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 ${\mathbb{C}}^n{⋉}_{\phi }N$ 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 $\partial \overline{\partial }$-Lemma is not strongly-closed under deformations of the complex structure.

References in corpus (7)

• On the $\partial \overline{\partial }$-Lemma and Bott-Chern cohomology
• Reformulating Supersymmetry with a Generalized Dolbeault Operator
• The cohomologies of the Iwasawa manifold and of its small deformations
• Geometry of nilmanifolds with left-invariant complex structure and deformations in the large
• On Cohomological Decomposability of Almost-Kähler Structures
• Cohomologies of certain orbifolds
• Degree of non-Kählerianity for 6-dimensional nilmanifolds

Cited by in corpus (17)

• On small deformations of balanced manifolds
• On the Structure of Double Complexes
• On local stabilities of $p$-Kähler structures
• Compact Complex Manifolds with Small Gauduchon Cone
• Dolbeault and Bott-Chern formalities: deformations and $\partial \overline{\partial }$-lemma
• Leray-Hirsch theorem and blow-up formula for Dolbeault cohomology
• Dolbeault and $J$-invariant cohomologies on almost complex manifolds
• On the deformed Bott-Chern cohomology
• Global stability of the Pluriclosed flow on compact simply-connected simple Lie groups of rank two
• On Bott-Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms
• On non-Kähler degrees of complex manifolds
• Deformed Aeppli cohomology: canonical deformations and jumping formulas
• Formality of the Dolbeault complex and deformations of holomorphic Poisson manifolds
• Geometric formalities along the Chern-Ricci flow
• The $\partial \bar{\partial }$-lemma under surjective maps
• SYZ mirror symmetry of solvmanifolds
• Some computations on trivial canonical-bundle solvmanifolds