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.vscode/ltex.dictionary.en-GB.txt vendored
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Albanese
Hirzebruch
multiplicitity
subvariety

233
01-intro.tex
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\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}
\section{The Albanese for compact manifolds}
\begin{thm}\label{thm:1}%
In Setting~\ref{set:1}, let $C \subset X$ be a rational curve. Then, the
Albanese morphism $\alb_x(X,0) : X \to \Alb_x(X,0)$ maps the curve $C$ to a
point.
\end{thm}
\begin{proof}
The normalization of $C$ yields a diagram
\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
\[
\begin{tikzcd}[column sep=2cm]
\bP¹ \ar[r, "n\text{, normalization}"'] \ar[rr, bend left=15, "\eta"] & C \ar[r, "\text{inclusion}"'] & X.
a : X → A, \quad x \mapsto 0_A
\]
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
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
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 $x = y = n(0_{\bP¹})$.
\end{defn}
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}
---
\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}
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{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{rem}
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
\[
\alb_x (X) : X → \Alb_x X.
\]
\end{rem}
\begin{thm}
Birational projective manifolds $X$ and $Y$ have canonically isomorphic
$\cC$-Albanese varieties.
\section{The Albanese for compact pairs with trivial boundary}
\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}
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%
% 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

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\input{01-intro}
\input{02-ratlCurves}
Test
\bibstyle{alpha}
\bibliographystyle{alpha}