Inequalities �� la Fr��licher and cohomological decompositions
Inequalities �� la Fr��licher and cohomological decompositions
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 �� la Fr��licher 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 �� la Fr��licher 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
- Scuola Normale Superiore Italy
- University of Florence Italy
- University of Parma Italy
Mathematics - Differential Geometry, symplectic, generalized-complex, Mathematics - Complex Variables, Symplectic manifolds (general theory), generalized complex structures, Aeppli cohomology, Aeppli cohomology; Bott-Chern cohomology; Generalized-complex; Symplectic; ∂∂-lemma; Geometry and Topology; Mathematical Physics; Algebra and Number Theory, 510, Compact complex \(n\)-folds, Compact Kähler manifolds: generalizations, classification, compact complex manifolds, partial derivative(partial derivative)over-bar-lemma, Differential Geometry (math.DG), Mathematics - Symplectic Geometry, 32Q99, 53D05, 53D18, Bott-Chern cohomology, FOS: Mathematics, symplectic manifolds, Symplectic Geometry (math.SG), Complex Variables (math.CV), Mathematics - Differential Geometry; Mathematics - Differential Geometry; Mathematics - Complex Variables; Mathematics - Symplectic Geometry; 32Q99, 53D05, 53D18, Generalized geometries (à la Hitchin), \(\partial \overline{\partial}\)-lemma
Mathematics - Differential Geometry, symplectic, generalized-complex, Mathematics - Complex Variables, Symplectic manifolds (general theory), generalized complex structures, Aeppli cohomology, Aeppli cohomology; Bott-Chern cohomology; Generalized-complex; Symplectic; ∂∂-lemma; Geometry and Topology; Mathematical Physics; Algebra and Number Theory, 510, Compact complex \(n\)-folds, Compact Kähler manifolds: generalizations, classification, compact complex manifolds, partial derivative(partial derivative)over-bar-lemma, Differential Geometry (math.DG), Mathematics - Symplectic Geometry, 32Q99, 53D05, 53D18, Bott-Chern cohomology, FOS: Mathematics, symplectic manifolds, Symplectic Geometry (math.SG), Complex Variables (math.CV), Mathematics - Differential Geometry; Mathematics - Differential Geometry; Mathematics - Complex Variables; Mathematics - Symplectic Geometry; 32Q99, 53D05, 53D18, Generalized geometries (à la Hitchin), \(\partial \overline{\partial}\)-lemma
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