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Is there an example of two different complex manifolds that have the same de Rham cohomology and Dolbeault cohomology but different Bott-Chern/Aeppli cohomology?

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    $\begingroup$ I don't know the answer, but I suggest looking in Angella's Cohomological Aspects in Complex non-Kähler Geometry. $\endgroup$ Jul 30, 2016 at 5:22
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    $\begingroup$ Thank you for the lead/hint. Angella does show an example of two different classes(which he calls iia and iib ) of deformations of the Iwasawa manifold in the table on p. 49 which have different Bott-Chern/Aeppli cohomology and the same DeRham and Dolbeault cohomologies. You've basically answered my question. $\endgroup$ Jul 30, 2016 at 16:27
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    $\begingroup$ An interesting question is whether this can happen already on compact complex surfaces. In all the cases, Bott-Chern/Aeppli is determined by DeRham and Dolbeault, except possibly for non-Kahler elliptic surfaces where in general the dimensions of Bott-Chern/Aeppli are not known. $\endgroup$
    – YangMills
    Jul 30, 2016 at 17:29
  • $\begingroup$ @Andrew McHugh: I think it would be useful to have an outline of the example in an answer below. Would you mind writing up such an answer? If not, I can do it. $\endgroup$ Jul 30, 2016 at 18:46
  • $\begingroup$ @MichaelAlbanese : If you would be so kind as to write up such an answer it would be appreciated. Thank you for your help. $\endgroup$ Jul 31, 2016 at 15:56

2 Answers 2

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I learnt all of the following from section $3.2$ of Angella's Cohomological Aspects in Complex Non-Kähler Geometry.


Let $R$ be a commutative ring with identity. The three-dimensional Heisenberg group over $R$ is

$$\mathbb{H}(3, R) = \left\{\begin{bmatrix} 1 & z^1 & z^3\\ 0 & 1 & z^2\\ 0 & 0 & 1\end{bmatrix} : z^1, z^2, z^3 \in R\right\}.$$

The Iwasawa manifold $\mathbb{I}_3$ is the quotient of $\mathbb{H}(3, \mathbb{C})$ by the discrete subgroup $\mathbb{H}(3, \mathbb{Z}[i])$ acting on the left, i.e. $\mathbb{I}_3 := \mathbb{H}(3, \mathbb{Z}[i])\setminus\mathbb{H}(3, \mathbb{C})$. It is a compact complex three-dimensional manifold which is holomorphically parallelisable and not of Kähler type.

The small deformations of the Iwasawa manifold were classified by Nakamura in Complex Parallelisable Manifolds and their Small Deformations. There are three classes, corresponding to the possible Hodge numbers of the deformations. Nakamura explicitly determined which small deformations gave rise to which class. Furthermore, the conjugate Hodge numbers are constant within a given class and they differ between classes.

Angella showed that the deformations of class (ii) further decompose into two subclasses: (ii.a) and (ii.b). The Bott-Chern numbers and Aeppli numbers are constant within a given subclass, but they differ between subclasses. In particular, if $X_a$ and $X_b$ are small deformations of $\mathbb{I}_3$ in subclasses (ii.a) and (ii.b) repectively, then

$$h^{2,2}_{\text{BC}}(X_a) = 7,\qquad h^{2,2}_{\text{BC}}(X_b) = 6,\qquad h^{p,q}_{\text{BC}}(X_a) = h^{p,q}_{\text{BC}}(X_b)\ \text{for}\ (p,q) \neq (2, 2).$$

The Hodge star operator gives rise to an isomorphism between $(p, q)$ Bott-Chern cohomology and $(n-q, n-p)$ Aeppli cohomology, so $h^{p,q}_{\text{BC}} = h^{n-q,n-p}_{\text{A}}$. Therefore, relationship between the Aeppli numbers of $X_a$ and $X_b$ is

$$h^{1,1}_{\text{A}}(X_a) = 7,\qquad h^{1,1}_{\text{A}}(X_b) = 6,\qquad h^{p,q}_{\text{A}}(X_a) = h^{p,q}_{\text{A}}(X_b)\ \text{for}\ (p,q) \neq (1, 1).$$

So, $X_a$ and $X_b$ have different Bott-Chern numbers and different Aeppli numbers, but as $X_a$ and $X_b$ are both deformations of class (ii), they have the same Hodge numbers (and conjugate Hodge numbers). Finally, by Ehresmann's Theorem, $X_a$ and $X_b$ are diffeomorphic so they have the same Betti numbers.

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As for compact complex surfaces, the Bott-Chern/Aeppli cohomologies are determined by de Rham/Dolbeault cohomologies:this is contained in Lemma 2.3 in: Teleman, A.: The pseudo-effective cone of a non-Kählerian surface and applications. Math. Ann. 335(4), 965–989 (2006), arXiv:0704.2948.

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