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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?

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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 at 10:45
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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 at 10:49

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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).

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