Vaisman manifolds are complex manifolds which can be endowed with a special type of a Hermitian structure, namely a locally conformally Kähler metric with parallel Lee form. The geometry of Vaisman manifolds is closely related to Kählerian geometry, as these manifolds come endowed with a natural transversally Kähler foliation. However, Vaisman manifolds do not satisfy the dd^c-lemma, therefore it is interesting to study their Bott-Chern cohomology, which is then a refined invariant.
In this talk, I will explain how one can express this cohomology in terms of the basic cohomology with respect to the foliation, and in particular infer that the Bott-Chern numbers and the Dolbeault numbers determine each other. At the same time however, the numerical obstructions to the dd^c-lemma can be arbitrarily high.- This is based on joint work with Alexandra Otiman.