Bott–Chern cohomology of solvmanifolds

Abstract

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...

Cohomological Aspects on Complex and Symplectic Manifolds

“…The analog result for the complex Bott-Chern cohomology is not true, see e.g., [4,Remark 3.6]. The previous Theorem lead us to the following quantitative characterization of the Hard Lefschetz condition in dimension 4.…”

“…Notice that there exist special classes of complex manifolds where the dimensions of the Bott-Chern (and by duality Aeppli) cohomology groups can be computed explicitly by means of suitable sub-complexes of the complex of forms (see [4]) making this result concrete in studying the ∂∂-lemma.…”

On local stabilities of -Kähler structures

“…From (10), we have where G BC and G A are the associated Green's operators of BC and A , respectively. Here BC is defined in (7) and A is the second Kodaira-Spencer operator (often also called Aeppli Laplacian) A = ∂ * ∂ * ∂∂ + ∂∂∂ * ∂ * + ∂∂ * ∂∂ * + ∂∂ * ∂∂ * + ∂∂ * + ∂∂ * .…”

Stability of holomorphically parallelizable manifolds

“…In fact, Kählerness for nilmanifolds is characterized by ∂∂-Lemma, see [11,Theorem 1,Corollary], which is in turn characterized in terms of Bott-Chern cohomology, [5,Theorem B]. On the other hand, several results concerning the computation of Bott-Chern cohomology for nilmanifolds and solvmanifolds are known, see, e.g., [4,2] and the references therein. When restricting to the class of nilmanifolds, Kählerness is a closed properties under deformations.…”