# ${\bar{\partial}}$-geometrically formal ?

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?

-
I think you can argue as in the paper D. Kotschick, S. Terzic: Geometric formality of homogeneous spaces and biquotients. Pacific J. Math. 249 (2011). On a homogeneous space with a homogeneous Hermitian metric, $\bar{\partial}$-harmonic forms are invariant. If the Dolbeault cohomology is an exterior algebra on two odd-degree generators, the only non-trivial wedge to check is a constant multiple of the volume form and hence $\bar{\partial}$-harmonic. This should imply the geometric $\bar{\partial}$-formality of Calabi-Eckmann manifolds $S^1\times S^{2n-1}$. – Matthias Wendt Nov 15 '14 at 10:45
For further examples related to (geometric) $\bar{\partial}$-formality, you can check the papers (D. Angella, G. Dloussky, A. Tomassini: On Bott-Chern cohomology of compact complex surfaces) and (L.A. Cordero, M. Fernandez, A. Gray, L. Ugarte: Dolbeault homotopy theory and compact nilmanifolds) – Matthias Wendt Nov 15 '14 at 10:49

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).