On the Bott-Chern and Aeppli cohomology

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 $\mathrm{\partial }\bar{\mathrm{\partial }}$-Lemma.