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\begin{document}
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\maketitle

\begin{abstract}
We give an analytic version of the injectivity theorem by using multiplier ideal sheaves,
and prove some extension theorems for the adjoint bundle of dlt pairs.
Moreover, by combining techniques of the minimal model program,
we obtain
some results for semi-ampleness related to
the abundance conjecture in birational geometry
and the Strominger-Yau-Zaslow conjecture
for hyperK\"ahler manifolds.

\end{abstract}

\section{Introduction}\label{real-intro}
Throughout this paper, we work over $\mathbb{C}$, the complex number field,
and  freely use the standard notation in
\cite{bchm}, \cite{Dem}, \cite{kamama}, and \cite{komo}.
Further we interchangeably use the words \lq \lq Cartier divisors", \lq \lq line bundles", and
\lq \lq invertible sheaves".

The following conjecture is one of
the most important conjectures in the classification theory of algebraic varieties:

\begin{conj}[Generalized abundance conjecture]\label{abun conj}Let $X$ be a normal projective variety and $\Delta$ be an effective $\mathbb{Q}$-divisor such that $(X,\Delta)$ is a klt pair.
Then
$\kappa(X, K_X+\Delta)=\kappa_{\sigma}(X, K_X+\Delta)$.   In particular, if $K_X+\Delta$ is nef, then it is semi-ample.
$($For the definition of $\kappa(\cdot)$ and $\kappa_{\sigma}(\cdot)$,
see  \cite{nakayama-zariski-abun}.$)$
\end{conj}

Toward the abundance conjecture,
we need to solve the non-vanishing conjecture
and the extension conjecture, see \cite{dhp}, \cite[Introduction]{fujino-abundance}, \cite[Section5]{fg3}.
One of the purposes of this paper is to study
the following extension conjecture for the adjoint bundle of dlt pairs
formulated in \cite[Conjecture 1.3]{dhp}:

\begin{conj}[{Extension conjecture for dlt pairs}]\label{c-dlt}
Let $X$ be a normal projective variety
and $S + B$ be an effective $\mathbb Q$-divisor
with the following assumptions:
\begin{itemize}
\item $(X, S+B)$ is a dlt pair.
\item $\lfloor S+B \rfloor =S$.
\item $K_X+S+B$ is nef.
\item $K_X+S+B$ is $\mathbb Q$-linearly equivalent to
an effective divisor $D$ with $S\subseteq \Supp D \subseteq \Supp\,(S+B)$.

\end{itemize}
Then the restriction map
$$H^0(X,\mathcal O _X(m(K_X+S+B)))\to H^0(S,  \mathcal O _S(m(K_X+S+B)))  $$
is surjective for all sufficiently divisible integers $m\geq 2$.
\end{conj}

When $S$ is a normal irreducible variety (namely (X, S+B) is a plt pair),
Demailly--Hacon--P\u{a}un proved the above conjecture in \cite{dhp}
by using technical methods based on a version of the Ohsawa-Takegoshi extension theorem.
This result can be seen as an extension theorem
for plt pairs.

In this paper,
we study the extension conjecture for {\textit{dlt pairs}}
by giving an analytic version of the injectivity theorem
instead of the Ohsawa-Takegoshi extension theorem.
Thanks to the injectivity theorem,
we can obtain some extension theorems for not only plt paris but also dlt pairs.
This is one of our advantages.
Our injectivity theorem is as follows:

\begin{thm}[A version of the injectivity theorem {=Theorem \ref{main-inj}}]
\label{intro-main-inj}
Let $(F, h_{F})$ and $(L, h_{L})$
be $($singular$)$ hermitian line bundles
with semi-positive curvature
on a compact K\"ahler manifold $X$.
Assume that there exists
an effective $\mathbb{R}$-divisor $\Delta$ with
\begin{align*}
h_{F}=h_{L}^{a}\cdot h_{\Delta},
\end{align*}
where $a$ is a positive real number and  $h_{\Delta}$  is
the singular metric defined by $\Delta$.

Then for a $($non-zero$)$ section $s$ of $L$
satisfying $\sup_{X} |s|_{h_{L}} < \infty$,
the multiplication map

\begin{equation*}
\Phi_{s}: H^{q}(X, K_{X} \otimes F \otimes \I{h_{F}})
\xrightarrow{\otimes s}
H^{q}(X, K_{X} \otimes F\otimes L \otimes \I{h_{F} h_{L}})
\end{equation*}
is $($well-defined and$)$ injective for any $q$.
Here $\I{h}$ denotes the multiplier ideal sheaf
associated to a singular metric $h$.
\end{thm}

The injectivity theorem has been
studied by several authors, for example, Tankeev \cite{tan}, Koll\'ar \cite{Kol86}, Enoki \cite{Eno90}, Ohsawa \cite{ohsawa}, Enault--Viehweg \cite{ev}, Fujino \cite{fujino-book1}, \cite{Fuj12-A}, \cite{F-fund}, \cite{fujino-book2}, and Ambro \cite{ambro-qlog}, \cite{ambro}.
Recently  the second author gave an injectivity theorem with multiplier ideal sheaves,
which corresponds to the case of  $\Delta=0$ (see \cite{Mat-inj}).
Our proof is a generalization of  the methods of \cite{Eno90}, \cite{ohsawa},
\cite{Fuj12-A}, and  \cite{Mat-inj}.

By applying
this injectivity theorem to an extension problem, we obtain the following extension theorem.
Even if $K_{X}+\Delta$ is semi-positive
(namely admits a smooth metric with semi-positive curvature),
it seems to be not able to  prove Conjecture \ref{c-dlt} for dlt pairs
by the Ohsawa-Takegoshi extension theorem in at least current situation,
and thus we need our injectivity theorem  (Theorem \ref{intro-main-inj}).

\begin{thm}[{=Theorem \ref{ext}}]\label{intro-ext}
Let $X$ be a compact K\"ahler manifold and
$S+B$ be an effective $\mathbb Q$-divisor with the
following assumptions:
\begin{itemize}
\item $S+B$ is a simple normal crossing divisor with
$0 \leq S+B \leq 1$ and $\lfloor S+B \rfloor =S$.
\item $K_{X}+S+B$ is ${\mathbb{Q}}$-linearly
equivalent to an effective divisor $D$
with $S \subseteq\mathrm{Supp}\, D$.
\item $K_{X}+S+B$ admits a $($singular$)$ metric $h$ with semi-positive curvature.
\end{itemize}
Then for an integer $m \geq 2$ with Cartier divisor $m(K_{X}+S+B)$
and any section $u \in H^{0}(S, \mathcal{O}_{S}(m(K_{X}+S+B)))$
that comes from
$H^{0}(S, \mathcal{O}_{S}(m(K_{X}+S+B))\otimes \I{h^{m-1}h_{B}} )$,
the section $u$ can be extended to a section in $H^{0}(X, \mathcal{O}_{X}(m(K_{X}+S+B)) )$.
\end{thm}

In particular, we can prove Conjecture \ref{c-dlt}
under the assumption that $K_{X}+ \Delta$ admits a (singular) metric
whose curvature is semi-positive and Lelong number is identically zero
(see Corollary \ref{ext-cor}).
This assumption is stronger than the assumption that $K_{X}+\Delta$ is nef, but
weaker than the assumption that $K_{X}+\Delta$ is semi-positive.
Remark that Verbitsky proved the non-vanishing conjecture on hyperK\"ahler manifolds
under the same assumption (see \cite{ver}).

As compared with Conjecture \ref{c-dlt}, one of our advantages is removing the condition  $\Supp\,D\subseteq \Supp (S+B) $ (this version of the extension conjecture  is  \cite[Conjecture 5.8]{fg3}).
Thanks to removing the condition  $\Supp\,D\subseteq \Supp (S+B) $ in Conjecture \ref{c-dlt}, we can run the MMP preserving the good condition for metrics
(cf. \cite[Section 8]{dhp} and \cite[Theorem 5.9]{fg3}).
By applying the above theorem and techniques of
the MMP, we obtain the following theorem related to
the abundance conjecture:

\begin{thm}[{=Theorem \ref{main-thm-semiample}}]\label{main-thm}
Assume that Conjecture \ref{abun conj} holds in dimension $n-1$.
Let $X$ be an $n$-dimensional normal projective variety and $\Delta$ be an $\mathbb{Q}$-divisor
with the following assumptions:
\begin{itemize}
\item $(X,\Delta)$ is a klt pair.
\item  There exists a projective birational morphism $\varphi:Y \to X$ such that $Y$ is smooth and $\varphi^*(\mathcal{O}_X(m(K_X+\Delta)))$ admits
a $($singular$)$ metric whose curvature
is semi-positive and Lelong number is identically zero.
Here $m$ is a positive integer such that $m(K_X+\Delta)$ is Cartier.
\end{itemize}
If $\kappa(K_X+\Delta) \geq 0$, then $K_X+\Delta$ is semi-ample.

\end{thm}

Finally, by combining Verbitsky's non-vanishing theorem
(\cite[Theorem 4.1]{ver})
on hyperK\"ahler manifolds
(holomorphic symplectic manifolds),
we obtain a result for semi-ampleness on four dimensional projective hyperK\"ahler manifolds,
which is closely related to the Strominger-Yau-Zaslow conjecture for hyperK\"ahler manifolds.
See \cite{ac} for recent related topics and \cite{cop} for non-algebraic cases.

\begin{thm}[{=Corollary \ref{cor-hk}}]\label{cor-hk-intro}Let $X$ be a $4$-dimensional projective hyperK\"ahler manifold and $L$ be a line bundle admitting a $($singular$)$ metric whose curvature
is semi-positive and Lelong number is identically zero.
Then $L$ is semi-ample.
\end{thm}

We summarize the contents of this paper.
In Section \ref{preliminaries},
we collect the basic definitions and facts needed later.
In Section \ref{proof of injectivity}, we prove
the injectivity theorem (Theorem \ref{intro-main-inj}).
In Section \ref{applications-inj}, we give
direct applications  of the injectivity theorem to
the extension theorem (Theorem \ref{intro-ext} and Corollary \ref{ext-cor}).
Section \ref{applications} is devoted to results for semi-ampleness
related to the abundance conjecture (Theorem \ref{main-thm} and Theorem  \ref{cor-hk-intro}).

\begin{ack}
The authors wish to express their deep gratitude to Professor Osamu Fujino for pointing out several mistakes in the draft and discussions,
and to Professor Mihai P\u{a}un for a discussion about Theorem \ref{ext}.
The first author thanks Professors Paolo Cascini, Christopher Hacon,  Keiji Oguiso, Vladimir Lazi\'c for discussions and comments.
The first author is partially supported by Grant-in-Aid for Young Scientists (A) $\sharp$26707002 from JSPS and the Research expense from the JRF fund.
The second author is partially supported by the Grant-in-Aid for
Young Scientists (B) $\sharp$25800051 from JSPS.
\end{ack}

\section{Preliminaries}\label{preliminaries}
In this section,
we first recall the definitions and basic properties
of singular metrics and multiplier ideal sheaves.
For more details, refer to \cite{Dem}, \cite{Dem-book}.

We denote by $X$ a compact complex manifold and
by $F$ a line bundle on $X$ unless otherwise mentioned.
Further we fix a smooth (hermitian) metric $g$ on $F$.

\begin{defi}[Singular metrics and their curvature currents]\label{s-met}
\ \vspace{0.2cm} \\
(1) For an $L^{1}$-function $\varphi$ on $X$,
the  metric $h$ defined by
\begin{equation*}
h:= g e^{-\varphi}
\end{equation*}
is called a \textit{singular metric} on $F$.
Further $\varphi$ is called the \textit{weight} of $h$
with respect to the fixed smooth metric $g$.
\vspace{0.2cm} \\
(2)
A (singular) metric $h$ on $F$ is said to have
{\textit{algebraic}} (resp. {\textit{analytic}}) {\textit{singularities}},
if there exists an ideal sheaf
$\mathcal{I} \subseteq\mathcal{O}_{X}$ such that
a weight $ \varphi$ of $h$ can be locally
written as
\begin{equation*}
\varphi = \frac{c}{2} \log \big(
|f_{1}|^{2} + |f_{2}|^{2} + \cdots + |f_{k}|^{2}\big) +v,
\end{equation*}
where $c$ is a positive rational number
(resp. real number),
$f_{1}, \dots f_{k}$ are local generators of $\mathcal{I}$,
and $v$ is a smooth function.
\vspace{0.2cm} \\
(3) The {\textit{curvature current}}
$\sqrt{-1} \Theta_{h}(F)$
associated to $h$ is defined by
\begin{equation*}
\sqrt{-1} \Theta_{h}(F) = \sqrt{-1} \Theta_{g}(F)+ \ddbar \varphi,
\end{equation*}
where $  \sqrt{-1} \Theta_{g}(F)$ is
the Chern curvature of $g$.
\end{defi}
In this paper, we simply abbreviate
the singular metric (resp. the curvature current)
to the metric (resp. the curvature).
The Levi form $\ddbar \varphi$ is taken in the sense of
distributions,
and thus the curvature is a $(1,1)$-current
but not always a smooth $(1,1)$-form.
The curvature $\sqrt{-1} \Theta_{h}(F)$ of $h$
is said to be {\textit{semi-positive}} if
$\sqrt{-1} \Theta_{h}(F) \geq 0$ in the sense of currents.

\begin{defi}[Multiplier ideal sheaves]\label{multiplier}
Let $h$ be a metric on $F$ such that
$\sqrt{-1} \Theta_{h}(F) \geq \gamma$
for some smooth $(1,1)$-form $\gamma$ on $X$.
The ideal sheaf $\I{h}$ defined
to be
\begin{equation*}
\I{h}(B):= \I{\varphi}(B):= \{f \in \mathcal{O}_{X}(B)\ \big|\
|f|\, e^{-\varphi} \in L^{2}_{\rm{loc}}(B) \}
\end{equation*}
for every open set $B \subseteq X$, is called
the \textit{multiplier ideal sheaf} associated to $h$.
\end{defi}
It is known that
multiplier ideal sheaves are coherent sheaves.
The following is a typical example of singular metrics
that often appear in algebraic geometry.

\begin{eg}
For given holomorphic sections $\{s_{i}\}_{i=1}^{N}$ of
the $m$-th tensor powers $F^{m}$ of $F$,
the (singular) metric  $g e^{-\varphi}$ can be defined by
\begin{align*}
\varphi:= \frac{1}{2m} \sum_{i=1}^{N}
\log |s_{i}|^{2}_{g^{m}}.
\end{align*}
Then the curvature of this metric is semi-positive,
and further the multiplier ideal sheaf
can be algebraically computed (see \cite{Dem}).
In particular, for an effective $\mathbb{R}$-divisor
$D$ on $X$,
the metric $h_{D}$ on $\mathcal{O}_{X}(D)$ can
be constructed by the natural section of $D$.
We can easily check $\I{h_{D}}=\mathcal{O}_{X}(-\lfloor D \rfloor )$ if $D$ is a simple normal crossing divisor.

\end{eg}

We recall the definition of the Lelong number of singular metrics and Skoda's lemma
which gives a relation between
the multiplier ideal sheaf
and the Lelong number of singular metrics.

\begin{defi}[Lelong numbers]
Let $\varphi$ be a (quasi-)psh function
on an open set $B$ in $\mathbb{C}^n$.
The {\textit{Lelong number}} $\nu(\varphi, x)$ of $\varphi$ at $x \in B$ is defined by
$$\nu(\varphi, x)=\liminf_{z \to x}\frac{\varphi(z)}{\mathrm{log}|z-x|}.$$
For a singular metric $h$ on $F$ such that
$\sqrt{-1} \Theta_{h}(F) \geq \gamma$
for some smooth $(1,1)$-form,
we define the Lelong number $\nu(h, x)$ of $h$ at $x \in X$
by $\nu(h, x):=\nu(\varphi, x)$,
where $\varphi$ is a weight of $h$.
\end{defi}

\begin{thm}[Skoda's lemma]\label{Skoda}
Let $\varphi$ be a $($quasi$)$-psh function on an open set $B$ in  $\mathbb{C}^n$.
\begin{itemize}
\item[$\bullet$] If $\nu(\varphi, x)<1$, then we have
$\I{\varphi}_{x}= \mathcal{O}_{B, x}$.
\item[$\bullet$] If $\nu(\varphi, x) \geq n+s$ for
some integer $s \geq 0$, then we have
$\I{\varphi}_{x}\subseteq\mathfrak{M}_{B, x}^{s+1}$,
where $\mathfrak{M}_{B, x}$ is the maximal ideal of
$\mathcal{O}_{B, x}$.
\end{itemize}
\end{thm}

Next we give the definition of singularities of pairs.

\begin{defi}[Singularities of pairs]
Let $X$ be a normal variety and $\Delta$ be an effective $\mathbb{Q}$-divisor
on $X$ such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier.
Let $\varphi:Y\rightarrow X$ be a log resolution of $(X,\Delta)$.
We set $$K_Y=\varphi^*(K_X+\Delta)+\sum a_iE_i,$$ where $E_i$ is a
prime divisor on $Y$ for every $i$.
The pair $(X,\Delta)$ is called
\begin{itemize}
\item[$\bullet$] \emph{kawamata log terminal} $($\emph{klt},
for short$)$ if $a_i > -1$ for all $i$,
\item[$\bullet$]\emph{log canonical} $($\emph{lc}, for short$)$ if $a_i \geq -1$ for all $i$.
\end{itemize}

\end{defi}

\begin{defi}[Semi-log canonical, {\cite[Definition 1.1]{fujino-abundance}}]\label{slc} Let $X$ be a reduced $S_2$-scheme. We assume that it is pure $n$-dimensional and is normal crossing in codimension $1$. Let $\Delta$ be an effective $\mathbb{Q}$-Weil divisor on $X$ such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier.

Let $X=\bigcup X_i$ be the decomposition into irreducible components, and $\nu:X':=\coprod X'_i \rightarrow X=\bigcup X_i$ the {\em normalization}, where the \emph{normalization} $\nu : X'=\coprod X'_i \rightarrow X=\bigcup X_i$ means that $\nu|_{X_i'} : X'_i \to X_i$ is the usual normalization for any $i$. We call $X$ a \emph{normal scheme} if $\nu$ is isomorphic. Define the $\mathbb{Q}$-divisor $\Theta$ on $X'$ by $K_{X'}+\Theta=\nu^*(K_X+\Delta)$ and set $\Theta_i=\Theta|_{X_i'}$.

We say that $(X,\Delta)$ is \emph{semi-log canonical} (for short, \emph{slc}) if $(X_i',\Theta_i)$ is an lc pair for every $i$.
 \end{defi}

\section{A Version of the Injectivity theorem}\label{proof of injectivity}
The purpose of this section is to give
an analytic version of
the injectivity theorem by using multiplier ideal sheaves.
This theorem is a generalization of \cite[Theorem 1.3]{Mat-inj}
and it is applied in order to obtain
the extension theorem (Theorem \ref{intro-ext}).
See \cite{Fuj12-A} and \cite{Mat-inj}
for relations of various injectivity theorems and vanishing theorems.

\begin{thm}[A version of the injectivity theorem]
\label{main-inj}
Let $(F, h_{F})$ and $(L, h_{L})$
be $($singular$)$ hermitian line bundles
with semi-positive curvature
on a compact K\"ahler manifold $X$.
Assume that there exists
an effective $\mathbb{R}$-divisor $\Delta$ with
\begin{align*}
h_{F}=h_{L}^{a}\cdot h_{\Delta},
\end{align*}
where $a$ is a positive real number and  $h_{\Delta}$  is
the singular metric defined by $\Delta$.

Then for a $($non-zero$)$ section $s$ of $L$
satisfying $\sup_{X} |s|_{h_{L}} < \infty$,
the multiplication map

\begin{equation*}
\Phi_{s}: H^{q}(X, K_{X} \otimes F \otimes \I{h_{F}})
\xrightarrow{\otimes s}
H^{q}(X, K_{X} \otimes F\otimes L \otimes \I{h_{F} h_{L}})
\end{equation*}
is $($well-defined and$)$ injective for any $q$.
Here $\I{h}$ denotes the multiplier ideal sheaf
associated to a singular metric $h$.
\end{thm}

\begin{rem}\label{rem-inj}
$(1)$ The multiplication map is well-defined
thanks to the assumption of $\sup_{X} |s|_{h_{L}} < \infty$.
When $h_{L}$ is a metric with minimal singularities on $L$, this assumption is always satisfied
for any section $s$ of $L$
$($see \cite{Dem} for the definition of metrics with minimal singularities$)$.
\vspace{0.1cm}\\
(2) The case of $\Delta=0$
corresponds to the main result in \cite{Mat-inj}.
To obtain the extension theorem $($Theorem \ref{intro-ext}$)$,
it is important to consider the case of $\Delta \not=0$.
\vspace{0.1cm}\\
$(3)$ If $h_{L}$ and $h_{F}$ are smooth on a Zariski open set,
the same conclusion holds under the weaker assumption of
$\sqrt{-1}\Theta_{h_{F}}(F) \geq a \sqrt{-1}\Theta_{h_{L}}(L)$
$($see \cite{Fuj12-A}, \cite{Mat-Nad}$)$.
\end{rem}

\begin{proof}
The proof is a generalization of the proof of the main result
in \cite{Mat-inj} which corresponds to the case of $\Delta=0$.
First of all, we recall Enoki's techniques to generalize
Kollar's injectivity theorem, which give a proof of
the special case where $h_{L}$ is smooth and $\Delta=0$.
In this case,
the cohomology group $H^{q}(X, K_{X} \otimes F)$
is isomorphic to the space
of the harmonic forms with respect to $h_{F}$
\begin{equation*}
\mathcal{H}^{n, q}(F)_{h_{F}}:= \{u \mid u
\text{ is a smooth $F$-valued $(n,q)$-form on } X
\text{ such that } \dbar u  = D^{''*}_{h_{F}} u =0.  \},
\end{equation*}
where  $D^{''*}_{h_{F}}$ is the adjoint operator of
the $\dbar$-operator.
For an arbitrary harmonic form
$u \in \mathcal{H}^{n, q}(F)_{h_{F}}$,
we can conclude that $D^{''*}_{h_{F}h_{L}} su =0$
thanks to semi-positivity of the curvature
and $h_{F}=h^{a}_{L}$.
This step strongly depends on semi-positivity
of the curvature.
Then the multiplication map $\Phi_{s}$
induces the map from $\mathcal{H}^{n, q}(F)_{h_{F}}$ to
$\mathcal{H}^{n, q}(F\otimes L)_{h_{F}h_{L}}$, and thus
the injectivity is obvious.

In our situation, we must consider
singular metrics with transcendental
(non-algebraic) singularities.
It is quite difficult
to directly handle transcendental singularities,
and thus in Step 1, we approximate a given metric $h_{F}$ by metrics
$\{h_{\e} \}_{\e>0}$ that are smooth on a Zariski open set.

Then we represent a given cohomology class in
$H^{q}(X, K_{X} \otimes F \otimes \I{h_{F}})$
by the associated harmonic form
$u_{\e}$ with respect to $h_{\e}$ on the Zariski open set.
We want to show that $s u_{\e}$ is also harmonic
by using the same method as Enoki's proof.
However, the same argument as Enoki's proof fails
since the curvature of $h_{\e}$ is not semi-positive.
For this reason, in Step2,
we investigate the asymptotic behavior of
the harmonic forms $u_{\e}$ with respect to
a family of the regularized metrics
$\{ h_{\e} \}_{\e>0}$.
Then we show that the $L^{2}$-norm
$\| D^{''*}_{ h_{\e} h_{L, \e} }su_{\e}\|$ converges to zero as letting $\e$ go to zero,
where $h_{L, \e}$ is a suitable approximation of $h_{L}$.
Moreover, In Step 3, we construct solutions $\gamma_{\e}$ of
the $\dbar$-equation $\dbar \gamma_{\e} = su_{\e}$
such that
the $L^{2}$-norm $\| \gamma_{\e} \|$
is uniformly bounded,
by applying the $\rm{\check{C}}$ech complex with
the topology induced by the local $L^{2}$-norms.
In Step 4, we see
\begin{equation*}
\| su_{\e} \| ^{2} =
\lla su_{\e}, \dbar\gamma_{\e} \rra
\leq \| D^{''*}_{h_{\e} h_{L, \e}} su_{\e}\|
\| \gamma_{\e} \| \to 0 \quad {\text{as }} \e \to 0.
\end{equation*}
From these observations, we can conclude  that
$u_{\e}$ converges to zero in a suitable sense,
which completes the proof.

\vspace{0.2cm} \\
{\bf{Step 1 (The equisingular approximation of $h_{F}$)}}
\vspace{0.1cm} \\
Throughout the proof, we fix a K\"ahler form $\omega$ on $X$.
For the proof,
we want to apply the theory of harmonic integrals,
but the metric $h_{F}$ may not be smooth.
For this reason, we approximate $h_{F}$ by metrics
$\{ h_{\e} \}_{\e>0}$ that are smooth on a Zariski open set.
By \cite[Theorem 2.3]{dps01}, we can obtain
metrics $\{ h_{\e} \}_{\e>0}$ on $F$ with the following properties:
\begin{itemize}
\item[(a)] $h_{\e}$ is smooth on $X \setminus Z_{\e}$, where $Z_{\e}$ is a subvariety on $X$.
\item[(b)]$h_{\e_{2}} \leq h_{\e_{1}} \leq h_{F}$
holds for any
$0< \e_{1} < \e_{2} $.
\item[(c)]$\I{h_{F}}= \I{h_{\e}}$.
\item[(d)]$\sqrt{-1} \Theta_{h_{\e}}(F) \geq -\e \omega$.
\end{itemize}
Since the point-wise norm $|s|_{h_{L}}$ is bounded on $X$
and $h_{F}=h^{a}_{L}h_{\Delta}$,
the set
$\{x \in X \mid \nu(h_{F}, x)>0 \}$
is contained in the subvariety $Z$ defined by
$Z:=s^{-1}(0) \cup {\rm{Supp}}\, \Delta$.
Therefore
we may assume a stronger property
than property (a) (for example see \cite[Theorem 2.3]{Mat-inj}), namely
\begin{itemize}
\item[(e)] $h_{\e}$ is smooth on $Y:=X \setminus Z$,
where $Z=s^{-1}(0) \cup {\rm{Supp}}\, \Delta$.
\end{itemize}

Now we construct a \lq \lq complete" K\"ahler form on $Y$
with suitable potential function.
Take a quasi-psh function $\psi$ on $X$ such that
$\psi$ has a logarithmic pole along $Z$
and $\psi$ is smooth on $Y$.
Since quasi-psh functions are upper semi-continuous,
the function $\psi$ is bounded above,
and thus we may assume $\psi \leq -e$.
Then we define the $(1,1)$-form $\ome$ on $Y$ by
\begin{equation*}
\ome:= \ell \omega + \ddbar \Psi,
\end{equation*}
where
$\ell$ is a positive number and $\Psi:=\frac{1}{\log(-\psi)}$.
We can show that
the $(1,1)$-form $\ome$ satisfies the following properties
for a sufficiently large $\ell >0$:
\begin{itemize}
\item[(A)] $\ome$ is a complete K\"ahler form on $Y$.
\item[(B)] $\Psi$ is bounded on $X$.
\item[(C)] $\ome \geq \omega $.
\end{itemize}
Indeed, properties (B), (C) are obvious
by the definition of $\Psi$ and $\ome$, and property (A) follows
from straightforward computations
(see \cite[Lemma 3.1]{Fuj12-A} for the precise proof of property (A)).

In the proof,
we mainly consider harmonic forms on $Y$
with respect to $h_{\e}$ and $\ome$.
Let $L_{(2)}^{n, q}(Y, F)_{h_{\e}, \ome}$ be
the space of the $L^{2}$-integrable $F$-valued $(n,q)$-forms
$\alpha$ with respect to the inner product $\|\cdot \|_{h_\e, \ome}$
 defined by
\begin{equation*}
\|\alpha \|^{2}_{h_\e, \ome}:= \int_{Y}
|\alpha |^{2}_{h_{\e}, \ome}\ \ome^{n}.
\end{equation*}
Then we can obtain the following orthogonal decomposition:
\begin{equation*}
L_{(2)}^{n, q}(Y, F)_{h_{\e}, \ome}
=
{\rm{Im}}\,\dbar
\oplus
\mathcal{H}^{n, q}(F)_{h_{\e}, \ome}
\oplus {\rm{Im}}\,D^{''*}_{h_{\e}}.
\end{equation*}
Here the operator $D^{'*}_{h_{\e}}$
(resp. $D^{''*}_{h_{\e}}$) denotes
the closed extension of the formal adjoint of the
$(1,0)$-part $D^{'}_{h_{\e}}$ (resp. $(0,1)$-part $D^{''}_{h_{\e}}=\dbar$)
of the Chern connection $D_{h_{\e}}=D^{'}_{h_{\e}}+ D^{''}_{h_{\e}}$.
Note that they coincide with
the Hilbert space adjoints since $\ome$ is complete.
Further $\mathcal{H}^{n, q}(F)_{h_{\e}, \ome}$ denotes
the space of the harmonic forms with respect to
$h_{\e}$ and $\ome$, namely
\begin{equation*}
\mathcal{H}^{n, q}(F)_{h_{\e}, \ome}:=
\{\alpha   \mid \alpha
\text{ is an } F\text{-valued } (n,q)\text{-form with  }
\dbar \alpha= D^{''*}_{h_{\e}}\alpha=0.    \}.
\end{equation*}
Harmonic forms in $ \mathcal{H}^{n, q}(F)_{h_{\e}, \ome}$
are smooth by the regularization theorem for elliptic operators.
These results are known to specialists.
The precise proof of them
can be found in \cite[Claim 1]{Fuj12-A}.

For every $(n, q)$-form $\beta$,
we have
$|\beta|^{2}_{\ome}\ \ome^{n} \leq
|\beta|^{2}_{\omega}\ \omega^{n}$ since
the inequality $\ome \geq \omega$ holds by property (C).
From this inequality and property (b) of $h_{\e}$,
we obtain
\begin{equation}\label{ine}
\|\alpha \|_{h_{\e}, \ome} \leq
\|\alpha \|_{h_{\e}, \omega} \leq
\|\alpha \|_{h_{F}, \omega}
\end{equation}
for an $F$-valued $(n,q)$-form $\alpha$,
which plays a crucial role in the proof.

Take an arbitrary cohomology class
$\{u \} \in H^{q}(X, K_{X} \otimes F \otimes \I{h_{F}})$
represented by an $F$-valued
$(n, q)$-form $u$ with $\|u \|_{h_{F}, \omega} < \infty$.
In order to prove that
the multiplication map $\Phi_{s}$ is injective,
we assume that the cohomology class of $su$
is zero in
$H^{q}(X, K_{X}\otimes F\otimes L \otimes \I{h_{F}h_{L}})$.
Our final goal is to show that
the cohomology class of $u$ is actually zero
under this assumption.

By inequality (\ref{ine}),
we have $\|u\|_{h_{\e}, \ome} < \infty$ for any $\e > 0$.
Therefore by the above orthogonal decomposition,
there exist
$u_{\e} \in \mathcal{H}^{n, q}(F)_{h_{\e}, \ome}$ and
$v_{\e} \in L_{(2)}^{n,q-1}(Y, F)_{h_{\e}, \ome}$ such that
\begin{equation*}
u=u_{\e}+\dbar v_{\e}.
\end{equation*}
Note that
the component of ${\rm{Im}} D^{''*}_{h_{\e}}$ is zero
since $u$ is $\dbar$-closed.

At the end of this step, we explain
the strategy of the proof.
In Step 2, we show that
$\|D^{''*}_{h_{\e} h_{L, \e}}
s u_{\e} \|_{ h_{\e} h_{L, \e}, \ome}$
converges to zero
as letting $\e$ go to zero.
Here $h_{L, \e}$ is the singular metric on $L$ defined by
\begin{align*}
h_{L, \e}:= h_{\e}^{1/a}\, h_{\Delta}^{-1/a}.
\end{align*}
Since the cohomology class of $su$ is zero,
there are solutions $\gamma_{\e}$ of the $\dbar$-equation
$\dbar \gamma_{\e} = s u_{\e}$.
For the proof, we need to obtain $L^{2}$-estimates of them.
In Step 3,
we construct solutions $\gamma_{\e}$ of the $\dbar$-equation
$\dbar \gamma_{\e} = s u_{\e}$ such that
the norm $\| \gamma_{\e} \|_{h_{\e} h_{L, \e}, \ome}$
is uniformly bounded.
Then we have
\begin{equation*}
\|su_{\e} \|^{2}_{h_{\e} h_{L, \e}, \ome} \leq
\|D^{''*}_{h_{\e} h_{L, \e}} s u_{\e} \|_{h_{\e} h_{L, \e}, \ome}
\| \gamma_{\e} \|_{h_{\e} h_{L, \e}, \ome}.
\end{equation*}
By Step 2 and Step 3, we can conclude that
the right hand side goes to zero as
letting $\e$ go to zero.
In Step 4, from this convergence, we prove that
$u_{\e}$ converges to zero
in a suitable sense,
which implies that the cohomology class of $u$ is zero.

\vspace{0.1cm}\\
\vspace{0.2cm} \\
{\bf{Step 2 (A generalization of Enoki's proof of the injectivity theorem)}}
\vspace{0.1cm}\\
The aim of this step is to prove the following proposition,
whose proof can be seen as
a generalization of Enoki's injectivity theorem.
\vspace{-0.2cm}
\begin{prop} \label{D''}
As letting $\e$ go to zero, the norm
$\|D^{''*}_{h_{\e} h_{L, \e}} s u_{\e} \|_{h_{\e} h_{L, \e}, \ome}$ converges
to zero.
\end{prop}
\hspace{-0.5cm}
\begin{proof}[Proof of Proposition \ref{D''}]
We have the following inequality:
\begin{equation}\label{ine2}
\|u_{\e} \|_{h_{\e}, \ome}
\leq \|u \|_{h_{\e}, \ome}
\leq \|u \|_{h, \omega}.
\end{equation}
This inequality is often used in the proof.
The first inequality follows from the definition of  $u_{\e}$
and  the second inequality follows from
inequality (\ref{ine}).
Remark the right hand side does not depend on $\e$.
By applying Nakano's identity and the density lemma
to $u_{\e}$ (for example see \cite[Proposition 2.4]{Mat-inj}),
we obtain
\begin{equation}\label{B-eq}
0 = \lla \sqrt{-1}\Theta_{h_{\e}}(F)
\Lambda_{\ome} u_{\e}, u_{\e}
  \rra_{h_{\e}, \ome} +
\|D^{'*}_{h_{\e}}u_{\e} \|^{2}_{h_{\e}, \ome}.
\end{equation}
Note that the left hand side is zero since $u_{\e}$ is harmonic.
Let $A_{\e}$ be the first term and $B_{\e}$ be
the second term of the right hand side of equality (\ref{B-eq}).
First,
we show that the first term $A_{\e}$ and
the second term $B_{\e}$ converge to zero.
For simplicity, we denote the integrand of $A_{\e}$ by $g_{\e}$,
namely
\begin{equation*}
g_{\e}:= \langle  \sqrt{-1}\Theta_{h_{\e}}(F)
\Lambda_{\ome} u_{\e}, u_{\e}
 \rangle_{h_{\e}, \ome}.
\end{equation*}
Then there exists a positive constant $C>0$ (independent of $\e$)
such that
\begin{equation}\label{ine3}
g_{\e} \geq -\e C |u_{\e}|^{2}_{h_{\e}, \ome}.
\end{equation}
It is easy to check this inequality.
Indeed, let $\lambda_{1}^{\e} \leq \lambda_{2}^{\e} \leq
\dots \leq \lambda_{n}^{\e} $ be the
eigenvalues of $\sqrt{-1}\Theta_{h_{\e}}(F)$ with respect to
$\ome$.
Then for any point $y \in Y$ there exists
a local coordinate $(z_{1}, z_{2}, \dots, z_{n})$
centered at $y$ such that
\begin{align*}
\sqrt{-1}\Theta_{h_{\e}}(F) = \sum_{j=1}^{n}
\lambda_{j}^{\e} dz_{j} \wedge d\overline{z_{j}}\quad \text{and} \quad
\ome = \sum_{j=1}^{n}
 dz_{j} \wedge d\overline{z_{j}}
\quad {\rm{ at}}\ y.
\end{align*}
When we locally write $u_{\e}$ as
$u_{\e} =\sum_{|K|=q} f_{K}^{\e}\ dz_{1}\wedge \dots \wedge dz_{n}
\wedge d\overline{z}_{K}$,
we have
\begin{equation*}
g_{\e}= \sum_{|K|=q}
\Big{(} \sum_{j \in K} \lambda_{j}^{\e} \Big{)}
|f_{K}^{\e}|^{2}_{h_{\e}}
\end{equation*}
by a straightforward computation.
On the other hand, from
property (C) of $\ome$ and property (d) of $h_{\e}$,
we have
$\sqrt{-1}\Theta_{h_{\e}}(F)
\geq -\e \omega
\geq -\e \ome$.
This implies $\lambda_{j}^{\e} \geq -\e$,
and thus we obtain inequality (\ref{ine3}).

From equality (\ref{B-eq}) and inequality (\ref{ine3}),
we obtain
\begin{align*}
0 \geq A_{\e} &= \int_{Y} g_{\e}\ \ome^{n} \\
& \geq -\e C \int_{Y} |u_{\e}|^{2}_{h_{\e}, \ome}\ \ome^{n}\\
& \geq -\e C \|u \|^{2}_{h_{F}, \omega}.
\end{align*}
The last inequality follows from inequality (\ref{ine2}).
Therefore $A_{\e}$ converges to zero, and further
we can conclude that $B_{\e}$ also converges to zero
by equality (\ref{B-eq}).

To apply Nakano's identity to $su_{\e}$ again,
we first check $su_{\e} \in
L_{(2)}^{n,q}(Y, F \otimes L)_{h_{\e} h_{L, \e}, \ome}$.
By the assumption,
the point-wise norm $|s|_{h_{L}}$ with respect to $h_{L}$ is
bounded, and further we have
$|s|_{h_{L, \e}} \leq |s|_{h_{L}}$
from property (b) of $h_{\e}$.
They imply
\begin{equation*}
\|s u_{\e} \|_{h_{\e} h_{L, \e}, \ome} \leq
\sup_{X} |s|_{h_{L, \e}}  \|u_{\e} \|_{h_{\e}, \ome}  \leq
\sup_{X} |s|_{h_{L}}  \|u \|_{h_{F}, \omega} < \infty.
\end{equation*}
Remark that the right hand side does not depend on $\e$.
By applying Nakano's identity to $su_{\e}$,
we obtain
\begin{align}
& \|D^{''*}_{h_{\e} h_{L, \e}}
su_{\e} \|^{2}_{h_{\e} h_{L, \e}, \ome} \nonumber \\
=& \lla \sqrt{-1}\Theta_{h_{\e} h_{L, \e}}(F\otimes L)
\Lambda_{\ome} su_{\e}, su_{\e}
  \rra_{h_{\e} h_{L, \e}, \ome} +
\|D^{'*}_{h_{\e} h_{L, \e}}su_{\e} \|^{2}_{h_{\e} h_{L, \e}, \ome}\label{B-eq2}.
\end{align}
Here we used $\dbar s u_{\e}=0$.
We first see that
the second term of the right hand side converges to zero.
Since $s$ is a holomorphic $(0, 0)$-form, we have
$D^{'*}_{h_{\e} h_{L, \e}}su_{\e} =
s D^{'*}_{h_{\e}}u_{\e}$.
Thus we have
\begin{equation*}
\|D^{'*}_{h_{\e} h_{L, \e}}su_{\e} \|^{2}_{h_{\e} h_{L, \e}, \ome} \leq
\sup_{X}|s|^{2}_{h_{L, \e}}  \int_{Y}
|D^{'*}_{h_{\e}}u_{\e} |^{2}_{h_{\e}, \ome}\ \ome^{n}
\leq \sup_{X}|s|^{2}_{h_{L}} B_{\e}.
\end{equation*}
Since $|s|^{2}_{h_{L}}$ is bounded and
$B_{\e}$ converges to zero,
the second term
$\|D^{'*}_{h_{\e} h_{L, \e}}su_{\e} \|_{h_{\e} h_{L, \e}, \ome}$ also converges to zero.

For the proof of the proposition,
it remains to show that
the first term of the right hand side of equality (\ref{B-eq2})
converges to zero.
We can easily see
$\sqrt{-1}\Theta_{h_{\e} h_{L, \e}}(F\otimes L) = {(1+1/a)}\sqrt{-1}\Theta_{h_{\e}}(F)$
from $\sqrt{-1} \Theta_{h_{\Delta}}=0$ on $Y$ and
the definition of $h_{L, \e}$.
Therefore we obtain
\begin{equation*}
\lla \sqrt{-1}\Theta_{h_{\e} h_{L, \e}}(F\otimes L)
\Lambda_{\ome} su_{\e}, su_{\e}
  \rra_{h_{\e} h_{L, \e}, \ome} =
(1+1/a) \int_{Y} |s|^{2}_{h_{L, \e}}
g_{\e}\ \ome^{n}
\end{equation*}
Now
we investigate $A_{\e}$ in details.
By the definition of $A_{\e}$, we have
\begin{equation*}
A_{\e}= \int_{\{ g_{\e} \geq 0 \}} g_{\e}\ \ome^{n} +
\int_{\{ g_{\e} \leq 0 \}} g_{\e}\ \ome^{n}.
\end{equation*}
It is easy to see that
the second term converges to zero.
Indeed, simple computations and inequality (\ref{ine3})
imply
\begin{align*}
0 \geq \int_{\{ g_{\e} \leq 0 \}} g_{\e}\ \ome^{n} &\geq
-\e C \int_{\{ g_{\e} \leq 0 \}}|u_{\e}|^{2}_{h_{\e}, \ome}\
\ome^{n} \\
&\geq -\e C \int_{Y}|u_{\e}|^{2}_{h_{\e}, \ome}\
\ome^{n} \\
&\geq -\e C \| u \|^{2} _{h_{F}, \omega}.
\end{align*}
Therefore the first term also converges to zero.
Now we have
\begin{align*}
\int_{Y} |s|^{2}_{h_{L, \e}}
g_{\e}\ \ome^{n} =  \Big\{ \int_{\{ g_{\e} \geq 0 \}}
|s|^{2}_{h_{L, \e}} g_{\e}\ \ome^{n} +
\int_{\{ g_{\e} \leq 0 \}}
|s|^{2}_{h_{L, \e}} g_{\e}\ \ome^{n}\Big\}.
\end{align*}
On the other hand, we have the inequalities
\begin{align*}
\bullet \hspace{0.3cm} 0 \leq \int_{\{ g_{\e} \geq 0 \}}
|s|^{2}_{h_{L, \e}} g_{\e}\ \ome^{n} &\leq  \sup_{X}|s|^{2}_{h_{L, \e}}
\int_{\{ g_{\e} \geq 0 \}}
g_{\e}\ \ome^{n} \\
& \leq   \sup_{X}|s|^{2}_{h_{L}}\ \int_{\{ g_{\e} \geq 0 \}} g_{\e}\ \ome^{n},\\
\bullet \hspace{0.3cm}  0 \geq \int_{\{ g_{\e} \leq 0 \}}
|s|^{2}_{h_{L, \e}} g_{\e}\ \ome^{n}_{\e} &\geq
\sup_{X}|s|^{2}_{h_{L, \e}} \int_{\{ g_{\e} \leq 0 \}}
 g_{\e}\ \ome^{n} \\
&\geq   \sup_{X}|s|^{2}_{h_{L}} \int_{\{ g_{\e} \leq 0 \}} g_{\e}\ \ome^{n}.
\end{align*}
Therefore the right hand side of equality (\ref{B-eq2})
converges to zero.
We obtain the conclusion of Proposition \ref{D''}.
\end{proof}
\
\vspace{0.2cm} \\
{\bf{Step 3 (A construction of
solutions of the $\dbar$-equation via
the $\bf\rm{\check{C}}$eck complex)}}
\vspace{0.1cm} \\
By the absolutely same method as \cite[Step 3]{Mat-inj},
we can prove the following proposition.
See \cite[Step 3]{Mat-inj} for the proof.

\begin{prop}\label{sol-1}
There exist $F$-valued $(n, q-1)$-forms $\alpha_{\e}$
on $Y$
with the following properties\,$:$
\begin{equation*}
{\rm{(1)}}\hspace{0.2cm} \dbar \alpha_{\e}=u-u_{\e}.
\quad
{\rm{(2)}}\hspace{0.2cm} \text{The norm }
\|\alpha_{\e} \|_{h_{\e}, \ome}
 \text{ is uniformly bounded}.
\end{equation*}
\end{prop}

\hspace{0.1cm}\\ \
{\bf{Step 4 (The limit of the harmonic forms)}}
\vspace{0.1cm}\\ \
In this step, we investigate the limit of $u_{\e}$ and
complete the proof of Theorem \ref{main-inj}.
First we prove the following proposition.
\begin{prop}\label{sol}
There exist $F\otimes L$-valued $(n, q-1)$-forms
$\gamma_{\e}$
on $Y $
with the following properties\,$:$
\begin{equation*}
{\rm{(1)}}\hspace{0.2cm} \dbar \gamma_{\e}=su_{\e}.
\quad
{\rm{(2)}}\hspace{0.2cm} \text{The norm }
\|\gamma_{\e} \|_{h_{\e} h_{L, \e}, \ome}
 \text{ is uniformly bounded}.
\end{equation*}
\end{prop}
\begin{proof}
There exists an $F\otimes L$-valued $(n, q-1)$-form $\gamma$
such that $\dbar \gamma =  su$ and
$\|\gamma \|_{h_{F} h_{L}, \omega} < \infty$.
(Recall that we are assuming that the cohomology class of $su$ is zero in
$H^{q}(X, K_{X}\otimes F\otimes L \otimes
\I{h_{F} h_{L}})$.)
If we take $\alpha_{\e}$ with the properties in Proposition \ref{sol-1}
and put $\gamma_{\e}:= -s \alpha_{\e} + \gamma$,
then we have $\dbar \gamma_{\e} = su_{\e}$.
Further an easy computation yields
\begin{align*}
\|\gamma_{\e}\|_{h_{\e} h_{L, \e}, \ome} &\leq
\|s \alpha_{\e}\|_{h_{\e} h_{L, \e}, \ome} +
\|\gamma \|_{h_{\e} h_{L, \e}, \ome}\\
&\leq \sup_{X}|s|_{h_{L}} \|\alpha_{\e} \|_{h_{\e}, \ome} +
\|\gamma \|_{h_{F}h_{L}, \ome}.
\end{align*}
Since $\| \gamma \|_{h_{F}h_{L}, \ome}
\leq \| \gamma \|_{h_{F}h_{L}, \omega} < \infty$ and the norm $\|\alpha_{\e} \|_{h_{\e}, \ome}$ is uniformly bounded,
the right hand side can be estimated by a constant independent of $\e$.
\end{proof}
We consider the limit of the norm
$\| s u_{\e}\|_{h_{\e}h_{L, \e}, \ome}$.

\begin{prop}\label{converge}
The norm $\| s u_{\e}\|_{h_{\e}h_{L, \e}, \ome}$
converges to zero as letting $\e$ go to zero.
\end{prop}
\begin{proof}
By taking
$\gamma_{\e} \in L_{(2)}^{n, q-1}
(Y, F \otimes L)_{h_{\e}h_{L, \e}, \ome}$ satisfying the properties
in Proposition \ref{sol},
we obtain
\begin{align*}
\| s u_{\e}\|_{h_{\e}h_{L, \e}, \ome}^{2}
&=\lla  s u_{\e}, \dbar \gamma_{\e}
 \rra_{h_{\e}h_{L, \e}, \ome} \\
&=
 \lla  D^{''*}_{h_{\e}h_{L, \e}}s u_{\e},\gamma_{\e}
 \rra_{h_{\e}h_{L, \e}, \ome} \\
&\leq
\|D^{''*}_{h_{\e}h_{L, \e}} s u_{\e}\| _{h_{\e}h_{L, \e}, \ome}
\|\gamma_{\e} \|_{h_{\e}h_{L, \e}, \ome}.
\end{align*}
By Proposition \ref{sol}, the norm
$\|\gamma_{\e} \|_{h_{\e}h_{L, \e}, \ome}$ is uniformly bounded.
On the other hand, the norm
$\|D^{''*}_{h_{\e}h_{L, \e}} s u_{\e}\| _{h_{\e}h_{L, \e}, \ome}$
converges to zero by Proposition \ref{D''}.
Therefore the norm $\| s u_{\e}\|_{h_{\e}, \ome}$ also converges to zero.
\end{proof}

Fix a small number $\e_{0}>0$.
Then for any positive number $\e$ with $0< \e < \e_{0}$,
by property (b) of $h_{\e}$, we obtain
\begin{equation*}
\|u_{\e} \|_{h_{\e_{0}}, \ome} \leq \|u_{\e} \|_{h_{\e}, \ome}
\leq  \|u \|_{h_{F}, \omega}.
\end{equation*}
It says that the norm of $u_{\e} $ with respect to
$h_{\e_{0}}$ is
uniformly bounded.
Therefore
there exists a subsequence of $\{ u_{\e} \}_{\e >0}$
that  converges to
$\alpha \in L_{(2)}^{n, q}(Y, F)_{h_{\e_{0}}, \ome}$
with respect to the weak $L^{2}$-topology.
For simplicity, we denote this subsequence
by the same notation $\{ u_{\e} \}_{\e >0}$.
Then we prove the following proposition.

\begin{prop}\label{zero}
The weak limit $\alpha$ of $\{ u_{\e} \}_{\e >0}$
in $L_{(2)}^{n, q}(Y, F)_{h_{\e_{0}}, \ome}$
is zero.
\end{prop}
\begin{proof}

For every positive number $\delta>0$,
we define the open subset $A_{\delta}$ of $Y$ by
$A_{\delta}:= \{x \in Y \mid  |s|^{2}_{h_{L,\e_{0}}} > \delta  \}$.
By an easy computation, we have
\begin{align*}
\| s u_{\e}  \|^{2}_{h_{\e}h_{L, \e}, \ome}
&\geq \| s u_{\e}  \|^{2}_{h_{\e_{0}}h_{L, \e_{0}}, \ome} \\
&\geq \int_{A_{\delta}} |s|^{2}_{ h_{L, \e_{0}} }
|u_{\e}|^{2}_{h_{\e_{0}}, \ome}\ \ome^{n} \\
&\geq \delta  \int_{A_{\delta}}
|u_{\e}|^{2}_{h_{\e_{0}}, \ome}\ \ome^{n} \geq 0
\end{align*}
for any $\delta>0$.
Since the left hand side converges to zero,
the norm $\|u_{\e} \|_{h_{\e_{0}}, \ome, A_{\delta}}$ on $A_{\delta}$
also converges to zero.
Notice that $u_{\e} |_{A_{\delta}}$
converges to $\alpha |_{A_{\delta}}$ with respect to
the weak $L^{2}$-topology in
$ L_{(2)}^{n, q}(A_{\delta}, F)_{h_{\e_{0}}, \ome}$.
Here  $u_{\e} |_{A_{\delta}}$ (resp. $\alpha |_{A_{\delta}}$) denotes
the restriction of  $u_{\e}$ (resp. $\alpha$) to $A_{\delta}$.
Indeed
for every
$\gamma \in L_{(2)}^{n, q}(A_{\delta}, F)_{h_{\e_{0}}, \ome}$,
the inner product
$\lla u_{\e} |_{A_{\delta}}, \gamma
\rra_{A_{\delta}}
= \lla u_{\e} , \widetilde{\gamma}
\rra_{Y} $
converges to
$\lla \alpha, \widetilde{\gamma}
\rra_{Y}
= \lla \alpha|_{A_{\delta}} , \gamma
\rra_{A_{\delta}} $, where $\widetilde{\gamma}$ denotes the zero extension of $\gamma$ to
$Y$.
Since $u_{\e} |_{A_{\delta}}$
converges to $\alpha |_{A_{\delta}}$,
we obtain
\begin{equation*}
\|\alpha |_{A_{\delta}} \|_{h_{\e_{0}}, \ome, A_{\delta}}
\leq
\liminf_{\e \to 0}\|u_{\e} |_{A_{\delta}} \|_{h_{\e_{0}},  \ome,
A_{\delta}}=0.
\end{equation*}
(Recall the norm of the weak limit can be
estimated by
the limit inferior of the norms of sequences.)
Therefore
we have $\alpha |_{A_{\delta}} = 0$ for any $\delta>0$.
By the definition of $A_{\delta}$, the union of
$\{A_{\delta} \}_{\delta >0}$ is equal to $Y=X \setminus Z$, which asserts
that the weak limit $\alpha $ is zero on $Y$.
\end{proof}

By using Proposition \ref{zero},
we complete the proof of Theorem \ref{main-inj}.
By the definition of $u_{\e}$,
we have
\begin{equation*}
u = u_{\e} + \dbar v_{\e}.
\end{equation*}
Proposition \ref{zero} says
that $\dbar v_{\e}$ converges to $u$ with respect to
the weak $L^{2}$-topology.
Then it is easy to see that
$u$ is a $\dbar$-exact form
(that is, $u \in {\rm{Im}} \,\dbar$).
This is because
the subspace ${\rm{Im}} \dbar$
is closed in
$L_{(2)}^{n, q}(Y, F)_{h_{\e_{0}}, \ome}$
with respect to the weak $L^{2}$-topology.
Indeed, for every
$\gamma = \gamma_{1} + D^{''*}_{h_{\e_{0}}}\gamma_{2} \in
\mathcal{H}^{n, q}(F)_{h_{\e_{0}}, \ome}
\oplus {\rm{Im}}\,D^{''*}_{h_{\e_{0}}}$, we have
$\lla u, \gamma  \rra =
\lim_{\e \to 0}\lla \dbar v_{\e}, \gamma_{1} + D^{''*}_{h_{\e_{0}}}\gamma_{2}
\rra =0$.
Therefore we can conclude $u \in {\rm{Im}}\,\dbar$.

In summary, we proved that $u$ is a $\dbar$-exact form in
$L_{(2)}^{n, q}(Y, F)_{h_{\e_{0}}, \ome}$, which
says that the cohomology class $\{u \}$ of $u$ is zero
in $H^{q}(X, K_{X} \otimes F \otimes \I{h_{\e_{0}}})$.
By property (c), we obtain the conclusion of Theorem \ref{main-inj}.

\end{proof}

\section{Proof of Corollaries related to the Extension conjecture}
\label{applications-inj}
The purpose of this section is to obtain
some extension theorems as applications of our injectivity theorem.
For this purpose,
by making use of the injectivity theorem (Theorem \ref{main-inj}),
we first prove the following extension theorem,
which can be seen as a special case of
the extension conjecture for dlt pairs.

\begin{thm}\label{ext}
Let $X$ be a compact K\"ahler manifold and
$\Delta=S+B$ be an effective $\mathbb Q$-divisor
with the following assumptions:
\begin{itemize}
\item[(1)] $\Delta$ is a simple normal crossing divisor with
$0 \leq \Delta \leq 1$ and $\lfloor \Delta \rfloor =S$.
\item[(2)]
$K_{X}+\Delta$ is $\mathbb Q$-linearly equivalent to
an effective divisor $D$ with $S\subset\Supp\,D$.
\item[(3)] $K_{X}+\Delta$ admits a $($singular$)$
metric $h$ with semi-positive curvature.
\end{itemize}
Then for an integer $m \geq 2$ with Cartier divisor $m(K_{X}+S+B)$
and any section $u \in H^{0}(S, \mathcal{O}_{S}(m(K_{X}+S+B)))$
that belongs to the image of
$H^{0}(S, \mathcal{O}_{S}(m(K_{X}+S+B))\otimes \I{h^{m-1}h_{B}} ) \to
H^{0}(S, \mathcal{O}_{S}(m(K_{X}+S+B))) $,
the section $u$ can be extended to a section in $H^{0}(X, \mathcal{O}_{X}(m(K_{X}+S+B)) )$.

\vspace{0.2cm}
Moreover if we assume that $h \leq C h_{D}$ for some $C>0$,
where $h_{D}$ is the singular metric induced by $D$,
then any cohomology class  $u \in
H^{q}(S, \mathcal{O}_{S}(m(K_{X}+\Delta))\otimes \I{h^{m-1}h_{B}} )$
can be extended to a class in
$H^{q}(X, \mathcal{O}_{X}(m(K_{X}+\Delta))\otimes \I{h^{m-1}h_{B}} )$ for any $q \geq 0$.

\end{thm}

\begin{rem}
$(1)$ If $X$ is projective and $S$ is smooth
$($namely $(X,\Delta)$ is plt$)$
and further if the divisor induced by a give
$u \in H^{0}(S, \mathcal{O}_{S}(m(K_{X}+S+B)))$
is larger than
the extension obstruction divisor
$($which is zero if $K_{X}+\Delta$ is nef$)$,
then $K_{X}+\Delta$ admits a $($singular$)$ metric $h$ with $\sup_{S} |u|_{h} < \infty$.
$($see \cite[Corollary 1.8]{dhp}$)$.

\vspace{0.1cm}\\
$(2)$ Even if $S$ has singularities,
we can construct a $($singular$)$ metric $h$ with $\sup_{S} |u|_{h} < \infty$
in the special case where
$u$ is zero along the singular locus of $S$,
by the same method as in \cite{dhp}.

\end{rem}

\begin{proof}
We may add
the additional assumption of
$h \leq h_{D}$,
where $h_{D}$ is the singular metric on $\mathcal{O}_{X}(K_{X}+\Delta)$
defined by the effective divisor $D$.
Indeed, for a smooth metric $g$ on $\mathcal{O}_{X}(K_{X}+\Delta)$ and
an $L^{1}$-function $\varphi$ (resp. $\varphi_{D}$)
with $h= g\, e^{-\varphi} $ (resp. $h_{D}= g\, e^{-\varphi_{D}} $),
the metric defined by $g\, e^{-\max( \varphi, \varphi_{D})}$
satisfies assumption $(3)$ again.

We prove only the first conclusion since
the second conclusion follows from the same argument as the first conclusion.
We put $G:=m(K_{X}+ \Delta)$,
and consider the following exact sequence:
\begin{equation*}
0 \rightarrow \mathcal{O}_{X}(G - S) \otimes I (h^{m-1}h_{B})
\rightarrow \mathcal{O}_{X}(G ) \otimes \I{h^{m-1}h_{B}}
\rightarrow \mathcal{O}_{S}(G ) \otimes \I{h^{m-1}h_{B}}
\rightarrow0.
\end{equation*}
We prove
the induced homomorphism
\begin{equation*}
H^{q}(X, \mathcal{O}_{X}(G - S) \otimes I (h^{m-1}h_{B}))
\to
H^{q}(X, \mathcal{O}_{X}(G) \otimes I (h^{m-1}h_{B}))
\end{equation*}
is injective by the injectivity theorem.
Then the conclusion follows from the induced long exact sequence.

By the assumption on the support of $D$, we can take
an integer $a>0$ such that $aD$ is a
Cartier divisor and $S \leq a D$.
Then we have the following commutative diagram:

\[\xymatrix{
& \hspace{-3.5cm} H^{q}(X, \mathcal{O}_{X}(G) \otimes I (h^{m-1}h_{B})) \supseteq \mathrm{Im}\,(+S)
 \ar[d]^{+ (aD -S)}
& \\
H^{q}(X, \mathcal{O}_{X}(G - S) \otimes I (h^{m-1}h_{B}))
\ar[ru]^{+ S}
\ar[r]_{\hspace{-1.0cm}+ aD}  &
H^{q}(X, \mathcal{O}_{X}(G - S + aD) \otimes I (h^{a+m-1}h_{B})), &
}\]
where the map $+S:H^{q}(X, \mathcal{O}_{X}(G - S) \otimes I (h^{m-1}h_{B}))  \to H^{q}(X, \mathcal{O}_{X}(G ) \otimes I (h^{m-1}h_{B})).  $
Our goal is to show
that the map to the upper right is injective.
For this goal, we show that the horizontal map is injective
as an application of Theorem \ref{main-inj}.

By the definition of $G$,
we have
\begin{equation*}
G -S =m(K_{X}+\Delta)-S=K_{X} + (m-1)(K_{X}+ \Delta) + B.
\end{equation*}
Then the line bundle $F:= \mathcal{O}_{X}((m-1)(K_{X}+ \Delta) + B)$
equipped with the metric $h_{F}:= h^{m-1} h_{B}$
and the line bundle $L:=\mathcal{O}_{X}(aD)$
equipped with the metric $h_{L}:= h^{{a}}$
satisfy the assumptions in Theorem \ref{main-inj}.
Indeed, we have $h_{F}=h_{L}^{(m-1)/a} h_{B}$
by the construction, and further
the point-wise norm $|s_{aD}|_{h_{L}}$ is bounded on $X$
by the inequality $h \leq h_{D}$,
where $s_{aD}$ is the natural section of $aD$.
Therefore the horizontal map is injective
by Theorem \ref{main-inj}.

\if0

\vspace{0.3cm}
\hspace{0.1cm}\\ \
{\bf{Step 2}}
\hspace{0.3cm}
Our goal of this step is to show that
$u^{k}$ belongs to
$H^{0}(S, \mathcal{O}_{S}(G ) \otimes I (h^{m-1}h_{B}))$.
We take a local extension $f$ of $u^{k}$
to a neighborhood of a given point in $S$.
The following lemma implies that
$|f|_{h^{m-1}h_{B}}$ is locally $L^{2}$-integrable,
and thus we know $u^{k} \in H^{0}(S, \mathcal{O}_{S}(G ) \otimes I (h^{m-1}h_{B}))$.
Indeed, a local weight $\varphi$ (resp. $\psi$) of $h$ (resp. $h_{B}$)
is a psh function,
and further $e^{-\psi}$ is
locally $L^{2}$-integrable
since $B$ is a simple normal crossing divisor with $B<1$.
Therefore the assumptions in the lemma is satisfied.

\begin{lem}
Let $\varphi$ and $\psi$ be psh functions and $f$
be a holomorphic function on the open ball of radius one $B(0;1)$ in $\mathbb{C}^{n}$.
If $e^{-\psi}$ is locally $L^{2}$-integrable,
and the value of $|f|^{}e^{-\varphi}$ is
finite at the origin,
then  $|f| e^{-\varphi-\psi}$ is
locally $L^{2}$-integrable
$($that is, $f$ is a local section of $\I{\varphi+\psi}$$)$.
\end{lem}
\begin{proof}

After we first show that $|f|^{p}e^{-p\varphi}$ is
locally $L^{2}$-integrable for any integer $p>0$,
we obtain the conclusion by H\"odler's inequality.
If $\varphi$ has analytic singularities,
it is obvious that
the function $|f|^{}e^{-\varphi}$ is
bounded on a neighborhood of the origin.
However, it is not true
when $\varphi$ has transcendental singularities.
For example when $\varphi$ is a psh function defined by
$\varphi(z):= \sum_{k=1}^{\infty} \e_{k} \log |z+1/k|$
where $\{\e_{k}\}_{k=1}^{\infty}$ is convergent series,
$e^{-\varphi}$ is finite at the origin,
but $e^{-\varphi}$ is not bounded on any neighborhood at the origin.
Note that $e^{-p\varphi}$ is locally $L^{2}$-integrable for any positive integer $p$ even in this case.

By using the approximation of $\varphi$ with
psh functions with analytic singularities.
we show that
$|f|^{p}e^{-p\varphi}$ is
locally $L^{2}$-integrable for any integer $p>0$.
Let $\varphi_{\ell}$ be the psh function induced by
the Bergman kernel the $L^{2}$-space of the holomorphic functions with
the $L^{2}$-norm defined by
\begin{equation*}
\| f \|^{2}:= \int _{B(0;1)} |f|^{2} e^{-2 \ell \varphi}.
\end{equation*}
Namely,
\begin{align*}
\varphi_{\ell}:=\frac{1}{2\ell} \log \sum_{k=1}^{\infty} |f_{k,\ell}|^{2},
\end{align*}
where $\{f_{k,l}  \}_{k=1}^{\infty}$ is
a complete orthonormal basis.
By the Ohsawa-Takegoshi $L^{2}$-extension theorem we obtain
$\varphi_{\ell} - C/\ell \leq \varphi_{\ell}$ for some constant $C$
(see \cite{Dem}[(13.2) Theorem]),
and thus we know that the value
$|f|e^{-\varphi_{\ell}}$ is finite at the origin.
In particular, $|f|e^{-\varphi_{\ell}}$ is bounded on
the open ball of a small radius
since $\varphi_{\ell}$ has analytic singularities.

Then for integers $p, \ell>0$,
straightforward computations implies
\begin{align*}
& \int_{B(0;r)} |f|^{2p}e^{-2p\varphi} \\
=&\int_{\{\varphi \leq \frac{\ell}{\ell-p} \varphi_{\ell}\}\cap B(0;r)} |f|^{2p}e^{-2p\varphi}
+
\int_{\{\varphi > \frac{\ell}{\ell-p} \varphi_{\ell}\}\cap B(0;r)} |f|^{2p}e^{-2p\varphi} \\
 \leq &
\sup  |f|^{2p} \int_{\{\varphi \leq \frac{\ell}{\ell-p} \varphi_{\ell}\}\cap B(0;r)}
e^{ -2\ell \varphi - 2(p-\ell)\varphi}
+
\int_{B(0;r)} |f|^{2p}e^{-2p \varphi_{\ell}
-2\frac{p^{2}}{\ell-p}\varphi_{\ell}}.
\end{align*}
First we see that the second term of the right hand side is finite.
By the H\"older's inequality,
for real numbers $x, y>1$ with $1/x+1/y=1$,
we have
\begin{align*}
\int_{B(0;r)} |f|^{2p}e^{-2p \varphi_{\ell} -2
\frac{p^{2}}{\ell-p}\varphi_{\ell}}
\leq
\Big( \int_{B(0;r)}  |f|^{2px}e^{-2px \varphi_{\ell}} \Big)^{1/x}
\cdot
\Big( \int_{B(0;r)}  e^{-2 \frac{p^{2}y}{\ell-p}\varphi_{\ell}} \Big)^{1/y}
\end{align*}
The Lelong number
\begin{equation*}
\nu(\frac{p^{2}y}{\ell-p}\varphi_{\ell}, 0)
\leq \frac{p^{2}y}{\ell-p} \nu(\varphi, 0)
\end{equation*}
is smaller than one for a sufficiently large $\ell$.
Further if we take a small $r$ (depending on $\ell$),
the function
$|f|^{2px}e^{-2px \varphi_{\ell}}$ is bounded on $B(0;r)$.
Therefore the second term is finite for a sufficiently small $r$.

Now we consider the first term.
By simple computations, we obtain
\begin{align*}
\int_{\{\varphi \leq \frac{\ell}{\ell-p} \varphi_{\ell}\}\cap B(0;r)}
e^{ -2\ell \varphi - 2(p-\ell)\varphi}
&\leq \int_{\{\varphi \leq \frac{\ell}{\ell-p}
\varphi_{\ell}\}\cap B(0;r)}
e^{ -2\ell \varphi + 2 \ell \varphi_{\ell}} \\
&\leq \int_{B(0;r)} (\sum_{k=1}^{\infty} |f_{k,\ell}|^{2})
e^{ -2\ell \varphi}.
\end{align*}
By the techniques in \cite[Theorem 2.3]{dps01},
we can conclude that the right hand side is finite
for a small $r$.
We consider the family of the ideal sheaves
$\mathcal{J}_{k_{0}}$ generated by
$\{ f_{k,\ell}(z) \cdot {\overline{f_{k,\ell}({\overline{w}})}} \}_{k \leq {k_{0}}}$ on $B(0;r) \times B(0;r)$.
By taking a sufficiently small $r$,
we may assume that
$\mathcal{J}_{k_{0}}$ is the maximal element of this family
by the Noetherian property,
and further we may assume that
\begin{equation*}
\sum_{k=1}^{N}  f_{k,\ell}(z)
\cdot {\overline{f_{k,\ell}({\overline{w}})}}
\end{equation*}
is uniformly convergent on $B(0;r) \times B(0;r)$.
The limit belongs to $\mathcal{J}_{k_{0}}$ since
the space of the sections of $\mathcal{J}_{k_{0}}$ is closed
under the topology of uniformly convergence
(see \cite[Section D, Chapter I\hspace{-.1em}I]{gr65}).
By the restriction to the conjugate diagonal,
we obtain
\begin{equation*}
\sum_{k=1}^{\infty}  |f_{k,\ell}(z)|^{2}
\leq C \sum_{k=1}^{k_{0}}  |f_{k,\ell}(z)|^{2}.
\end{equation*}
\end{proof}

Finally we obtain the conclusion by H\"odler's inequality.
There exists a number $q>1$ such that $e^{-q\psi}$
is locally $L^{2}$-integrable by the (weak) openness conjecture
which is first proved in \cite{Ber13}
(see the strong openness conjecture \cite{GZ13}).
By taking $p$ with $1/p +1/q=1$,
we have
\begin{align*}
\int_{B(0;r)} |f|^{2} e^{-2\varphi-2\psi} \leq
\Big(\int_{B(0;r)} |f|^{2p} e^{-2p\varphi} \Big)^{1/p}\cdot
\Big(\int_{B(0;r)} e^{-2q \psi} \Big)^{1/q}.
\end{align*}
By the first half of the proof and the choice of $q$,
this is finite if $r$ is sufficiently small.

\fi

\end{proof}

To obtain some results related to the abundance conjecture
(Theorem \ref{main-thm-semiample} and Corollary \ref{sm semi positive}),
we need the following corollary,
which is a slight generalization of Theorem \ref{ext}.

\begin{cor}\label{ext-cor2}
Under the same situation as in Theorem \ref{ext},
instead of assumption $(3)$,
we assume the following assumption:
\begin{itemize}
\item[$(3')$] There exist effective $\mathbb{Q}$-divisors
$E$ and $F$ and a $($singular$)$ metric $h$ on $\mathcal{O}_{X}(F)$
with semi-positive curvature such that
\begin{itemize}
\item[$\bullet$] $K_{X}+\Delta \sim_{\mathbb{Q}} E+F$,
\item[$\bullet$] $E+B$ is simple normal crossing and $E$ has no common component with $S$,
\item[$\bullet$] $\nu(h, x) = 0$ at every point $x \in S$.
\end{itemize}
\end{itemize}
Let $\widetilde{s}$ be a section on $X$ with ${\rm{div}}\, \widetilde{s} = mE$.
Then for a section $u \in H^{0}(S, \mathcal{O}_{S}(mF))$,
the section $u \cdot \widetilde{s} \in
H^{0}(S, \mathcal{O}_{S}(m(K_{X}+\Delta)))$ can be extended to
a section in $H^{0}(X, \mathcal{O}_{X}(m(K_{X}+\Delta)))$.
\end{cor}

\begin{proof}
Let $h_{E}$ be the singular metric on $\mathcal{O}_{X}(E)$
induced by the section $\widetilde{s} \in H^{0}(X,\mathcal{O}_{X}(mE))$.
By the definition, the metric $h_{E}$ satisfies
$\sqrt{-1}\Theta_{h_E} \geq 0$ and $\sup |\widetilde{s}|_{h^{m}_{E}} < \infty$.
The product $h \cdot  h_{E}$ determines the singular metric
on $K_{X}+\Delta$ with semi-positive curvature.
Therefore it is sufficient to show that
$u \cdot \widetilde{s}$ belongs to
$H^{0}(S, \mathcal{O}_{S}(m(K_{X}+\Delta))
\otimes  \mathcal{J})$,
where we put $\mathcal{J}:= \I{ h^{m-1}h^{m-1}_{E}h_{B} }$ for simplicity.

In the first step,
we see that
\begin{equation*}
\mathcal{J}_{x} = \I{h^{m-1}_{E}h_{B}}_{x}
\end{equation*}
for every $x \in S$,
where $\mathcal{F}_{x}$ denotes the stalk of a sheaf $\mathcal{F}$ at $x$.
Let $f$ be a holomorphic function
on an open neighborhood $U_{x}$ of $x \in S$ with $ f \in \I{h^{m-1}_{E}h_{B}}_{x} $,
and let $\varphi$ (resp. $\varphi_{E}$, $\varphi_{B}$) be
a local weight of $h$ (resp. $h_{E}$, $h_{B}$).
By taking a real number $p>1$
with $\I{h^{p(m-1)}_{E}h^{p}_{B}}=\I{h^{m-1}_{E}h_{B}}$,
we may assume that $|f| e^{ -p(m-1)\varphi_{E} - p\varphi_{B}}$
is $L^{2}$-integrable on $U_{x}$.
Then,
by taking the positive number $q$ with $1/p + 1/q = 1$,
we obtain
\begin{align*}
\int_{U_{x}} |f|^{2}
e^{-2(m-1)\varphi -2(m-1)\varphi_{E} - 2\varphi_{B}}
&\leq
\Big(\int_{U_{x}} |f|^{2p}
e^{ -2p(m-1)\varphi_{E} - 2p\varphi_{B}} \Big)^{1/p}\cdot
\Big(\int_{U_{x}} e^{-2q (m-1)\varphi} \Big)^{1/q}\\
\end{align*}
by H\"older's inequality.
The function $e^{-2q (m-1)\varphi}$ is locally $L^{2}$-integrable
for any $q>0$
by Skoda's lemma and the assumption of the Lelong number.
On the other hand, as mentioned above,
$|f|^{p} e^{ -p(m-1)\varphi_{E} - p\varphi_{B}}$ is also locally $L^{2}$-integrable.
Therefore we have $\mathcal{J}_{x} = \I{h^{m-1}_{E}h_{B}}_{x} $
for every $x \in S$.

In the second step,
we prove
\begin{equation*}
u\cdot  \widetilde{s} \in H^{0}(S, \mathcal{O}_{S}(m(K_{X}+\Delta))
\otimes \mathcal{J}|_{S}),
\end{equation*}
where $\mathcal{J}|_{S}$ is the restriction of $\mathcal{J}$ defined by
\begin{equation*}
\mathcal{J}|_{S} :=
\mathcal{J}\cdot \mathcal{O}_{S}=
\mathcal{J}/(\mathcal{J}\cap \mathcal{I}_{S}).
\end{equation*}
Let $\widetilde{u}$ be a local extension of $u$
on an open neighborhood $U_{x}$ of $x \in S$.
By the klt condition of $B$,
we can take a real number $p>1$
with $\I{h^{p}_{B}}=\mathcal{O}_{X}$.
Then for the holomorphic function $g:=\widetilde{u}\cdot  \widetilde{s}$,
by taking the positive number $q$ with $1/p + 1/q = 1$,
we obtain
\begin{align*}
\int_{U_{x}} |g|^{2}
e^{-2(m-1)\varphi_{E} - 2\varphi_{B}}
&\leq
\Big(\int_{U_{x}}  |g|^{2p} e^{ -2p(m-1)\varphi_{E}- 2p\varphi_{B}} \Big)^{1/p}\cdot
\Big(\int_{U_{x}} 1 \Big)^{1/q} \\
&\leq
\sup_{U_{x}} |g|^{2p} e^{ -2p(m-1)\varphi_{E}}
\Big(\int_{U_{x}}  e^{- 2p\varphi_{B}} \Big)^{1/p}\cdot
\Big(\int_{U_{x}} 1 \Big)^{1/q}
\end{align*}
by H\"older's inequality again.
The point-wise norm
$|g|^{2p} e^{ -2p(m-1)\varphi_{E}}$ is bounded by the choice of $h_{E}$,
and
$e^{-2q (m-1)\varphi}$ is locally $L^{2}$-integrable
for any $q>0$
by the assumption on the Lelong number.
It implies that
$u\cdot  \widetilde{s} $ (locally) belongs to $\I{h^{m-1}_{E}h_{B}}|_{S}=
\mathcal{}\mathcal{J}|_{S}$.

Finally we show
\begin{align*}
u \cdot \widetilde{s} &\in H^{0}(S, \mathcal{O}_{S}(m(K_{X}+\Delta))\otimes \mathcal{J} ).

\end{align*}
By simple computations
we have $\mathcal{O}_{S}\otimes \mathcal{J}=
\mathcal{O}_{X}\otimes \mathcal{J}/(\mathcal{J} \cdot \mathcal{I}_{S})$,
and thus, by the second step, it is sufficient to see
$$\mathcal{J}\cap \mathcal{I}_{S}
= \mathcal{J} \cdot \mathcal{I}_{S}.$$
Here $\mathcal{I}_{S}$ denotes the ideal sheaf defined by $S$.
By the first step and the assumption on the support of $E+B$.
we have
\begin{equation*}
\mathcal{J}_{x} = \I{h^{m-1}_{E}h_{B}}_{x} =\mathcal{O}_{X}(-\lfloor(m-1)E+B \rfloor )
\end{equation*}
for every $x \in S$.
Since the divisors
$S$ and $E+B$ has no common component by the assumption on $E$,
we can easily see
$\mathcal{J}\cap \mathcal{I}_{S}
= \mathcal{J} \cdot \mathcal{I}_{S}.$
Therefore $u \cdot \widetilde{s}$ actually belongs to
$H^{0}(S, \mathcal{O}_{S}(m(K_{X}+\Delta))
\otimes \mathcal{J})$.
The conclusion follows from Theorem \ref{ext}.
\end{proof}

\begin{rem}
When we apply the injectivity theorem in order to extend sections,
we need to handle $\mathcal{O}_{S} \otimes \I{\varphi }$ $($not $\I{\varphi }|_{S}$$)$.
When we apply the Ohsawa-Takegoshi extension theorem,
we usually use the restriction of multiplier ideal sheaves $\I{\varphi }|_{S} $.
It is relatively difficult to handle $\mathcal{O}_{S} \otimes \I{\varphi }$.
However the support condition of $E$ $($the second assumption of the above corollary$)$
fortunately appears in the proof of the applications related to the abundance conjecture,
which asserts $\mathcal{O}_{S} \otimes \I{\varphi }= \I{\varphi }|_{S}$.
\end{rem}

In a special case of the above corollary when $E=\mathcal{O}_X$ and $\widetilde{s}=1 \in H^0(X, \mathcal{O}_X)$, we obtain the following:

\begin{cor}\label{ext-cor}
Under the same situation as in Theorem \ref{ext},
instead of assumption $(3)$,
we assume the following assumption:
\begin{itemize}
\item[$(3'')$] $K_{X}+\Delta$ admits a $($singular$)$
metric $h$ such that $\sqrt{-1}\Theta_{h} \geq 0$ and
$\nu(h, x) = 0$ at every point $x \in S$.
\end{itemize}
Then for an integer $m \geq 2$
with Cartier divisor $m(K_{X}+\Delta)$,
a section $u \in
H^{0}(S, \mathcal{O}_{S}(m(K_{X}+\Delta)) )$
can be extended to a section in
$H^{0}(X, \mathcal{O}_{X}(m(K_{X}+\Delta)) )$.
\end{cor}

For further applications of the above theorems,
we prepare the following lemma.

\begin{lem}\label{pull-min}
Let $\varphi$ be a $($quasi$)$-psh function
on a complex manifold $X$.
If the Lelong number $\nu(\varphi, x_{0})$ is zero at $x_{0} \in X$,
then for any modification
$\pi: Y \to X$, the Lelong number
$\nu(\pi^{*} \varphi, y)$ is zero at every point $y \in \pi^{-1}(x_{0})$.
\end{lem}
\begin{proof}
For a contradiction, we assume that
$\nu(\pi^{*} \varphi, y_{0})>0$
for some point $y_{0} \in \pi^{-1}(x_{0})$.
By Skoda's lemma (see Theorem \ref{Skoda}),
we can take a sufficiently large number $m>0$ such that
$\pi^{*}dV_{X} e^{-2m \pi^{*}\varphi}$ is
not integrable on a neighborhood of $y_{0}$,
where $dV_{X}$ is a standard volume form on a neighborhood
$U$ of $x_{0}$.
By the change of variable formula,
we have
\begin{align*}
\int_{U} e^{-2m \varphi}dV_{X} =
\int_{\pi^{-1}(U)}  e^{-2m \pi^{*}\varphi} \pi^{*}dV_{X}.
\end{align*}
By the assumption of $\nu(\varphi, x_{0})=0$,
the left hand side if finite for a sufficiently small $U$.
It is a contradiction to the choice of $m$.
Therefore $\nu(\pi^{*} \varphi, y)=0$ at every point $y \in \pi^{-1}(x_{0})$.

\end{proof}

\section{Proof of Corollaries related to the abundance conjecture}
\label{applications}

In this section,
we prove some applications related to the abundance conjecture. The proof of the following theorem is based on \cite[Section 8]{dhp} and \cite[Theorem 5.9]{fg3}.
We use the different MMP from them to preserve metric conditions.

\begin{thm}\label{main-thm-semiample}Assume that Conjecture \ref{abun conj} holds in dimension $n-1$.
Let $X$ be an $n$-dimensional normal projective variety and $\Delta$ be an $\mathbb{Q}$-divisor
with the following assumptions:
\begin{itemize}
\item $(X,\Delta)$ is a klt pair such that there exists an effective $\mathbb{Q}$-divisor $D$ such that $K_X+\Delta \sim_{\mathbb{Q}}D$.

\item  There exists a projective birational morphism $\varphi:Y \to X$ such that $Y$ is smooth and $\varphi^*(\mathcal{O}_X(m(K_X+\Delta)))$ admits
a $($singular$)$ metric whose curvature
is semi-positive and Lelong number is identically zero on $\mathrm{Supp}\,\varphi^*D$.
Here $m$ is a positive integer such that $m(K_X+\Delta)$ is Cartier.
\end{itemize}
Then $K_X+\Delta$ is semi-ample.

\end{thm}

\begin{proof}Note that Conjecture \ref{abun conj}  in dimension $n-1$ implies that the existence of good minimal model for $(n-1)$-dimensional klt pairs by \cite[Theorem 4.3]{gl} or \cite[Remark 2.6]{dhp}. First we may assume that $\kappa(K_X+\Delta)=0$ by Kawamata's theorem \cite[Theorem 7.3]{kawamata_pluri} (see also \cite[5.6 Lemma]{kemamc}).
There exists the effective $\mathbb{Q}$-divisor $D$ such  that
$$D \sim_{\mathbb{Q}}K_X+\Delta$$
by the assumption. For a contradiction, assume  $D \not =0$.
We may assume that $\varphi$ is a log resolution of $(X,\Delta+D)$.  We write
$$K_Y+B=\varphi^*(K_X+\Delta)\sim_{\mathbb{Q}} \varphi^*D.
$$
Let $l= \mathrm{lct(\varphi^*D; Y, B)}$. Set effective $\mathbb{Q}$-divisors $C$, $S$, and  $G$ such that $C+S=(B+l\varphi^*D)^{\geq0}$, $\lfloor C+S\rfloor=S$, and $G=(B+l\varphi^*D)^{\leq0}$. Then there are no common components of any two divisors of $C$, $S$, and $G$, and we have
$$K_Y+C+S\sim_{\mathbb{Q}} (1+l)\varphi^*D+G,
$$

$$S+C-G=B+l\varphi^*D.$$
 Note that $S\not =0$ and $G$ is $\varphi$-exceptional.
Then we see that $S \subseteq \mathrm{Supp}\,\varphi^*D$. Let us consider the MMP for $(Y,C+S)$ over $X$ by \cite{bchm} and \cite[Theorem 2.3]{fujino-ss}. Then this program contracts only the divisor $G$. Let $f:Y \dashrightarrow Y'$ be a minimal model of this program and $\varphi':Y' \to X$ the induced morphism. Then $S$ is not contracted by $f$ since $S$ and $G$ have no common component. Denote $C'$ and $S' (\not =0)$ by the strict transforms on $Y$ of $C$ and $S$. Then it follows that $$K_{Y'}+C'+S' \sim_{\mathbb{Q}} (1+l) \varphi'^*D.
$$
Thus $K_{Y'}+C'+S' $ is nef.

\begin{cl}\label{claim-ex}For a sufficiently large and divisible  $m'  \in \mathbb{Z}$,
the restriction map
$$H^0(Y', m'(K_{Y'}+C'+S' )) \to H^0(S', m'(K_{Y'}+C'+S' ))
$$
 is surjective

\end{cl}
\begin{proof}[Proof of Claim \ref{claim-ex}]\label{Proof of claim-ex}
 Let $u$ be a non-zero section in $H^0(S', m'(K_{Y'}+C'+S' ))$ and let

\begin{equation*}
\xymatrix{ & W\ar[dl]_{\alpha} \ar[dr]^{\beta}\\
 Y\ar@{-->}[rr]_{f}  & & Y'}
\end{equation*}
be common log resolutions such that $\beta$ is isomorphic over the generic point of every lc center of  $(Y,' C'+S')$. Then we write
$$K_{W}+S_W+F\sim_{\mathbb{Q}}(1+l)\beta^*\varphi'^*D+E,
$$
where $S_W$ is the strict transform of $S'$, and $E$ and $F$ are effective $\mathbb{Q}$-divisor such that $\lfloor F\rfloor=0$ and any two divisors of  $S_W$, $F$ and $E$ have no common component. Note that $m'F$ and $m'E$ are Cartier since $m'$ is sufficiently large and divisible. Let $u_{m'E}$ and $h_{m'E}$ be  the global section  and the metric associated to $m'E$ respectively. Then by the assumption we have a semi-positive singular metric $h$ on $m' (1+l)\beta^*\varphi'^*D$ induced by the pullback (we use the same symbol since it contains no confusion).

By applying Corollary \ref{ext-cor2} for $F=(1+l)\beta^*\varphi'^*D$  (cf. Lemma\,\ref{pull-min}) and  $\widetilde{s}=u_{m'E}$,  we have a section $U \in H^0(m'(K_W+S_W+F))$ such that $U_{|_{S_W}}=\beta^*u \otimes (u_{m'E})|_{S_W} $. Thus we see that
$$H^0(Y', m'(K_{Y'}+C'+S' )) \to H^0(S', m'(K_{Y'}+C'+S' ))
$$
 is surjective. Indeed we see that
 $$H^0(W, m'(K_W+S_W+F)) \to H^0(Y', m'(K_{Y'}+C'+S'))
 $$
  is isomorphic by mapping $s \mapsto \beta^*s \otimes u_{m'E}$ for a section $s \in H^0(W, m'(K_W+S_W+F)),$
  and $$H^0(S', m'(K_{Y'}+C'+S')) \to H^0(S_W, m'(K_{S_W}+F|_{S_W}))
 $$
 is injection by mapping  $t \mapsto \beta^*t \otimes (u_{m'E})|_{S_W}$ for a section $t \in H^0(S', m'(K_{Y'}+C'+S'))$ by $\beta_*\mathcal{O}_{S_W}=\mathcal{O}_{S'}$ from Koll\'ar--Shokurov's connectedness Theorem  \cite[17.4 Theorem]{KolFlip}.
We finish the proof of Claim \ref{claim-ex}.

\end{proof}
On the other hand, this restriction map is zero map since
$$\kappa(K_X+\Delta)=\kappa(K_{Y'}+C'+S')=0$$
 and $S' \subseteq \mathrm{Supp}\,\varphi'^*D$.  The pair $(S', C'_{S'})$ is a projective semi-log canonical pair such that $K_{S'}+C'_{S'}$ is nef, where $(K_{Y'}+C'+S')|_{S'}=(K_{S'}+C'_{S'})$. This implies $K_{S'}+C'_{S'}$ is semi-ample, because of  the  abundance conjecture (Conjecture \ref{abun conj}) holds in dimension $n-1$ and \cite[Theorem 1.5]{fg3} or \cite{hx} (here we need the assumption of projectivity). This is a contradiction to Claim \ref{claim-ex}. Thus $D=0$. We finish the proof.
\end{proof}

By combining the abundance theorem in dimension $3$ (\cite[Theorem 1.1]{k1}, \cite[1.1 Theorem]{kemamc}, \cite[Theorem 0.1]{fujino-abundance}), we obtain the following results:

\begin{cor}\label{sm semi positive}Let $(X,\Delta)$ be a $4$-dimensional projective klt pair.
Assume that there exists a projective birational morphism $\varphi:Y \to X$ on to each components such that $Y$ is smooth and $\varphi^*(\mathcal{O}_X(m(K_X+\Delta)))$ admits
a singular metric whose curvature is semi-positive and Lelong number is identically zero
(in particular it is satisfied if $h$ is smooth).
Here $m$ is an integer with Cartier divisor $m(K_X+\Delta)$.
If $\kappa(K_X+\Delta) \geq 0$, then $K_X+\Delta$ is semi-ample.

\end{cor}

\begin{rem}\label{rem-smooth}When $h$ is smooth,  we can show Corollary \ref{sm semi positive} without using our injectivity theorem. By replacing Theorem  \ref{intro-main-inj} to the generalized Enoki's injectivity theorem after Fujino \cite[Theorem 1.2 and Corollary1.3]{Fuj12-A} in the proof of Theorems \ref{ext}, \ref{main-thm-semiample},  and Corollary \ref{ext-cor2}, we can obtain it.

\end{rem}

Finally we give a result for semi-ampleness
by combining with Verbitsky's non-vanishing theorem (\cite[Theorem 4.1]{ver}).

\begin{cor}\label{cor-hk}Let $X$ be a $4$-dimensional projective hyperK\"ahler manifold
and $L$ be a line bundle admitting a $($singular$)$
metric whose curvature is semi-positive and Lelong number is identically zero
$($in particular it is satisfied if $h$ is smooth$)$.
Then $L$ is semi-ample.

\end{cor}

\begin{proof} It is enough to show $\kappa(L) \geq 0$ by Corollary \ref{sm semi positive} since if there exists an effective $\mathbb{Q}$-divisor such that $D \sim_{\mathbb{Q}}L$,  the pair $(X, \varepsilon D)$ is  klt and $K_X+\varepsilon D \sim_{\mathbb{Q}}\varepsilon L$ for sufficiently small $\varepsilon>0$.  If $q(L,L)>0$, then $L$ is big, where $q(\cdot, \cdot)$ is the Bogomolov--Beauville--Fujiki form. In the case of $q(L,L)=L^{\mathrm{dim}\,X}=0$, then $\kappa(L) \geq 0$ follows from  \cite[Theorem 4.1]{ver}.

\end{proof}

config: arxiv


TITLE: Solving the Integral $\int_0^{\infty} \, \mathrm{arcsinh} \left(\frac{a}{\sqrt{x^2+y^2}} \right) \, \mathrm{cos}(b \, x) \,\mathrm{d}x$
QUESTION [5 upvotes]: Is there any possibily to solve the following integral
$$\int_0^{\infty} \, \mathrm{arcsinh} \left(\frac{a}{\sqrt{x^2+y^2}} \right) \, \mathrm{cos}(b \, x) \,\mathrm{d}x$$
with $a>0$, $y>0$ and $-\pi/2<\mathrm{arg}(b)<0$.
I assume, the result is connected to Bessel and Struve functions. Thank you.
Edit:
Using integration by parts with
$$\int\mathrm{cos}(b \, x)\,\mathrm{d}x=\frac{\mathrm{sin}(b \, x)}{b}$$
$$\frac{\mathrm{d}}{\mathrm{d}x}( \mathrm{arcsinh} \left(\frac{a}{\sqrt{x^2+y^2}} \right))= - \frac{a \,x}{(x^2+y^2) \, \sqrt{x^2+y^2+a^2}}$$
and the limits
$$ \lim_{x\to0}  \frac{\mathrm{sin}(b \, x)}{b} \, \mathrm{arcsinh} \left(\frac{a}{\sqrt{x^2+y^2}} \right) = 0$$
$$ \lim_{x\to\infty}  \frac{\mathrm{sin}(b \, x)}{b} \, \mathrm{arcsinh} \left(\frac{a}{\sqrt{x^2+y^2}} \right) = 0 $$
Gives the integral
$$\int_0^{\infty} \, \frac{a \,x}{(x^2+y^2) \, \sqrt{x^2+y^2+a^2}} \, \frac{\mathrm{sin}(b \, x)}{b} \,\mathrm{d}x$$
if that makes anything simpler...
Edit2:
Mathematica tells me that the last integrand can be presented as the product of three G-functions. Inhere it is said that the integral of the product of three G-functions can be computed under certain restrictions. Sadly it is not mentioned which restrictions. Does anybody know anything about this?
It would be:
$$\frac{1}{x^2+y^2+a^2} = \frac{1}{\sqrt{\pi} \, \sqrt{y^2+a^2}} \,\mathrm{MeijerG}\left[\left\{\{\tfrac{1}{2} \},\{ \} \right\},\left\{\{0 \},\{ \} \right\},\tfrac{x^2}{y^2+a^2}\right]$$
$$\frac{1}{x^2+y^2} = \frac{1}{y^2} \,\mathrm{MeijerG}\left[\left\{\{0 \},\{ \} \right\},\left\{\{0 \},\{ \} \right\},\tfrac{x^2}{y^2}\right]$$
$$\mathrm{sin}(b\,x)= \sqrt{\pi} \, \mathrm{MeijerG}\left[\left\{\{ \},\{ \} \right\},\left\{\{\tfrac{1}{2} \},\{ 0\} \right\},\tfrac{x^2 \, b^2}{4}\right]$$
which finally results in
$\frac{a}{y^2 \, \sqrt{y^2+a^2}} \, \int_0^{\infty} \, x \, \mathrm{MeijerG}\left[\left\{\{\tfrac{1}{2} \},\{ \} \right\},\left\{\{0 \},\{ \} \right\},\tfrac{x^2}{y^2+a^2}\right] \, \mathrm{MeijerG}\left[\left\{\{0 \},\{ \} \right\},\left\{\{0 \},\{ \} \right\},\tfrac{x^2}{y^2}\right] \, \mathrm{MeijerG}\left[\left\{\{ \},\{ \} \right\},\left\{\{\tfrac{1}{2} \},\{0 \} \right\},\tfrac{x^2 \, b^2}{4}\right] \, \mathrm{d}x$
The $\mathrm{MeijerG}$ are defined according to Mathematica syntax. Or (I hope I converted this correctly)
$$\frac{a}{y^2 \, \sqrt{y^2+a^2}} \, \int_0^{\infty} \, x \, G^{1,1}_{1,1}\left(\begin{array}{c|c}\begin{matrix}\frac{1}{2}\\ 0 \end{matrix}&\frac{x^2}{y^2+a^2}\end{array}\right) \, G^{1,1}_{1,1}\left(\begin{array}{c|c}\begin{matrix}0\\ 0 \end{matrix}&\frac{x^2}{y^2}\end{array}\right)\, G^{1,0}_{0,2}\left(\begin{array}{c|c}\begin{matrix}-\\\frac{1}{2},\, 0\end{matrix}&\frac{x^2 \,b^2}{4}\end{array}\right) \, \mathrm{d}x$$

REPLY [1 votes]: First, we simplify the parameters:
$$\int_0^{\infty} \, \frac{a \,x}{(x^2+y^2) \, \sqrt{x^2+y^2+a^2}} \, \frac{\mathrm{sin}(b \, x)}{b} \,\mathrm{d}x= \frac{a}{by} \int_0^\infty \frac{t \sin rt ~dt}{(1+t^2) \sqrt{c^2+t^2}}$$
$$c^2=1+\frac{a^2}{y^2}, \qquad r=b y$$
I assume $b$ is real for this calculation (which may not be the case for the OP).
So we want to find:
$$f(r,c)=\int_0^\infty \frac{t \sin rt ~dt}{(1+t^2) \sqrt{c^2+t^2}}$$
For $c=1$ (which means $a=0$), we have a closed form as a modified Bessel function:
$$f(r,1)=r K_0(r)$$
This gives the following idea. Write:
$$\sqrt{c^2+t^2}=\sqrt{1+\frac{a^2}{y^2}+t^2}=\sqrt{1+t^2} \sqrt{1+\frac{a^2}{y^2(1+t^2)}}$$
And expand the latter as a binomial series. Then we have:
$$f=\sum_{k=0}^\infty \frac{(-1)^k (2k)!}{k!^2} \left( \frac{a}{2y} \right)^{2k} \int_0^\infty \frac{t \sin rt ~dt}{(1+t^2)^{3/2+k} }$$
Amazingly enough all the integral have a closed form:
$$\int_0^\infty \frac{t \sin rt ~dt}{(1+t^2)^{3/2+k} }=\frac{r^{k+1}}{(2k+1)!!} K_k (r)$$
Here $!!$ means double factorial: $(2k+1)!!=1 \cdot 3 \cdot 5 \cdots (2k+1)$.
Finally we have for the original integral:

$$\int_0^{\infty} \, \frac{a \,x}{(x^2+y^2) \, \sqrt{x^2+y^2+a^2}} \, \frac{\mathrm{sin}(b \, x)}{b} \,\mathrm{d}x= a \sum_{k=0}^\infty \frac{(-1)^k (2k)!}{k!^2 (2k+1)!!} \left( \frac{a  }{2}\sqrt{\frac{b }{y}} \right)^{2k} K_k (b y)$$

We could also attempt residues on the integral:
$$\int_{-\infty}^\infty \frac{t \sin rt ~dt}{(1+t^2) \sqrt{c^2+t^2}}$$
but it has a square root and so will involve the branch cuts, I'm not sure how to deal with it.


TITLE: How to sketch $(y^2-2)^2+(x^2-2)^2=2$?
QUESTION [0 upvotes]: I get the overall idea, but why is the shape of these figures not a circle?

REPLY [3 votes]: Consider new frame with coordinate axes defined as $X=x^2$ and $Y=y^2$.
In this new frame of reference, the equation you mentioned will be a perfect circle with center at (2,2) and radius of $\sqrt 2$.
But every point on this new frame of reference will have, 4 copies in the original $x,y$ frame. Why?
A point $(X_0,Y_0)$ in new frame will have $4$ copies which will be $(\sqrt {X_0},\sqrt {Y_0}),(\sqrt {X_0},-\sqrt {Y_0}),(-\sqrt {X_0},\sqrt {Y_0}),(-\sqrt {X_0},-\sqrt {Y_0}) $.
Since the circle in new reference frame lies completely in 1st quadrant, therefore the circle will be replicated into $4$ figures in $4$ different quadrants in original frame of reference.
This can also be verified by viewing the graph here.


TITLE: Is there a difference between compact and closed?
QUESTION [7 upvotes]: Off course there is a difference between 'compact' and 'closed', but
here I am asking for something deeper. Whether a set in a topological
space deserves label 'closed' is somehow depending not only on the
set itself but also on its surroundings. Set $F$ can be closed when
you look at it as a subset of $A$ while it can loose this label again
when it is looked at as a subset of $B$. This can be disturbing.
With label 'compact' that problem does not exists. "I
am compact" declares the set firmly,.."and
nobody can take that from me!" I like that attitude. When
I deal with 'closed' then I tend to use 'set' and when I deal with
'compact' then 'space'. The concepts 'compact' and 'closed' are only
used here to make myself clear. It is not more than an example. For the rest I rely on your intuition.
Is there some mathematical/topological theory in which this distinction is caught?

REPLY [1 votes]: In some sense, a compact space is universally closed:

A Tychonoff space $X$ is compact iff for every Hausdorff space $Y$ and every continuous map $f : X \to Y$ then $f(X)$ is closed in $Y$.

The implication is clear since a continuous map sends compact sets on compact sets. Conversely, taking $Y=\beta X$ we deduce that $X$ is closed in its Stone-Čech compactification so $X= \beta X$ is compact.


TITLE: Need help with this question.
QUESTION [1 upvotes]: I was trying to solve this question but got stuck. If we further solve it we get $I'(e)=0 $ which does no help to find the value of the integral. I know an alternate way to take $7$ as variable but I want to know if the process I used can help in anyway to evaluate this integral. How will we find $I(e)$ if $I'(e)$ comes 0 here? To be more clear I want to know how to evaluate this question using Leibnitz rule and differentiating with respect to $e$. I finally got $ I(e)=0+c$
Further I tried to find $ I(1) $ and I found it's giving 0 that would imply my $ c=0 $ that means answer should be $0$ but it's $ ln7 $. Please correct me where am I wrong.

REPLY [1 votes]: Using Double Integration
Let $\displaystyle I = \int^{\infty}_{0}\frac{e^{-x}-e^{-7x}}{x}dx = \int^{\infty}_{0}\bigg(\int^{7}_{1}e^{-tx}dt\bigg)dx$
So $\displaystyle I = \int^{7}_{1}\bigg(\int^{\infty}_{0}e^{-xt}dx\bigg)dt = \int^{7}_{1}\frac{1}{t}dt=\ln(t)\bigg|^{7}_{1} = \ln(7).$
Using Differentiation under integral sign
Let $\displaystyle I(\alpha) = \int^{\infty}_{0}\frac{e^{-\alpha x}-e^{-x}}{x}dx,$ Then $\displaystyle I'(\alpha) = -\int^{\infty}_{0}\frac{x\cdot e^{-\alpha x}}{x}dx = -\frac{e^{-\alpha x}}{\alpha}\bigg|^{\infty}_{0}=\frac{1}{\alpha}$
So $\displaystyle I (\alpha) = \ln(\alpha)+\mathcal{C}.$ Put $\alpha =1,$ We get $I(1)=0$
So we gave $\displaystyle I (\alpha) = \int^{\infty}_{0}\frac{e^{-\alpha x}-e^{-x}}{x}dx= \ln(\alpha).$
So $\displaystyle I(7)=\int^{\infty}_{0}\frac{e^{-7x}-e^{-x}}{x}dx = \ln(7).$


TITLE: annuity interest payments
QUESTION [1 upvotes]: Someone takes a loan of $S_1$ as an annuity at an annual interest rate $i$, paid back in monthly installments of $M$. Now I wonder how I can get the annual interest payments for each year $t$.
I can calculate the remaining debt after each year using
$$ S_{t}=\frac{M}{\frac{i}{12}} - \bigg(1+\frac{i}{12}\bigg)^{12}\times \bigg(\frac{M}{\frac{i}{12}}-S_{t-1}\bigg) $$
(e.g., wikipedia, adapted to have monthly payments). Using this, I can then calculate the interest numerically as $I_t = 12 \times M - (S_t - S_{t-1})$ but this approach fails in the last period when $M$ is paid only for a few months until $S_1$ is fully repaid. Therefore, I wonder whether anyone is aware of an equation to get annual interest payments analytically.

REPLY [0 votes]: The loan $S=Ma_{\overline{n}|r}$ where $r=i/12$ in $n$ total months.
If $S$ and $M$ are known we can easily find tne number of months to repay the loan
$$
S=Ma_{\overline{n}|r}=M\frac{1-(1+r)^{-n}}{r}\quad\Longrightarrow\; n=\frac{\log\left(\frac{M}{M-rS}\right)}{\log(1+r)}
$$
and the number of years is $m=n/12$.
At any time $k$ (month) the interest payment is
$$
I_k=rMa_{\overline{n-k+1}|r}=M\left(1-v^{n-k+1} \right)
$$
with $v=\frac{1}{1+r}$ and the principal payment is
$$
P_k=M v^{n-k+1}
$$
so that $M=P_k+I_k$ and the oustanding balance is
$$
B_k=Ma_{\overline{n-k}|r}
$$
If the outstanding balance at month $k-1$ is known, i.e. what is left to be paid back, and the payment $M$, we can find the interest payed at month $k$ as
$$
I_k=rB_{k-1}
$$
and $P_k=M-I_k$.
The monthly repayment of the loan is equivalent to the yearly repayment of the loan with effective annual interest $$j=\left(1+\frac{i}{12}\right)^{12}-1=(1+r)^{12}-1$$
for $m=n/12$ years:
$$
S=Ma_{\overline{n}|r}=M\frac{1-\frac{1}{(1+r)^n}}{r}=M\frac{1-\frac{1}{(1+j)^m}}{r}=M\cdot\frac{j}{r}\cdot \frac{1-\frac{1}{(1+j)^m}}{j}=A a_{\overline{m}|j}
$$
observing that $(1+r)^n=(1+r)^{12\times\frac{n}{12}}=(1+j)^m$ and the yearly installment is $A=M\times\frac{j}{r}$. Observe that $A$ is the future vale at the end of the year of 12 installment of value $M$ at monthly interest $r$:
$$
A=M\times\frac{j}{r}=M\times\frac{(1+r)^{12}-1}{r}=M\cdot s_{\overline{12}|r}
$$
and expressing $r=i/12$ we have that
$$
A=M\times\frac{j}{r}=M\times\frac{j}{i/12}=12M\times\frac{j}{i}
$$
that is $A>12M$: the installment that we have to pay annually is greater than the sum of the monthly installments in one year, because the annual effective interest $j$ is greater than the monthly interest $i$:
$$
\frac{j}{i}=\frac{(1+i/12)^{12}-1}{i}>1\quad \text{for } i>0
$$
At any time $t$ (year) the interest payment is
$$
I_t=jAa_{\overline{m-t+1}|j}=A\left(1-\nu^{m-t+1} \right)
$$
with $\nu=\frac{1}{1+j}$ and the principal payment is
$$
P_t=A \nu^{m-t+1}
$$
so that $A =P_t+I_t$ and the oustanding balance is
$$
B_t=Aa_{\overline{m-t}|j}
$$
If the outstanding balance at year $t-1$ is known, i.e. what is left to be paid back, and the payment $A=Mj/r$ we can find the interest payed at year $t$ as
$$
I_t=jB_{t-1}
$$
and $P_t=A-I_t$.


TITLE: Proving the Cauchy-Schwarz inequality, spivak's calculus
QUESTION [3 upvotes]: I'm working on a problem concerning the Cauchy-Schwarz inequality from Spivak's calculus 3rd edition. The problem consists of completing three equivalent proofs of the inequality. In one of the proofs the inequality can be deduced from the fact $(x_1^2+x_2^2)(y_1^2+y_2^2)=(x_1y_1+x_2y_2)^2+(x_1y_2-x_2y_1)^2$ for real numbers $x_1,x_2,y_1$ and $y_2$. Manipulation of this equation together with facts about inequalities leads one to $|x_1y_1+x_2y_2|\leq \sqrt{x_1^2+x_2^2}\sqrt{y_1^2+y_2^2}$ . The other proof uses $2xy\leq x^2+y^2$ and the substitutions $x:=\frac{x_A}{\sqrt{x_1^2+x_2^2}}$ and $y:=\frac{y_A}{\sqrt{y_1^2+y_2^2}}$, first for $A=1$ and then for $A=2$. Here's where my difficulties arise: when doing the second proof, I can arrive at the inequality, but not for the absolute value of $x_1y_1+x_2y_2$. Can anyone work the second proof and see how the absolute value comes into play?
Thanks

REPLY [3 votes]: The same proof procedure can also be applied for $-2xy\le x^2+y^2$
You will find
$$x_1y_1+x_2y_2 \leq \sqrt{x_1^2+x_2^2}\sqrt{y_1^2+y_2^2}$$
and
$$-(x_1y_1+x_2y_2)\leq \sqrt{x_1^2+x_2^2}\sqrt{y_1^2+y_2^2}$$
so
$$|x_1y_1+x_2y_2|\leq \sqrt{x_1^2+x_2^2}\sqrt{y_1^2+y_2^2}
$$


TITLE: $\mathbb {R}^{\omega}$ in the box topology is not first countable.
QUESTION [6 upvotes]: I'm trying to show that $\mathbb {R}^{\omega}$ in the box topology is not first countable. But I cannot come up with a contradiction by assuming that for each $x \in \mathbb {R}^{\omega}$, there is a countable basis. My intuition is that I need to use the fact that a countable product of countable sets is uncountable, so I need to show that for any local basis, I can come up with a basis consisting of the product of countable open intervals in each coordinate. Can anyone help me through?

REPLY [3 votes]: If it is 1st countable then for any subset A with a limit point x then there should exist a sequence $x_n$ converging to x. Now take $A=\{(y_n)_n: y_n>0 \forall n\}$. 0 is a limit point of A but there is no sequence in A converging to 0. If possible let $a_n$ is any such sequence where $a_n=(a_{n1},a_{n2},\ldots)$ then the open set $(-a_{11},a_{11})\times (-a_{22}, a_{22})\times\cdots$ contains no points in the sequence.


TITLE: Help with a Michael Spivak Calculus problem about sequence
QUESTION [2 upvotes]: Consider a sequence of closed intervals $I_1 = [a_1, b_1], I_2 = [a_2, b_2]...$ Suppose $a_{n} \le a_{n+1}$ and $b_{n+1} \le b_n$ for all $n$. Prove that there is a point $x$, which is in every $I_n$.
I need some hints to begin. I could simply make the argument $x = a_n$ or $x = b_n$ since it is a closed-interval, but I want something more rigorous?

REPLY [1 votes]: Let's get one thing clear: setting $x=a_n$ does not yet prove that there exists some value $x$ which is in every $I_n$. For example, if $$I_n=\left[-\frac{1}{n}, \frac 1n\right],$$ then $a_n=-\frac1n$ and therefore, no matter what the value of $n$ is, the statement

$a_n$ is in $I_n$ for all values of $n$

is false, so your "proof" is not even an informal proof.
 Still, in this example, it is clear that $0$ is in all of the sets $I_n$, so this only shows your proof is wrong, the statement itself.

As for hints on how to begin, I advise you to consider the following points:

The sequence $a_1, a_2, \dots$ is an increasing sequence
The sequence $a_1, a_2, \dots$ is bounded above by $b_1$ (why? What would happen if there existed one such $a_m$ that $a_m > b_1$? Consider the relation between $b_1$ and $b_m$ in that case).
The sequence $b_1, b_2, \dots$ is a decreasing sequence
The sequence $b_1, b_2\dots$ is bounded below by $a_1$ (why)?

Now, what can you say about the two sequences? Do they converge? If yes, what can you say about their limits?


TITLE: Why some dynamic systems can undergo sudden changes?
QUESTION [2 upvotes]: Everybody has observed that the weather may change from beautiful sunshine to extremely bad weather (heavy rain, stormy winds, ...) within less than half hour. What is the fundamental reason for this?
And there are many other natural systems that change extremely rapidly. Even in rock climbing, for example, if you have placed the feet on a hold, they may slip off in a fraction of a second in case of improper foot placement. What are the reasons for ultra-rapid changes in time, i.e. that for a (field) quantity $X(t)$, the value
$$S_X(t) := \lim_{\epsilon \rightarrow 0} (X(t+\epsilon)-X(t))$$
does not tend to zero?

REPLY [3 votes]: In the context of dynamical systems, there are at least three main mechanisms that can lead to sudden changes:

chaos;
multistability;
bifurcation.

In the first case, chaos, your $S_X$ diverges trivially, from the definition itself $-$ in the chaotic regime nearby trajectories in phase space diverge exponentially. That's probably the explanation most closely related to the weather "instability". The prototypical scheme of two trajectories diverging exponentially in phase space is:

where $\lambda$ is the Lyapunov exponent.
Multistability means that more then one outcome is allowed by the dynamics, with the system's initial condition determining which one is eventually attained. The region in phase space whose points evolve to a given attractor is called its basin of attraction. In the double well potential, for instance, each of the two local minima is an attractor in the presence of dissipation, with their basins of attraction looking something like:

This situation can lead to sudden changes when the system finds itself close to the boundary between two basins of attraction. And when the basin of the desired outcome is relatively small, it's hard not to be close to the boundary. Not falling, in the rock climbing example, could be seen as the desired basin, and it's a rather small one, given the typical safety margins of a few millimeters or less.
In a bifurcation, the system experiences a qualitative change in behavior with the variation of a parameter. In some sense it's then not exactly the same system as before. For example, if in a $1$-D system, the point $0$ is an attractor for negative values of its parameter $r$, but a repellor for $r>0$ (where two other attractors arise), the system undergoes a bifurcation at $r=0$ and its bifurcation diagram (attractors and repellors as a function of a parameter) can look like:

Thus, in a system close to a bifurcation, a small change in the parameter can lead to a sudden global change. In the weather, or rather climate example, that could correspond to, say, an ice age being precipitated by a small change in the concentration of a gas in the atmosphere or a small change in Earth's orbit.


TITLE: List the elements of $\mathbb Z_2 \times \mathbb Z_3$ and write its operation table (the notation is additive).
QUESTION [2 upvotes]: $\mathbb Z_2 \times \mathbb Z_3  = \{(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2)\}.$
$$
\begin{array}{c|lcr}
+ & (0, 0) & (0, 1) & (0, 2) & (1, 0) & (1, 1) & (1, 2)\\
\hline
(0, 0) & (0, 0) & (0, 1) & (0, 2) & (1, 0) & (1, 1) & (1, 2) \\
(0, 1) & (0, 1) & (0, 2) & (0, 3) & (1, 1) & (1, 2) & (1, 3) \\
(0, 2)  & (0, 2) & (0, 3) & (0, 4) & (1, 2) & (1, 3) & (1, 4) \\
(1, 0) & (1, 0) & (1, 1) & (1, 2) & (2, 0) & (2, 1) & (2, 3) \\
(1, 1)  & (1, 1) & (1, 2) & (1, 3) & (2, 1) & (2, 2) & (2, 3) \\
(1, 2)  & (1, 2) & (1, 3) & (1, 4) & (2, 2) & (2, 3) & (2, 4) \\
\end{array}
$$
Please, check my work.

REPLY [0 votes]: As Omne Bonum mentions, adding two members of a group must yield a member of the group. Using the notation given in the question, in $\mathbb Z_2 \times \mathbb Z_3$,
$$(a, b) + (c, d) = ((a+c)\mod 2, \ (b+d)\mod 3)$$
Here is the corrected addition table.
$$\begin{array}{c|lcr}
+ & (0, 0) & (0, 1) & (0, 2) & (1, 0) & (1, 1) & (1, 2)\\
\hline
(0, 0) & (0, 0) & (0, 1) & (0, 2) & (1, 0) & (1, 1) & (1, 2) \\
(0, 1) & (0, 1) & (0, 2) & (0, 0) & (1, 1) & (1, 2) & (1, 0) \\
(0, 2) & (0, 2) & (0, 0) & (0, 1) & (1, 2) & (1, 0) & (1, 1) \\
(1, 0) & (1, 0) & (1, 1) & (1, 2) & (0, 0) & (0, 1) & (0, 2) \\
(1, 1) & (1, 1) & (1, 2) & (1, 0) & (0, 1) & (0, 2) & (0, 0) \\
(1, 2) & (1, 2) & (1, 0) & (1, 1) & (0, 2) & (0, 0) & (0, 1) \\
\end{array}$$


TITLE: explaining complex geometrically
QUESTION [0 upvotes]: Questions from past test that I need help on.
1) If $f(z) = z+1$ and $g(z) = \frac{−1}{z}$, show that $gfg^{−1}(z) = \frac{z}{1−z}$
For this function I was having trouble finding $g^{-1}$ so Ive kind of cheated a bit so first I computed
$fg$ to get $\frac{-(z+1)}{z} g^{-1} = \frac{z}{1−z} \rightarrow g^{-1} = \frac{z^{2}}{-(z+1)(1-z)}$
and then through some computing I got
$g^{-1} = \frac{z^2}{1-z^2}$
I wanted to know is there any other "right" way to approach this problem?
2) Describe in words what $g^{-1}, f, g$, in succession do to the half plane, and hence explain geometrically why $gfg^{-1}$ has fixed point $0$.
For this one, I was thinking that $g$ and $g^{-1}$ are reflections about the $y$ axis and the unit circle, and $f$ reflects the plane over the $y$ axis. I'm just not sure if that's correct, and if so why would the origin remain fixed even though $f$ translated it?
Thanks in advance!
I really need help in these questions please and thank you

REPLY [1 votes]: Hint:
$$(1)\;\;\;\;g^2(z)=g\left(\frac{-1}{z}\right)=\frac{-1}{\frac{-1}{z}}=z\Longrightarrow g^{-1}=g\ldots$$
$$(2)\;\;\;\;\overline{\frac{-1}{z}}=\frac{-1}{\overline z}=-\frac{z}{|z|^2}$$
so $\,g\,,\,g^{-1}\,$ "normalize" $\,z\,$ (i.e., maps it to the unit circle) but change its sign...
$\,f\,$ is simply a translation...

REPLY [1 votes]: Hint: $gfg^{-1}(z)=g(f(g^{-1}(z)))$. To get $g^{-1}(z)$, we let $w=-\frac{1}{z}$, and resolve $z$, which gives $z=g^{-1}(w)$.