Working…

00-abstract.tex

@@ -9,4 +9,4 @@ agrees with notions from the literature in the smooth case, but it is better
 behaved in the singular setting, perhaps more conceptual, and has functorial
 properties that relate it to minimal model theory.
 
-% !TEX root = orbiAlb1
+% !TEX root = orbiAlb4

01-intro.tex

@@ -9,18 +9,10 @@
 \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
+  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}
@@ -30,16 +22,11 @@
 \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
+  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
   \[
-    \alb_x (X) : X → \Alb_x X.
+    \alb (X) : X → \Alb X.
   \]
 \end{rem}
 
@@ -49,17 +36,15 @@
 \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
+\begin{defn}[The $\cC$-Albanese of a compact pair with trivial boundary]\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
   \[
-    a : (X,0) → (A, Δ_A), \quad x \mapsto a
+    a : (X,0) → (A, Δ_A),
   \]
-  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,
+  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).
@@ -68,35 +53,17 @@
 \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.  
+  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{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.
+  Let $X$ be a compact Kähler manifold.  If $q^+_{\Alb}(X,0) < ∞$, then an
+  Albanese of $(X,0)$ exists.
 \end{thm}
 
 

02-ratlCurves.tex

@@ -5,34 +5,14 @@
 \svnid{$Id: 01-intro.tex 727 2024-05-06 20:00:54Z rousseau $}
 \selectlanguage{british}
 
-\section{The $\cC$-Albanese morphism in the presence of rational curves}
+\section{The Albanese morphism in the presence of rational curves}
 \subversionInfo
 
-\begin{setting}\label{set:1}
-  Let $X$ be a compact Kähler manifold and let $x \in X$ be any point. Assume
-  that an Albanese of the $\cC$-pair $(X,0)$ exists.
+\begin{setting}\label{set:1}%
+  Let $X$ be a compact Kähler manifold. Assume that an Albanese of the
+  $\cC$-pair $(X,0)$ exists.
 \end{setting}  
 
-\begin{rem}[Mapping subvarieties to a point]
-  Assume Setting~\ref{set:1}.  If $x_1, x_2 \in X$ are any two points, it
-  follows from the universal property of the Albanese that the varieties
-  $\Alb_{x_1}(X,0)$ and $\Alb_{x_2}(X,0)$ are isomorphic. To be more precise,
-  there exists a unique Lie group isomorphism $t$ that makes the following
-  diagram commute,
-  \[
-    \begin{tikzcd}[column sep=2cm]
-      X \ar[r, "\alb_{x_1}(X{,}0)"] \ar[d, equal] & \Alb_{x_1}(X,0) \ar[d, two heads, hook, "t_{x_1x_2}"] \\
-      X \ar[r, "\alb_{x_2}(X{,}0)"'] & \Alb_{x_2}(X,0).
-    \end{tikzcd}
-  \]
-  If $Y \subseteq X$ is any subvariety, then the following two statements are equivalent.
-  \begin{itemize}
-    \item The morphism $\alb_{x_1}(X,0)$ maps $Y$ to a point.
-    \item The morphism $\alb_{x_2}(X,0)$ maps $Y$ to a point.
-  \end{itemize}
-  If the conditions are satisfied, then say that \emph{the Albanese morphism of
-  $(X,0)$ maps $Y$ to a point}.
-\end{rem}
 
 \begin{thm}\label{thm:1}%
   Assume Setting~\ref{set:1}.  Then, the Albanese morphism of $(X,0)$ maps all
@@ -50,22 +30,42 @@
   
   Given that $X$ is smooth, recall from \cite{orbiAlb1} that $\eta$ is a
   $\cC$-morphism, between $\cC$-pairs $(\bP¹, 0)$ and $(X, 0)$. Since
-  $\Alb_0(\bP¹,0)$ exists, the universal property of the Albanese yields a
+  $\Alb(\bP¹,0)$ exists, the universal property of the Albanese yields a
   diagram
   \[
     \begin{tikzcd}[column sep=2cm]
-      \bP¹ \ar[r, "\alb_0(\bP¹{,}0)"] \ar[d, "n"'] & \Alb_0(\bP¹,0) \ar[d] \\
-      X \ar[r, "\alb_{x'}(X{,}0)"'] & \Alb_{x'}(X,0).
+      \bP¹ \ar[r, "\alb(\bP¹{,}0)"] \ar[d, "n"'] & \Alb(\bP¹,0) \ar[d] \\
+      X \ar[r, "\alb(X{,}0)"'] & \Alb(X,0).
     \end{tikzcd}
   \]
-  The claim follows immediately once we observe that $\Alb_0(\bP¹,0)$ is a point.
+  The claim follows immediately once we observe that $\Alb(\bP¹,0)$ is a point.
 \end{proof}
 
+\begin{cor}\label{cor:2}%
+  In Setting~\ref{set:1}, let $\mu : X \to Y$ be a morphism to a normal
+  projective variety.  If all fibres of $\mu$ are rationally chain connected,
+  then $\alb(X,0)$ factors via $\mu$,
+  \[
+    \begin{tikzcd}
+      X \ar[r, "\mu"'] \ar[rr, bend left=15, "\alb(X{,}0)"] & Y \ar[r, "\exists!\:\beta"'] & \Alb(X,0).
+    \end{tikzcd}
+  \]
+\end{cor}
+
 \begin{cor}\label{cor:3}%
   In Setting~\ref{set:1}, assume that $X$ is rationally connected.  Then,
-  $\Alb_x(X,0)$ is a point.
+  $\Alb(X,0)$ is a point.
 \end{cor}
 
+\todo{
+  \begin{itemize}
+    \item Factorization via minimal model.
+    \item Independence of bimeromorphic model.
+    \item Factorization via MRC quotient.
+  \end{itemize}
+
+}
+
 \begin{example}[Theorem~\ref{thm:1} is wrong for singular spaces]
   Let $\pi : S \to \bP¹$ be one of the rational ruled ``Hirzebruch'' surfaces.
   Let $C_S \subset S$ be any section.  Construct a commutative diagram as
@@ -112,17 +112,6 @@
   \item \todo{Need example where Theorem~\ref{thm:1} fails if $X$ is singular.}
 \end{itemize}
   
-\begin{cor}\label{cor:2}
-  Let $X$ be a projective manifold and let $x \in X$ be any point.  Let $\mu : X
-  \to Y$ be a morphism to a normal projective variety.  If all fibres of $\mu$
-  are rationally chain connected, then $\alb_x(X,0)$ factors via $\mu$,
-  \[
-    \begin{tikzcd}
-      X \ar[r, "\mu"'] \ar[rr, bend left=15, "\alb_x(X{,}0)"] & Y \ar[r, "\exists!\:\beta"'] & \Alb_x(X,0).
-    \end{tikzcd}
-  \]
-\end{cor}
-  
 \begin{rem}
   ---
   \begin{itemize}