Bott-Chern-Aeppli, Dolbeault and Frolicher on Compact Complex 3-folds

We give the complete Bott-Chern-Aeppli cohomology for compact complex 3-folds in terms of Dolbeault, Frolicher, a bi-degree DeRham-like type of cohomology, \(K^{p,q}\), defined as $$ K^{p,q}=\frac{ker( \partial ) \cap ker( {\bar{\partial}}) }{im( \partial )\cap ker( {\bar{\partial}} )+im( {\bar{\partial}})\cap ker( \partial )}$$ and \({\check{H}}^1({\mathcal{PH}})\). (Here \(\mathcal{PH}\) is the sheaf of phuri-harmonic functions.) We then work out the complete Bott-Chern-Aeppli cohomology in some examples. We give the Bott-Chern-Aeppli cohomology for a hypothetical complex structure on \(S^6\) in terms of Dolbeault and Frolicher. We also give the Bott-Chern-Aeppli cohomology on a Calabi-Eckman 3-fold concurring with the calculations of Angella and Tomassini\cite{AngellaAndTomassini}. Finally, we show agreement of our results with the calculation by Angella\cite{Angella} of the Bott-Chern-Aeppli cohomology for small Kuranishi deformations of the Iwasawa manifold.