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Suppose $X\subset Y$ is a complex submanifold. Suppose also $Y$ satisfies the $\partial\bar\partial$-Lemma, or equivalently the $dd^c$-Lemma. Does $X$ satisfy the $\partial\bar\partial$-Lemma, or equivalently the $dd^c$-Lemma?

When $Y$ is Kähler this is obvious true since then $X$ is Kähler too. Hence we should ask this question for general non-Kähler manifolds.

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  • $\begingroup$ 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? $\endgroup$
    – Najib Idrissi
    Commented Jul 9, 2021 at 8:45
  • $\begingroup$ @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. $\endgroup$
    – Michael Albanese
    Commented Jul 11, 2021 at 12:17
  • $\begingroup$ @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". $\endgroup$
    – Najib Idrissi
    Commented Jul 12, 2021 at 7:40
  • $\begingroup$ 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. $\endgroup$
    – Najib Idrissi
    Commented Jul 12, 2021 at 7:42
  • $\begingroup$ @NajibIdrissi: I see, thanks for clarifying. $\endgroup$
    – Michael Albanese
    Commented Jul 12, 2021 at 14:36

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