On a compact complex manifold X, we prove a Frölicher-type inequality for Bott-Chern cohomology and we show that the equality holds if and only if X satisfies the ∂∂-Lemma.

Abstract. We prove that, for some classes of complex nilmanifolds, the Bott-Chern cohomology is completely determined by the Lie algebra associated to 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 in [26] by M. Schweitzer.

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We study conditions under which sub-complexes of a double complex of vector spaces allow to compute the Bott-Chern cohomology. We are especially aimed at studying the Bott-Chern cohomology of special classes of solvmanifolds, namely, complex parallelizable solvmanifolds and solvmanifolds of splitting type. More precisely, we can construct explicit finite-dimensional double complexes that allow to compute the Bott-Chern cohomology of compact quotients of complex Lie groups, respectively, of some Lie groups of the type C n ⋉ϕ N where N is nilpotent. As an application, we compute the Bott-Chern cohomology of the complex parallelizable Nakamura manifold and of the completely-solvable Nakamura manifold. In particular, the latter shows that the property of satisfying the ∂∂-Lemma is not strongly-closed under deformations of the complex structure. Contents2010 Mathematics Subject Classification. 22E25; 53C30; 57T15; 53C55; 53D05. to be a completing useful tool besides the de Rham and the Dolbeault cohomologies. In this spirit, in [10], it is shown that an inequalityà la Frölicher, involving just the dimensions of the Bott-Chern cohomology and of the de Rham cohomology, holds true on any compact complex manifold, and further allows to characterize the validity of the ∂∂-Lemma (namely, the very special cohomological property that every ∂-closed ∂-closed d-exact form is ∂∂-exact too, see, e.g. [30]).A compact complex manifold satisfies the ∂∂-Lemma if and only if the Bott-Chern cohomology is naturally isomorphic to the Dolbeault cohomology, [30, Remark 5.16]. Therefore, since compact Kähler manifolds satisfy the ∂∂-Lemma because of the Kähler identities, [30, Lemma 5.11], the Bott-Chern cohomology is particularly interesting in studying complex non-Kähler manifolds.In non-Kähler geometry, a very fruitful source of examples is provided by the class of nilmanifolds and solvmanifolds, namely, compact quotients of connected simply-connected nilpotent, respectively solvable, Lie groups by co-compact discrete subgroups. For instance, the geometry of nilmanifolds can be often reduced to the study of the associated Lie algebras, [22,61,14]. On the other hand, nilmanifolds do not admit too strong geometric structures, [15,36]. More precisely, on a nilmanifold, the finite-dimensional sub-complex of left-invariant forms (namely, the forms being invariant for the action of the Lie group on itself given by left-translations) suffices in computing the de Rham cohomology, [56,38]. Whenever the nilmanifold is endowed with a suitable left-invariant complex structure, also the Dolbeault cohomology, [62,26,23,60,61], and the Bott-Chern cohomology, [4], can be computed by means of just left-invariant forms.Instead, for solvmanifolds, the left-invariant forms are usually not enough to recover the whole de Rham cohomology: an example is the non-completely-solvable solvmanifold provided in [28, Corollary 4.2]. The de Rham cohomology of solvmanifolds has been studied by several authors, e.g. A. Hattori [38], G. D. Mostow [54], S. Console and A. F...

Abstract. We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class VII and for compact complex surfaces diffeomorphic to solvmanifolds.