71 lines
2.4 KiB
TeX
71 lines
2.4 KiB
TeX
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% Do not edit the following line. The text is automatically updated by
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% subversion.
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%
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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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\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. An Albanese of the manifold $X$ is a
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compact torus $A$ and a morphism $a : X → A$, such that the following
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universal property holds: If $S$ is any other compact torus and if $s : X →
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S$, is any morphism, then there exists a unique morphism $c$ making the
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following diagram commutative,
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\[
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\begin{tikzcd}
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X \ar[r, "a"'] \ar[rr, "s", bend left=20] & A \ar[r, "∃!c"'] & S.
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\end{tikzcd}
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\]
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\end{defn}
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\begin{rem}
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If it exists, 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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\[
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\alb (X) : X → \Alb X.
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\]
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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}[\protect{The Albanese of a compact pair with trivial boundary, \cite[Def.~9.1]{orbiAlb2}}]\label{def:1-2}%
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Let $X$ be a compact Kähler manifold. An Albanese of the $\cC$-pair $(X,0)$
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is a torus quotient $(A, Δ_A)$ and a $\cC$-morphism
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\[
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a : (X,0) → (A, Δ_A),
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\]
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such that the following universal property holds: If $(S, Δ_S)$ is any other
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torus quotient and if $s : (X,0) → (S, Δ_S)$ is any $\cC$-morphism, then there
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exists a unique $\cC$-morphism $c$ making 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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If it exists, the universal property guarantees that the Albanese of
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Definition~\ref{def:1-2} 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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\[
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\alb (X,0) : (X,0 → \Alb (X,0).
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\]
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\end{rem}
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\begin{thm}[\protect{Existence of the Albanese, \cite[Thm.~9.2]{orbiAlb2}}]\label{thm:22-1} %
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Let $X$ be a compact Kähler manifold. If $q^+_{\Alb}(X,0) < ∞$, then an
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Albanese of $(X,0)$ exists. \qed
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\end{thm}
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% !TEX root = orbiAlb1
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