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
Received: 20 November 2018
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.
Authors address:
[a]: Pennsylvania State University-Harrisburg, 777 W. Harrisburg Pike, Middletown, PA 17057 USA.
Full Text (PDF)
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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