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Stefan Kebekus
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Albanese
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Hirzebruch
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multiplicitity
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subvariety
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231
01-intro.tex
231
01-intro.tex
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@ -5,168 +5,99 @@
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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 $\cC$-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 and let $x \in X$ be any point. Assume
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that an Albanese of the $\cC$-pair $(X,0)$ exists.
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\end{setting}
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\section{The Albanese for compact manifolds}
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\begin{thm}\label{thm:1}%
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In Setting~\ref{set:1}, let $C \subset X$ be a rational curve. Then, the
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Albanese morphism $\alb_x(X,0) : X \to \Alb_x(X,0)$ maps the curve $C$ to a
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point.
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\end{thm}
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\begin{proof}
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The normalization of $C$ yields a diagram
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\begin{defn}[The Albanese of a compact Kähler manifold]\label{def:1-1}%
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Let $X$ be a compact Kähler manifold and let $x ∈ X$ be any point. An
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Albanese of the pointed manifold $X$, $x \in X$ is a compact torus quotient
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$A$ and a pointed $\cC$-morphism
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\[
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\begin{tikzcd}[column sep=2cm]
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\bP¹ \ar[r, "n\text{, normalization}"'] \ar[rr, bend left=15, "\eta"] & C \ar[r, "\text{inclusion}"'] & X.
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a : X → A, \quad x \mapsto 0_A
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\]
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such that the following universal property holds: If $S$ is any other compact
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torus and if
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\[
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s : X → S, \quad x \mapsto 0_S
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\]
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is any pointed morphism, then there exists a unique morphism $c$ making the
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following diagram commutative,
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\[
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\begin{tikzcd}[column sep=2.4cm]
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X \ar[r, "a"'] \ar[rr, "s", bend left=10] & A \ar[r, "∃!c"'] & S.
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\end{tikzcd}
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\]
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Consider the point $y := n(0_{\bP¹}) \in X$. It follows from the universal
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property of the Albanese that the Albanese varieties $\Alb_x(X,0)$ and
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$\Alb_y(X,0)$ are isomorphic. To be more precise, there exists a unique Lie
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group isomorphism $t$ that makes the following diagram commute,
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\[
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\begin{tikzcd}[column sep=2cm]
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X \ar[r, "\alb_x(X{,}0)"] \ar[d, equal] & \Alb_x(X,0) \ar[d, two heads, hook, "t"] \\
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X \ar[r, "\alb_y(X{,}0)"'] & \Alb_y(X,0).
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\end{tikzcd}
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\]
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We may therefore assume without loss of generality that $x = y = n(0_{\bP¹})$.
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Given that $X$ is smooth, recall from \cite{orbiAlb1} that $\eta$ is a
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$\cC$-morphism, between $\cC$-pairs $(\bP¹, 0)$ and $(X, 0)$. Invoking the
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universal property of the Albanese once more, we find an analogous diagram
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\[
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\begin{tikzcd}[column sep=2cm]
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\bP¹ \ar[r, "\alb_0(\bP¹{,}0)"] \ar[d, equal] & \Alb_0(\bP¹,0) \ar[d] \\
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X \ar[r, "\alb_x(X{,}0)"'] & \Alb_x(X,0).
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\end{tikzcd}
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\]
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The claim follows immediately once we observe that $\Alb_0(\bP¹,0)$ is a point.
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\end{proof}
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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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$\Alb_x(X,0)$ is a point.
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\end{cor}
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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{cor}\label{cor:2}
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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 a morphism to a normal projective variety. If all fibres of $\mu$
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are rationally chain connected, then $\alb_x(X,0)$ factors via $\mu$,
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\[
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\begin{tikzcd}
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X \ar[r, "\mu"'] \ar[rr, bend left=15, "\alb_x(X{,}0)"] & Y \ar[r, "\exists!\:\beta"'] & \Alb_x(X,0).
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\end{tikzcd}
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\]
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\end{cor}
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\end{defn}
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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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The morphism $c$ of Definition~\ref{def:1-1} maps $0_A$ to $0_S$ and is
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therefore a Lie group morphism.
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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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\begin{rem}
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The universal property guarantees that the Albanese of
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Definition~\ref{def:1-1} is unique up to unique morphism, allowing us to speak
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of ``the Albanese''. When precision is required, we denote the Albanese as
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\[
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\alb_x (X) : X → \Alb_x X.
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\]
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\end{rem}
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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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\section{The Albanese for compact pairs with trivial boundary}
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\todo{define torus quotient}
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\begin{defn}[The $\cC$-Albanese of a compact Kähler manifold]\label{def:1-2}%
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Let $X$ be a compact Kähler manifold and let $x ∈ X$ be any point. An
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Albanese of the pointed $\cC$-pair $(X,0)$, $x \in X$ is a pointed torus
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quotient $(A, Δ_A)$, $a \in A$ and a pointed $\cC$-morphism
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\[
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a : (X,0) → (A, Δ_A), \quad x \mapsto a
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\]
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such that the following universal property holds: If $(S, Δ_S)$, $s ∈ S$ is
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any other pointed torus quotient and if $s : (X,0) → (S, Δ_S)$ is any pointed
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$\cC$-morphism, then there exists a unique pointed $\cC$-morphism $c$ making
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the following diagram commutative,
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\[
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\begin{tikzcd}[column sep=2.4cm]
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(X, 0) \ar[r, "a"'] \ar[rr, "s", bend left=10] & (A, Δ_A) \ar[r, "∃!c"'] & (S, Δ_S).
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\end{tikzcd}
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\]
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\end{defn}
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\begin{rem}
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The $\cC$-morphism $c$ of Definition~\ref{def:1-2} maps $a$ to $s$ and is
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therefore a morphism of pointed pairs.
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\end{rem}
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\begin{defn}[The $\cC$-Albanese of a compact Kähler manifold]\label{def:1-1}%
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Let $X$ be a compact Kähler manifold and let $x ∈ X$ be any point. An
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Albanese of $(X,0)$ is a pointed torus quotient $\bigl(\Alb_x(X,0),
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Δ_{\Alb_x(X,0)}\bigr)$, $a \in \Alb_x(X,0)$ and a $\cC$-morphism
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\[
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\alb_x(X,0) : (X,0) → \bigl(\Alb_x(X,0), Δ_{\Alb_x(X,0)}\bigr)
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\]
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such that the following holds.
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\begin{enumerate}
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\item The morphism $\alb_x(X,0)$ sends $x$ to $a$.
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\item If $(S, Δ_S)$, $s ∈ S$ is any other pointed torus quotient and if $s :
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(X,0) → (S, Δ_S)$ is any $\cC$-morphism that sends $x$ to $s$, then $s$
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factors uniquely as
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\[
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\begin{tikzcd}[column sep=2.4cm]
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(X, 0) \ar[r, "\alb_x(X{,}D)"'] \ar[rr, "s", bend left=10] & \bigl(\Alb_x(X,0), Δ_{\Alb_x(X,0)}\bigr) \ar[r, "∃!c"'] & (S, Δ_S).
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\end{tikzcd}
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\]
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\end{enumerate}
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\end{defn}
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\begin{thm}[The Albanese of a $\cC$-pair]\label{thm:22-1} %
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Let $(X, D)$ be a $\cC$-pair where $X$ is compact Kähler. If $q⁺_{\Alb}(X,D)
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< ∞$, then an Albanese of $(X,D)$ exists.
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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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@ -0,0 +1,184 @@
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%
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% Do not edit the following line. The text is automatically updated by
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% subversion.
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%
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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 $\cC$-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 and let $x \in X$ be any point. Assume
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that an Albanese of the $\cC$-pair $(X,0)$ exists.
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\end{setting}
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\begin{rem}[Mapping subvarieties to a point]
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Assume Setting~\ref{set:1}. If $x_1, x_2 \in X$ are any two points, it
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follows from the universal property of the Albanese that the varieties
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$\Alb_{x_1}(X,0)$ and $\Alb_{x_2}(X,0)$ are isomorphic. To be more precise,
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there exists a unique Lie group isomorphism $t$ that makes the following
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diagram commute,
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\[
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\begin{tikzcd}[column sep=2cm]
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X \ar[r, "\alb_{x_1}(X{,}0)"] \ar[d, equal] & \Alb_{x_1}(X,0) \ar[d, two heads, hook, "t_{x_1x_2}"] \\
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X \ar[r, "\alb_{x_2}(X{,}0)"'] & \Alb_{x_2}(X,0).
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\end{tikzcd}
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\]
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If $Y \subseteq X$ is any subvariety, then the following two statements are equivalent.
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\begin{itemize}
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\item The morphism $\alb_{x_1}(X,0)$ maps $Y$ to a point.
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\item The morphism $\alb_{x_2}(X,0)$ maps $Y$ to a point.
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\end{itemize}
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If the conditions are satisfied, then say that \emph{the Albanese morphism of
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$(X,0)$ maps $Y$ to a point}.
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\end{rem}
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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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Consider the points $0 \in \bP¹$ and $x' := n(0) \in X$.
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Given that $X$ is smooth, recall from \cite{orbiAlb1} that $\eta$ is a
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$\cC$-morphism, between $\cC$-pairs $(\bP¹, 0)$ and $(X, 0)$. Since
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$\Alb_0(\bP¹,0)$ exists, the universal property of the Albanese yields a
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diagram
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\[
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\begin{tikzcd}[column sep=2cm]
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\bP¹ \ar[r, "\alb_0(\bP¹{,}0)"] \ar[d, "n"'] & \Alb_0(\bP¹,0) \ar[d] \\
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X \ar[r, "\alb_{x'}(X{,}0)"'] & \Alb_{x'}(X,0).
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\end{tikzcd}
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\]
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The claim follows immediately once we observe that $\Alb_0(\bP¹,0)$ is a point.
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\end{proof}
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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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$\Alb_x(X,0)$ is a point.
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\end{cor}
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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
|
||||
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
|
||||
|
|
@ -133,8 +133,8 @@
|
|||
|
||||
|
||||
\input{01-intro}
|
||||
\input{02-ratlCurves}
|
||||
|
||||
Test
|
||||
|
||||
\bibstyle{alpha}
|
||||
\bibliographystyle{alpha}
|
||||
|
|
|
|||
Loading…
Reference in New Issue