Inequalities à la Frölicher and cohomological decompositions

Abstract

We study Bott-Chern and Aeppli cohomologies of a vector space endowed with two anti-commuting endomorphisms whose square is zero. In particular, we prove an inequality à la Frölicher relating the dimensions of the Bott-Chern and Aeppli cohomologies to the dimensions of the Dolbeault cohomologies. We prove that the equality in such an inequality à la Frölicher characterizes the validity of the so-called cohomological property of satisfying the $\partial \overline{\partial }$-Lemma. As an application, we study cohomological properties of compact either complex, or symplectic, or, more in general, generalized-complex manifolds.

to appear in J. Noncommut. Geom

References in corpus (5)

• On the $\partial \overline{\partial }$-Lemma and Bott-Chern cohomology
• Autour de la cohomologie de Bott-Chern
• Reformulating Supersymmetry with a Generalized Dolbeault Operator
• New aspects of the ddc-lemma
• Generalized geometry and the Hodge decomposition

Cited by in corpus (13)

• Hodge theory for twisted differentials
• The Frolicher-type inequalities of foliations
• On the deformed Bott-Chern cohomology
• On the cohomology of almost complex and symplectic manifolds and proper surjective maps
• Cohomologies of locally conformally symplectic manifolds and solvmanifolds
• On the Bott-Chern and Aeppli cohomology
• Symplectic cohomologies and deformations
• On non-Kähler degrees of complex manifolds
• Quaternionic Bott-Chern cohomology and existence of HKT metrics
• Formality of the Dolbeault complex and deformations of holomorphic Poisson manifolds
• On second non-HLC degree of closed symplectic manifold
• Residues and currents from singular forms on complex manifolds
• Some Morse-type inequalities for symplectic manifolds