173 lines
7.0 KiB
TeX
173 lines
7.0 KiB
TeX
%
|
|
% 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{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
|
|
\[
|
|
\begin{tikzcd}[column sep=2cm]
|
|
\bP¹ \ar[r, "n\text{, normalization}"'] \ar[rr, bend left=15, "\eta"] & C \ar[r, "\text{inclusion}"'] & X.
|
|
\end{tikzcd}
|
|
\]
|
|
Consider the 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 $x = y = n(0_{\bP¹})$.
|
|
|
|
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}
|
|
\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
|