Inserted: 8 nov 2016
Last Updated: 17 nov 2017
Journal: Bielefeld Geometry & Topology Days
This survey summarizes the results discussed in a talk at "Bielefeld Geometry & Topology Days" held at Bielefeld University in July 2015. We are interested in quantitative and qualitative properties of Bott-Chern cohomology. We announce new results obtained in D. Angella, N. Tardini, Quantitative and qualitative cohomological properties for non-K\"ahler manifolds, arXiv:1507.07108 jointly with Nicoletta Tardini. In particular, we prove an upper bound of the dimensions of Bott-Chern cohomology in terms of Hodge numbers. We also introduce a notion, called Schweitzer qualitative property, which encodes the existence of a non-degenerate pairing in Bott-Chern cohomology, like the Poincar\'e duality for the de Rham cohomology. We show that this property characterizes the $\partial\overline\partial$-Lemma.