From 3be7549eb83d2ce8f38a7b4aad1d36ce1f4d5c5d Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Wed, 5 Jun 2024 13:50:15 +0200 Subject: [PATCH] =?UTF-8?q?working=E2=80=A6?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .vscode/ltex.dictionary.en-GB.txt | 1 + 01-intro.tex | 233 +++++++++++------------------- 02-ratlCurves.tex | 184 +++++++++++++++++++++++ orbiAlb4.tex | 2 +- 4 files changed, 268 insertions(+), 152 deletions(-) create mode 100644 02-ratlCurves.tex diff --git a/.vscode/ltex.dictionary.en-GB.txt b/.vscode/ltex.dictionary.en-GB.txt index a880930..4f3d278 100644 --- a/.vscode/ltex.dictionary.en-GB.txt +++ b/.vscode/ltex.dictionary.en-GB.txt @@ -26,3 +26,4 @@ Kebekus Albanese Hirzebruch multiplicitity +subvariety diff --git a/01-intro.tex b/01-intro.tex index 62671e7..b335bd0 100644 --- a/01-intro.tex +++ b/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} - 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} % !TEX root = orbiAlb1 diff --git a/02-ratlCurves.tex b/02-ratlCurves.tex new file mode 100644 index 0000000..07d8ea2 --- /dev/null +++ b/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 diff --git a/orbiAlb4.tex b/orbiAlb4.tex index 57d4712..c30c4ff 100644 --- a/orbiAlb4.tex +++ b/orbiAlb4.tex @@ -133,8 +133,8 @@ \input{01-intro} +\input{02-ratlCurves} -Test \bibstyle{alpha} \bibliographystyle{alpha}