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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.  An Albanese of the manifold $X$ is a
  compact torus $A$ and a morphism $a : X → A$, such that the following
  universal property holds:  If $S$ is any other compact torus and if $s : X →
  S$, is any morphism, then there exists a unique morphism $c$ making the
  following diagram commutative,
  \[
    \begin{tikzcd}
      X \ar[r, "a"'] \ar[rr, "s", bend left=20] & A \ar[r, "∃!c"'] & S.
    \end{tikzcd}
  \]
\end{defn}

\begin{rem}
  If it exists, 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
  \[
    \alb (X) : X → \Alb X.
  \]
\end{rem}


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

\todo{define torus quotient}


\begin{defn}[\protect{The Albanese of a compact pair with trivial boundary, \cite[Def.~9.1]{orbiAlb2}}]\label{def:1-2}%
  Let $X$ be a compact Kähler manifold.  An Albanese of the $\cC$-pair $(X,0)$
  is a torus quotient $(A, Δ_A)$ and a $\cC$-morphism
  \[
    a : (X,0) → (A, Δ_A),
  \]
  such that the following universal property holds:  If $(S, Δ_S)$ is any other
  torus quotient and if $s : (X,0) → (S, Δ_S)$ is any $\cC$-morphism, then there
  exists a unique $\cC$-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}
  If it exists, the universal property guarantees that the Albanese of
  Definition~\ref{def:1-2} is unique up to unique morphism, allowing us to speak
  of ``the Albanese''.  When precision is required, we denote the Albanese as
  \[
    \alb (X,0) : (X,0 → \Alb (X,0).
  \]
\end{rem}

\begin{thm}[\protect{Existence of the Albanese, \cite[Thm.~9.2]{orbiAlb2}}]\label{thm:22-1} %
  Let $X$ be a compact Kähler manifold.  If $q^+_{\Alb}(X,0) < ∞$, then an
  Albanese of $(X,0)$ exists.  \qed
\end{thm}


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