Timeline for Completion/Compactification of a Kähler metric on $\mathbb C^2$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 29, 2020 at 16:41 | comment | added | Martin de Borbon | I realized that you add some other points when w grows like z to the power mu. So I don't know how to answer your question, sorry. | |
| Jul 29, 2020 at 13:53 | comment | added | Martin de Borbon | Btw, your metric is toric, so one could look at the moment polytope and see it does not corresponds to a manifold | |
| Jul 29, 2020 at 13:50 | comment | added | Martin de Borbon | If you restrict to z=0 you add a point and get a sphere. If you restrict to w=0 you add another point and get a sphere (scaled by mu). I think, if mu is not 1, those are the only two points. Let me restrict to mu>1. Look at the distance from $(z_0, w_0)$ to $(z_0, 0)$ where $(z_0, w_0)$ belongs to the line $z= \lambda w$ for some fixed nonzero $\lambda$. Restrict to the line $z=z_0$, you get a sphere and the distance squared from $(z_0, w_0)$ to $(z_0, 0)$ is roughly $|z_0|^{1-\mu}$ so goes to zero as $z_0$ goes to infinity | |
| Jul 29, 2020 at 12:24 | comment | added | Robbixmaths | @MartindeBorbon Thank you for the answer! But I did not understand your argument, could you elaborate it, please? | |
| Jul 29, 2020 at 8:24 | comment | added | Martin de Borbon | I think that if mu is not one, then the metric completion adds only two points, so it's not a manifold | |
| Jul 29, 2020 at 8:00 | comment | added | AmorFati | This is an interesting problem, what is the motivation? | |
| Jul 26, 2020 at 19:39 | history | edited | Robbixmaths | CC BY-SA 4.0 |
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| Jul 26, 2020 at 18:56 | history | edited | Robbixmaths | CC BY-SA 4.0 |
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| Jul 26, 2020 at 18:51 | review | First posts | |||
| Jul 26, 2020 at 18:56 | |||||
| Jul 26, 2020 at 18:49 | history | asked | Robbixmaths | CC BY-SA 4.0 |