Timeline for If $Y$ satisfies the $\partial\bar{\partial}$-lemma and $X$ is a complex submanifold of $Y$, does $X$ satisfy the $\partial\bar{\partial}$-lemma?
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13 events
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| S Jul 16, 2021 at 21:04 | history | bounty ended | CommunityBot | ||
| S Jul 16, 2021 at 21:04 | history | notice removed | CommunityBot | ||
| Jul 14, 2021 at 2:41 | comment | added | YangMills | I will assume $X,Y$ compact. Then pretty much all known examples of $\partial\overline{\partial}$-manifolds come from small deformations of class $\mathcal{C}$ manifolds. Now by Fujiki and Varouchas if $X$ is class $\mathcal{C}$ then $Y$ is also class $\mathcal{C}$, hence satisfies the $\partial\overline{\partial}$-lemma. So the first place to look for counterexamples would be to small deformations of class $\mathcal{C}$ manifolds which are not class $\mathcal{C}$ themselves. Such examples were constructed by LeBrun-Poon and Campana. I don't know whether anyone has looked at submanifolds... | |
| Jul 12, 2021 at 14:36 | comment | added | Michael Albanese | @NajibIdrissi: I see, thanks for clarifying. | |
| Jul 12, 2021 at 7:42 | comment | added | Najib Idrissi | Anyway, yes, my point is that the question has an obvious answer of "no" when $X$ is not assumed to be compact. The question is five years old and Ryan Du has not been online since 2017, so we may never know if compactness is part of the assumptions. | |
| Jul 12, 2021 at 7:40 | comment | added | Najib Idrissi | @MichaelAlbanese Sorry, I didn't write my comment correctly since I wanted to give the benefit of doubt. I should have written: "Among Kähler manifolds, only compact ones are known to satisfy the $dd^c$ lemma for sure. Other ones might too, but the proof I know is for compact manifolds. And $\Sigma_g^2 \setminus \Delta$ does not satisfy it." Even though the statement of "Main theorem" of DGMS says that the manifold must be compact, note that the proof of the formality from the $dd^c$ lemma doesn't use compactness anywhere. Look at the section "First proof". | |
| Jul 11, 2021 at 12:17 | comment | added | Michael Albanese | @NajibIdrissi: There are other manifolds which satisfy the $dd^c$ lemma, such as compact Moishezon manifolds, see Corollary 5.23 of Real Homotopy Theory of Kähler Manifolds by Deligne, Griffiths, Morgan, and Sullivan. Also, the implication $dd^c$ lemma implies formal requires compactness as far as I know (at least, this is required in the DGMS paper, see the statement of the Main Theorem in section 6). For these reasons, my interpretation of the question is that $X$ and $Y$ are compact. | |
| Jul 9, 2021 at 8:45 | comment | added | Najib Idrissi | I may be mistaken but I thought that only compact Kähler manifolds are known to satidfy the $dd^c$ lemma? A nonformal Kähler manifold (e.g. $\Sigma_g^2 \setminus \Delta \subset \Sigma_g^2$) would do the the trick I guess? | |
| S Jul 8, 2021 at 19:09 | history | bounty started | Tom | ||
| S Jul 8, 2021 at 19:09 | history | notice added | Tom | Draw attention | |
| Aug 25, 2016 at 17:51 | history | edited | Michael Albanese |
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| Aug 25, 2016 at 14:56 | history | edited | Michael Albanese | CC BY-SA 3.0 |
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| Aug 25, 2016 at 14:51 | history | asked | Ryan Du | CC BY-SA 3.0 |