working…

.vscode/ltex.dictionary.en-GB.txt

@@ -26,3 +26,4 @@ Kebekus
 Albanese
 Hirzebruch
 multiplicitity
+subvariety

01-intro.tex

@@ -5,168 +5,99 @@
 \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}
-\subversionInfo
 
-\begin{setting}\label{set:1}
+\section{The Albanese for compact manifolds}
-  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.
-\end{setting}  
 
-\begin{thm}\label{thm:1}%
+\begin{defn}[The Albanese of a compact Kähler manifold]\label{def:1-1}%
-  In Setting~\ref{set:1}, let $C \subset X$ be a rational curve.  Then, the
+  Let $X$ be a compact Kähler manifold and let $x ∈ X$ be any point.  An
-  Albanese morphism $\alb_x(X,0) : X \to \Alb_x(X,0)$ maps the curve $C$ to a
+  Albanese of the pointed manifold $X$, $x \in X$ is a compact torus quotient
-  point.
+  $A$ and a pointed $\cC$-morphism
-\end{thm}
-\begin{proof}
-  The normalization of $C$ yields a diagram 
   \[
-    \begin{tikzcd}[column sep=2cm]
+    a : X → A, \quad x \mapsto 0_A
-      \bP¹ \ar[r, "n\text{, normalization}"'] \ar[rr, bend left=15, "\eta"] & C  \ar[r, "\text{inclusion}"'] & X.
+  \]
+  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
+  following diagram commutative,
+  \[
+    \begin{tikzcd}[column sep=2.4cm]
+      X \ar[r, "a"'] \ar[rr, "s", bend left=10] & A \ar[r, "∃!c"'] & S.
     \end{tikzcd}
   \]
-  Consider the point $y := n(0_{\bP¹}) \in X$.  It follows from the universal
+\end{defn}
-  property of the Albanese that the Albanese varieties $\Alb_x(X,0)$ and
-  $\Alb_y(X,0)$ are isomorphic. To be more precise, there exists a unique Lie
-  group isomorphism $t$ that makes the following diagram commute,
-  \[
-    \begin{tikzcd}[column sep=2cm]
-      X \ar[r, "\alb_x(X{,}0)"] \ar[d, equal] & \Alb_x(X,0) \ar[d, two heads, hook, "t"] \\
-      X \ar[r, "\alb_y(X{,}0)"'] & \Alb_y(X,0).
-    \end{tikzcd}
-  \]
-  We may therefore assume without loss of generality that $x = y = n(0_{\bP¹})$.
-
-  Given that $X$ is smooth, recall from \cite{orbiAlb1} that $\eta$ is a
-  $\cC$-morphism, between $\cC$-pairs $(\bP¹, 0)$ and $(X, 0)$. Invoking the
-  universal property of the Albanese once more, we find an analogous diagram
-  \[
-    \begin{tikzcd}[column sep=2cm]
-      \bP¹ \ar[r, "\alb_0(\bP¹{,}0)"] \ar[d, equal] & \Alb_0(\bP¹,0) \ar[d] \\
-      X \ar[r, "\alb_x(X{,}0)"'] & \Alb_x(X,0).
-    \end{tikzcd}
-  \]
-  The claim follows immediately once we observe that $\Alb_0(\bP¹,0)$ is a point.
-\end{proof}
-
-\begin{cor}\label{cor:3}%
-  In Setting~\ref{set:1}, assume that $X$ is rationally connected.  Then,
-  $\Alb_x(X,0)$ is a point.
-\end{cor}
-
-\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
-  follows,
-  \[
-    \begin{tikzcd}[column sep=2cm]
-      & S_2 \ar[r, "\alpha\text{, blow-up}"] \ar[d, "\gamma\text{, contraction}"'] & S_1 \ar[r, "\beta\text{, blow-up}"] & S \ar[r, "\pi\text{, fibre bundle}"] & \bP¹ \ar[d, equal] \\
-      C \ar[r, "\iota"'] & X \ar[rrr, "\rho\text{, rational fibration}"'] & & & \bP¹.
-    \end{tikzcd}
-  \]
-  \begin{itemize}
-    \item Choose four distinct points $x_1, …, x_4 \in \bP¹$.
-    \item Choose four points $s_\bullet \in \pi^{-1}(x_\bullet) \in \bP¹$.
-    \item Let $\beta$ be the blow-up up of the four points $s_\bullet$.  
-    \item The surface $S_1$ is smooth.  The fibres $F_{1\bullet} :=
-    (\pi\circ\beta)^{-1}(x_\bullet)$ are reduced. Each fibre $F_{1\bullet}$
-    consists of two $(-1)$-curves, meeting transversally in a point
-    $s_{1\bullet}$.
-    \item Let $\alpha$ be the blow-up up of the four points $s_{1\bullet}$.  
-    \item The surface $S_2$ is smooth but the fibres $F_{2\bullet} :=
-    (\pi\circ\beta\circ\alpha)^{-1}(x_\bullet)$ are no longer reduced. Each
-    fibre $F_{2\bullet}$ consists of two reduced $(-1)$-curves and one
-    $(-2)$-curve $F'_{2\bullet}$ of multiplicity two.
-    \item Let $\gamma$ be the contraction of the four points disjoint
-    $(-2)$-curves $F'_{2\bullet}$.  The map $\pi\circ\beta\circ\alpha$  factors
-    via the contraction map because we contract fibre components only.
-    \item Let $C \subset X$ be the strict transform of the section $C_S$.
-  \end{itemize}
-  The surface $X$ is then singular, with four quotient singularities of type
-  $A_1$ over the $x_\bullet$. All fibres of $\rho$ are supported on smooth
-  rational curves, but the fibres over $x_\bullet$ have multiplicitity two and
-  pass through the singularities.
-
-  The criterion for $\cC$-morphism spelled out in \cite{orbiAlb1} quickly
-  implies that $\rho$ is a $\cC$-morphism between the pair $(X,0)$ and the torus
-  quotient $(\bP¹, \frac{1}{2}·\sum_i x_i)$.  The universal property of the
-  Albanese immediately implies that the map $\rho$  factors via $\alb_x(X,0)$. A
-  more detailed analysis, applying Theorem~\ref{thm:1} to the smooth fibres of
-  $\rho$, shows that the torus quotient $(\bP¹, \frac{1}{2}·\sum_i x_i)$ is
-  equal to the Albanese and that $\rho$ is the Albanese map.
-\end{example}
-
-\begin{itemize}
-  \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}
-  ---
+  The morphism $c$ of Definition~\ref{def:1-1} maps $0_A$ to $0_S$ and is
-  \begin{itemize}
+  therefore a Lie group morphism.
-    \item Corollary~\ref{cor:2} does not equip $Y$ with the structure of a $\cC$-pair.
-    \item Corollary~\ref{cor:2} does not assume that $\mu$ is a morphism of $\cC$-pairs.
-    \item Even if $\mu$ is a morphism of $\cC$-pairs, Corollary~\ref{cor:2}
-    does not claim that $\beta$ is a morphism of $\cC$-pairs.
-    \item Corollary~\ref{cor:2} neither gives a morphism between $\Alb_x(X,0)$
-    and $\Alb_{\mu(x)}(Y,0)$ nor does it claim that these are isomorphic.
-  \end{itemize}
 \end{rem}
 
-\begin{itemize}
+\begin{rem}
-  \item \todo{Kummer K3s are nice examples where the Albanese grows when we
+  The universal property guarantees that the Albanese of
-  contract rational curves.}
+  Definition~\ref{def:1-1} is unique up to unique morphism, allowing us to speak
-  \item \todo{Want more examples to showcase all the things that can go wrong.}
+  of ``the Albanese''.  When precision is required, we denote the Albanese as
-  \item \todo{Corollary~\ref{cor:2} implies that the $\cC$-Albanese map factors
+  \[
-  via the MRC fibration of $X$, and via any map from $X$ to one of its minimal
+    \alb_x (X) : X → \Alb_x X.
-  models.  This should be exploitable in geometrically meaningful situations.}
+  \]
-\end{itemize}
+\end{rem}
 
-\todo{There are settings where the factorization of Corollary~\ref{cor:2} is a
-factorization into morphisms of $\cC$-pairs.}
 
-\begin{thm}
+\section{The Albanese for compact pairs with trivial boundary}
-  Birational projective manifolds $X$ and $Y$ have canonically isomorphic
+
-  $\cC$-Albanese varieties.
+\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
+  \[
+    a : (X,0) → (A, Δ_A), \quad x \mapsto a
+  \]
+  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,
+  \[
+    \begin{tikzcd}[column sep=2.4cm]
+      (X, 0) \ar[r, "a"'] \ar[rr, "s", bend left=10] & (A, Δ_A) \ar[r, "∃!c"'] & (S, Δ_S).
+    \end{tikzcd}
+  \]
+\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.  
+\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.
 \end{thm}
-\begin{proof}
-  \todo{PENDING}
-\end{proof}
 
-\begin{thm}
-  Let $X$ be a projective manifold and let $x \in X$ be any point.  Let $\mu : X
-  \to Y$ be an MRC fibration of $X$, where $Y$ is again a projective manifold.
-  Then, the $\cC$-pairs $\Alb_x(X,0)$ and $\Alb_{\mu(x)}(Y,0)$ are naturally
-  isomorphic.
-\end{thm}
-\begin{proof}
-  \todo{PENDING}
-\end{proof}
-
-
-\section{Examples}
-
-\begin{itemize}
-  \item \todo{Discuss the Stoppino-example: general type, simply-connected,
-  augmented irregularity zero, but has a non-trivial $\cC$-Albanse.}
-\end{itemize}
-
-
-\section{The $\cC$-Albanese morphism for special manifolds}
-
-\begin{itemize}
-  \item \todo{Discuss special surfaces.}
-  \item \todo{Figure out what we can say for special threefolds.}
-\end{itemize}
 
 % !TEX root = orbiAlb1

02-ratlCurves.tex

@@ -0,0 +1,184 @@
+%
+% Do not edit the following line.  The text is automatically updated by
+% subversion.
+%
+\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}
+\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.
+\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
+  rational curves to points.
+\end{thm}
+\begin{proof}
+  Let $C \subseteq X$ be any rational curve. The normalization of $C$ yields a
+  diagram 
+  \[
+    \begin{tikzcd}[column sep=2cm]
+      \bP¹ \ar[r, "n\text{, normalization}"'] \ar[rr, bend left=15, "\eta"] & C  \ar[r, "\text{inclusion}"'] & X.
+    \end{tikzcd}
+  \]
+  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
+  $\Alb_0(\bP¹,0)$ exists, the universal property of the Albanese yields a
+  diagram
+  \[
+    \begin{tikzcd}[column sep=2cm]
+      \bP¹ \ar[r, "\alb_0(\bP¹{,}0)"] \ar[d, "n"'] & \Alb_0(\bP¹,0) \ar[d] \\
+      X \ar[r, "\alb_{x'}(X{,}0)"'] & \Alb_{x'}(X,0).
+    \end{tikzcd}
+  \]
+  The claim follows immediately once we observe that $\Alb_0(\bP¹,0)$ is a point.
+\end{proof}
+
+\begin{cor}\label{cor:3}%
+  In Setting~\ref{set:1}, assume that $X$ is rationally connected.  Then,
+  $\Alb_x(X,0)$ is a point.
+\end{cor}
+
+\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
+  follows,
+  \[
+    \begin{tikzcd}[column sep=2cm]
+      & S_2 \ar[r, "\alpha\text{, blow-up}"] \ar[d, "\gamma\text{, contraction}"'] & S_1 \ar[r, "\beta\text{, blow-up}"] & S \ar[r, "\pi\text{, fibre bundle}"] & \bP¹ \ar[d, equal] \\
+      C \ar[r, "\iota"'] & X \ar[rrr, "\rho\text{, rational fibration}"'] & & & \bP¹.
+    \end{tikzcd}
+  \]
+  \begin{itemize}
+    \item Choose four distinct points $x_1, …, x_4 \in \bP¹$.
+    \item Choose four points $s_\bullet \in \pi^{-1}(x_\bullet) \in \bP¹$.
+    \item Let $\beta$ be the blow-up up of the four points $s_\bullet$.  
+    \item The surface $S_1$ is smooth.  The fibres $F_{1\bullet} :=
+    (\pi\circ\beta)^{-1}(x_\bullet)$ are reduced. Each fibre $F_{1\bullet}$
+    consists of two $(-1)$-curves, meeting transversally in a point
+    $s_{1\bullet}$.
+    \item Let $\alpha$ be the blow-up up of the four points $s_{1\bullet}$.  
+    \item The surface $S_2$ is smooth but the fibres $F_{2\bullet} :=
+    (\pi\circ\beta\circ\alpha)^{-1}(x_\bullet)$ are no longer reduced. Each
+    fibre $F_{2\bullet}$ consists of two reduced $(-1)$-curves and one
+    $(-2)$-curve $F'_{2\bullet}$ of multiplicity two.
+    \item Let $\gamma$ be the contraction of the four points disjoint
+    $(-2)$-curves $F'_{2\bullet}$.  The map $\pi\circ\beta\circ\alpha$  factors
+    via the contraction map because we contract fibre components only.
+    \item Let $C \subset X$ be the strict transform of the section $C_S$.
+  \end{itemize}
+  The surface $X$ is then singular, with four quotient singularities of type
+  $A_1$ over the $x_\bullet$. All fibres of $\rho$ are supported on smooth
+  rational curves, but the fibres over $x_\bullet$ have multiplicitity two and
+  pass through the singularities.
+
+  The criterion for $\cC$-morphism spelled out in \cite{orbiAlb1} quickly
+  implies that $\rho$ is a $\cC$-morphism between the pair $(X,0)$ and the torus
+  quotient $(\bP¹, \frac{1}{2}·\sum_i x_i)$.  The universal property of the
+  Albanese immediately implies that the map $\rho$  factors via $\alb_x(X,0)$. A
+  more detailed analysis, applying Theorem~\ref{thm:1} to the smooth fibres of
+  $\rho$, shows that the torus quotient $(\bP¹, \frac{1}{2}·\sum_i x_i)$ is
+  equal to the Albanese and that $\rho$ is the Albanese map.
+\end{example}
+
+\begin{itemize}
+  \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}
+    \item Corollary~\ref{cor:2} does not equip $Y$ with the structure of a $\cC$-pair.
+    \item Corollary~\ref{cor:2} does not assume that $\mu$ is a morphism of $\cC$-pairs.
+    \item Even if $\mu$ is a morphism of $\cC$-pairs, Corollary~\ref{cor:2}
+    does not claim that $\beta$ is a morphism of $\cC$-pairs.
+    \item Corollary~\ref{cor:2} neither gives a morphism between $\Alb_x(X,0)$
+    and $\Alb_{\mu(x)}(Y,0)$ nor does it claim that these are isomorphic.
+  \end{itemize}
+\end{rem}
+
+\begin{itemize}
+  \item \todo{Kummer K3s are nice examples where the Albanese grows when we
+  contract rational curves.}
+  \item \todo{Want more examples to showcase all the things that can go wrong.}
+  \item \todo{Corollary~\ref{cor:2} implies that the $\cC$-Albanese map factors
+  via the MRC fibration of $X$, and via any map from $X$ to one of its minimal
+  models.  This should be exploitable in geometrically meaningful situations.}
+\end{itemize}
+  
+\todo{There are settings where the factorization of Corollary~\ref{cor:2} is a
+factorization into morphisms of $\cC$-pairs.}
+
+\begin{thm}
+  Birational projective manifolds $X$ and $Y$ have canonically isomorphic
+  $\cC$-Albanese varieties.
+\end{thm}
+\begin{proof}
+  \todo{PENDING}
+\end{proof}
+
+\begin{thm}
+  Let $X$ be a projective manifold and let $x \in X$ be any point.  Let $\mu : X
+  \to Y$ be an MRC fibration of $X$, where $Y$ is again a projective manifold.
+  Then, the $\cC$-pairs $\Alb_x(X,0)$ and $\Alb_{\mu(x)}(Y,0)$ are naturally
+  isomorphic.
+\end{thm}
+\begin{proof}
+  \todo{PENDING}
+\end{proof}
+
+
+\section{Examples}
+
+\begin{itemize}
+  \item \todo{Discuss the Stoppino-example: general type, simply-connected,
+  augmented irregularity zero, but has a non-trivial $\cC$-Albanse.}
+\end{itemize}
+
+
+\section{The $\cC$-Albanese morphism for special manifolds}
+
+\begin{itemize}
+  \item \todo{Discuss special surfaces.}
+  \item \todo{Figure out what we can say for special threefolds.}
+\end{itemize}
+
+% !TEX root = orbiAlb1

orbiAlb4.tex

@@ -133,8 +133,8 @@
 
 
 \input{01-intro}
+\input{02-ratlCurves}
 
-Test
 
 \bibstyle{alpha}
 \bibliographystyle{alpha}