# Comptes Rendus
Mathématique

Ananyo Dan and Agustín Romano-Velázquez

## Examples of non-flat bundles of rank one

Volume 361 (2023), p. 965-968

Published online: 7 September 2023

https://doi.org/10.5802/crmath.459

<figure><img src="image_1.png" /></figure>

This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
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Complex analysis and geometry / *Analyse et géométrie complexes*

# Examples of non-flat bundles of rank one

*Exemples de fibrés en droites qui ne sont pas plats*

Ananyo Dan*, $a$ and Agustín Romano-Velázquez$^{b, c}$

$^a$ School of Mathematics and Statistics, University of Sheffield, Hicks building, Hounsfield Road, S3 7RH, UK

$^b$ Alfréd Rényi Institute Of Mathematics, Hungarian Academy Of Sciences, Reáltanoda Utca 13-15, H-1053, Budapest, Hungary

$^c$ Universidad Nacional Autónoma de México Avenida Universidad s/n, Colonia Lomas de Chamilpa CP 62210, Cuernavaca, Morelos Mexico

E-mails: a.dan@sheffield.ac.uk, agustin@renyi.hu, agustin.romano@im.unam.mx

**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.

**Résumé.** 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.

**2020 Mathematics Subject Classification.** 14B05, 13C14.

**Funding.** The first author is funded by EPSRC grant number EP/T019379/1. The second author is funded by OTKA 126683 and Lendület 30001.

Manuscript received 12 November 2022, accepted 21 December 2022.

## 1. Introduction

Existence of connections on modules defined over isolated surface singularities have been extensively studied (see [1, 3–6]). 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 [5, p. 106], [4, p. 1562], [1, p. 903]), even with the help of a computer (see [3, p. 322]). 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:

<footnote>* Corresponding author.</footnote>
**Theorem 1.** 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.

Note that, given a line bundle $\mathcal{L}$ on $X \setminus \{x\}$, the pushforward $i_*\mathcal{L}$ is a reflexive sheaf on $X$, where $i$ is the inclusion of $X \setminus \{x\}$ into $X$ (see [7, Proposition 1.6]). 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.

**Corollary 2.** 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 $\mathbb{C}^3$ 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 connection.

The study of the obstruction to the existence of flat connection on MCM modules has applications in Lie-Rinehart cohomology (see [6]) and Chern-Simmons theory (see [1]).

## Acknowledgement

We thank Prof. Javier F. de Bobadilla and Prof. Duco van Straten for their interest in this problem and helpful comments. 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.

## 2. 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 [7] for basic properties of reflexive sheaves.

### 2.1. Brieskorn-Pham surfaces

Given positive integers $(p, q, r)$, denote by $G(p, q, r) \subset \mathbb{C}^3$ the zero locus of the polynomial $X^p + Y^q + Z^r$. The resulting surface is the *Brieskorn-Pham type surface*. The origin $0$ is the only singularity of the surface. Denote by

$$ \mathcal{S} := \{C \subset G(p, q, r) \mid C \text{ is a non-singular curve passing through the origin } 0\}. $$

The following theorem proves the existence of smooth curves through the singularity of $G(p, q, r)$.

**Theorem 3.** Assume $(p, q, r) = 1$ and $p \le q \le r$. Then, $\mathcal{S} \ne \emptyset$ if and only if at least one of the following conditions hold:

(1) 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,

(2) the inequality $r > pq/\gcd(p, q)$ holds.

**Proof.** See [8, Theorem 3]. □
## 2.2. *Proof of Theorem 1*

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 $\mathrm{GL}_1(\mathbb{C}) = \mathbb{C}^*$ 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 $\mathcal{I}_C$ the ideal sheaf of $C$. We now show that the restriction $M_U$ of $M := \mathcal{H}\mathrm{om}_X(\mathcal{I}_C, \mathcal{O}_X)$ 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:

$$0 \to \mathcal{I}_C \to \mathcal{O}_X \to \mathcal{O}_C \to 0 \quad (1)$$

By the depth comparison in exact sequences (see [2, Proposition 1.2.9]), we have that $\mathcal{I}_C$ is a reflexive $\mathcal{O}_X$-module. This implies that

$$M^{\vee} \cong \mathcal{I}_C^{\vee\vee} \cong \mathcal{I}_C.$$

This implies, $M_U$ is trivial if and only if $\mathcal{I}_C$ is trivial. By (1), $\mathcal{I}_C$ 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 $f \in \mathcal{O}_X$ be a function defining $C$ (the existence of $f$ is guaranteed as $C$ is Cartier) and $m$ (resp. $m'$) the maximal ideal of $\mathcal{O}_X$ (resp. $\mathcal{O}_X/(f)$). We then have the following short exact sequence of $\mathcal{O}_X$-modules:

$$0 \to (f) \to m \to m' \to 0. \quad (2)$$

Since $C$ is a smooth curve, the dimension as a complex vector space of $m'/(m')^2$ is one. Let $c' \in m'$ be a generator of $m'/(m')^2$. By the surjectivity of (2), there exists an element $c \in m$ that is mapped onto $c'$. We will now show that $m/(m)^2$ is generated as a $\mathbb{C}$-vector space by $f$ and $c$. Indeed, let $g$ be any element in $m$. If $g$ maps to 0 in $m'$ then by the exact sequence (2), $g = \beta f$ for some $\beta \in \mathcal{O}_X$. If $g$ maps to a non-zero element in $m'$, then there exists $\alpha \in \mathcal{O}_X$ such that $g - \alpha c$ is mapped to zero in $m'$. Hence by the exactness of (2), there exists $\beta \in \mathcal{O}_X$ such that $\beta f = g - \alpha c$. Therefore, $g$ is a $\mathcal{O}_X$-linear combination of $f$ and $c$ in both cases. Since $g$ was arbitrary, this implies that the dimension as a complex vector space of $m/(m)^2$ 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. $\square$

**Proof of Corollary 2.** Let $(p, q, r)$ be a triple of positive integers that are pairwise coprime. By [9, p. 7], the link of $(G(p, q, r), 0)$ is an integer homology sphere. Hence, the fundamental group of the link is perfect. By Theorem 3 there exists a smooth curve in $G(p, q, r)$ passing through the origin. The corollary then follows from Theorem 1. $\square$

**Example 4.** Let $f = x^2 + y^3 + z^7$. By Corollary 2, the hypersurface in $\mathbb{C}^3$ defined by $f$ admits MCM modules of rank one that cannot be equipped with a flat connection.
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