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\TopicFR{Analyse et géométrie complexes}
\TopicEN{Complex analysis and geometry}

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\title{Examples of non-flat bundles of rank one}
\alttitle{Exemples de fibrés en droites qui ne sont pas plats}

\author{\firstname{Ananyo} \lastname{Dan}\IsCorresp}
\address{School of Mathematics and Statistics, University of Sheffield, Hicks building, Hounsfield Road, S3 7RH, UK}
\email{a.dan@sheffield.ac.uk}

\author{\firstname{Agust\'in} \lastname{Romano-Vel\'azquez}}
\address{Alfréd Rényi Institute Of Mathematics, Hungarian Academy Of Sciences, Reáltanoda Utca 13-15, H-1053, Budapest, Hungary}
\address{Universidad Nacional Aut\'onoma de M\'exico Avenida Universidad s/n, Colonia Lomas de Chamilpa CP 62210, Cuernavaca, Morelos Mexico}
\email{agustin@renyi.hu}
\email{agustin.romano@im.unam.mx}

\thanks{The first author is funded by EPSRC grant number EP/T019379/1. The second author is funded by OTKA 126683 and Lend\"ulet 30001.}
\CDRGrant{EPSRC EP/T019379/1}
\CDRGrant{OTKA 126683}
\CDRGrant{Lend\"ulet 30001}

\subjclass{14B05, 13C14}
%~\keywords{Flat connections, Brieskorn--Pham surfaces, Maximal Cohen--Macaulay modules, Isolated surface singularities}


\begin{altabstract}
%~On s’attend à ce qu'il existe des fibrés en droites sur une surface non-singulière quasi-affine qui n'admettent pas de connexion plate. Cependant, de notre connaissance, il n'y a pas d'exemple connu d'un tel fibré en droites. Dans cet article, nous donnons plusieurs exemples de fibrés en droites sur certaines surfaces non singulières, quasi-affines, qui ne peuvent être équipées d'une connexion plate.
On s’attend à ce qu'il existe des fibrés en droites sur une surface non-singulière quasi-affine qui n'admettent pas de connexion plate mais, à notre connaissance, aucun exemple d'un tel fibré n'est connu. Dans cet article, nous en donnons plusieurs exemples explicites.
\end{altabstract}

\begin{abstract}
%~It is expected that there exist line bundles on a quasi-affine non-singular surface which do not admit a flat connection. However, to the best of our knowledge there is no known example of such a line bundle. In this article we give several explicit examples of line bundles on certain non-singular, quasi-affine surfaces that cannot be equipped with a flat connection.
It is expected that there exist line bundles on a quasi-affine non-singular surface which do not admit a flat connection. However, to the best of our knowledge there is no known example of such a line bundle. In this article we give several explicit examples of line bundles on certain non-singular, quasi-affine surfaces that cannot be equipped with a flat connection.
\end{abstract}



\begin{document}
\maketitle

\section{Introduction}

Existence of connections on modules defined over isolated surface singularities have been extensively studied (see~\cite{erik1, erik2, gus1, erik3, blo}). However, in these literatures the authors stress that they cannot produce a single example of a maximal Cohen--Macaulay (MCM) module over a surface singularity that does not admit a flat connection (see~\cite[p.~106]{gus1}, \cite[p.~1562]{erik1}, \cite[p.~903]{blo}), even with the help of a computer (see~\cite[p.~322]{erik2}). By flat connection, we mean a connection with zero curvature. In this article we produce numerous examples of maximal Cohen--Macaulay modules over certain isolated surface singularities that cannot be equipped with a flat connection. In particular, we prove:

\begin{theo}\label{thm}
Let $(X,x)$ be the germ of a normal surface singularity such that the fundamental group of the link is perfect. Suppose that $(X,x)$ contains a smooth curve passing through $x$. Then, there exists a line bundle on $X\setminus \{x\}$ that cannot be equipped with a flat connection. Moreover, one can associate to any such smooth curve (i.e., passing through $x$), an unique (up to isomorphism) line bundle on $X\setminus \{x\}$ that cannot be equipped with a flat connection.
\end{theo}

Note that, given a line bundle $\cL$ on $X\setminus \{x\}$, the pushforward $i_*\cL$ is a reflexive sheaf on $X$, where $i$ is the inclusion of $X\setminus \{x\}$ into $X$ (see~\cite[Proposition~1.6]{stabhar}). Furthermore, if $X$ is a normal, integral surface, then reflexive sheaves on it are maximal Cohen--Macaulay. Therefore, as a corollary we produce explicit examples of maximal Cohen--Macaulay modules that cannot be equipped with a flat connection.

\begin{coro}\label{cor}
Let $(p,q,r)$ be a triple of positive integers that are pairwise coprime and $r>pq$. Denote by $G(p,q,r)$ the surface in $\C^3$ defined by the polynomial $X^p+Y^q+Z^r$ and by $U(p,q,r)$ the regular locus of $G(p,q,r)$. Then, there exists line bundles on $U(p,q,r)$ that cannot be equipped with a flat connections.
\end{coro}

The study of the obstruction to the existence of flat connection on MCM modules has applications in Lie--Rinehart cohomology (see~\cite{erik3}) and Chern--Simmons theory (see~\cite{blo}).

\subsection*{Acknowledgement}

We thank Prof. Javier F. de Bobadilla and Prof. Duco van Straten for their interest in this problem and helpful comments.
%The first author is funded by EPSRC grant number EP/T019379/1. The second author is funded by OTKA 126683 and Lend\"ulet 30001.
The second author thanks CIRM, Luminy, for its hospitality and for providing a perfect work environment.
He also thanks Prof. Javier F. de Bobadilla, the 2021 semester 2 Jean-Morlet Chair, for the invitation.


\section{Example of non-flat invertible sheaves}

We give examples of rank $1$ invertible sheaves which cannot be equipped with a flat connection. We will assume familiarity with reflexive sheaves. See~\cite{stabhar} for basic properties of reflexive sheaves.

\subsection{Brieskorn--Pham surfaces}

Given positive integers $(p,q,r)$, denote by $G(p,q,r) \subset \C^3$ the zero locus of the polynomial $X^p+Y^q+Z^r$. The resulting surface is the \emph{Brieskorn--Pham type surface}. The origin $0$ is the only singularity of the surface. Denote by
\[
\cS:=\{C \subset G(p,q,r)\, |\, C \text{ is a non-singular curve passing through the origin }0\}.
\]
The following theorem proves the existence of smooth curves through the singularity of $G(p,q,r)$.

\begin{theo}\label{thm:exist}
Assume $(p,q,r)=1$ and $p \le q \le r$. Then, $\cS \not= \emptyset$ if and only if at least one of the following conditions hold:
\begin{enumerate}
\item two of the three integers $(p,q,r)$ are coprime and the other one is divisible by at least one of the two coprime numbers,
\item the inequality $r>pq/\gcd(p,q)$ holds.
\end{enumerate}
\end{theo}

\begin{proof}
See~\cite[Theorem~3]{jiang}.
\end{proof}


\subsection{Proof of Theorem~\ref{thm}}

Denote by $U$ the regular locus of $(X,x)$, i.e. $U= X\setminus \{x\}$. Note that the fundamental group $\pi_1(U)$ is the same as the fundamental group of the link $L$ of $(X,x)$. By hypothesis, the fundamental group of $L$ is perfect. This means that the abelianization $\pi_1(U)^{\ab}$ of the fundamental group $\pi_1(U)$ is trivial. Since $\GL_1(\C)=\C^*$ is abelian, any $1$-dimensional group representation of $\pi_1(U)$ factors through $\pi_1(U)^{\ab}$, which is trivial. Therefore, every $1$-dimensional representation of $\pi_1(U)$ is trivial. By the Riemann--Hilbert correspondence, this implies that there does not exist any non-trivial line bundle on $U$ that can be equipped with a flat connection. Therefore, it suffices to show the existence of non-trivial line bundles on $U$.

By hypothesis, there exists $C \subset X$ a smooth curve contained in the surface $X$ passing through the singular point. Denote by $\I_C$ the ideal sheaf of $C$. We now show that the restriction $M_U$ of $M:=\Hc_X(\I_C,\mo_X)$ to $U$ is a non-trivial line bundle. Indeed, $M_U$ is trivial if and only if $i_*(M_U) \cong M$ is trivial ($M$ is reflexive, and reflexive sheaves are uniquely determined by their restriction to the open subset $U$), where $i: U \to X$ is the natural inclusion. Furthermore, since $M$ is a reflexive module, $M$ is trivial if and only if $M^{\vee}$ is trivial (double dual of a reflexive module is isomorphic to itself). Consider the short exact sequence:
\begin{equation}\label{eq01}
0 \to \I_C \to \mo_X \to \mo_C \to 0
\end{equation}
By the depth comparison in exact sequences (see~\cite[Proposition~1.2.9]{brun1}), we have that $\I_C$ is a reflexive $\mo_X$-module. This implies that
\[
M^{\vee} \cong \I_C^{\vee \vee} \cong \I_C.
\]
This implies, $M_U$ is trivial if and only if $\I_C$ is trivial. By~\eqref{eq01}, $\I_C$ is trivial if and only if $C$ is a Cartier divisor. So, it suffices to show that $C$ is a Weil divisor which is not Cartier.

We prove this by contradiction. In particular, suppose that $C$ is a Cartier divisor. We are going to give a contradiction to the non-smoothness of $X$. Let $f \in \mo_X$ be a function defining $C$ (the existence of $f$ is guaranteed as $C$ is Cartier) and $\fm$ (resp. $\fm'$) the maximal ideal of $\mo_X$ (resp. $\mo_X/(f)$). We then have the following short exact sequence of $\mo_X$-modules:
\begin{equation}\label{eq03}
0 \to (f) \to \fm \to \fm' \to 0.
\end{equation}
Since $C$ is a smooth curve, the dimension as a complex vector space of $\fm'/(\fm')^2$ is one. Let $c' \in \fm'$ be a generator of $\fm'/(\fm')^2$. By the surjectivity of~\eqref{eq03}, there exists an element $c \in \fm$ that is mapped onto $c'$. We will now show that $\fm/(\fm)^2$ is generated as a $\C$-vector space by $f$ and $c$. Indeed, let $g$ be any element in $\fm$. If $g$ maps to $0$ in $\fm'$ then by the exact sequence~\eqref{eq03}, $g=\beta f$ for some $\beta \in \mo_X$. If $g$ maps to a non-zero element in $\fm'$, then there exists $\alpha \in \mo_X$ such that $g-\alpha c$ is mapped to zero in $\fm'$. Hence by the exactness of~\eqref{eq03}, there exists $\beta \in \mo_X$ such that $\beta f=g -\alpha c$. Therefore, $g$ is a $\mo_X$-linear combination of $f$ and $c$ in both cases. Since $g$ was arbitrary, this implies that the dimension as a complex vector space of $\fm/(\fm)^2$ is two (generated by $f$ and $c$). But this will mean $X$ is smooth at the origin, which is a contradiction. Hence, $C$ cannot by a Cartier divisor. This proves the theorem.\qed


\begin{proof}[Proof of Corollary~\ref{cor}]
Let $(p,q,r)$ be a triple of positive integers that are pairwise coprime. By~\cite[p.~7]{N99}, the link of $(G(p,q,r),0)$ is an integer homology sphere. Hence, the fundamental group of the link is perfect. By Theorem~\ref{thm:exist} there exists a smooth curve in $G(p,q,r)$ passing through the origin. The corollary then follows from Theorem~\ref{thm}.
\end{proof}

\begin{exem}
Let $f= x^2 + y^3 + z^7$. By Corollary~\ref{cor}, the hypersurface in $\C^3$ defined by $f$ admits MCM modules of rank one that cannot be equipped with a flat connection.
\end{exem}

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