213 lines
7.9 KiB
TeX
213 lines
7.9 KiB
TeX
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\svnid{$Id: 01-intro.tex 727 2024-05-06 20:00:54Z rousseau $}
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\selectlanguage{british}
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\section{The Albanese morphism in the presence of rational curves}
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\subversionInfo
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\begin{setting}\label{set:1}%
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Let $X$ be a compact Kähler manifold. Assume that an Albanese of the
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$\cC$-pair $(X,0)$ exists.
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\end{setting}
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\begin{thm}\label{thm:1}%
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Assume Setting~\ref{set:1}. Then, the Albanese morphism of $(X,0)$ maps all
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rational curves to points.
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\end{thm}
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\begin{proof}
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Let $C \subseteq X$ be any rational curve. The normalization of $C$ yields a
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diagram
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\[
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\begin{tikzcd}[column sep=2cm]
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\bP¹ \ar[r, "n\text{, normalization}"'] \ar[rr, bend left=15, "\eta"] & C \ar[r, "\text{inclusion}"'] & X.
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\end{tikzcd}
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\]
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Given that $X$ is smooth, recall from \cite[Ex.~8.6]{orbiAlb1} that $\eta$ is
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a $\cC$-morphism, between $\cC$-pairs $(\bP¹, 0)$ and $(X, 0)$. Since
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$\Alb(\bP¹,0)$ exists, the universal property of the Albanese yields a diagram
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\[
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\begin{tikzcd}[column sep=2cm]
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\bP¹ \ar[r, "\alb(\bP¹{,}0)"] \ar[d, "n"'] & \Alb(\bP¹,0) \ar[d] \\
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X \ar[r, "\alb(X{,}0)"'] & \Alb(X,0).
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\end{tikzcd}
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\]
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The claim follows immediately once we observe that $\Alb(\bP¹,0)$ is a point.
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\end{proof}
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\todo{There is nothing special about $\bP¹$ here. This works for every space with nontrivial $\cC$-Albanese.}
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\begin{cor}\label{cor:2}%
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In Setting~\ref{set:1}, let $\mu : X \to Y$ be a morphism to a normal analytic
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variety. If all fibres of $\mu$ are rationally chain connected, then
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$\alb(X,0)$ factors via $\mu$,
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\[
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\begin{tikzcd}
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X \ar[r, "\mu"'] \ar[rr, bend left=15, "\alb(X{,}0)"] & Y \ar[r, "\exists!\:\beta"'] & \Alb(X,0).
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\end{tikzcd}
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\]
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\qed
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\end{cor}
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\begin{cor}\label{cor:3}%
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In Setting~\ref{set:1}, assume that $X$ is rationally connected. Then,
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$\Alb(X,0)$ is a point.
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\end{cor}
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\begin{cor}\label{cor:4}%
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In Setting~\ref{set:1}, let $\mu : X \to Y$ be a bimeromorphic modification of
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a compact manifold $Y$. Then, $\alb(X,0)$ factors via $\mu$,
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\[
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\begin{tikzcd}
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X \ar[r, "\mu"'] \ar[rr, bend left=15, "\alb(X{,}0)"] & Y \ar[r, "\exists!\:\beta"'] & \Alb(X,0),
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\end{tikzcd}
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\]
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the morphism $\beta$ is a $\cC$-morphism between the pairs $(Y,0)$ and
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$\Alb(X,0)$, and $\beta : (Y,0) \to \Alb(Y,0)$ is an Albanese of $(Y,0)$.
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\end{cor}
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\begin{proof}
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\todo{PENDING}
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\end{proof}
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\begin{cor}\label{cor:5}%
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In Setting~\ref{set:1}, let $Y$ be a compact Kähler manifold bimeromorphic to
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$X$. Then, an Albanese of $(Y,0)$ exists. If $f : X \dasharrow Y$ is
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bimeromorphic, then there exists a unique morphism of $\cC$-pairs rendering
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the following diagram commutative,
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\[
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\begin{tikzcd}
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X \ar[d, "\alb(X{,}0)"'] \ar[r, dashed, "f"] & Y \ar[d, "\alb(Y{,}0)"'] \\
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\Alb(X,0) \ar[r, "\exists! \alb(f)"'] & \Alb(Y,0)
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\end{tikzcd}
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\]
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\end{cor}
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\begin{proof}
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\todo{PENDING}
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\end{proof}
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\begin{cor}\label{cor:6}%
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In Setting~\ref{set:1}, the automorphism group of $X$ and the Cremona group
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act on $\Alb(X,0)$ in a way that makes the morphism $\alb(X,0)$ equivariant.
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\end{cor}
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\begin{proof}
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\todo{PENDING}
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\end{proof}
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\todo{
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\begin{itemize}
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\item Need example where a rational variety has a
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\item Factorization via minimal model.
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\item For varieties of general type, factorization via the canonical.
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\end{itemize}
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}
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\begin{example}[Theorem~\ref{thm:1} is wrong for singular spaces]
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Let $\pi : S \to \bP¹$ be one of the rational ruled ``Hirzebruch'' surfaces.
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Let $C_S \subset S$ be any section. Construct a commutative diagram as
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follows,
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\[
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\begin{tikzcd}[column sep=2cm]
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& S_2 \ar[r, "\alpha\text{, blow-up}"] \ar[d, "\gamma\text{, contraction}"'] & S_1 \ar[r, "\beta\text{, blow-up}"] & S \ar[r, "\pi\text{, fibre bundle}"] & \bP¹ \ar[d, equal] \\
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C \ar[r, "\iota"'] & X \ar[rrr, "\rho\text{, rational fibration}"'] & & & \bP¹.
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\end{tikzcd}
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\]
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\begin{itemize}
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\item Choose four distinct points $x_1, …, x_4 \in \bP¹$.
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\item Choose four points $s_\bullet \in \pi^{-1}(x_\bullet) \in \bP¹$.
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\item Let $\beta$ be the blow-up up of the four points $s_\bullet$.
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\item The surface $S_1$ is smooth. The fibres $F_{1\bullet} :=
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(\pi\circ\beta)^{-1}(x_\bullet)$ are reduced. Each fibre $F_{1\bullet}$
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consists of two $(-1)$-curves, meeting transversally in a point
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$s_{1\bullet}$.
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\item Let $\alpha$ be the blow-up up of the four points $s_{1\bullet}$.
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\item The surface $S_2$ is smooth but the fibres $F_{2\bullet} :=
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(\pi\circ\beta\circ\alpha)^{-1}(x_\bullet)$ are no longer reduced. Each
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fibre $F_{2\bullet}$ consists of two reduced $(-1)$-curves and one
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$(-2)$-curve $F'_{2\bullet}$ of multiplicity two.
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\item Let $\gamma$ be the contraction of the four points disjoint
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$(-2)$-curves $F'_{2\bullet}$. The map $\pi\circ\beta\circ\alpha$ factors
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via the contraction map because we contract fibre components only.
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\item Let $C \subset X$ be the strict transform of the section $C_S$.
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\end{itemize}
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The surface $X$ is then singular, with four quotient singularities of type
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$A_1$ over the $x_\bullet$. All fibres of $\rho$ are supported on smooth
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rational curves, but the fibres over $x_\bullet$ have multiplicitity two and
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pass through the singularities.
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The criterion for $\cC$-morphism spelled out in \cite{orbiAlb1} quickly
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implies that $\rho$ is a $\cC$-morphism between the pair $(X,0)$ and the torus
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quotient $(\bP¹, \frac{1}{2}·\sum_i x_i)$. The universal property of the
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Albanese immediately implies that the map $\rho$ factors via $\alb_x(X,0)$. A
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more detailed analysis, applying Theorem~\ref{thm:1} to the smooth fibres of
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$\rho$, shows that the torus quotient $(\bP¹, \frac{1}{2}·\sum_i x_i)$ is
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equal to the Albanese and that $\rho$ is the Albanese map.
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\end{example}
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\begin{itemize}
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\item \todo{Need example where Theorem~\ref{thm:1} fails if $X$ is singular.}
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\end{itemize}
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\begin{rem}
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---
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\begin{itemize}
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\item Corollary~\ref{cor:2} does not equip $Y$ with the structure of a $\cC$-pair.
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\item Corollary~\ref{cor:2} does not assume that $\mu$ is a morphism of $\cC$-pairs.
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\item Even if $\mu$ is a morphism of $\cC$-pairs, Corollary~\ref{cor:2}
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does not claim that $\beta$ is a morphism of $\cC$-pairs.
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\item Corollary~\ref{cor:2} neither gives a morphism between $\Alb_x(X,0)$
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and $\Alb_{\mu(x)}(Y,0)$ nor does it claim that these are isomorphic.
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\end{itemize}
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\end{rem}
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\begin{itemize}
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\item \todo{Kummer K3s are nice examples where the Albanese grows when we
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contract rational curves.}
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\item \todo{Want more examples to showcase all the things that can go wrong.}
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\item \todo{Corollary~\ref{cor:2} implies that the $\cC$-Albanese map factors
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via the MRC fibration of $X$, and via any map from $X$ to one of its minimal
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models. This should be exploitable in geometrically meaningful situations.}
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\end{itemize}
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\todo{There are settings where the factorization of Corollary~\ref{cor:2} is a
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factorization into morphisms of $\cC$-pairs.}
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\begin{thm}
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Birational projective manifolds $X$ and $Y$ have canonically isomorphic
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$\cC$-Albanese varieties.
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\end{thm}
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\begin{proof}
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\todo{PENDING}
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\end{proof}
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\begin{thm}
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Let $X$ be a projective manifold and let $x \in X$ be any point. Let $\mu : X
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\to Y$ be an MRC fibration of $X$, where $Y$ is again a projective manifold.
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Then, the $\cC$-pairs $\Alb_x(X,0)$ and $\Alb_{\mu(x)}(Y,0)$ are naturally
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isomorphic.
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\end{thm}
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\begin{proof}
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\todo{PENDING}
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\end{proof}
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\section{Examples}
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\begin{itemize}
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\item \todo{Discuss the Stoppino-example: general type, simply-connected,
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augmented irregularity zero, but has a non-trivial $\cC$-Albanse.}
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\end{itemize}
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\section{The $\cC$-Albanese morphism for special manifolds}
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\begin{itemize}
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\item \todo{Discuss special surfaces.}
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\item \todo{Figure out what we can say for special threefolds.}
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\end{itemize}
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% !TEX root = orbiAlb1
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