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%\makeatletter
%%\def\multiciterangedelim{--}
%\def\TITREspecial{\relax}
%\def\cdr@specialtitle@english{Complex algebraic geometry, in memory of Jean-Pierre Demailly}
%\def\cdr@specialtitle@french{\hbox{Géométrie\,algébrique\,complexe,\,en\,mémoire\,de\,Jean-Pierre\,Demailly}}
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\title[Good Minimal Models for K\"ahler Varieties]{Existence of Good Minimal Models for K\"ahler varieties of Maximal Albanese Dimension}
\alttitle{Existence d'un bon modèle minimal pour les variétés kählériennes de dimension d'Albanese maximale}

\author{\firstname{Omprokash} \lastname{Das}}
\address{School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Navy Nagar, Colaba, Mumbai 400005}
\email{omdas@math.tifr.res.in}
\email{omprokash@gmail.com}

\author{\firstname{Christopher} \lastname{Hacon}\IsCorresp}
\address{Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, Utah 84112, USA}
\email{hacon@math.utah.edu}

\thanks{Omprokash Das is supported by the Start--Up Research Grant (SRG), Grant No. \# SRG/2020/000348 of the Science and Engineering Research Board (SERB), Govt. Of India. Christopher Hacon was partially supported by the NSF research grants no: DMS-1952522, DMS-1801851, DMS-2301374 and by a grant from the Simons Foundation; Award Number: 256202.}

\CDRGrant[Start--Up Research Grant]{\# SRG/2020/000348}
\CDRGrant[NSF]{DMS-1952522}
\CDRGrant[NSF]{DMS-1801851}
\CDRGrant[NSF]{DMS-2301374}
\CDRGrant[Simons Foundation]{256202}

\begin{abstract}
In this short article we show that if $(X, B)$ is a compact K\"ahler klt pair of maximal Albanese dimension, then it has a good minimal model, i.e. there is a bimeromorphic contraction $\phi:X\dashrightarrow X'$ such that $K_{X'}+B'$ is semi-ample.
\end{abstract}

\begin{altabstract}
Dans ce court article, nous montrons que si $(X,B)$ est une paire kählérienne compacte klt de dimension d'Albanese maximale, $(X,B)$ admet un bon modèle minimal, c'est-à-dire qu'il existe une contraction biméromorphe $\phi: X\dashrightarrow X'$ telle que $K_{X'}+B'$ est semi-ample.
\end{altabstract}

\begin{document}
\maketitle

\section{Introduction}

The main result of this paper is the following

\begin{theo}\label{thm:main}
Let $(X,B)$ be a compact K\"ahler klt pair of maximal Albanese dimension. Then $(X,B)$ has a good minimal model.
\end{theo}

This generalizes the main result of~\cite {Fuj15} from the projective case to the K\"ahler case. The main idea is to observe that replacing $X$ by an appropriate resolution, then the Albanese morphism $X\to A$ is projective and so by~\cite{DHP22} and~\cite{Fuj22} we may run the relative MMP over $A$. Thus we may assume that $K_X+B$ is nef over $A$. If $X$ is projective and $K_X+B$ is not nef, then by the cone theorem, $X$ must contain a $K_X+B$ negative rational curve $C$. Since $A$ contains no rational curves, then $C$ is vertical over $A$, contradicting the fact that $K_X+B$ is nef over $A$~\cite{Fuj15}. Unluckily the cone theorem is not known for K\"ahler varieties and so we pursue a different argument. It would be interesting to find an alternative proof based on the approach of~\cite{CH20}.


\section{Preliminaries}

An \emph{analytic variety} or simply a \emph{variety} is a reduced irreducible complex space. Let $X$ be a compact K\"ahler manifold and $\Alb(X)$ is the \emph{Albanese torus} (not necessarily an Abelian variety). Then by $a:X\to \Alb(X)$ we will denote the \emph{Albanese morphism}. This morphism can also be characterized via the following universal property: $a:X\to \Alb(X)$ is the Albanese morphism if for every morphism $b:X\to T$ to a complex torus $T$ there is a unique morphism $\phi:\Alb(X)\to T$ such that $b=\phi\circ a$.

The Albanese dimension of $X$ is defined as $\dim a(X)$. We say that $X$ has maximal Albanese dimension if $\dim a(X)=\dim X$ or equivalently, the Albanese morphism $a:X\to \Alb(X)$ is \emph{generically finite} onto its image. For the definition of \emph{singular} K\"ahler space see~\cite{DH20} or~\cite{HP16}.

A compact analytic variety $X$ is said to be in \emph{Fujiki's class} $\mcC$ if $X$ is bimeromorphic to a compact K\"ahler manifold $Y$. In particular, there is a resolution of singularities $f:Y\to X$ such that $Y$ is a compact K\"ahler manifold.

\begin{defi}\label{def:albanese-morphism}
Let $X$ be a compact analytic variety in Fujiki's class $\mcC$. Assume that $X$ has rational singularities. Choose a resolution of singularities $\mu:Y\to X$ such that $Y$ is a K\"ahler manifold and let $a_Y:Y\to \Alb(Y)$ be the Albanese morphism of $Y$. Then from the proof of~\cite[Lemma~8.1]{Kaw85} it follows that $a_Y\circ \mu^{-1}:X\dashrightarrow \Alb(Y)$ extends to a unique morphism $a:X\to \Alb(X):=\Alb(Y)$. We call this morphism the Albanese morphism of $X$. Observe that $a:X\to \Alb(X)$ satisfies the universal property stated above. The Albanese dimension of $X$ is defined as above. Note that if $X$ is a compact analytic variety with rational singularities, bimeromorphic to a complex torus $A$, then $A\cong\Alb(X)$ and $X\to A$ is a bimeromorphic morphism.
\end{defi}

The following result is well known, however, for a lack of an appropriate reference and for the convenience of the reader we give a complete proof here.

\begin{lemm}\label{lem:effective-k-dim}
Let $A$ be a complex torus and $X\subset A$ is an analytic subvariety. Then for any resolution of singularities $\mu:Y\to X$, $H^0(Y, \omega_Y)\neq \{0\}$.
\end{lemm}

\begin{proof}
Let $\mu:Y\to X$ be a resolution of singularities of $X$. If $d=\dim X$, then the map $\mu ^*\Omega ^d_A\to \Omega ^d_Y$ is generically surjective. Since $\Omega_A^d$ is a trivial vector bundle, it is globally generated and hence there is a non-zero section in the image of $\mu ^*:H^0(\Omega ^d_A)\to H^0(\Omega ^d_Y)$.
\end{proof}

\begin{coro}\label{cor:maximal-albanese-kappa}
Let $X$ be a compact analytic variety in Fujiki's class $\mcC$ with canonical singularities. If $X$ has maximal Albanese dimension, then $\kappa(X)\geq0$.
\end{coro}

\begin{proof}
First note that if $f:W\to X$ is a proper bimeromorphic morphism, then $\kappa(X)\geq0$ if and only if $\kappa(W)\geq0$, since $X$ has canonical singularities. Now let $a:X\to \Alb(X)$ be the Albanese morphism, $Y:=a(X)$, and $\pi:Z\to Y$ is a resolution of singularities of $Y$. Then $\kappa(Z)\geq0$ by Lemma~\ref{lem:effective-k-dim}. Note that there is a generically finite meromorphic map $\phi:X\bir Z$; resolving the graph of $\phi$ we may assume that $X$ is smooth and $\phi:X\to Z$ is a morphism. Then $K_X=\phi^*K_Z+E$, where $E\geq0$ is an effective divisor. Therefore $\kappa(X)\geq0$, since $\kappa(Z)\geq0$.
\end{proof}

\subsection{Fourier-Mukai transform}

Let $T$ be a complex torus of dimension $g$ and $\hat T=\Pic^0(T)$ its dual torus. Let $p_T: T\times\hat T\to T$ and $p_{\hat T}:T\times\hat T\to \hat T$ be the projections, and $\mcP$ the normalized Poincar\'e line bundle on $T\times\hat T$ so that $\mcP|_{T\times\{0\}}\cong \mcO_T$ and $\mcP|_{\{0\}\times \hat T}\cong\mcO_{\hat T}$. Let $\hat S$ be the functor from the category of $\mcO_T$-sheaves to the category of $\mcO_{\hat T}$-sheaves, defined by
\[
\hat S(\msF):=p_{\hat T, *}(p_T^*\msF\otimes\mcP),
\]
where $\msF$ is a sheaf of $\mcO_T$-modules. Similarly, $S$ is a functor from the category of $\mcO_{\hat T}$-sheaves to the category of $\mcO_T$-sheaves, defined as
\[
S(\msG):=p_{T, *}(p_{\hat T}^*\msG\otimes\mcP),
\]
where $\msG$ is a sheaf of $\mcO_{\hat T}$-modules.

The corresponding derived functors are
\[
\bfR\hat S({\,\boldsymbol{\cdot}\,}):=\bfR p_{\hat T, *}(p_T^*({\,\boldsymbol{\cdot}\,})\otimes\mcP) \text{ and } \bfR S({\,\boldsymbol{\cdot}\,}):=\bfR p_{T, *}(p_{\hat T}^*({\,\boldsymbol{\cdot}\,})\otimes\mcP).
\]
Recall the following fundamental result of Mukai~\cite[Theorem~2.2, and (3.8)]{Mu81}, \cite[Theorem~13.1]{PPS17}

\begin{theo}\label{thm:fm-isomorphism}
With notations and hypothesis as above, there are isomorphisms of functors (on the bounded derived category of coherent sheaves)
\begin{equation*}
\bfR\hat S\circ \bfR S \cong (-1)_{\hat T}^*[-g], \qquad
\bfR S\circ\bfR\hat S \cong (-1)_T^*[-g],
\end{equation*}
\begin{equation*}
\bDelta_T \circ \bfR S=((-1_T)^*\circ \bfR S \circ \bDelta _{\hat T})[-g].
\end{equation*}
\end{theo}

Recall that $\bDelta_T({\,\boldsymbol{\cdot}\,}):=\bfR{\mathcal Hom}(\,\boldsymbol{\cdot}\,,\mathcal O _T)[g]$ is the dualizing functor.

\begin{defi}\label{def:h-vector-bundle}
Let $A$ be a complex torus. For $a\in A$, let $t_a:A\to A$ be the usual translation morphism defined by $a$. A vector bundle $\mcE$ on $A$ is called \emph{homogeneous}, if $t_a^*\mcE\cong\mcE$ for all $a\in A$.
\end{defi}

\begin{rema}\label{rmk:h-vector-bundle}
Let $A$ be a complex torus, $\hat A$ the dual torus and $\dim A=\dim \hat A=g$. Then from the proof of~\cite[Example~3.2]{Mu81} it follows that $R^g\hat S$ gives an equivalence of categories
\begin{align*}
\bfH_A&:=\{\text{Homogeneous vector bundles on } A\},\\ \text{and}\quad
\bfC^f_{\hat A}&:=\{\text{Coherent sheaves on } \hat A \text{ supported at finitely many points} \}.
\end{align*}
Note that in~\cite{Mu81} the results are all stated for abelian varieties, however, we observe that in the proof of~\cite[Example~3.2]{Mu81} the main arguments follow from Theorem~\ref{thm:fm-isomorphism} and the isomorphisms in~\cite[(3.1), p.~158]{Mu81}, both of which hold for complex tori. In particular, \cite[Example~3.2]{Mu81} holds for complex tori.
\end{rema}

We will need the following result on the rational singularity of (log) canonical models of klt pairs.

\begin{prop}\label{pro:rational-singularities}
Let $(X, B)$ be a klt pair, where $X$ is a compact analytic variety in Fujiki's class $\mcC$. Assume that the Kodaira dimension $\kappa(X, K_X+B)\geq0$. Then $R(X, K_X+B):=\oplus_{m\geq0} H^0(X, m(K_X+B))$ is a finitely generated $\mbC$-algebra and
\[
\bar Z=\Proj R(X, K_X+B)
\]
has rational singularities.
\end{prop}

\begin{proof}
The finite generation of $R(X, K_X+B)$ follows from~\cite[Theorem~1.3]{DHP22} and~\cite[Theorem~5.1]{Fuj15}. Let $f:X\bir Z$ be the Iitaka fibration of $K_X+B$. Resolving $Z, f$ and $X$, we may assume that $X$ is a compact K\"ahler manifold, $B$ has SNC support, $Z$ is a smooth projective variety and $f$ is a morphism. Then from the proof of~\cite[Theorem~5.1]{Fuj15} it follows that there is a smooth projective variety $Z'$ which is birational to $Z$ and an effective $\mbQ$-divisor $B_{Z'}\geq0$ such that $(Z', B_{Z'})$ is klt, $K_{Z'}+B_{Z'}$ is big and the following holds
\[
R(X, K_X+B)^{(d)}\cong R(Z', K_{Z'}+B_{Z'})^{(d')},
\]
where the superscripts $d$ and $d'$ represent the corresponding $d$ and $d'$-Veronese subrings.

Thus $\bar Z=\Proj R(X, K_X+B)\cong \Proj R(Z', K_{Z'}+B_{Z'})$ is the log-canonical model of $(Z', B_{Z'})$. If $(Z'', B_{Z''})$ is a minimal model of $(Z', B_{Z'})$ as in~\cite[Theorem~1.2$\MK$(2)]{BCHM10}, then by the base-point free theorem, there is a birational morphism $\phi:Z''\to \bar Z$ such that $K_{Z''}+B_{Z''}=\phi^*(K_{\bar Z}+B_{\bar Z})$, where $B_{\bar Z}:=\phi_*B_{Z''}\geq0$. Thus $(\bar Z, B_{\bar Z})$ is a klt pair, and hence $\bar Z$ has rational singularities.
\end{proof}


\section{Main Theorem}

In this section we will prove our main theorem. We begin with some preparation.

\begin{defi}\label{def:plurigenera}
Let $X$ be a smooth compact analytic variety. Then the $m$-th \emph{plurigenera} of $X$ is defined as
\[
P_m(X):=\dim_{\mbC} H^0(X, \omega^m_X).
\]
\end{defi}

The next result is one of our main tools in the proof of the main theorem, it is also of independent interest. It follows immediately from the main results of~\cite{PPS17}.

\begin{theo}\label{thm:maximal-albanese-torus}
Let $X$ be a compact K\"ahler variety with terminal singularities. Assume that $X$ has maximal Albanese dimension and $\kappa (X)=0$. Then $X$ is bimeromorphic to a torus. Additionally, if $K_X$ is also nef, then $X$ is isomorphic to a torus.
\end{theo}

\begin{rema}
Note that the above result holds if we simply assume that $X$ is in Fujiki's class $\mathcal C$. Indeed, if $X'\to X$ is a resolution of singularities such that $X'$ is K\"ahler, then $\kappa (X')=0$ and so $X'\to \Alb(X')$ is bimeromorphic, and hence so is $X\to \Alb(X')$. Note also that if $X$ is a complex manifold of maximal Albanese dimension, then $X$ is automatically in Fujiki's class $\mathcal C$. To see this, consider the Stein factorization $X\to Y\to A$. Then $Y\to A$ is finite and so $Y$ is also K\"ahler (see~\cite[Proposition~1.3.1$\MK$(v) and (vi), p.~24]{Var89}). Let $X'\to X$ be a resolution of sungularities such that $X'\to Y$ is projective, then $X'$ is K\"ahler and so $X$ is in Fujiki's class $\mathcal C$.
\end{rema}

\begin{proof}[Proof of Theorem~\ref{thm:maximal-albanese-torus}]
Since $X$ is terminal, it has rational singularities, and thus by Definition~\ref{def:albanese-morphism} the Albanese morphism $a:X\to \Alb(X)$ exists. Let $\pi:\tilde{X}\to X$ be a resolution of singularities of $X$. Then $a\circ\pi:\tilde X\to \Alb(X)$ is the Albanese morphism of $\tilde X$. Moreover, since $X$ has terminal singularities, $\kappa(\tilde X)=\kappa(X)=0$. Thus replacing $X$ by $\tilde X$, we may assume that $X$ is a compact K\"ahler manifold. Let $d=\dim X$ and pick a general element $\Theta \in H^0(\Omega _A^d)$, where $A=\Alb(X)$. Then $0\ne a^*\Theta \in H^0(\Omega _X^d)$ and so $P_1(X)>0$. It follows that $P_k(X)=h^0(X, \omega^k_X)>0$ for all $k>0$. Since $\kappa (X)=0$, we have $P_1(X)=P_2(X)=1$. Thus by~\cite[Theorem~19.1]{PPS17}, $X\to A$ is surjective, and hence $\dim X=\dim A=h^{1,0}(X)$. Thus by~\cite[Theorem~B]{PPS17}, $X$ is bimeromorphic to a complex torus and so $a:X\to A$ is (surjective and) bimeromorphic.

Assume now that $X$ has terminal singularities and $K_X$ is nef. Let $a:X\to A$ be the Albanese morphism. By what we have seen above, this morphism is bimeromorphic. Thus $K_X\num a^*K_A+E\equiv E$, where $E\geq0$ is an effective Cartier divisor such that $\Supp(E)=\Ex(a)$ (since $A$ is smooth). By the negativity lemma (see~\cite[Lemma~1.3]{Wang21}) we have $E=0$, and hence $a$ is an isomorphism.
\end{proof}

\begin{coro}\label{cor:maximal-albanese-torus}
Let $(X, B)$ be a compact K\"ahler klt pair. Assume that $X$ has maximal Albanese dimension and $\kappa(X, K_X+B)=0$. Then $X$ is bimeromorphic to a torus. Additionally, if $K_X+B\sim_{\mbQ} 0$, then $X$ is isomorphic to a torus.
\end{coro}

\begin{proof}
Passing to a terminalization by running an appropriate MMP over $X$ (using~\cite[Theorem~1.4]{DHP22}) we may assume that $(X, B)$ has $\mbQ$-factorial terminal singularities. Now since $\kappa(X)\geq0$ by Corollary~\ref{cor:maximal-albanese-kappa}, $\kappa(X, K_X+B)=0$ implies that $\kappa(X, K_X)=0$. Thus by Theorem~\ref{thm:maximal-albanese-torus}, $a:X\to A:=\Alb(X)$ is a surjective bimeromorphic morphism. Now assume that $K_X+B\sim_{\mbQ} 0$. Then $K_X+B=a^*K_A+E+B\sim_{\mbQ} B+E$, where $E\geq0$ is an effective Cartier divisor such that $\Supp (E)=\Ex(a)$, since $A$ is smooth. Thus $(B+E)\sim_{\mbQ} 0$, as $K_X+B\sim_{\mbQ} 0$, and hence $B=E=0$ (as $X$ is K\"ahler). In particular, $a:X\to A$ is an isomorphism.
\end{proof}

Now we are ready to prove our main theorem.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Let $a:X\to A$ be the Albanese morphism. Since $X$ has maximal Albanese dimension, $a$ is generically finite over its image $a(X)$. By the relative Chow lemma (see~\cite[Corollary~2]{Hir75} and~\cite[Theorem~2.16]{DH20}) there is a log resolution $\mu :X'\to X$ of $(X, B)$ such that the Albanese morphism $a'=a\circ \mu :X'\to A$ is projective. Let $K_{X'}+B'=\mu^*(K_X+B)+F$, where $ F\geq 0$ such that $\Supp (F)=\Ex(\mu)$, and $(X', B')$ has klt singularities. Note that if $(X',B')$ has a good minimal model $\psi:X '\dasharrow X^m$, then $\psi$ contracts every component of $F$ and the induced bimeromorphic map $X\dasharrow X^m$ is a good minimal model of $(X,B)$ (see~\cite[Lemmas~2.5 and 2.4]{HX13} and their proofs). Thus, we may replace $(X,B)$ by $(X',B')$ and assume that $(X, B)$ is a log smooth pair and $X\to A$ is a projective morphism. From Corollary~\ref{cor:maximal-albanese-kappa} it follows that $\kappa(X)\geq0$. In particular, $\kappa(X, K_X+B)\geq0$. Now we split the proof into two parts. In Step 1 we deal with the $\kappa(X, K_X+B)=0$ case, and the remaining cases are dealt with in Step 2.

\begin{proof}[Step 1]
Suppose that $\kappa (X, K_X+B)=0$. Then by Theorem~\ref{thm:maximal-albanese-torus}, the Albanese morphism $a:X\to A:=\Alb(X)$ is bimeromorphic. Let $D$ be an irreducible component of the unique effective divisor $G\in | m(K_X+B)|$ for $m>0$ sufficiently divisible. We make the following claim.

\begin{enonce}{Claim}\label{clm:a-exceptional}
$D$ is $a$-exceptional; in particular, $G$ is $a$-exceptional.
\end{enonce}

\begin{proof}
First passing to a higher model of $X$ we may assume that $D$ has SNC support. Consider the short exact sequence
\[
0\to \omega _X\to \omega _X(D)\to \omega _D\to 0.
\]
Let $V^0(\omega _D):=\{ P\in \Pic^0(A)\;|\; h^0(D, \omega _D\otimes a^* P)\ne 0\}$. If $\dim V^0(\omega _D)> 0$, then it contains a subvariety $K+P$, where $P$ is torsion in $\Pic^0(A)$ and $K$ is a subtorus of $\Pic^0(A)$ with $\dim K>0$ (see~\cite[Corollary~17.1]{PPS17}). Since $a:X\to A$ is surjective and bimeromorphic, we have $H^i(X,a^*Q)=H^i(A,Q)=0$ for any $\OO_A\ne Q\in \Pic^0(A)$; in particular, $H^1(X, \omega_X\otimes a^*Q)=H^{n-1}(X, a^*Q^{-1})^\vee=0$, where $n=\dim X$. Thus $H^0(X, \omega _X(D)\otimes a^*Q)\to H^0(D, \omega _D\otimes a^*Q)$ is surjective for all $\OO_A\ne Q\in \Pic^0(A)$, and so $h^0(X, \omega _X(D)\otimes a^*Q)>0$ for all $\OO_A\ne Q\in P+K$. Since $P$ is torsion, $\ell P=0$ for some $\ell>0$. Consider the morphism
\begin{equation}\label{eqn:non-empty-system}
|K_X+D+P+Q_1|\times \cdots \times |K_X+D+P+Q_\ell|\to |\ell(K_X+D)|,
\end{equation}
where $Q_i\in K$ such that $\sum _{i=1}^\ell Q_i=0$.

Since $\dim K>0$, for $\ell\geq 2$, the $Q_1,\ldots, Q_\ell$ vary in the subvariety $\mathcal K\subset K^{\times \ell}$ defined by the equation $\sum _{i=1}^\ell Q_i=0$. Thus $\dim \mathcal K \geq \ell\cdot (\dim K)-1\geq \ell-1\geq 1$. Therefore $\dim |\ell(K_X+D)|>0$, i.e. $ h^0(X, \ell(K_X+D))>1$. Since $D$ is contained in the support of $G$, we have $(r-\ell)G\geq \ell D$ for some $r>0$. Then $h^0(X, rm(K_X+B))\geq h^0(X, \ell (K_X+D))>1$, which is a contradiction. Therefore, $\dim V^0(\omega _D)\leq 0$. By~\cite[Theorem~A]{PPS17}, $a_*\omega _D$ is a GV sheaf so that $\bfR\hat S \bDelta _A(a_*\omega _D)=\bfR^0\hat S\bDelta _A(a_*\omega _D)$. If $\dim V^0(\omega _D)= 0$, then $\bfR^0\hat S (\bDelta _A(a_* \omega _D))$ is an Artinian sheaf of modules on $A$, and hence by Theorem~\ref{thm:fm-isomorphism} and Remark~\ref{rmk:h-vector-bundle}
\[
\bDelta _A(a_*\omega _D)=(-1_A)^*\bfR S(\bfR\hat S \bDelta _A(a_*\omega _D))[g]=(-1_A)^*\bfR S(\bfR^0\hat S \bDelta _A (a_*\omega _D))[g]
\]
is a shift of a homogeneous vector bundle which we denote by $\mcE$ (see Remark~\ref{rmk:h-vector-bundle}). But then
\[
a_*\omega _D=\bDelta _A(\bDelta _A(a_*\omega _D))=\mcE^\vee
\]
is also a homogeneous vector bundle and hence its support is either empty or entire $A$. The latter is clearly impossible, since $\Supp(a_*\omega _D)\ne A$, and hence $V^0(\omega _D)=\emptyset$. Thus by~\cite[Proposition~13.6$\MK$(b)]{PPS17}, $a_*\omega _D=0$; in particular $D$ is $a$-exceptional.
\end{proof}

Now by~\cite[Theorem~1.4]{DHP22} and~\cite[Theorem~1.1]{Fuj22} we can run the relative minimal model program over $A$ and hence may assume that $K_X+B$ is nef over $A$. From our claim above we know that $K_X+B\sim _{\mathbb Q}E\geq 0$ for some effective $a$-exceptional divisor $E\geq0$. Then by the negativity lemma we have $E=0$; thus $\mcO_X(m(K_X+B))\cong\mcO_X$ for sufficiently divisible $m>0$, and hence we have a good minimal model.
\let\qed\relax
\end{proof}

\begin{proof}[Step 2]
Suppose now that $\kappa (X, K_X+B)\geq 1$ and let $f:X\bir Z$ be the Iitaka fibration. Note that the ring $R(X, K_X+B):=\oplus_{m\geq0}H^0(X, \mcO_X(\lrd m(K_X+B)\rrd))$ is a finitely generated $\mbC$-algebra by~\cite[Theorem~1.3]{DHP22}. Define $\bar Z:=\Proj R(X, K_X+B)$. Then $Z\bir \bar Z$ is a birational map of projective varieties. Resolving the graph of $Z\bir \bar Z$ we may assume that $Z$ is a smooth projective variety and $\nu:Z\to \bar Z$ is a birational morphism. Then passing to a resolution of $X$ we may assume that $f$ is a morphism and $(X, B)$ is a log smooth pair. Write $K_F+B_F=(K_X+B)|_F$, where $F$ is a very general fiber of $f$, so that $\kappa (F, K_F+B_F)=0$. Note that $a|_F$ is also generically finite (as $F$ is a very general fiber of $f$) and thus $F$ has maximal Albanese dimension. In particular, $(F,B_F)$ has a good minimal model by {Step 1}. Let $\psi : F\bir F'$ be this minimal model; then $K_{F'}+B_{F'}\sim_{\mbQ} 0$. Thus by Corollary~\ref{cor:maximal-albanese-torus}, $F'$ is a torus and $B_{F'}=0$; in particular, $\psi:F\to F'$ is the Albanese morphism. Thus $a|_F:F\to A$ factors through $\psi:F\to F'$; let $\alpha:F'\to A$ be the induced morphism. Let $K:=\alpha(F')$; then $K$ is a torus, and $\alpha$ is \'etale over $K$, as $F'$ and $K$ are both homogeneous varieties. Now since $A$ contains at most countably many subtori and $F$ is a very general fiber, $K$ is independent of the very general points $z\in Z$, and hence so is $F'$. Define $A':=A/K$, then $A'$ is again a torus. Since the composite morphism $X\to A'$ contracts $F$ and $\dim F=\dim K$, from the rigidity lemma (see~\cite[Lemma~4.1.13]{BS95}) and dimension count it follows that there is a meromorphic map $Z\bir A'$ generically finite onto its image. Since $Z$ is smooth, we may assume that $Z\to A'$ is a morphism (see~\cite[Lemma~8.1]{Kaw85}). Similarly, since $\bar Z$ has rational singularities by Proposition~\ref{pro:rational-singularities}, again from~\cite[Lemma~8.1]{Kaw85} it follows that $\bar Z\to A'$ is a morphism.

Since $\bar Z=\Proj R(X,K_X+B)$, we may choose an ample $\mbQ$-divisor $\bar H$ on $\bar Z$ such that if $H_X$ is its pull-back to $X$, then $K_X+B\sim _{\mathbb Q} H_X+E$ and $|k(K_X+B)|=|k H_X|+kE$ for any sufficiently large and divisible integer $k>0$, where $E\geq0$ is effective (it suffices to pick $k$ so that $k(K_X+B)$ and $kH_X$ are Cartier and $R(X,K_X+B)$ is generated in degree $k$).

Now let $\bar A :=\bar Z \times _{A'}A$. Observe that there is a unique morphism $\bar a:X\to \bar A$ determined by the universal property of fiber products. We claim that $E$ is exceptional over $\bar A$. If not, then let $D$ be a component of $E$ which is not exceptional over $\bar A$. Let $h:X\to \bar Z$ be the composite morphism $X\to Z\to \bar Z$ and $W:=h(D)$. Choose a sufficiently divisible and large positive integer $s>0$ such that $s\bar H$ is very ample, $r(K_X+B)$ is Cartier, $rE\geq D$ and $|r(K_X+B)|=|rH_X|+rE$, where $r=(n+1)s$ and $n=\dim X$.
\begin{equation}
\begin{aligned}
\xymatrixcolsep{2pc}\xymatrixrowsep{2pc}\xymatrix{ X\ar@/_1pc/[dd]^f\ar[dr]_{\bar a}\ar@/^2pc/[rrd]^a \\
& \bar A:=\bar Z\times_{A'} A\ar[d]\ar[r] & A\ar[d]\\
Z\ar[r] & \bar Z\ar[r] & A':=A/K }
\end{aligned}
\end{equation}

\begin{enonce}{Claim}\label{clm:non-vanishing}
$|K_D+(n+1)sH_D|\ne \emptyset$, where $H_D=H_X|_{D}$.
\end{enonce}

\begin{proof}
Let $D_i=G_1\cap \ldots \cap G_i$ be the intersection of general divisors $G_1,\ldots, G_m\in |sH_D|$, where $0\leq i\leq m:=\dim W$ and $D_0:=D$. Let $M:=K_D+(n+1)sH_D$, then we have the short exact sequences
\[
0\to \mathcal O _{D_i}(M-G_{i+1}) \to \mathcal O _{D_i}(M)\to \mathcal O _{D_{i+1}}(M)\to 0.
\]
Recall that $h:X\to \bar Z$ is the given morphism; let $h_i:=h|_{D_i}$. Then
\begin{align*}
(M-G_{i+1})|_{D_i} &\sim (K_D+nsH_D)|_{D_i}\\
&\sim \Bigl(K_D+\sum_{j=1}^i G_j+(n-i)sH_D\Bigr)\Big|_{D_i}\\
&\sim K_{D_i}+(n-i)sH_{D_i}\\
&\sim K_{D_i}+h_i^* (n-i)s\bar H,
\end{align*}
where $H_{D_i}:=H_X|_{D_i}$. By~\cite[Theorem~3.1$\MK$(i)]{Fuj22b} the only associated subvarieties of
\[
R^1h_{i,*}\mcO_{D_i}(M-G_{i+1})=R^1h_{i,*} \mcO_{D_i}(K_{D_i})\otimes \mcO _{\bar Z}((n-i)s\bar H)
\]
are $W_i:=h(D_i)\subset \bar Z$, i.e. $R^1h_{i,*}\mcO_{D_i}(M-G_{i+1})$ is a torsion free sheaf on $W_i$. Therefore, the induced homomorphism $h_{i,*}\mcO_{D_{i+1}}(M)\to R^1h_{i,*}\mcO_{D_i}(M-G_{i+1})$ is zero and we have the following exact sequence
\[
 0 \to h_{i,*}\mcO_{D_i}(M-G_{i+1})\to h_{i,*}\mcO_{D_i}(M)\to h_{i,*}\mcO_{D_{i+1}}(M)\to 0.
\]
By~\cite[Theorem~3.1$\MK$(ii)]{Fuj22b} we have
\[
H^1(\bar Z, h_{i,*}\mathcal O _{D_i}(M-G_{i+1}))=H^1(\bar Z, h_{i,*} \mcO_{D_i}(K_{D_i})\otimes \mcO _{\bar Z}((n-i)s\bar H))=0,
\]
and thus we have the following surjections
\begin{equation}\label{eqn:surjections}
H^0(D,\OO _{D}(M))\to H^0(D_1,\OO _{D_1}(M_{D_1}))\to \cdots \to H^0(D_m,\OO _{D_m}(M_{D_m}))\to H^0(G, \OO _G(M|_{G})),
\end{equation}
where $G$ is a connected (and hence irreducible, as $D_m$ is smooth) component of $D_m$. Note that $G$ is a general fiber of $D\to W$, since $H_{D}$ is a pullback from $W$ and $m=\dim W$.

Let $w:=h(G)\in W\subset \bar Z$. Then $G\to \bar G:=\bar a(G)$ is generically finite (as so is $D\to \bar a (D)$ by our assumption), and $\bar G \to a(G)$ is an isomorphism, since $\bar A _w\to K\subset A$ is an isomorphism, as $\bar A _w=(A\times _{A'}\bar Z)_w=A\times _{A'}\{w\}\cong K$. In particular, $G$ has maximal Albanese dimension, and hence $h^0(G, K_G)>0$ by Lemma~\ref{lem:effective-k-dim}. Now since $M|_{G}\sim K_G$, from the surjections in~\eqref{eqn:surjections} it follows that $|M|=|K_D+(n+1)sH_D|\ne \emptyset$, and hence the claim follows.
\end{proof}

Now consider the short exact sequence
\[
0\to \omega _X(L)\to \omega _X(L+D)\to \omega _D(L)\to 0,
\]
where $L= rH_X$. Then by~\cite[Theorem~3.1$\MK$(i)]{Fuj22b}, $R^1h_* \omega _X(L)=R^1h_* \omega _X\otimes \OO _{\bar Z}(r\bar H)$ is torsion free, and hence $h_* \omega _X(L+D)\to h_* \omega _D(L)$ is surjective. Again by~\cite[Theorem~3.1$\MK$(ii)]{Fuj22b}, $H^1(\bar Z,h_* \omega _X(L))=H^1(\bar Z,h_* \omega _X\otimes \OO _{\bar Z}(r\bar H))=0$, and so $H^0(X,\omega _X(L+D))\to H^0(D,\omega _D(L))$ is surjective. Since $|K_D+L|_D|\ne 0$ by Claim~\ref{clm:non-vanishing}, $D$ is not contained in the base locus of $|K_X+L+D|$. Let $0\leq b:=\mult_D(B)<1$ and $e:=\mult_D(E)>0$. Then $\sigma E+B-D\geq 0$ and $\mult_D(\sigma E+B-D)=0$ for $\sigma =\frac {1-b} e>0$. We may assume that $\sigma \leq r$ (as $r$ is sufficiently large and divisible). Adding $rE+B-D$ to a general divisor $G\in|K_X+L+D|$ we get
\[
\Gamma:=rE+B-D+G\sim_{\mbQ} (r+1)(K_X+B)\sim_\Q (r+1)(H_X+E).
\]
Then for any sufficiently divisible $m>0$ we have
\[
{\mult}_D(m\Gamma) = m(r-\sigma){\mult}_D(E)<m(r+1){\mult}_D(E),
\]
which is a contradiction to the fact that $|k(K_X+B)|=|kH_X|+kE$ for sufficiently divisible $k=m(r+1)>0$. Thus $D$ is exceptional over $\bar A$.

Let $n=\dim X$. We will run a relative $(K_{X}+B+(2n+3)s H_{X})$-MMP over $A$. Note that since $|(2n+3)sH_X|$ is a base-point free linear system on a smooth compact variety $X$, by Sard's theorem there is an effective $\mbQ$-divisor $H'\geq0$ such that $(2n+3)sH_X\sim_{\mbQ} H'$ and $(X, B+H')$ has klt singularities. Thus $K_X+B+(2n+3)sH_X\sim_{\mbQ} K_X+B+H'$ and we can run a $(K_X+B+(2n+3)sH_X)$-MMP over $A$ by~\cite[Theorem~1.4]{DHP22}, and obtain $X\dasharrow X'$ so that $K_{X'}+B'+(2n+3)s H_{X'}\sim_{\mbQ} ((2n+3)s+1)H_{X'}+E'$ is nef over $A$. Note that if $R$ is a $(K_{X}+B+(2n+3)s H_{X})$-negative extremal ray over $A$, then it is also $(K_{X}+B)$-negative and so it is spanned by a rational curve $C$ such that $0>(K_{X}+B)\cdot C\geq -2n$ (see~\cite[Theorem~2.46]{DHP22}). But then $C$ is vertical over $\bar Z$, otherwise $(K_{X}+B+(2n+3)s H_{X})\cdot C>0$, as $H_X$ is the pullback of an ample divisor $\bar H$ on $\bar Z$, this is a contradiction. Thus it follows that every step of this MMP is also a step of an MMP over $\bar Z$, and hence there is an induced morphism $\mu:X'\to \bar A:=\bar Z \times _{A'}A$. It follows that
\[
K_{X'}+B'\sim _\Q \mu ^* H_{\bar A}+E'\sim _{\Q, \bar A}E'\geq 0,
\]
where $H_{\bar A}$ is the pullback of the ample divisor $\bar H$ by the projection $\bar A\to \bar Z$.

Then $E'$ is nef and exceptional over $\bar A$, and hence by the negativity lemma, $E'=0$. But then $K_{X'}+B'\sim _\Q \mu ^* H_{\bar A}$ and since $H_{\bar A}$ is semi-ample, so is $K_{X'}+B'$.
\end{proof}
\let\qed\relax
\end{proof}

\begin{coro}
Let $(X,B)$ be a compact K\"ahler klt pair of maximal Albanese dimension such that $a:X\to A:=\Alb(X)$ is a projective morphism. Then we can run a $(K_X+B)$-Minimal Model Program which ends with a good minimal model.
\end{coro}

\begin{proof}
Note that since $a:X\to A$ is generically finite over image, $K_X+B$ is relatively big over $a(X)$. Thus by~\cite[Theorem~1.4]{DHP22} and~\cite[Theorem~1.8]{Fuj22}, we can run a $(K_X+B)$-Minimal Model Program over $A$. Notice that each step of this MMP is also a step of the $(K_X+B)$-MMP. Therefore, we may assume that $K_X+B$ is nef over $A$ and we must check that it is indeed nef on $X$. Let $(\bar X, \bar B)$ be a good minimal model of $(X,B)$, which exists by Theorem~\ref{thm:main}. By what we have seen, $(\bar X, \bar B)$ is also a minimal model over $A$. But then $\phi:(X,B)\dasharrow (\bar X, \bar B)$ is an isomorphism in codimension~$1$. If $p:Y\to X$ and $q:Y\to \bar X$ is a common resolution, then $p^*(K_X+B)-q^*(K_{\bar X}+\bar B)$ is exceptional over $X$ (resp. $\bar X$) and nef over $\bar X$ (resp. anti-nef over $X$). From the negativity lemma, it follows that $p^*(K_X+B)=q^*(K_{\bar X}+\bar B)$. In particular, $p^*(K_X+B)$ is semi-ample, and hence so is $K_X+B$. Thus $(X,B)$ is a good minimal model.
\end{proof}

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\end{document}