QUANTITATIVE AND QUALITATIVE COHOMOLOGICAL PROPERTIES FOR NON-KÄHLER MANIFOLDS

We introduce a “qualitative property” for Bott-Chern cohomology of complex non-Kähler manifolds, which is motivated in view of the study of the algebraic structure of Bott-Chern cohomology. We prove that such a property characterizes the validity of the ∂ ∂ ¯ -Lemma. This follows from a quantitative study of Bott-Chern cohomology. In this context, we also prove a new bound on the dimension of the Bott-Chern cohomology in terms of the Hodge numbers. We also give a generalization of this upper bound, with applications to symplectic cohomologies.