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

TL;DR

This paper computes the Bott-Chern-Aeppli cohomology for compact complex 3-folds using Dolbeault, Frolicher, and a new bi-degree cohomology, providing explicit examples including hypothetical structures on $S^6$ and deformations of the Iwasawa manifold.

Contribution

It introduces a comprehensive method to determine Bott-Chern-Aeppli cohomology on complex 3-folds, connecting it with Dolbeault, Frolicher, and a novel bi-degree cohomology, and verifies results with known examples.

Findings

Computed Bott-Chern-Aeppli cohomology for specific 3-folds.

Established relations between Bott-Chern-Aeppli, Dolbeault, and Frolicher cohomologies.

Validated results against existing calculations for deformations of the Iwasawa manifold.

Abstract

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.