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Stefan Kebekus
@@ -9,4 +9,4 @@ agrees with notions from the literature in the smooth case, but it is better
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behaved in the singular setting, perhaps more conceptual, and has functorial
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properties that relate it to minimal model theory.
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% !TEX root = orbiAlb1
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% !TEX root = orbiAlb4
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75
01-intro.tex
75
01-intro.tex
@@ -9,18 +9,10 @@
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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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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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@@ -30,16 +22,11 @@
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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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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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\[
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\alb_x (X) : X → \Alb_x X.
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\alb (X) : X → \Alb X.
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\]
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\end{rem}
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@@ -49,17 +36,15 @@
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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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\begin{defn}[The $\cC$-Albanese of a compact pair with trivial boundary]\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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\[
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a : (X,0) → (A, Δ_A), \quad x \mapsto a
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a : (X,0) → (A, Δ_A),
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\]
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such that the following universal property holds: If $(S, Δ_S)$, $s ∈ S$ 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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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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@@ -68,35 +53,17 @@
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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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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{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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Let $X$ be a compact Kähler manifold. If $q^+_{\Alb}(X,0) < ∞$, then an
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Albanese of $(X,0)$ exists.
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\end{thm}
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@@ -5,34 +5,14 @@
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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 $\cC$-Albanese morphism in the presence of rational curves}
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\section{The Albanese morphism in the presence of rational curves}
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\subversionInfo
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\begin{setting}\label{set:1}
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Let $X$ be a compact Kähler manifold and let $x \in X$ be any point. Assume
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that an Albanese of the $\cC$-pair $(X,0)$ exists.
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\begin{setting}\label{set:1}%
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Let $X$ be a compact Kähler manifold. Assume that an Albanese of the
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$\cC$-pair $(X,0)$ exists.
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\end{setting}
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\begin{rem}[Mapping subvarieties to a point]
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Assume Setting~\ref{set:1}. If $x_1, x_2 \in X$ are any two points, it
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follows from the universal property of the Albanese that the varieties
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$\Alb_{x_1}(X,0)$ and $\Alb_{x_2}(X,0)$ are isomorphic. To be more precise,
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there exists a unique Lie group isomorphism $t$ that makes the following
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diagram commute,
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\[
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\begin{tikzcd}[column sep=2cm]
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X \ar[r, "\alb_{x_1}(X{,}0)"] \ar[d, equal] & \Alb_{x_1}(X,0) \ar[d, two heads, hook, "t_{x_1x_2}"] \\
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X \ar[r, "\alb_{x_2}(X{,}0)"'] & \Alb_{x_2}(X,0).
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\end{tikzcd}
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\]
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If $Y \subseteq X$ is any subvariety, then the following two statements are equivalent.
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\begin{itemize}
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\item The morphism $\alb_{x_1}(X,0)$ maps $Y$ to a point.
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\item The morphism $\alb_{x_2}(X,0)$ maps $Y$ to a point.
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\end{itemize}
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If the conditions are satisfied, then say that \emph{the Albanese morphism of
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$(X,0)$ maps $Y$ to a point}.
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\end{rem}
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\begin{thm}\label{thm:1}%
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Assume Setting~\ref{set:1}. Then, the Albanese morphism of $(X,0)$ maps all
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@@ -50,22 +30,42 @@
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Given that $X$ is smooth, recall from \cite{orbiAlb1} that $\eta$ is a
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$\cC$-morphism, between $\cC$-pairs $(\bP¹, 0)$ and $(X, 0)$. Since
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$\Alb_0(\bP¹,0)$ exists, the universal property of the Albanese yields a
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$\Alb(\bP¹,0)$ exists, the universal property of the Albanese yields a
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diagram
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\[
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\begin{tikzcd}[column sep=2cm]
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\bP¹ \ar[r, "\alb_0(\bP¹{,}0)"] \ar[d, "n"'] & \Alb_0(\bP¹,0) \ar[d] \\
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X \ar[r, "\alb_{x'}(X{,}0)"'] & \Alb_{x'}(X,0).
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\bP¹ \ar[r, "\alb(\bP¹{,}0)"] \ar[d, "n"'] & \Alb(\bP¹,0) \ar[d] \\
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X \ar[r, "\alb(X{,}0)"'] & \Alb(X,0).
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\end{tikzcd}
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\]
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The claim follows immediately once we observe that $\Alb_0(\bP¹,0)$ is a point.
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The claim follows immediately once we observe that $\Alb(\bP¹,0)$ is a point.
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\end{proof}
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\begin{cor}\label{cor:2}%
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In Setting~\ref{set:1}, let $\mu : X \to Y$ be a morphism to a normal
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projective variety. If all fibres of $\mu$ are rationally chain connected,
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then $\alb(X,0)$ factors via $\mu$,
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\[
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\begin{tikzcd}
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X \ar[r, "\mu"'] \ar[rr, bend left=15, "\alb(X{,}0)"] & Y \ar[r, "\exists!\:\beta"'] & \Alb(X,0).
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\end{tikzcd}
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\]
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\end{cor}
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\begin{cor}\label{cor:3}%
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In Setting~\ref{set:1}, assume that $X$ is rationally connected. Then,
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$\Alb_x(X,0)$ is a point.
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$\Alb(X,0)$ is a point.
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\end{cor}
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\todo{
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\begin{itemize}
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\item Factorization via minimal model.
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\item Independence of bimeromorphic model.
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\item Factorization via MRC quotient.
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\end{itemize}
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}
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\begin{example}[Theorem~\ref{thm:1} is wrong for singular spaces]
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Let $\pi : S \to \bP¹$ be one of the rational ruled ``Hirzebruch'' surfaces.
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Let $C_S \subset S$ be any section. Construct a commutative diagram as
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@@ -111,17 +111,6 @@
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\begin{itemize}
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\item \todo{Need example where Theorem~\ref{thm:1} fails if $X$ is singular.}
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\end{itemize}
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\begin{cor}\label{cor:2}
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Let $X$ be a projective manifold and let $x \in X$ be any point. Let $\mu : X
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\to Y$ be a morphism to a normal projective variety. If all fibres of $\mu$
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are rationally chain connected, then $\alb_x(X,0)$ factors via $\mu$,
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\[
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\begin{tikzcd}
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X \ar[r, "\mu"'] \ar[rr, bend left=15, "\alb_x(X{,}0)"] & Y \ar[r, "\exists!\:\beta"'] & \Alb_x(X,0).
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\end{tikzcd}
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\]
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\end{cor}
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\begin{rem}
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---
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