\begin{document}
\begin{abstract}
  In this work, we investigate the positivity of logarithmic and orbifold cotangent bundles along hyperplane arrangements in projective spaces. We show that a very interesting example given by Noguchi (as early as in 1986) can be pushed further to a very great extent. Key ingredients of our approach are the use of Fermat covers and the production of explicit global symmetric differentials. This allows us to obtain some new results in the vein of several classical results of the literature on hyperplane arrangements. These seem very natural using the modern point of view of augmented base loci, and working in Campana's orbifold category.
  As an application of our results, we derive two new orbifold hyperbolicity results, going beyond some classical results of value distribution theory.
\end{abstract}
\maketitle

\section{Introduction}
\input{0.intro}
\setcounter{alphtheo}{0} 
\section{Ampleness modulo boundary of the logarithmic cotangent bundle}
\label{se:lognog}
\input{1.lognog}
\section{Positivity of orbifold cotangent bundles}
\label{se:campana}
\input{2.campana}
\section{Ampleness modulo boundary of the orbifold cotangent bundle}
\label{se:orbinog}
\input{3.orbinog}
\section{Bigness of the orbifold cotangent bundle with low multiplicities}
\label{se:orbibig}
\input{4.orbibig}
\section{Applications to complex hyperbolicity}
\label{se:hyperb}
\input{5.hyperb}

\bigskip

\paragraph{Acknowledgments}
L.D. and E.R. would like to thank \emph{Joël Merker} for interesting discussions on explicit orbifold sections and around resultant which helped a lot to find the right attack angle for our problem.

L.D. would like to thank \emph{Henri Guenancia} for his help on augmented base loci and particularly around Lemma~\ref{lemm:bplus}, which plays an important role in the reformulation of Noguchi's result. L.D. would also like to thank \emph{Mikhail Zaidenberg} for many interesting discussions over the years and for making him aware of Theorem~\ref{theo:optimal}.
These interactions took place during the conference Alkage hosted by \emph{Jean-Pierre Demailly}, which gave L.D. a great opportunity to present a preliminary version of this work to a distinguished audience.

E.R. would like to thank \emph{Stefan Kebekus} and \emph{Tanuj Gomez} for discussions on the positivity of orbifold cotangent bundles, and \emph{Eric Riedl} for discussions on bigness of cotangent bundles.

L.D. would like to thank \emph{Frédéric Han} for identifying the condition of general position with respect to quadrics.

Lastly, L.D. would like to warmly thank \emph{Damian Brotbek} for introducing him in much detail to his work on explicit symmetric differential forms during the supervision of his postdoc in Strasbourg, and for the fruitful collaboration that followed.

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\bibliography{noguchi}
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\end{document}
{"config": "arxiv", "file": "2006.13515/noguchi.tex"}