From 8585a0877948c241b4f3b87149c5097c50bda781 Mon Sep 17 00:00:00 2001
From: Stefan Kebekus <kebekus@users.noreply.github.com>
Date: Fri, 7 Jun 2024 14:28:04 +0200
Subject: [PATCH] Done for today.

---
 .vscode/ltex.dictionary.en-GB.txt |  3 ++
 01-intro.tex                      |  6 +--
 02-ratlCurves.tex                 | 63 +++++++++++++++++++++++++------
 3 files changed, 57 insertions(+), 15 deletions(-)

diff --git a/.vscode/ltex.dictionary.en-GB.txt b/.vscode/ltex.dictionary.en-GB.txt
index 4f3d278..f43a70e 100644
--- a/.vscode/ltex.dictionary.en-GB.txt
+++ b/.vscode/ltex.dictionary.en-GB.txt
@@ -27,3 +27,6 @@ Albanese
 Hirzebruch
 multiplicitity
 subvariety
+Cremona
+equivariant
+bimeromorphic
diff --git a/01-intro.tex b/01-intro.tex
index f64ec6a..468f141 100644
--- a/01-intro.tex
+++ b/01-intro.tex
@@ -36,7 +36,7 @@
 \todo{define torus quotient}
 
 
-\begin{defn}[The $\cC$-Albanese of a compact pair with trivial boundary]\label{def:1-2}%
+\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
   \[
@@ -61,9 +61,9 @@
   \]
 \end{rem}
 
-\begin{thm}[The Albanese of a $\cC$-pair]\label{thm:22-1} %
+\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.
+  Albanese of $(X,0)$ exists.  \qed
 \end{thm}
 
 
diff --git a/02-ratlCurves.tex b/02-ratlCurves.tex
index 4b239b8..e33a3f5 100644
--- a/02-ratlCurves.tex
+++ b/02-ratlCurves.tex
@@ -26,12 +26,9 @@
       \bP¹ \ar[r, "n\text{, normalization}"'] \ar[rr, bend left=15, "\eta"] & C  \ar[r, "\text{inclusion}"'] & X.
     \end{tikzcd}
   \]
-  Consider the points $0 \in \bP¹$ and $x' := n(0) \in X$.  
-  
-  Given that $X$ is smooth, recall from \cite{orbiAlb1} that $\eta$ is a
-  $\cC$-morphism, between $\cC$-pairs $(\bP¹, 0)$ and $(X, 0)$. Since
-  $\Alb(\bP¹,0)$ exists, the universal property of the Albanese yields a
-  diagram
+  Given that $X$ is smooth, recall from \cite[Ex.~8.6]{orbiAlb1} that $\eta$ is
+  a $\cC$-morphism, between $\cC$-pairs $(\bP¹, 0)$ and $(X, 0)$. Since
+  $\Alb(\bP¹,0)$ exists, the universal property of the Albanese yields a diagram
   \[
     \begin{tikzcd}[column sep=2cm]
       \bP¹ \ar[r, "\alb(\bP¹{,}0)"] \ar[d, "n"'] & \Alb(\bP¹,0) \ar[d] \\
@@ -41,15 +38,18 @@
   The claim follows immediately once we observe that $\Alb(\bP¹,0)$ is a point.
 \end{proof}
 
+\todo{There is nothing special about $\bP¹$ here. This works for every space with nontrivial $\cC$-Albanese.}
+
 \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 $\mu$,
+  In Setting~\ref{set:1}, let $\mu : X \to Y$ be a morphism to a normal analytic
+  variety.  If all fibres of $\mu$ are rationally chain connected, then
+  $\alb(X,0)$ factors via $\mu$,
   \[
     \begin{tikzcd}
       X \ar[r, "\mu"'] \ar[rr, bend left=15, "\alb(X{,}0)"] & Y \ar[r, "\exists!\:\beta"'] & \Alb(X,0).
     \end{tikzcd}
   \]
+  \qed  
 \end{cor}
 
 \begin{cor}\label{cor:3}%
@@ -57,13 +57,52 @@
   $\Alb(X,0)$ is a point.
 \end{cor}
 
+
+\begin{cor}\label{cor:4}%
+  In Setting~\ref{set:1}, let $\mu : X \to Y$ be a bimeromorphic modification of
+  a compact manifold $Y$.  Then, $\alb(X,0)$ factors via $\mu$,
+  \[
+    \begin{tikzcd}
+      X \ar[r, "\mu"'] \ar[rr, bend left=15, "\alb(X{,}0)"] & Y \ar[r, "\exists!\:\beta"'] & \Alb(X,0),
+    \end{tikzcd}
+  \]
+  the morphism $\beta$ is a $\cC$-morphism between the pairs $(Y,0)$ and
+  $\Alb(X,0)$, and $\beta : (Y,0) \to \Alb(Y,0)$ is an Albanese of $(Y,0)$.
+\end{cor}
+\begin{proof}
+  \todo{PENDING}
+\end{proof}
+
+\begin{cor}\label{cor:5}%
+  In Setting~\ref{set:1}, let $Y$ be a compact Kähler manifold bimeromorphic to
+  $X$. Then, an Albanese of $(Y,0)$ exists.  If $f : X \dasharrow Y$ is
+  bimeromorphic, then there exists a unique morphism of $\cC$-pairs rendering
+  the following diagram commutative,
+  \[
+    \begin{tikzcd}
+      X \ar[d, "\alb(X{,}0)"'] \ar[r, dashed, "f"] & Y \ar[d, "\alb(Y{,}0)"'] \\
+      \Alb(X,0) \ar[r, "\exists! \alb(f)"'] & \Alb(Y,0)
+    \end{tikzcd}
+  \]
+\end{cor}
+\begin{proof}
+  \todo{PENDING}
+\end{proof}
+
+\begin{cor}\label{cor:6}%
+  In Setting~\ref{set:1}, the automorphism group of $X$ and the Cremona group
+  act on $\Alb(X,0)$ in a way that makes the morphism $\alb(X,0)$ equivariant.
+\end{cor}
+\begin{proof}
+  \todo{PENDING}
+\end{proof}
+
 \todo{
   \begin{itemize}
+    \item Need example where a rational variety has a 
     \item Factorization via minimal model.
-    \item Independence of bimeromorphic model.
-    \item Factorization via MRC quotient.
+    \item For varieties of general type, factorization via the canonical.
   \end{itemize}
-
 }
 
 \begin{example}[Theorem~\ref{thm:1} is wrong for singular spaces]