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 SM and on the product of a Kodaira surface with an Inoue surface.