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.
Accepted:
Published online:
Ananyo Dan 1; Agustín Romano-Velázquez 2, 3
@article{CRMATH_2023__361_G6_965_0, author = {Ananyo Dan and Agust{\'\i}n Romano-Vel\'azquez}, title = {Examples of non-flat bundles of rank one}, journal = {Comptes Rendus. Math\'ematique}, pages = {965--968}, publisher = {Acad\'emie des sciences, Paris}, volume = {361}, year = {2023}, doi = {10.5802/crmath.459}, language = {en}, }
Ananyo Dan; Agustín Romano-Velázquez. Examples of non-flat bundles of rank one. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 965-968. doi : 10.5802/crmath.459. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.459/
[1] Algebraic Chern–Simons theory, Am. J. Math., Volume 119 (1997) no. 4, pp. 903-952 | DOI | MR | Zbl
[2] Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, 1998 | DOI
[3] Computing obstructions for existence of connections on modules, J. Symb. Comput., Volume 42 (2007) no. 3, pp. 313-323 | DOI | MR | Zbl
[4] 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] 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] 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] Stable reflexive sheaves, Math. Ann. (1980), pp. 121-176 | DOI | MR | Zbl
[8] Lines on Brieskorn–Pham surfaces, Kodai Math. J., Volume 23 (2000) no. 2, pp. 214-223 | MR | Zbl
[9] 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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