enumerate}
\smallskip

\item[\rm (1)] The metric $h_L$ has analytic singularities, and let $Z$ be the support of the set $(h_L= \infty)$.

\item[\rm (2)] The integrals
\begin{equation}\nonumber
  \int_X|v|^2_\omega e^{-\varphi_L}dV_\omega,\qquad  \int_X
  |D' v|^2 e^{-\varphi_L}dV_{\omega}
\end{equation}
are convergent.
\smallskip

\item[\rm (3)] There exists a complete metric $\omega_Z$ on $X\setminus Z$ such that we have
\begin{equation}\nonumber
  \int_X|v|^2_{\omega_Z} e^{-\varphi_L}dV_{\omega_Z}< \infty
  \end{equation}

\end{enumerate}

\noindent Then the equation \eqref{int2}
has a solution $u$ such that
\begin{equation}\label{int4}
\int_X|u|^2e^{-\varphi_L}\leq \int_X|v|^2_{\omega}e^{-\varphi_L}dV_{\omega}.
\end{equation}
\end{thm}
\medskip

\begin{remark}
For applications, it would be very important to remove the hypothesis (3) in Theorem \ref{L2II}. However, we are not sure whether
the statement is still correct.
\end{remark}

\subsection{A remark: first order extension}\label{firstorderex}
The first result that we establish here is that one can always
extend a pluricanonical section $s$ defined on $X$ to the first infinitesimal neighbourhood. This is  established in the paper \cite{Lev83} by different methods.

We are in the context of \eqref{eq35} and let $\lambda_1 :=\frac{\Lambda_1}{dt} |_X$ be the restriction on the central fiber.
By applying the Lie derivative $\mathcal{L}_\Xi$ to \eqref{eq35}, as we have seen in the section \ref{alg}, this can be rewritten as in \eqref{eq38}.
The restriction of this expression to the central fiber gives
\begin{equation}\label{eq42}
\frac{1}{m}\lambda_1= \bar \partial a_1 + D'_X\big(b_1\big)
\end{equation}
on $X\setminus Z$, where we use the (abuse of...) notation
\begin{equation}\label{eq43}
a_1:= \frac{\mathcal{L}_\Xi(s_1)}{dt}\Big|_X,\qquad b_1:= \rho_1|_X
\end{equation}
and $\rho_1$ is the form already appearing in Lemma \ref{linalg}.
\medskip

\noindent The form $D'_X\big(b_1\big)$ is certainly closed on $X\setminus Z$. In order to apply
Theorem \ref{Hodge}, we have to check the $L^2$ hypothesis. We recall that here the data is
\begin{equation}\label{eq44}
L:= (m-1)K_X, \qquad \varphi_L:= \frac{m-1}{m}\log |s_i|^2
\end{equation}
where $s_i$ is the local expression of of the section $s$ of $mK_X$.
\smallskip

\noindent Given the explicit expression of $a_1$ and $b_1$, the $L^2$ hypothesis in (ii) of Theorem \ref{Hodge} are easy to check. ``For punishment", we will give the details for $b_1$ and its derivative.

With respect to the local coordinates in \eqref{eq34}, we have
\begin{equation}\label{eq45}
b_1|_U\simeq s_i\sum_\alpha \widehat {dz_i^\alpha}\wedge \bar \partial v_i^\alpha
\end{equation}
and this is clearly $L^2$ with respect to the weight in \eqref{eq44}. The symbol
$\simeq$ is \eqref{eq45} means that the restriction of $b_1$ to $U$ equals the RHS
with respect to the coordinates $(z_i)$.

Moreover, the fact that $D'_X(b_1)$ is equally $L^2$ boils down to the convergence of the
integral
\begin{equation}\label{eq46}
\int_{U_i}\frac{|df|^2}{|f|^{2\frac{m-1}{m}}}d\lambda
\end{equation}
where $f$ is a holomorphic function defined in the open set of $\mathbb{C}^n$ containing $U$.
This in turn is quickly verified by a change of variables formula.
\smallskip

\noindent In conclusion, we have
\begin{equation}\label{eq47}
\lambda_1= \bar \partial \gamma_1  \qquad\text{on }  X .
\end{equation}

Now we can construct the $2$-jet extension as follows.
Let $\cup U_i$ be a Stein cover of $\mathcal{X}$ and $\{\phi_i \}_i$ be a partition of unity.
On every $U_i$, we can thus find a $\gamma_{1, i}$ such that $\Lambda_1 |_{U_i} = \bar \partial \gamma_{1,i}$ and $\frac{\gamma_{1,i}}{dt } |_{U_i \cap X} = \gamma_1$. Therefore $\gamma_{1,i} - \gamma_{1,j} = t \cdot \gamma_{i,j}$ since the difference is holomorphic.
Set $s_2 := s_1 - t \sum \phi_i \gamma_{1,i} $. Then $s_2$ is a smooth section of the bundle
$K_{\mathcal{X}}+ L$ such that
\begin{equation}\label{eq48}
s_2|_X= s, \qquad \bar \partial s_2= t^2\Lambda_2.
\end{equation}
\medskip

\begin{remark}\label{rk1}
This first step in the extension of the section $s$ is somehow misleading, i.e. too simple in some sense. Some of the real difficulties one has to deal with are appearing during the
extension to the second infinitesimal nbd, see the subsection \ref{2nd} below.
\end{remark}

\subsection{Extension of sections whose zero set is non-singular}\label{smoothdiv} In order to extend our section $s$ to the first infinitesimal neighbourhood, we have used the Lie derivative
with respect to an arbitrary vector field $\Xi$. For higher order extension, this does not seem to be possible because of the singularities of the operator $D'_\mathcal{X}$. In the case of a section $s$ whose zero set $(s=0)\subset X$ is non-singular (treated in \cite{Lev83}), this is done as follows.
\smallskip

\noindent We assume that the extension $s_k$ to the $k^{\rm th}$ infinitesimal neighbourhood has already been constructed.
For each trivialising open set $\Omega_i\subset \mathcal{X}$ together with fixed coordinates functions
$\displaystyle (z_i^{\alpha})_{\alpha=1,\dots, n}$ we denote by
\begin{equation}\label{ideal3}
f_i
\end{equation}
the holomorphic function corresponding to the $k^{\rm th}$-jet of $s_k$. The set $(f_i= 0)\subset \Omega_i$ is
non-singular and transversal to the central fiber $X$. Then we define
a new set of local coordinates
\begin{equation}\label{ideal4}
(t, w_i^1,\dots, w_i^n)
\end{equation}
on $\Omega_i$ such that $w_i^1= f_i$ --here we use the fact that at each point of
$\Omega_i$ we can find an index $\alpha$ such that the form
\begin{equation}\label{ideal9}
df_i\wedge dt\wedge  \widehat{dz_i^\alpha}\neq 0
\end{equation}
is non-vanishing at the said point. Strictly speaking we have shrink eventually the
set $\Omega_i$; however, given that the map $p:\mathcal{X}\to \mathbb D$ is proper we
can assume that the coordinates \eqref{ideal4} are defined on  $\Omega_i$ itself.
\smallskip

\noindent Therefore the equality
\begin{equation}\label{ideal5}
w_i^1= g_{i j}(t, w_j)\cdot w_j^1+ t^{k+1}\tau_{ij}(t, w_j)
\end{equation}
is valid on the overlapping subsets $\Omega_i\cap \Omega_j$, where $(g_{ij})$ are the
transition functions for the bundle $K_\mathcal{X}+ L$.
\smallskip

\noindent We introduce next the following vector field
\begin{equation}\label{ideal6}
\Xi_k:= \sum \theta_i\frac{\partial}{\partial t}\Big|_{\Omega_i}
\end{equation}
corresponding to the covering $\big(\Omega_i, (t, w_i)\big)$.

\noindent By the transition relation \eqref{ideal5}, we have
\begin{equation}\label{ideal7}
\Xi_k|_{\Omega_i}= \frac{\partial}{\partial t}\Big|_{\Omega_i}+ (a_i w_i^1+ b_i t^{k})\frac{\partial}{\partial w_i^1}+
\sum _{\alpha\geq 2}v_i^\alpha\frac{\partial}{\partial w_i^\alpha}
\end{equation}
where $a_i, b_i$ and $v_i^\alpha$ are smooth. We notice that the vector field
$\Xi_k$ have the following important properties.

\begin{enumerate}
\item[(i)] Its coefficients are smooth.

\item[(ii)] The projection $dp(\Xi_k)$ equals $\displaystyle \frac{\partial}{\partial t}$.

\item[(iii)] Modulo the ideal $(t^k)$, it is tangent to the set $f_i=0$.
\end{enumerate}

\smallskip
\noindent The connection we are using on the bundle $(m_0-1)K_{\mathcal{X}}$ is induced by
the $\mathcal C^\infty$ section $s_k$. This means that with respect to our local coordinates in \eqref{ideal4} we have
\begin{equation}\label{ideal10}
\widetilde f_i(t, w)= w_i^1h_i(t, w_i)+ t^{k+1}g_i(t, w_i)
\end{equation}
for some smooth functions $g_i, h_i$. Here $\widetilde f_i(t, w)$ is the local expression of the section $s_k$.

\medskip

\noindent Given these considerations, we infer the following.

\begin{lemme}\label{trivca}
Let $k\geq 0$ be a positive integer, and assume that the extension $s_k$ such that
\begin{equation}\label{ideal100}
\bar \partial s_k= t^{k+1}\Lambda_k
\end{equation}has been already constructed. We use the vector field $\Xi_k$ introduced in \eqref{ideal6} and consider the Lie derivative $\displaystyle\mathcal{L}_{\Xi_k}$ induced by it. Lemma \ref{linalg} provides us with forms $\alpha_k$ and $\beta_k$ such that
\begin{equation}\label{ideal101}
\Lambda_k= \bar \partial \alpha_k+ D'_\mathcal{X}\big(dt\wedge \beta_k)
\end{equation}
modulo the ideal generated by $t$. Then the restrictions $\alpha_k|_{X}$ and $\beta_k|_X$
are smooth.
\end{lemme}
\medskip

\noindent Before explaining the proof, we note that Lemma \ref{trivca} combined with Theorem \ref{Hodge} show that the restriction
\begin{equation}\label{ideal102}
\frac{\Lambda_k}{dt}\Big|_X
\end{equation}
is $\bar \partial$-exact. Thus we can extend $s$ one step further, given that there exist
forms $\mu_k$ and $\Lambda_{k+1}$ of type $(n+1, 0)$ and $(n+ 1, 1)$ respectively,
such that
\begin{equation}\label{ideal103}
\Lambda_k= \bar \partial \mu_k+ t\Lambda_{k+1}.
\end{equation}
By combining \eqref{ideal100} and \eqref{ideal103} we infer the existence of
$s_{k+1}$ such that
\begin{equation}\label{ideal104}
\bar \partial s_{k+1}= t^{k+2}\Lambda_{k+1}.
\end{equation}
We can repeat this procedure inductively, showing that the section $s$ admits an extension to
the infinitesimal nbd of an arbitrary order.
The formal arguments in \cite{Lev83} are implying that $s$ extends to a topological nbd of $X$ in $\mathcal{X}$.
\smallskip

\noindent We turn now to the proof of Lemma \ref{trivca}.

\begin{proof}
Considering the preparation we have done in the previous sections, the arguments which follow should be clear: we will proceed by induction, by using the fact that for each $k$ we have the relations \eqref{eq55}.

To this end, it would be helpful to remark that we have the following
while computing the obstruction to extend the section modulo $t^{k+2}$, we have to deal with quantities as
\begin{equation}\label{ideal22}
\mathcal{L}_{\Xi_k} s_k, \qquad D' _\mathcal{X} \bigg( dt\wedge \Big( \Xi  \rfloor D' _\mathcal{X} \big(\Xi \rfloor (\bar \partial \Xi\rfloor s_k)\big) \Big)\bigg)
\end{equation}
and their iterations. Locally we can write $s_k= f\sigma$, where $f$ is the function given by the expression like
\eqref{ideal10}, and $\sigma$ is a top form with values in $(m-1)K_\mathcal{X}$. We have
\begin{equation}\label{ideal23}
\mathcal{L}_{\Xi_k} s_k= \Xi(f)\sigma+ f \mathcal{L}_{\Xi_k} \sigma
\end{equation}
in which the second term has a -potentially- singular component
\begin{equation}\label{ideal24}
f \mathcal{L}_{\Xi_k} \sigma \equiv -\frac{m-1}{m}f\cdot \frac{df}{f}\wedge \left(\Xi_k\rfloor \sigma\right)= -\frac{m-1}{m}\Xi_k(f)\sigma.
\end{equation}
\smallskip

\noindent The point here is that the vector field $\Xi_k$ was constructed in such a way
that $\Xi_k(f)$ is a \emph{multiple} of $f$, plus some power of $t$.
More precisely, we have
\begin{equation}\label{ideal29}
\Xi_k\cdot (f, t^{k+1})\subset (f, t^{k})
\end{equation}
where we remark that the ideal $(f, t^{k+1})$ is in fact globally defined on $\mathcal{X}$.

\noindent Therefore we can
take as many times the Lie derivative as the power of $t$ allows, the result will still be of the same type. We show next that the same is true for the second term in \eqref{ideal22}.

Indeed we have
\begin{equation}\label{ideal25}
D' _\mathcal{X} \big(\Xi_k \rfloor (\bar \partial \Xi_k\rfloor s_k)\big)= \partial f\wedge \Xi_k \rfloor (\bar \partial \Xi_k\rfloor \sigma)+
f D' _\mathcal{X} \big(\Xi_k \rfloor (\bar \partial \Xi_k\rfloor \sigma)\big)
\end{equation}
and a further contraction with $\Xi$ gives
\begin{equation}\label{ideal26}
\Xi_k\rfloor \left(\partial f\wedge \Xi_k \rfloor (\bar \partial \Xi_k\rfloor \sigma)\right)= \Xi_k(f)\cdot \Xi_k\rfloor (\bar \partial \Xi_k\rfloor \sigma)
\end{equation}
for the first term in \eqref{ideal25}. The singular part of the second one is
\begin{equation}\label{ideal27}
f\cdot \frac{df}{f}\wedge \Xi_k \rfloor (\bar \partial \Xi_k\rfloor \sigma)
\end{equation}
and when we contract it with $\Xi_k$ the result will be the RHS of \eqref{ideal26}.
\medskip

\noindent In conclusion, after the first derivative of the relation \eqref{eq339} the forms
$u_2$ and $v_2$ of \eqref{eq55} have their coefficients in the ideal $(f, t^k)$.
We can therefore iterate this procedure, and obtain the conclusion.
\end{proof}

\smallskip

\begin{remark} We note that at each step $k$ we choose a vector field $\Xi_k$ adapted to the corresponding extension $s_k$.
\end{remark}
\smallskip

\begin{remark} If the zero set $(s=0)$ of our initial section is singular, then one can still
construct a vector field adapted to it as in the proof just finished,
but the difference is that the new $\Xi$ will be singular along $s=0, ds= 0$.
\end{remark}
\subsection{A sufficient condition}\label{sufcond} In this subsection we assume that the divisor $\mathrm{Div}(s)$ corresponding to the section $s\in H^0(X, mK_X)$ has the following property.
\begin{hyp}\label{cond}
There exists a divisor $\Sigma$ on $\mathcal{X}$ such that we have.
\begin{enumerate}
\smallskip

\item[(i)] The divisor $\Sigma+ X$ is snc.
\smallskip

\item[(ii)] The support of $\mathrm{Div}(s)$ is contained in the restriction $\displaystyle \Sigma|_X$.
\end{enumerate}
\end{hyp}
\medskip

\noindent Then we show that the following holds true.

\begin{thm}\label{hypo} Let $s\in H^0(X, mK_X)$ be a pluricanonical section such that Hypothesis \ref{cond} is satisfied. Then $s$ admits a holomorphic extension to $\mathcal{X}$.
\end{thm}

\begin{proof} The role of the two conditions (i) and (ii) above is to furnish a special coordinate system on $\mathcal{X}$ which we will use during all the proof.

\noindent As consequence of these requirements we have a finite covering $\displaystyle \big(\Omega, (t, z_i)\big)_{i\in I}$ of an open subset of the central fiber $X$ of $p$ such that for each $i\in I$ we have $\Sigma\cap \Omega_i= (z_i^1\dots z_i^r= 0)$ together with
the transition functions
\begin{equation}\label{suff1}
z_i^\gamma= g_{ij}^\alpha(t, z_j)z_j^\gamma,\qquad z_i^\beta= \phi_{ij}^\beta(t, z_j)
\end{equation}
for each $\gamma= 1,\dots, r$ and $\beta= r+1,\dots, n$. We denote the resulting function by $\Phi$.
\smallskip

\noindent These coordinates are inducing a trivialisation of the canonical bundle of $K_\mathcal{X}$ with respect to which our section $s$ corresponds locally to holomorphic functions
\begin{equation}\label{suff2}
f_{i0}(z_i)= g_i(z_i)\prod_{q=1}^r(z_i^q)^{m_q}
\end{equation}
where $g_i$ is a nowhere vanishing holomorphic function on $\Omega_i\cap X$ and $m_q\geq 0$.
\medskip

\noindent We formulate the next statement.
\begin{claim}\label{blb}
For every order $k\geq 0$ there exists a section
$s_k\in H^0(\mathcal{X}, \mathcal{F}_k)$ --with $\mathcal{L}= (m-1)K_\mathcal{X}$-- such that $\displaystyle s_k|_X= s$ and such that it can be written locally as follows
\begin{equation}\label{suff3}
f_i(t, z_i)= \sum_{\alpha=0}^k \frac{t^\alpha}{\alpha!}f_{i\alpha}(z_i)
\end{equation}
where $f_{i\alpha}$ are holomorphic, such that for every positive real $\varepsilon> 0$ and for every $\alpha=0,\dots, k$ we have
$\displaystyle \int_{X\cap \Omega_i}\frac{|f_{i\alpha}|^2}{|f_{i0}|^{2(1-\varepsilon)\frac{m-1}{m}}}< \infty$.
\end{claim}
\medskip

\noindent We prove next Claim \ref{blb} by induction on $k$. For $k=0$ this is simply the hypothesis, so let us assume that a section $s_k$ as in \eqref{suff3} exists. We are constructing next $s_{k+1}$. If we denote by $\xi_{ij}$ the transition functions of the bundle $mK_\mathcal{X}$, then we have
\begin{equation}\label{suff4}
f_i(t, z_i)= \xi_{ij}(t, z_j)\left(f_j(t, z_j)+ t^{k+1}\Lambda_{ij}(t, z_j)\right)
\end{equation}
on the overlapping sets $\Omega_i\cap \Omega_j$. By the equalities \eqref{suff1} it follows that
\begin{equation}\label{suff5}
\frac{\partial^l f_{i\alpha}\big(\Phi(t, z_j)\big)}{\partial t^l}\Big|_{t=0}
\end{equation}
is $L^2$ integrable with respect to the measure $\displaystyle \frac{d\lambda(z_i)}{|f_{i0}|^{2(1-\varepsilon)\frac{m-1}{m}}}$ for any positive integer $l$ and for every positive $\varepsilon> 0$. Indeed, given \eqref{suff1} if some monomial, say, $\displaystyle (z_i^1)^{\delta_1}$ divides $f_{i\alpha}$, the same is true for any derivative with respect to $t$ as in \eqref{suff5} since the effect of the transition functions is to replace this monomial by  $\displaystyle (g_{ij}^1(t, z_j)z_j^1)^{\delta_1}$. Thus the $L^2$ condition that we are imposing to each component $f_{i\alpha}$ is preserved.

\noindent The conclusion is that we have
\begin{equation}\label{suff6}
 \int_{X\cap \Omega_{ij}}\frac{|\Lambda_{ij}(0, z_j)|^2}{|f_{j0}(z_j)|^{2(1-\varepsilon)\frac{m-1}{m}}}d\lambda(z_j)< \infty
\end{equation}
for every positive $\varepsilon> 0$ which means precisely that the $(n, 1)$ form $(m-1)K_X$-valued form $\lambda$ associated to the cocycle
$(\Lambda_{ij}|_X)_{i,j}$ is verifies the hypothesis of Theorem \ref{MT}.
\smallskip

\noindent The proof of Theorem \ref{MT} shows that $\displaystyle \frac{\lambda}{\prod s_{Y_i}^{\nu_q}}$ is $\bar \partial$-exact, where $Y_i$ are the components of $\Sigma|_X$ and
$\displaystyle \nu_q:= \left\lfloor \frac{m-1}{m} m_q\right\rfloor$
if $\displaystyle \frac{m-1}{m} m_q\not\in \mathbb{Z}$ and $\displaystyle \nu_q:= \frac{m-1}{m}m_q- 1$ if this number is an integer.
\smallskip

\noindent This can be re-interpreted in terms of cocycles as follows: there exists holomorphic $L^2$ functions $f_{j k+1}(z_j)$ such that we have
\begin{equation}\label{suff7}
\Lambda_{ij}(0, z_j)= f_{j k+1}(z_j)- \xi_{ji}(0, z_i)f_{i k+1}(z_i),
\end{equation}
such that the holomorphic functions $f_{j k+1}$ verify the $L^2$ requirement in the Claim. When combined with \eqref{suff4}, this completes the proof Theorem \ref{hypo}.
\end{proof}
\medskip

\subsection{Extension to the second infinitesimal neighbourhood}\label{2nd} We have already mentioned that the techniques we are developing in this paper are allowing us -in some particular cases- to extend the section $s$ to the second infinitesimal nbd of the central fiber. We present the arguments in this section.
\smallskip

\noindent Thanks to the subsection \ref{firstorderex}, we can always find a $1$-order extension $s_2$, i.e.
$$s_2 |_X =s , \qquad \bar \partial s_2 =t^2 \Lambda_2 .$$
The main observation is that if we write
\begin{equation}\label{part1}
\Lambda_2= \bar \partial \alpha_2+ D'_\mathcal{X}\big(dt\wedge \rho_2),
\end{equation}
as in Lemma \ref{trivca}, then the restriction $\displaystyle \rho_2|_X$ is
\emph{automatically} in $L^2$ under the assumption of  Theorem \ref{order2}.
Here the properties \eqref{stupid_cond1} of the vector field $\Xi$ used in order to define the Lie derivative. This is the content of our next statement.

\begin{lemme}Under the hypothesis of Theorem \ref{order2}, we have
\begin{equation}\label{part2}
\int_X|\rho_2|^2_{\omega_E}e^{-\varphi_L}dV_{\omega_E}< \infty.
\end{equation}
\end{lemme}

\begin{proof} This will be done by an explicit evaluation of $\rho_2$. Our first claim is
that it is enough to show the convergence of the integrals
\begin{equation}\label{part3}
\int_{(\mathbb{C}^n, 0)}|f\widehat{dz^\alpha}\wedge \bar \partial v|^2_{\omega_E}\frac{dV_{\omega_E}}{|f|^{2\frac{m-1}{m}}}
\end{equation}
and
\begin{equation}\label{part4}
 \int_{(\mathbb{C}^n, 0)}|\partial f\wedge \widehat{dz^{\alpha, \beta_k}}\wedge\bar \partial v|^2_{\omega_E}\frac{dV_{\omega_E}}{|f|^{2\frac{m-1}{m}}},
\end{equation}
as well as
\begin{equation}\label{part4t}
 \int_{(\mathbb{C}^n, 0)}|\partial_t f|^2|\bar \partial \Xi|^2_{\omega_E}\frac{d\lambda}{|f|^{2\frac{m-1}{m}}}.
\end{equation}
This is a consequence of formula
\eqref{eq55}, the computations are as follows. We have
\begin{equation}\label{calc3}
\bar \partial\Xi\rfloor \mathcal{L}_\Xi(s)\simeq \frac{1}{m}\Xi(f)\cdot \bar \partial\Xi\rfloor (dt\wedge dz)
\end{equation}
modulo a term divisible by $f$. On the other hand, we have
\begin{equation}\label{calc4}
D'\big(\Xi\rfloor(\bar \partial\Xi\rfloor s)\big)\simeq \frac{1}{m} \partial f\wedge \Big(\Xi\rfloor \big(\bar \partial\Xi\rfloor(dt\wedge dz)\big)\Big)
\end{equation}
again modulo a multiple of $f$. A further contraction with $\Xi$ gives
\begin{equation}\label{calc5}
\Xi\rfloor \Big(D'\big(\Xi\rfloor(\bar \partial\Xi\rfloor s)\big)\Big)\simeq
\frac{1}{m} \Xi(f)\cdot \Xi\rfloor \big(\bar \partial\Xi\rfloor(dt\wedge dz)\big).
\end{equation}
Finally, we apply $dt\wedge \cdot$ in \eqref{calc5}, and what we get is the same as
the RHS of \eqref{calc3}. Now given the formula which computes
$dt\wedge \rho_2$ our claim follows.
\smallskip

Coming back to the integrals above, for \eqref{part3} things are clear because the volume of $(X, \omega_E)$ is finite.
Modulo the blow-up map $\pi$ in Section \ref{hodge}, the integral \eqref{part4} reduces to the evaluation of the next quantity
\begin{equation}\label{part4mm}
 \sum_{\beta_k=1}^p\int_{(\mathbb{C}^n, 0)} \prod_{\alpha=1}^p |z_\alpha|^{2\delta_\alpha}\left|\frac{\widehat {dz_\beta}}{z_\beta}\right|^2_{ \omega_E} dV_{\omega_E}.
 \end{equation}
 where $\delta_\alpha> 0$ are positive rational numbers. The convergence of \eqref{part4mm}
 follows, given that we know the singularities of the metric $\omega_E$.

 \noindent For the expression \eqref{part4t} we use the fact that after the first order extension cf. Section \ref{firstorderex},  we have
\begin{equation}\label{part200}
 \int_{(\mathbb{C}^n, 0)}|\partial_t f|^2\frac{d\lambda}{|f|^{2\frac{m-1}{m}}}< \infty.
\end{equation}
But then it follows that we also have
\begin{equation}\label{part201}
 \int_{(\mathbb{C}^n, 0)}|\partial_t f|^2\frac{d\lambda}{|f|^{2\frac{m-1}{m}+ \varepsilon_0}}< \infty
\end{equation}
for some positive real $\varepsilon_0> 0$. This is enough to absorb the term arising from $|\bar \partial \Xi|^2_{\omega_E}$, because of the hypothesis \eqref{stupid_cond1}.
 \end{proof}
 \medskip

 \noindent In conclusion, we find ourselves in the following situation i.e. the setting of Theorem \ref{L2I}: let $(L, h_L)$ be a holomorphic line bundle on $X$ and the possible singular metric $h_L$ is of analytic singularities. Let $Z$ be the singular locus of $h_L$ and let $\omega_E$ be a Poincaré type metric on $X$ with poles along the $Z$. We have a $L$-valued $L^2$ form $\rho$ of type $(n-1,1)$ on $X$, such that
 \begin{equation}
 \tau:= D'_X\rho
 \end{equation}
 is $\bar \partial$ closed. If $\tau$ would be $L^2$, then we can apply directly
 Theorem \ref{deprim} and conclude that $\tau$ is $\bar \partial$-exact. However, we do not poses
 this information, and we will follow a different path.
 \smallskip

 \noindent We define the linear form $T_\rho$ by the formula
\begin{equation}
T_\rho (\psi):= \int_X\langle \rho, (D' _X)^\star \psi\rangle_{\omega_E}e^{-\varphi_L}
dV_{\omega_E}
\end{equation}
where $\psi$ is a $L$-valued smooth form of $(n, 1)$-type with compact support in $X\setminus Z$.

\noindent The current $T_\rho$ has the following properties.
\begin{lemme}\label{ort} Let $\psi$ be a test form as above, and consider the decomposition
\begin{equation}\label{part6}
\psi= \xi_1+ \xi_2
\end{equation}
according to $\mathrm{Ker}(\bar \partial)$ and its orthogonal. Then $T_\rho(\xi_2)= 0$.
\end{lemme}

\begin{proof} Note that $T_\rho(\xi_2)$ is indeed well-defined, thanks to Friedrichs lemma
(cf. e.g. \cite{bookJP}): the $L^2$ form $\xi_2$ belongs to the domain of $\bar \partial$, hence it is a limit in graph norm of smooth forms $(\xi_{2,k})$ with compact support. It follows that we have
\begin{equation}\label{part7}
(D' _X)^\star (\xi_{2,k})\to (D' _X)^\star (\xi_{2})
\end{equation}
in $L^2$ as $k\to\infty$. This is a consequence of Bochner formula
\begin{equation}\label{part8}
\int_X |(D' _X)^\star (\xi_{2,k}- \xi_{2})|^2dV\leq \int_X |\bar \partial (\xi_{2,k}- \xi_{2})|^2dV+
\int_X |\bar \partial^\star (\xi_{2,k}- \xi_{2})|^2dV
\end{equation}
(here semi-positive curvature is sufficient) and by Friedrichs lemma the RHS of \eqref{part8} tends to zero as $k\to\infty$.
\smallskip

\noindent It follows that $T_\rho(\xi_{2})$ is well-defined and moreover we have
\begin{equation}\label{part12}
T_\rho(\xi_{2})= \int_X\langle \rho, (D' _X)^\star \xi_{2}\rangle_{\omega_E}e^{-\varphi_L}.
\end{equation}
Thanks to the $L^2$ hypothesis on $\rho$, we have
\begin{equation}\label{part13}
\int_X\langle \rho, (D' _X)^\star \xi_{2}\rangle_{\omega_E}e^{-\varphi_L}=
\lim_{\varepsilon\to 0}\int_X\langle \rho, (D' _X)^\star (\mu_\varepsilon\xi_{2})\rangle_{\omega_E}e^{-\varphi_L}
\end{equation}
where $\displaystyle (\mu_\varepsilon)_{(\varepsilon> 0)}$ is the family of cut-off functions
adapted to Poincar\'e metric, whose construction was recalled in Lemma \ref{cutoff}.

\noindent Indeed, \eqref{part13} follows since we have
\begin{equation}\label{part14}
D'^\star (\mu_\varepsilon\xi_{2})= \mu_\varepsilon D'^\star (\xi_{2})+ {\partial \mu_\varepsilon}\rfloor \xi_2
\end{equation}
and again by the $L^2$ condition, we have
\begin{equation}
\lim_{\varepsilon\to 0}\int_X\langle \rho, D'^\star (\mu_\varepsilon\xi_{2})\rangle_{\omega_E}e^{-\varphi_L}=
\lim_{\varepsilon\to 0}\int_X\langle \rho, \mu_\varepsilon D'^\star (\xi_{2})\rangle_{\omega_E}e^{-\varphi_L}.
\end{equation}

\noindent Therefore, we infer that
\begin{equation}
\int_X\langle \rho, D'^\star \xi_{2}\rangle_{\omega_E}e^{-\varphi_L}=
\lim_{\varepsilon\to 0}\int_X\langle \mu_\varepsilon D'\rho, \xi_{2}\rangle_{\omega_E}e^{-\varphi_L}.
\end{equation}
On the other hand, $\xi_2= \bar \partial^\star \lambda$ for some $L^2$ form $\lambda$
which can be assumed to be $\bar \partial$-closed
and then we write
\begin{equation}\label{part171}
\int_X\langle \mu_\varepsilon D'\rho, \xi_{2}\rangle_{\omega_E}e^{-\varphi_L}=
\int_X\langle \bar \partial \mu_\varepsilon \wedge D'\rho, \lambda\rangle_{\omega_E}e^{-\varphi_L}
\end{equation}
because $D'\rho$ is $\bar \partial$-closed. The RHS term in \eqref{part171} is equal to
\begin{equation}\label{part181}
\int_X\langle D'(\bar \partial \mu_\varepsilon \wedge \rho), \lambda\rangle_{\omega_E}e^{-\varphi_L}
+ \int_X\langle \partial\bar{\partial} \mu_\varepsilon \wedge \rho, \lambda\rangle_{\omega_E}e^{-\varphi_L}
\end{equation}
up to a sign. The second term of \eqref{part181} rends to zero as $\varepsilon\to 0$, and so does
the first one, because by Bochner formula we have
\begin{equation}\label{part191}
\int_X|D'^{\star}(\lambda)|^2dV\leq \int_X|\bar \partial^\star\lambda|^2dV= \int_X|\xi_2|^2dV
\end{equation}
(because we assume that $\bar \partial\lambda=0$), and the RHS of \eqref{part191}
is convergent. Note again that here the semi-positivity of the curvature is enough, given that $\lambda$ is of type $(n,2)$.
\smallskip

\noindent In conclusion, the $T_\rho(\xi_{2})=0$.
 \end{proof}
 \smallskip

 \noindent Now we are ready to prove Theorem \ref{order2} and Theorem \ref{L2I}.

 \begin{proof}[Proof of Theorem \ref{L2I}]
Let $\psi$ be an $L$-valued smooth form of $(n, 1)$-type with compact support in $X\setminus Z$. Consider the Hodge decomposition \eqref{part6}; then we have
\begin{equation}
T_{v}(\psi)= T_v (\xi_1)
\end{equation}
by Lemma \ref{ort}. Then it follows by Cauchy-Schwarz inequality combined with Bochner formula and
the usual $L^2$ theory that there exists a $u$ such that
\begin{equation}
T_v (\psi)= \int_X\langle u, \bar \partial^\star \psi\rangle e^{-\varphi_L}
\end{equation}
in other words we infer that
\begin{equation}
D' _X (v)=  \bar \partial u  .
\end{equation}
 \end{proof}

  \begin{proof}[Proof of Theorem \ref{order2}]
 Our aim is to prove that the $\Lambda_2  |_X$ in \eqref{part1} is $\bar \partial$-exact,
which is equivalent to prove that the $\bar \partial$-closed form $D' _X ( \rho_2 |_X)$ is $\bar \partial$-exact.
It is a direct consequence of  Lemma \ref{ort} and Theorem \ref{L2I}.
  \end{proof}

\medskip

\noindent We prove next Theorem \ref{L2II}. Actually the motivation for this
result is that the form $\displaystyle \rho_2|_X$ is automatically $L^2$ \emph{if} we are using a non-singular metric on $X$.

\begin{proof}[Proof of Theorem \ref{L2II}]
  The main technical difficulty here is that the metric $\omega$ on $X\setminus Z$ is not complete. Usually this is bypassed by using a sequence of complete metrics
\begin{equation}\label{part15}
\omega_\varepsilon:= \omega+ \varepsilon \omega_Z
\end{equation}
and invoke the usual arguments in $L^2$ theory. This works perfectly for forms of type $(n, q)$ (because in this case, a monotonicity argument can be used) but in our case the form $v$ is of type $(n-1,1)$, hence in the absence of hypothesis (3), it is not necessarily $L^2$ with respect to $\omega_\delta$ above. We proceed as follows.
\smallskip

\noindent We recall that the weight $\varphi_L$ of the metric $h_L$ is assumed to have log poles along $Z$.
Let $\xi$ be an $L$-valued $(n, 1)$-form whose support is contained in $X\setminus Z$. We have to evaluate the quantity
\begin{equation}\label{part16}
\int_X\langle D'v, \xi\rangle e^{-\varphi_L}dV_\omega.
\end{equation}
The form $\xi$ can be written as
\begin{equation}\label{part18}
\xi= \xi_{1}+ \xi_{2}
\end{equation}
according to $\mathrm{Ker} (\bar \partial)$ and its orthogonal. Since $D'v$ is $L^2$ and $\bar \partial$-closed, we have
\begin{equation}\label{part19}
\int_X\langle D'v, \xi_{2}\rangle e^{-\varphi_{L}}dV_\omega= 0 .
\end{equation}
So \eqref{part16} equals the expression
\begin{equation}\label{part20}
\int_X\langle D'v, \xi_{1}\rangle e^{-\varphi_{L}}dV_\omega.
\end{equation}

\bigskip

\noindent Note that we have
\begin{equation}\label{part21}
\int_X\langle D'v, \xi_{1}\rangle e^{-\varphi_{L}}dV_\omega=
\lim_{\varepsilon\to 0}\int_X\langle D'v, \xi_{1, \varepsilon}\rangle e^{-\varphi_{L}}dV_{\omega_\varepsilon}.
\end{equation}
In \eqref{part21}, the notations are as follows. The metric $\omega_\varepsilon$ was introduced in
\eqref{part15}, and $\xi_{1, \varepsilon}$ is the orthogonal projection of $\xi$ on the $\mathrm{Ker}(\bar \partial)$
with respect to $(X\setminus Z, \omega_\varepsilon)$ and $(L, h_L)$, respectively.
The equality \eqref{part21} follows from the fact that $\xi_{1, \varepsilon}\to \xi_1$ uniformly on the compact sets of $X\setminus Z$.
\smallskip

\noindent It is at this point that we have to use the hypothesis (3): as a consequence of it, we can write
\begin{equation}\label{part22}
\int_X\langle D'v, \xi_{1, \varepsilon}\rangle e^{-\varphi_{L}}dV_{\omega_\varepsilon}= \int_X\langle v, D'^\star(\xi_{1, \varepsilon})\rangle e^{-\varphi_{L}}dV_{\omega_\varepsilon}.
\end{equation}
Moreover we have
\begin{equation}\label{part233}
\lim_{\varepsilon\to 0}\int_X|v|_{\omega_\varepsilon}^2dV_{\omega_\varepsilon}= \int_X|v|_{\omega}^2dV_{\omega}
\end{equation}
because if we write in coordinates
\begin{equation}\label{part24}
v= \sum v_{p\overline q}\widehat{dz_p}\wedge \overline{dz_q},
\end{equation}
then we have
\begin{equation}\label{part25}
|v|_{\omega_\varepsilon}^2dV_{\omega_\varepsilon}= \sum |v_{p\overline q}|^2\frac{1+ \varepsilon\lambda_p}{1+ \varepsilon\lambda_q}dz \wedge d\overline z
\end{equation}
and the trivial inequality $\displaystyle \frac{1+ \varepsilon\lambda_p}{1+ \varepsilon\lambda_q}< 1+ \frac{\lambda_p}{\lambda_q}$ implies that $|v|_{\omega_\varepsilon}^2dV_{\omega_\varepsilon} \leq
|v|_{\omega}^2dV_{\omega} + |v|_{\omega_Z}^2dV_{\omega_Z}$.
This allows us to use dominated convergence theorem and infer \eqref{part233}.
\smallskip

The $L^2$ norm of the form $D'^\star(\xi_{1, \varepsilon})$ is smaller than
\begin{equation}\label{part26}
\int_X|\bar \partial^\star(\xi_{1, \varepsilon})|^2e^{-\varphi_{L}}dV_{\omega_\varepsilon}= \int_X|\bar \partial^\star \xi|^2e^{-\varphi_{L}}dV_{\omega_\varepsilon}
\end{equation}
and since we assume from the
beginning that the support of the form $\xi$ is contained in $X\setminus Z$, the limit as
$\varepsilon\to 0$ of the RHS of \eqref{part26} is precisely
\begin{equation}\label{part27}
\int_X|\bar \partial^\star\xi|^2e^{-\varphi_{L}}dV_{\omega},
\end{equation}
and we are done.
\end{proof}

redpajama_set_name: RedPajamaArXiv