Timeline for Classification of (complex algebraic) vector bundles on punctured affine space
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| S Mar 8, 2017 at 12:04 | history | suggested | evgeny | CC BY-SA 3.0 |
corrected according to comments by D. Speyer below
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| Mar 8, 2017 at 11:53 | review | Suggested edits | |||
| S Mar 8, 2017 at 12:04 | |||||
| Mar 6, 2017 at 18:23 | comment | added | David E Speyer | Analytically, $\exp(x^{-1} y^{-1})$ is an example of a non-vanishing holomorphic function on $\mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$ which cannot be written as a product of a function which extends to $\mathbb{C} \times \mathbb{C}^{\ast}$ and one which extends to $\mathbb{C}^{\ast} \times \mathbb{C}$, so it gives a line bundle which is holomorphically nontrivial. | |
| Mar 6, 2017 at 18:22 | comment | added | Jorge Vitório Pereira | Every algebraic vector bundle on $\mathbb C^2- 0$ extends to an algebraic vector bundle on $\mathbb C^2$ (see Arapura's answer to mathoverflow.net/questions/35788/…) and therefore must be trivial. | |
| Mar 6, 2017 at 18:22 | comment | added | David E Speyer | Algebraically, every such function is of the form $\alpha x^i y^j$ for some scalar $\alpha$, and thus can be written as the product of $\alpha x^i$, which extends to $\mathbb{C}^{\ast} \times \mathbb{C}$, and $y^j$, which extends to $\mathbb{C}^{\ast} \times \mathbb{C}$. Thus, this line bundle is trivial. | |
| Mar 6, 2017 at 18:22 | comment | added | David E Speyer | Indeed, the analytic and algebraic cases are very different here; any algebraic line bundle on $\mathbb{A}^2 - 0$ is trivial. Both algebraically and analytically, any line bundle on $\mathbb{A}^2 - 0$ trivializes on $\mathbb{C} \times \mathbb{C}^{\ast}$ and $\mathbb{C}^{\ast} \times \mathbb{C}$, so it is determined by the gluing function, a non-vanishing function on $\mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$. | |
| Mar 6, 2017 at 18:14 | comment | added | Qfwfq | Wait but, if I understand correctly, Serre does the case of analytic vector bundles, which on a nonprojective variety might be rather different from the algebraic ones. | |
| Mar 6, 2017 at 18:11 | comment | added | Qfwfq | This is now (after I edited the question) a partial answer to question 1. for the case of line bundles. | |
| Mar 6, 2017 at 18:06 | history | edited | Ben McKay | CC BY-SA 3.0 |
added link
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| Mar 6, 2017 at 17:55 | history | answered | Ben McKay | CC BY-SA 3.0 |