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Is there an example of compact complex manifold, which is formal, but does not admit complex structure satisfying $dd^c$-lemma?

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The Hopf surface $S^1 \times S^3$ is formal but does not admit a complex structure satisfying the $dd^c$ lemma. Indeed, a closed four-manifold admitting a complex structure satisfying the $dd^c$ lemma (equivalently the $\partial \bar \partial$ lemma) admits a Kähler structure and thus has even first Betti number.

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  • $\begingroup$ (The most obvious things to look at usually fall out of your head.) Are there simply connected ones? $\endgroup$
    – Denis T
    Commented Jul 30, 2018 at 22:13
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    $\begingroup$ Such examples do not exist among 4-manifolds (since any simply connected four-manifold admitting a complex structure also admits a Kähler structure). Next, in dimension 6, it helps that any simply connected six-manifold is formal, but I do not know any examples that do not admit a complex structure satisfying the $dd^c$ lemma (nor do I know an example in still higher dimensions). $\endgroup$
    – Aleksandar Milivojević
    Commented Jul 30, 2018 at 22:28

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