Timeline for Which holomorphic curves can be leaves of a non-singular holomorphic foliation of $\mathbb C^2$?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Oct 27, 2023 at 14:19 | comment | added | LSpice | Thanks, @C7X!.. | |
| Oct 26, 2023 at 13:07 | comment | added | Kevin Casto | @TomGoodwillie Sure yeah, that example is about the holomorphic case -- the exponential exact sequence only applies in the holomorphic setting, not algebraic (since it uses the exponential map!) Apparently even Quillen-Suslin is just a calculation in the holomorphic setting. | |
| Oct 26, 2023 at 12:19 | comment | added | Tom Goodwillie | @Kevin Casto: But this is holomorphic, not algebraic. Is the answer the same? | |
| Oct 26, 2023 at 3:02 | comment | added | Kevin Casto | @TomGoodwillie for your first question yes, this is just the statement that the Picard group of $\mathbb C^2$ vanishes; see e.g. the fourth example on wikipedia en.wikipedia.org/wiki/Picard_group | |
| S Oct 26, 2023 at 2:53 | history | suggested | C7X | CC BY-SA 4.0 |
MathJaxify
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| Oct 26, 2023 at 0:33 | review | Suggested edits | |||
| S Oct 26, 2023 at 2:53 | |||||
| Oct 25, 2023 at 23:29 | comment | added | Tom Goodwillie | Ignorant question: is it true that every holomorphic line-bundle on $\mathbb C^2$ is trivial? If so, then it is true (as one might imagine you are thinking) that every foliation is spanned by some (nowhere vanishing) vector field. Also, if so, then every foliation is determined by a nowhere vanishing $1$-form. | |
| Oct 25, 2023 at 23:01 | comment | added | Daniel Asimov | I would if I knew how, but I don't. | |
| Oct 25, 2023 at 22:22 | comment | added | LSpice | It was pointed out to me, I think by @TheAmplitwist, that using Unicode like ℂ instead of TeX like $\mathbb C$ makes questions unsearchable—I guess the search engine only sees ASCII. So I think that it is friendlier to use TeX at the very least in the body. | |
| Oct 25, 2023 at 22:15 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Changed "ℂ-action" to "local ℂ-action".
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| Oct 25, 2023 at 22:14 | comment | added | Daniel Asimov | Good point; it may not be all of ℂ, just a local ℂ-action. (Though there is always a holomorphic foliation induced.) I will modify the question. | |
| Oct 25, 2023 at 20:39 | comment | added | Tom Goodwillie | How do you know that $V$ induces an action of the additive group $\mathbb C$? Don't you just get a (holomorphic) local flow in general, rather than a flow? | |
| Oct 25, 2023 at 19:41 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Added information about a holomorphic vector field on ℂ^2.
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| Oct 25, 2023 at 18:01 | history | asked | Daniel Asimov | CC BY-SA 4.0 |