On cohomological and formal properties of Strong Kähler with torsion and astheno-Kähler metrics

We provide families of compact astheno-Kähler nilmanifolds and we study the behaviour of the complex blowup of such manifolds. We prove that the existence of an astheno-Kähler metric satisfying an extra differential condition is not preserved by blowup. We also study the interplay between Strong Kähler with torsion metrics and geometrically Bott-Chern metrics. We show that Fino-Parton-Salamon nilmanifolds are geometrically-Bott-Chern-formal, whereas we obtain negative results on the product of two copies of primary Kodaira surface, Inoue surface of type ${\mathcal{S}}_M$ and on the product of a Kodaira surface with an Inoue surface.