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$M$ is non-Kähler complex manifold. Assume that $\omega$ is $\partial$-exact and $\bar\partial$-exact $(p,q)$-form.

Question. Is $\omega$ also $\partial\bar\partial$-exact?

Based on fabulous David Speyer's answer to this MO question I suspect that the answer is elementary or very hard.

$M$ is non-Kähler complex manifold. Assume that $\omega$ is $\partial$-exact and $\bar\partial$-exact $(p,q)$-form.

Question. Is $\omega$ also $\partial\bar\partial$-exact?

Based on fabulous David Speyer's answer to this MO question I suspect that the answer is elementary or very hard.

$M$ is non-Kähler complex manifold. Assume that $\omega$ is $\partial$-exact and $\bar\partial$-exact $(p,q)$-form.

Question. Is $\omega$ also $\partial\bar\partial$-exact?

Based on fabulous David Speyer's answer to this MO question I suspect that the answer is elementary or very hard.

$M$ is non-Kähler complex manifold. Assume that $\omega$ is $\partial$-exact and $\bar\partial$-exact $(p,q)$-form.

Question. Is $\omega$ also $\partial\bar\partial$-exact?

Based on fabulous David Speyer's answer to this MO question I suspect that the answer is elementary or very hard.

$M$ is non-Kahler complex manifold. Assume that $\omega$ is $\partial$-exact and $\bar\partial$-exact $(p,q)$-form.

Question. Is $\omega$ also $\partial\bar\partial$-exact?

Based on fabulous David Speyers answer to this MO question I suspect that the answer is elementary or very hard.

$M$ is non-Kähler complex manifold. Assume that $\omega$ is $\partial$-exact and $\bar\partial$-exact $(p,q)$-form.

Question. Is $\omega$ also $\partial\bar\partial$-exact?

Based on fabulous David Speyer's answer to this MO question I suspect that the answer is elementary or very hard.