Timeline for Analogue of Quillen-Suslin theorem for affine varieties
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 15, 2019 at 18:23 | comment | added | Earthliŋ | To summarize this answer in the smooth case, topologically trivial vector bundles over smooth affine varieties are known to be trivial for $\mathbb C^n \times (\mathbb C^∗)^m$, the case $m=0$ being the classical Quillen-Suslin theorem and $m \neq 0$ being the slightly more general affine toric case of Gubeladze. | |
| May 14, 2019 at 23:02 | comment | added | cgodfrey | @Qfwfq I wasn't sure if that ("holomorphic" homotopic to trivial bundle implies holomorphically trivial) was true so I stopped short of making that claim. Although based on the reference provided in the comments by the OP everything would make sense if it was true. And yes I decided to delete that first paragraph since it was not really related to the question and at best confusing. Certainly Grauert's Oka principle is better explained elsewhere. | |
| May 14, 2019 at 19:50 | comment | added | Qfwfq | @cgodfrey: sure, you didn't write that "holomorphic" homotopic to trivial bundle implies holomorphically trivial; though, since the OP's question asked for the latter, I thought I was missing some implication. (now you've deleted that paragraph, anyway) - (Also, yes, all these things are not equivalent in the algebraic and holomorphic settings) | |
| May 14, 2019 at 16:56 | history | edited | cgodfrey | CC BY-SA 4.0 |
removed confusing rephrasing of grauert's principle...
|
| May 14, 2019 at 16:33 | comment | added | cgodfrey | I appear to have accidentally invoked GAGA for an affine variety -- for $X$ the curve of positive genus minus a point all hlomorphic line bundles are trivial even though many algebraic ones are non-trivial! Thanks again for your comments, revisions forthcoming. | |
| May 14, 2019 at 11:06 | comment | added | Earthliŋ | I had in mind Cor. 3.3 of J. Leiterer's chapter "Holomorphic vector bundles and the Oka-Grauert principle" in Several Complex Variables IV which states: If $E$ and $F$ are holomorphic vector bundles of a Stein analytic space $X$ which are continuously isomorphic, then $E$ and $F$ are also holomorphically isomorphic. | |
| May 14, 2019 at 9:17 | comment | added | Earthliŋ | Thank you very much for your answer. Your "homotopy" point of view of Grauert's theorem seems to be at odds with the interpretation I stated in my question. In particular, making a continuous family (over $I$) between two holomorphic vector bundles into a holomorphic family doesn't say anything about the isomorphism type of the bundles at $0$ and $1$, as I guess @Qfwfq was wondering. | |
| May 14, 2019 at 8:50 | comment | added | cgodfrey | @Qfwfq Looking again I realize what I wrote was less than clear -- I also hope I didn't break math. | |
| May 14, 2019 at 8:47 | comment | added | cgodfrey | Oh no, I don't think it implies $E$ is holomorphically trivial. But being topologically trivial is equivalent to a continuous at every time$\tilde{E}$ and Graurt says we can make it holomorphic at every time. This is still far from being holomorphically trivial as seen in the example. | |
| May 13, 2019 at 23:59 | comment | added | Qfwfq | Does the existence of $\tilde{E}$ on $X\times I$ imply that $E$ is holomorphically trivial on $X$? Is there some sort of theorem that says that holomorphic v.b. on $X$ are in bijection with holomorphic-at-every-time-homotopy classes of holomorphic maps from $X$ into the Grassmannian? | |
| May 13, 2019 at 23:27 | history | edited | cgodfrey | CC BY-SA 4.0 |
Decided it was lame to say "some clarification"
|
| May 13, 2019 at 18:10 | history | answered | cgodfrey | CC BY-SA 4.0 |