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

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