Riv. Mat. Univ. Parma, Vol. 10, No. 1, 2019

Andrew McHugh[a]

Bott-Chern-Aeppli, Dolbeault, and Frölicher on compact complex $$3$$-folds

Pages: 25-62
Accepted in revised form: 31 Januay 2019
Mathematics Subject Classification (2010): Dolbeault cohomology, Hodge numbers, Aeppli cohomology, Bott-Chern cohomology, Frölicher spectral sequence.
Keywords: 32Q99, 53C55, 32C35.
[a]: Pennsylvania State University-Harrisburg, 777 W. Harrisburg Pike, Middletown, PA 17057 USA.

Abstract: We give the complete Bott-Chern-Aeppli cohomology for compact complex $$3$$-folds in terms of Dolbeault, Frölicher, a bi-degree de Rham-like type of cohomology, $$K^{p,q}$$, defined as

$$K^{p,q}=\displaystyle\frac{ker( \partial ) \cap ker( {\bar{\partial}}) }{im( \partial )\cap ker( {\bar{\partial}} )+im( {\bar{\partial}})\cap ker( \partial )}$$

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 Frölicher. We also give the Bott-Chern-Aeppli cohomology on a Calabi-Eckmann $$3$$-fold concurring with the calculations of Angella and Tomassini [5]. Finally, we show agreement of our results with the calculation by Angella [3] of the Bott-Chern-Aeppli cohomology for small Kuranishi deformations of the Iwasawa manifold.

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