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\svnid{$Id: 01-intro.tex 727 2024-05-06 20:00:54Z rousseau $}
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\section{The Albanese for compact manifolds}

\begin{defn}[The Albanese of a compact Kähler manifold]\label{def:1-1}%
  Let $X$ be a compact Kähler manifold and let $x ∈ X$ be any point.  An
  Albanese of the pointed manifold $X$, $x \in X$ is a compact torus quotient
  $A$ and a pointed $\cC$-morphism
  \[
    a : X → A, \quad x \mapsto 0_A
  \]
  such that the following universal property holds:  If $S$ is any other compact
  torus and if
  \[
    s : X → S, \quad x \mapsto 0_S
  \]
  is any pointed morphism, then there exists a unique morphism $c$ making the
  following diagram commutative,
  \[
    \begin{tikzcd}[column sep=2.4cm]
      X \ar[r, "a"'] \ar[rr, "s", bend left=10] & A \ar[r, "∃!c"'] & S.
    \end{tikzcd}
  \]
\end{defn}

\begin{rem}
  The morphism $c$ of Definition~\ref{def:1-1} maps $0_A$ to $0_S$ and is
  therefore a Lie group morphism.
\end{rem}

\begin{rem}
  The universal property guarantees that the Albanese of
  Definition~\ref{def:1-1} is unique up to unique morphism, allowing us to speak
  of ``the Albanese''.  When precision is required, we denote the Albanese as
  \[
    \alb_x (X) : X → \Alb_x X.
  \]
\end{rem}


\section{The Albanese for compact pairs with trivial boundary}

\todo{define torus quotient}


\begin{defn}[The $\cC$-Albanese of a compact Kähler manifold]\label{def:1-2}%
  Let $X$ be a compact Kähler manifold and let $x ∈ X$ be any point.  An
  Albanese of the pointed $\cC$-pair $(X,0)$, $x \in X$ is a pointed torus
  quotient $(A, Δ_A)$, $a \in A$ and a pointed $\cC$-morphism
  \[
    a : (X,0) → (A, Δ_A), \quad x \mapsto a
  \]
  such that the following universal property holds:  If $(S, Δ_S)$, $s ∈ S$ is
  any other pointed torus quotient and if $s : (X,0) → (S, Δ_S)$ is any pointed
  $\cC$-morphism, then there exists a unique pointed $\cC$-morphism $c$ making
  the following diagram commutative,
  \[
    \begin{tikzcd}[column sep=2.4cm]
      (X, 0) \ar[r, "a"'] \ar[rr, "s", bend left=10] & (A, Δ_A) \ar[r, "∃!c"'] & (S, Δ_S).
    \end{tikzcd}
  \]
\end{defn}

\begin{rem}
  The $\cC$-morphism $c$ of Definition~\ref{def:1-2} maps $a$ to $s$ and is
  therefore a morphism of pointed pairs.  
\end{rem}

\begin{defn}[The $\cC$-Albanese of a compact Kähler manifold]\label{def:1-1}%
  Let $X$ be a compact Kähler manifold and let $x ∈ X$ be any point.  An
  Albanese of $(X,0)$ is a pointed torus quotient $\bigl(\Alb_x(X,0),
  Δ_{\Alb_x(X,0)}\bigr)$, $a \in \Alb_x(X,0)$ and  a $\cC$-morphism
  \[
    \alb_x(X,0) : (X,0) → \bigl(\Alb_x(X,0), Δ_{\Alb_x(X,0)}\bigr)
  \]
  such that the following holds.
  \begin{enumerate}
    \item The morphism $\alb_x(X,0)$ sends $x$ to $a$.

    \item If $(S, Δ_S)$, $s ∈ S$ is any other pointed torus quotient and if $s :
    (X,0) → (S, Δ_S)$ is any $\cC$-morphism that sends $x$ to $s$, then $s$
    factors uniquely as
    \[
      \begin{tikzcd}[column sep=2.4cm]
        (X, 0) \ar[r, "\alb_x(X{,}D)"'] \ar[rr, "s", bend left=10] & \bigl(\Alb_x(X,0), Δ_{\Alb_x(X,0)}\bigr) \ar[r, "∃!c"'] & (S, Δ_S).
      \end{tikzcd}
    \]
  \end{enumerate}
\end{defn}

\begin{thm}[The Albanese of a $\cC$-pair]\label{thm:22-1} %
  Let $(X, D)$ be a $\cC$-pair where $X$ is compact Kähler.  If $q⁺_{\Alb}(X,D)
  < ∞$, then an Albanese of $(X,D)$ exists.
\end{thm}


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