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Stefan Kebekus
2024-05-27 11:22:23 +02:00
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*.aux
*.bbl
*.blg
*.brf
*.fdb_latexmk
*.fls
*.idx
*.ilg
*.ind
*.loa
*.lof
*.log
*.lot
*.out
*.pdf
*.synctex(busy)
*.synctex.gz
*.toc

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Campana
orbifolds
fibrations
fibration
Kähler
orbifold
Zariski
pseudometric
Miyaoka
Miyaoka-Yau
semipositivity
nc
uniformizable
Iitaka
Fujino
Weil
Bräunling
Brion
Soergel
Demleitner
Commelin
Winkelmann
Patakfalvi
Zsolt
Kebekus
Albanese

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\selectlanguage{british}
This paper surveys Campana's theory of $\cC$-pairs (or ``geometric orbifolds'')
in the complex-analytic setting, to serve as a reference for future work.
Written with a view towards applications in hyperbolicity, rational points, and
entire curves, it introduces the fundamental definitions of $\cC$-pair-theory
systematically and establishes a new notion of ``morphism''. The new definition
agrees with notions from the literature in the smooth case, but it is better
behaved in the singular setting, perhaps more conceptual, and has functorial
properties that relate it to minimal model theory.
% !TEX root = orbiAlb1

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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{thm}\label{thm:1}
Let $X$ be a projective manifold and let $x \in X$ be any point. If $C \subset
X$ is a rational curve, then the Albanese morphism of the $\cC$-pair $(X,0)$,
\[
\alb_x(X,0) : X \to \Alb_x(X,0),
\]
maps the curve $C$ to a point.
\end{thm}
\begin{proof}
\todo{PENDING}
\end{proof}
\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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2
gfx/set-externals.sh Executable file
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STR=$'gfx svn://vcs.mathematik.uni-freiburg.de/kebekus/gfx\nbibliography svn://vcs.mathematik.uni-freiburg.de/kebekus/bibliography'
svn propset svn:externals "$STR" .

20
gfx/setupProject.py Executable file
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#!/usr/bin/python3
import os
import subprocess
projectName = os.getcwd().split("/")[-1]
print("Working on project "+projectName)
print("Setup directory structure")
subprocess.call("svn rm branches tags trunk", shell=True)
subprocess.call("svn propset svn:externals 'gfx svn://vcs.mathematik.uni-freiburg.de/kebekus/gfx\nbibliography svn://vcs.mathematik.uni-freiburg.de/kebekus/bibliography' .", shell=True)
subprocess.call("svn up", shell=True)
print("Setup templates")
subprocess.call("cp gfx/templates/*.tex .", shell=True)
subprocess.call("mv example.tex "+projectName+".tex", shell=True)
subprocess.call("sed 's/example/"+projectName+"/g' <01.tex >01new.tex", shell=True)
subprocess.call("mv 01new.tex 01.tex", shell=True)
subprocess.call("svn add *.tex", shell=True)
subprocess.call("bash gfx/svn-propset.sh", shell=True)

378
gfx/stdPreamble.tex Normal file
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%
% PACKAGES
%
% Standard Packages
\usepackage{babel}
\usepackage{enumitem}
\usepackage{hyperref}
\usepackage[utf8]{inputenc}
\usepackage{newunicodechar}
\usepackage{mathtools}
\usepackage{varioref}
\usepackage[arrow,curve,matrix]{xy}
% Graphics Packages
\usepackage{colortbl}
\usepackage{graphicx}
\usepackage{tikz}
% Font packages
\usepackage{mathrsfs}
%
% GENERAL TYPESETTING
%
% Colours for hyperlinks
\definecolor{linkred}{rgb}{0.7,0.2,0.2}
\definecolor{linkblue}{rgb}{0,0.2,0.6}
% Limit table of contents to section titles
\setcounter{tocdepth}{1}
% Numbering of figures (see below for numbering of equations)
\numberwithin{figure}{section}
% Add an uparrow to the bibliography entries, just before the back-list of references
\usepackage[hyperpageref]{backref}
\renewcommand{\backref}[1]{$\uparrow$~#1}
% Numbering of parts in roman numbers
\renewcommand\thepart{\rm \Roman{part}}
% Sloppy formatting -- often looks better
\sloppy
% Changes the layout of descriptions and itemized lists. The indent specified in
% the original amsart style is too much for my taste.
\setdescription{labelindent=\parindent, leftmargin=2\parindent}
\setitemize[1]{labelindent=\parindent, leftmargin=2\parindent}
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#!/bin/bash
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%
% Do not edit the following line. The text is automatically updated by
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\svnid{$Id: 01-intro.tex 64 2013-12-04 07:33:02Z kebekus $}
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\documentclass[a4paper, british]{amsart}
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\author{Stefan Kebekus}
\address{Stefan Kebekus, Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104 Freiburg im Breisgau, Germany and University of Strasbourg Institute for Advanced Study (USIAS), Strasbourg, France}
\email{\href{mailto:stefan.kebekus@math.uni-freiburg.de}{stefan.kebekus@math.uni-freiburg.de}}
\urladdr{\href{http://home.mathematik.uni-freiburg.de/kebekus}{http://home.mathematik.uni-freiburg.de/kebekus}}
\thanks{Stefan Kebekus was supported in part by the DFG-Forschergruppe 790
``Classification of Algebraic Surfaces and Compact Complex Manifolds''. He
gratefully acknowledges the support through a joint fellowship of the Freiburg
Institute of Advanced Studies (FRIAS) and the University of Strasbourg
Institute for Advanced Study (USIAS).}
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\documentclass[a4paper, british]{amsart}
%
% Local font definitions -- need to come first
%
\usepackage{libertine}
\usepackage[libertine]{newtxmath}
%
% Standard macro packages
%
\input{../gfx/stdPreamble}
\input{../gfx/paperVersion-working}
\input{../gfx/varphi}
\graphicspath{{../}}
\usepackage{cancel}
\usepackage{tikz-cd}
\tikzset{commutative diagrams/arrow style=tikz}
\selectlanguage{british}
%
% Author info
%
\author{Daniel Greb}
\address{Daniel Greb, Essener Seminar für Algebraische Geometrie und Arithmetik, Fakultät für Ma\-the\-matik, Universität Duisburg--Essen, 45117 Essen, Germany}
\email{\href{mailto:daniel.greb@uni-due.de}{daniel.greb@uni-due.de}}
\urladdr{\href{https://www.esaga.uni-due.de/daniel.greb}{https://www.esaga.uni-due.de/daniel.greb}}
\author{Stefan Kebekus}
\address{Stefan Kebekus, Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Straße 1, 79104 Freiburg im Breisgau, Germany}
\email{\href{mailto:stefan.kebekus@math.uni-freiburg.de}{stefan.kebekus@math.uni-freiburg.de}}
\urladdr{\href{https://cplx.vm.uni-freiburg.de}{https://cplx.vm.uni-freiburg.de}}
\author{Thomas Peternell}
\address{Thomas Peternell, Mathematisches Institut, Universität
Bayreuth, 95440~Bayreuth, Germany}
\email{\href{mailto:thomas.peternell@uni-bayreuth.de}{thomas.peternell@uni-bayreuth.de}}
\urladdr{\href{http://www.komplexe-analysis.uni-bayreuth.de}{http://www.komplexe-analysis.uni-bayreuth.de}}
\thanks{\todo{FUNDING INFO PENDING}}
\keywords{$\cC$-pairs, morphisms of $\cC$-pairs, \foreignlanguage{french}{orbifoldes géométriques}, Campana constellations}
\subjclass[2020]{32C99, 32H99}
\title{The $\cC$-Albanese for projective manifolds}
\date{\today}
\makeindex
\makeatletter
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\begin{document}
%\begin{abstract}
% \input{00-abstract}
%\end{abstract}
\maketitle
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