Timeline for Non-isomorphic compact Kähler manifolds that are biholomorphic, symplectomorphic and isometric
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Sep 8, 2020 at 14:04 | comment | added | LSpice | Since yes / no answers can be confusing at first, I'm just putting the question being answered (as "no, not necessarily") here: "[Under the stated conditions,] Is there a diffeomorphism $\psi \colon M \to N$ such that $\psi^*(\omega_N) = \omega_M$ and $\psi^*(J_N) = J_N$?" | |
| Sep 8, 2020 at 12:34 | comment | added | Robert Bryant | @JoeT: Oh, I didn't notice/remember that; it seems like a peculiar, unmotivated restriction. Of course, the $\mathbb{CP}^2$ example still works, and moreover, the observation about embedding extends to $\mathbb{CP}^2$. All one needs to do is find a $\mathbb{CP}^2$ embedded in some $\mathbb{CP}^n$ in such a way that the metric induced from the ambient $\mathbb{CP}^n$'s Fubini-Study metric has no nontrivial isometries, which is easy to do. It won't have the same volume as $\mathbb{CP}^2$'s Fubini-Study metric, but some (high?) integer multiple of it. | |
| Sep 8, 2020 at 11:47 | comment | added | user164740 | I edited the question to require the isometry to be orientation-preserving. | |
| Sep 8, 2020 at 11:32 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Added a simpler example with better properties.
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| Sep 5, 2020 at 13:37 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Fixed some typos.
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| Sep 5, 2020 at 13:34 | comment | added | Robert Bryant | @DanielPomerleano: Oh, yes, you are right, those are typos. I'll fix them now. | |
| Sep 5, 2020 at 13:07 | comment | added | Daniel Pomerleano | @RobertBryant "...there is a symplectomorphism between $(M,J_M)$ and $(M,\omega_0)$." I think that should read $(M,\omega_M)$? Similarly in the next sentence. | |
| Sep 5, 2020 at 11:17 | vote | accept | CommunityBot | ||
| Sep 5, 2020 at 10:11 | comment | added | Robert Bryant | @JoeT: No. In the first place, as constructed above, $g_M$ will generically not be real-analytic in any coordinate system, and hence cannot be the pullback of a real-analytic metric via a real-analytic embedding. More directly, $g_M$ has the same total volume as the Fubini-Study metric on $\mathbb{CP}^2$, so any holomorphic embedding into $\mathbb{CP}^n$ that induced this metric would have to be a linear embedding and hence be isometric to the Fubini-Study metric on $\mathbb{CP}^2$ itself. | |
| Sep 5, 2020 at 8:33 | comment | added | user164740 | is there a holomorphic embedding $\mathbb{C}P^2\to \mathbb{C}P^n$ that pulls back the Fubini-Study metric to $g_M$? | |
| Sep 5, 2020 at 4:06 | comment | added | AmorFati | You sir, are a gift to MO. | |
| Sep 5, 2020 at 0:28 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Cleaned up the exposition a little bit.
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| Sep 5, 2020 at 0:21 | history | answered | Robert Bryant | CC BY-SA 4.0 |