Bott–Chern Formality and Massey Products on Strong Kähler with Torsion and Kähler Solvmanifolds

We study the interplay between geometrically-Bott–Chern-formal metrics and SKT metrics. We prove that a 6-dimensional nilmanifold endowed with a invariant complex structure admits an SKT metric if and only if it is geometrically-Bott–Chern-formal. We also provide some partial results in higher dimensions for nilmanifolds endowed with a class of suitable complex structures. Furthermore, we prove that any Kähler solvmanifold is geometrically formal. Finally, we explicitly construct lattices for a complex solvable Lie group in the list of Nakamura (J Differ Geom 10:85–112, 1975) on which we provide a non vanishing quadruple ABC-Massey product.