Timeline for Is the complex structure of $\mathbb CP^n$ unique?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 10, 2021 at 13:59 | history | edited | Tom | CC BY-SA 4.0 |
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| Jan 29, 2021 at 6:33 | answer | added | Nick L | timeline score: 10 | |
| Jan 28, 2021 at 18:47 | comment | added | YangMills | Since you mention large deformations of $\mathbb{P}^n$, these were shown to be isomorphic to $\mathbb{P}^n$ by Siu (Crelle 89, erratum Crelle 92) | |
| Jan 28, 2021 at 16:16 | vote | accept | Tom | ||
| Jan 28, 2021 at 16:10 | answer | added | diverietti | timeline score: 27 | |
| Jan 28, 2021 at 16:01 | comment | added | Tom | @abx, is there a reference say something detailed about "a Kähler manifold homeomorphic to $\mathbb P^n$ is isomorphic to $\mathbb P^n$"? | |
| Jan 28, 2021 at 15:39 | comment | added | abx | Precisely: if a complex manifold $M$ is diffeomorphic to $\mathbb{S}^6$, the blow up of a point in $M$ is diffeomorphic to $\mathbb{CP}^3$. | |
| Jan 28, 2021 at 15:32 | comment | added | Tom | @aglearner, you mean blow up a point of $S^6$, we can get $\mathbb CP^3$? | |
| Jan 28, 2021 at 14:37 | comment | added | abx | Note however that a Kähler manifold homeomorphic to $\mathbb{P}^n$ is isomorphic to $\mathbb{P}^n$ — this follows from Yau's theorem, plus some preious work of Kobayashi-Ochiai. | |
| Jan 28, 2021 at 14:32 | comment | added | aglearner | This is a well-known open problem, without any real progress. At the current moment no one knows if there is an exotic complex structure on $\mathbb CP^n$ for $n>2$. The question is even more famous for $\mathbb CP^3$, because it would have an exotic complex structure, would $S^6$ have (just blow up a point). And the latter is again not known: mathoverflow.net/questions/1973/…. | |
| Jan 28, 2021 at 14:10 | history | asked | Tom | CC BY-SA 4.0 |