A compact Riemannian manifold is called geometrically formal if the wedge product of two $d$-harmonic forms is $d$-harmonic. Are there any known results for when a non-Kahler compact complex manifold admits a hermitian metric such that the wedge product of two ${\bar{\partial}}$-harmonic forms is ${\bar{\partial}}$-harmonic?



${\bar{\partial}}$-geometrically formal ?
In a paper by S. Torelli and A. Tomassini, "On Dolbeault formality and small deformations" (to appear in Internat. J. Math.), the authors study (geometrically) Dolbeault formality. In particular, they investigate the behaviour of (geometrically) Dolbeault formality under small deformations of the complex structure.


In particular, they prove that these properties are not stable under small deformations. An example is provided on the Nakamura manifold (that is, one of the simplest non-nilpotent solvmanifolds in complex dimension $3$). It is geometrically Dolbeault formal (and so also Dolbeault formal); and it admits small deformations for which there exist non-trivial Dolbeault Massey products (and so they are non-Dolbeault formal).


