Abstract
We prove that, for some classes of complex nilmanifolds, the Bott–Chern cohomology is completely determined by the Lie algebra associated with the nilmanifold with the induced complex structure. We use these tools to compute the Bott–Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations, completing the computations by M. Schweitzer (arXiv:0709.3528v1 [math.AG]).
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Acknowledgements
The author would like to thank Adriano Tomassini and Jean-Pierre Demailly for their constant encouragement, their support and for many useful conversations. He would like also to thank Institut Fourier, Université de Grenoble i, for its warm hospitality. Very interesting conversations with Sönke Rollenske at cirm in Luminy and with Greg Kuperberg at Institut Fourier in Grenoble gave great motivations for looking at further results on this subject. Many thanks to Sönke are due also for his comments and remarks which improved the presentation of this paper. The author is very grateful to the anonymous referee for his/her careful reading and for many suggestions and remarks that highly improved the presentation of the paper.
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Communicated by Marco Abate.
This work was supported by GNSAGA of INdAM.
Appendix: Dimensions of the Cohomologies of the Iwasawa Manifold and of Its Small Deformations
Appendix: Dimensions of the Cohomologies of the Iwasawa Manifold and of Its Small Deformations
\(\mathbf{H}^{\mathbf{\bullet}}_{\mathbf{dR}}\) | b 1 | b 2 | b 3 | b 4 | b 5 |
|---|---|---|---|---|---|
\(\mathbb{I}_{3}\) and (i), (ii), (iii) | 4 | 8 | 10 | 8 | 4 |
\(\mathbf{H}^{\mathbf{\bullet,\bullet}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{1,0}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{0,1}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{2,0}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{1,1}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{0,2}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{3,0}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{2,1}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{1,2}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{0,3}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{3,1}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{2,2}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{1,3}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{3,2}}_{\overline {\partial }}\) | \(\mathbf{h}^{\mathbf{2,3}}_{\overline {\partial }}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\mathbb {I}_{3}\) and (i) | 3 | 2 | 3 | 6 | 2 | 1 | 6 | 6 | 1 | 2 | 6 | 3 | 2 | 3 |
(ii) | 2 | 2 | 2 | 5 | 2 | 1 | 5 | 5 | 1 | 2 | 5 | 2 | 2 | 2 |
(iii) | 2 | 2 | 1 | 5 | 2 | 1 | 4 | 4 | 1 | 2 | 5 | 1 | 2 | 2 |
\(\mathbf{H}^{\mathbf{\bullet,\bullet}}_{\textrm{BC}}\) | \(\mathbf{h}^{\mathbf{1,0}}_{\textrm{BC}}\) | \(\mathbf{h}^{\mathbf{0,1}}_{\textrm{BC}}\) | \(\mathbf{h}^{\mathbf{2,0}}_{\textrm {BC}}\) | \(\mathbf{h}^{\mathbf{1,1}}_{\textrm{BC}}\) | \(\mathbf{h}^{\mathbf{0,2}}_{\textrm{BC}}\) | \(\mathbf{h}^{\mathbf{3,0}}_{\textrm{BC}}\) | \(\mathbf{h}^{\mathbf{2,1}}_{\textrm {BC}}\) | \(\mathbf{h}^{\mathbf{1,2}}_{\textrm{BC}}\) | \(\mathbf{h}^{\mathbf{0,3}}_{\textrm {BC}}\) | \(\mathbf{h}^{\mathbf{3,1}}_{\textrm{BC}}\) | \(\mathbf{h}^{\mathbf{2,2}}_{\textrm {BC}}\) | \(\mathbf{h}^{\mathbf{1,3}}_{\textrm{BC}}\) | \(\mathbf{h}^{\mathbf{3,2}}_{\textrm {BC}}\) | \(\mathbf{h}^{\mathbf{2,3}}_{\textrm{BC}}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\mathbb {I}_{3}\) and (i) | 2 | 2 | 3 | 4 | 3 | 1 | 6 | 6 | 1 | 2 | 8 | 2 | 3 | 3 |
(ii.a) | 2 | 2 | 2 | 4 | 2 | 1 | 6 | 6 | 1 | 2 | 7 | 2 | 3 | 3 |
(ii.b) | 2 | 2 | 2 | 4 | 2 | 1 | 6 | 6 | 1 | 2 | 6 | 2 | 3 | 3 |
(iii.a) | 2 | 2 | 1 | 4 | 1 | 1 | 6 | 6 | 1 | 2 | 7 | 2 | 3 | 3 |
(iii.b) | 2 | 2 | 1 | 4 | 1 | 1 | 6 | 6 | 1 | 2 | 6 | 2 | 3 | 3 |
\(\mathbf{H}^{\mathbf{\bullet,\bullet}}_{\textrm{A}}\) | \(\mathbf{h}^{\mathbf{1,0}}_{\textrm {A}}\) | \(\mathbf{h}^{\mathbf{0,1}}_{\textrm{A}}\) | \(\mathbf{h}^{\mathbf{2,0}}_{\textrm {A}}\) | \(\mathbf{h}^{\mathbf{1,1}}_{\textrm{A}}\) | \(\mathbf{h}^{\mathbf{0,2}}_{\textrm {A}}\) | \(\mathbf{h}^{\mathbf{3,0}}_{\textrm{A}}\) | \(\mathbf{h}^{\mathbf{2,1}}_{\textrm {A}}\) | \(\mathbf{h}^{\mathbf{1,2}}_{\textrm{A}}\) | \(\mathbf{h}^{\mathbf{0,3}}_{\textrm {A}}\) | \(\mathbf{h}^{\mathbf{3,1}}_{\textrm{A}}\) | \(\mathbf{h}^{\mathbf{2,2}}_{\textrm {A}}\) | \(\mathbf{h}^{\mathbf{1,3}}_{\textrm{A}}\) | \(\mathbf{h}^{\mathbf{3,2}}_{\textrm {A}}\) | \(\mathbf{h}^{\mathbf{2,3}}_{\textrm{A}}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\mathbb {I}_{3}\) and (i) | 3 | 3 | 2 | 8 | 2 | 1 | 6 | 6 | 1 | 3 | 4 | 3 | 2 | 2 |
(ii.a) | 3 | 3 | 2 | 7 | 2 | 1 | 6 | 6 | 1 | 2 | 4 | 2 | 2 | 2 |
(ii.b) | 3 | 3 | 2 | 6 | 2 | 1 | 6 | 6 | 1 | 2 | 4 | 2 | 2 | 2 |
(iii.a) | 3 | 3 | 2 | 7 | 2 | 1 | 6 | 6 | 1 | 1 | 4 | 1 | 2 | 2 |
(iii.b) | 3 | 3 | 2 | 6 | 2 | 1 | 6 | 6 | 1 | 1 | 4 | 1 | 2 | 2 |
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Angella, D. The Cohomologies of the Iwasawa Manifold and of Its Small Deformations. J Geom Anal 23, 1355–1378 (2013). https://doi.org/10.1007/s12220-011-9291-z
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DOI: https://doi.org/10.1007/s12220-011-9291-z