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\svnid{$Id: 01-intro.tex 727 2024-05-06 20:00:54Z rousseau $}
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\selectlanguage{british}
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2024-06-05 13:50:15 +02:00
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\section{The Albanese for compact manifolds}
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\begin{defn}[The Albanese of a compact Kähler manifold]\label{def:1-1}%
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Let $X$ be a compact Kähler manifold and let $x ∈ X$ be any point. An
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Albanese of the pointed manifold $X$, $x \in X$ is a compact torus quotient
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$A$ and a pointed $\cC$-morphism
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\[
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a : X → A, \quad x \mapsto 0_A
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\]
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such that the following universal property holds: If $S$ is any other compact
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torus and if
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\[
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s : X → S, \quad x \mapsto 0_S
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\]
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is any pointed morphism, then there exists a unique morphism $c$ making the
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following diagram commutative,
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\[
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\begin{tikzcd}[column sep=2.4cm]
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X \ar[r, "a"'] \ar[rr, "s", bend left=10] & A \ar[r, "∃!c"'] & S.
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\end{tikzcd}
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\]
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\end{defn}
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\begin{rem}
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The morphism $c$ of Definition~\ref{def:1-1} maps $0_A$ to $0_S$ and is
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therefore a Lie group morphism.
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\end{rem}
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\begin{rem}
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The universal property guarantees that the Albanese of
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Definition~\ref{def:1-1} is unique up to unique morphism, allowing us to speak
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of ``the Albanese''. When precision is required, we denote the Albanese as
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\[
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\alb_x (X) : X → \Alb_x X.
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\]
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\end{rem}
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\section{The Albanese for compact pairs with trivial boundary}
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\todo{define torus quotient}
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\begin{defn}[The $\cC$-Albanese of a compact Kähler manifold]\label{def:1-2}%
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Let $X$ be a compact Kähler manifold and let $x ∈ X$ be any point. An
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Albanese of the pointed $\cC$-pair $(X,0)$, $x \in X$ is a pointed torus
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quotient $(A, Δ_A)$, $a \in A$ and a pointed $\cC$-morphism
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\[
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a : (X,0) → (A, Δ_A), \quad x \mapsto a
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\]
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such that the following universal property holds: If $(S, Δ_S)$, $s ∈ S$ is
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any other pointed torus quotient and if $s : (X,0) → (S, Δ_S)$ is any pointed
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$\cC$-morphism, then there exists a unique pointed $\cC$-morphism $c$ making
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the following diagram commutative,
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\[
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\begin{tikzcd}[column sep=2.4cm]
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(X, 0) \ar[r, "a"'] \ar[rr, "s", bend left=10] & (A, Δ_A) \ar[r, "∃!c"'] & (S, Δ_S).
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\end{tikzcd}
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\]
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\end{defn}
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\begin{rem}
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The $\cC$-morphism $c$ of Definition~\ref{def:1-2} maps $a$ to $s$ and is
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therefore a morphism of pointed pairs.
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\end{rem}
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\begin{defn}[The $\cC$-Albanese of a compact Kähler manifold]\label{def:1-1}%
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Let $X$ be a compact Kähler manifold and let $x ∈ X$ be any point. An
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Albanese of $(X,0)$ is a pointed torus quotient $\bigl(\Alb_x(X,0),
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Δ_{\Alb_x(X,0)}\bigr)$, $a \in \Alb_x(X,0)$ and a $\cC$-morphism
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\[
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\alb_x(X,0) : (X,0) → \bigl(\Alb_x(X,0), Δ_{\Alb_x(X,0)}\bigr)
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\]
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such that the following holds.
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\begin{enumerate}
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\item The morphism $\alb_x(X,0)$ sends $x$ to $a$.
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\item If $(S, Δ_S)$, $s ∈ S$ is any other pointed torus quotient and if $s :
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(X,0) → (S, Δ_S)$ is any $\cC$-morphism that sends $x$ to $s$, then $s$
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factors uniquely as
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\[
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\begin{tikzcd}[column sep=2.4cm]
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(X, 0) \ar[r, "\alb_x(X{,}D)"'] \ar[rr, "s", bend left=10] & \bigl(\Alb_x(X,0), Δ_{\Alb_x(X,0)}\bigr) \ar[r, "∃!c"'] & (S, Δ_S).
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\end{tikzcd}
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\]
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\end{enumerate}
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\end{defn}
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\begin{thm}[The Albanese of a $\cC$-pair]\label{thm:22-1} %
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Let $(X, D)$ be a $\cC$-pair where $X$ is compact Kähler. If $q⁺_{\Alb}(X,D)
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< ∞$, then an Albanese of $(X,D)$ exists.
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\end{thm}
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% !TEX root = orbiAlb1
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