Let $X$ be a compact complex manifold. Is the space of $(p,p)$ $d$-closed currents modulo $\partial\bar{\partial}$-exact ones naturally isomorphic to the Bott-Chern cohomology (made in the same way with smooth forms)? I can prove this statement for $p=1$ or for any $p$ on Kähler manifolds (via the $\partial\bar{\partial}$-lemma for currents), but I can not prove it for $p > 1$ on a general manifold. Does anyone know how to prove it?
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$\begingroup$ When $\partial\bar\partial$ lemma holds, Bott Chern cohomology and Dolbeault cohomology are the same, what is the your exact question? $\endgroup$– user21574Commented Mar 31, 2016 at 13:15
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$\begingroup$ @HassanJolany I think, we wants to see what happens for general complex manifolds. $\endgroup$– Sebastian GoetteCommented Mar 31, 2016 at 13:16
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2 Answers
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When $X$ is a compact complex manifold, its Bott-Chern cohomology groups can be computed either by smooth forms or by currents. The proof of this fact can be found for instance in Demailly's book (link here), page 326, considerations after the proof of Lemma 12.2.
Demailly derives there this result from hypercohomological considerations.
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This is a theorem which was first proved by Bruno Bigolin in this paper:
Osservazioni sulla coomologia del $\partial \overline{\partial}$, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Sér. 3, 24 no. 3 (1970), 571-583,
stated there as Proposition 2.2. The argument is pretty much the same as the one in Demailly's book that diverietti mentions.
