no. G6
Analyse et géométrie complexes
Examples of non-flat bundles of rank one
[Exemples de fibrés en droites qui ne sont pas plats]
Dan, Ananyo1  ; Romano-Velázquez, Agustín2, 3
1School of Mathematics and Statistics, University of Sheffield, Hicks building, Hounsfield Road, S3 7RH, UK
2Alfréd Rényi Institute Of Mathematics, Hungarian Academy Of Sciences, Reáltanoda Utca 13-15, H-1053, Budapest, Hungary
3Universidad Nacional Autónoma de México Avenida Universidad s/n, Colonia Lomas de Chamilpa CP 62210, Cuernavaca, Morelos Mexico
Comptes Rendus. Mathématique, Tome 361 (2023) no. G6, pp. 965-968

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.459
Classification : 14B05, 13C14

Dan, Ananyo  1   ; Romano-Velázquez, Agustín  2 , 3

1 School of Mathematics and Statistics, University of Sheffield, Hicks building, Hounsfield Road, S3 7RH, UK
2 Alfréd Rényi Institute Of Mathematics, Hungarian Academy Of Sciences, Reáltanoda Utca 13-15, H-1053, Budapest, Hungary
3 Universidad Nacional Autónoma de México Avenida Universidad s/n, Colonia Lomas de Chamilpa CP 62210, Cuernavaca, Morelos Mexico
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Examples of non-flat bundles of rank one},
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Dan, Ananyo; Romano-Velázquez, Agustín. Examples of non-flat bundles of rank one. Comptes Rendus. Mathématique, Tome 361 (2023) no. G6, pp. 965-968. doi: 10.5802/crmath.459

[1] Bloch, Spencer; Esnault, Hélène Algebraic Chern–Simons theory, Am. J. Math., Volume 119 (1997) no. 4, pp. 903-952 | Zbl | DOI | MR

[2] Bruns, Winfried; Herzog, Jürgen Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, 1998 | DOI

[3] Eriksen, Eivind; Gustavsen, Trond Stølen Computing obstructions for existence of connections on modules, J. Symb. Comput., Volume 42 (2007) no. 3, pp. 313-323 | DOI | MR | Zbl

[4] Eriksen, Eivind; Gustavsen, Trond Stølen Connections on modules over singularities of finite CM representation type, J. Pure Appl. Algebra, Volume 212 (2008) no. 7, pp. 1561-1574 | DOI | MR | Zbl

[5] Eriksen, Eivind; Gustavsen, Trond Stølen Connections on modules over singularities of finite and tame CM representation type, Generalized Lie Theory in Mathematics, Physics and Beyond, Springer, 2009, pp. 99-108 | DOI | Zbl

[6] Eriksen, Eivind; Gustavsen, Trond Stølen Lie–Rinehart cohomology and integrable connections on modules of rank one, J. Algebra, Volume 322 (2009) no. 12, pp. 4283-4294 | DOI | MR | Zbl

[7] Hartshorne, Robin Stable reflexive sheaves, Math. Ann. (1980), pp. 121-176 | DOI | MR | Zbl

[8] Jiang, Guangfeng; Oka, Mutsuo; Pho, Duc Tai; Siersma, Dirk Lines on Brieskorn–Pham surfaces, Kodai Math. J., Volume 23 (2000) no. 2, pp. 214-223 | MR | Zbl

[9] Némethi, András Five lectures on normal surface singularities, Low dimensional topology. Proceedings of the summer school, Budapest, Hungary, August 3–14, 1998 (Bolyai Society Mathematical Studies), Volume 8, János Bolyai Mathematical Society, 1998, pp. 269-351 | Zbl

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