Timeline for Non-trivial line bundle on $\mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jan 24, 2022 at 13:23 | comment | added | abx | @Piotr Achinger: Oops, you are right of course. I delete my comment. | |
| Jan 24, 2022 at 13:00 | comment | added | Richard Lärkäng | If you take a product of annuli instead, then a concrete example of such a divisor is described nicely in Section VI.5.2 ("Oka's counterexample") of Ranges book "Holomorphic Functions and Integral Representations in Several Complex Variables". | |
| Jan 24, 2022 at 12:51 | comment | added | Laurent Moret-Bailly | Note that every algebraic line bundle on $X$ (viewed as the torus $\mathbb{G}_{m,\mathbb{C}}^2$) is trivial since $\mathbb{C}[x^{\pm1},y^{\pm1}]$ is a UFD. | |
| Jan 24, 2022 at 11:50 | comment | added | Piotr Achinger | Take a look at the first chapter of Mumford's "Abelian varieties", it will give you a hint how to classify line bundles on this torus using the universal cover and some group cohomology. | |
| Jan 24, 2022 at 11:45 | comment | added | Piotr Achinger | @abx I'm confused. How is that possible if $\Delta=\{x=y\}$ is a principal divisor? | |
| Jan 24, 2022 at 11:05 | history | asked | ugosugo | CC BY-SA 4.0 |