Let $X$ be a compact complex manifold. A Hermitian metric $\omega$ is balanced if $d\omega^{n-1}=0$, where $n=\dim_{\mathbf{C}} X$. By a theorem of Alessandrini-Basanelli, this class of Hermitian manifolds is preserved under bimeromorphic modification. In particular, manifolds in the Fujiki class $\mathcal{C}$ (i.e., bimeromorphic to compact Kähler manifolds) and hence, Moishezon manifolds, support balanced metrics.
Explicit examples of balanced manifolds include the Iwasawa threefold and Hironaka's example (which is Moishezon).
An older result of Deligne-Griffiths-Morgan-Sullivan tells us that the $dd^c$-lemma is preserved under bimeromorphic modification. Finally, a theorem of Gauduchon informs us that the $dd^c$-lemma holds on $(1,1)$-forms if and only if $b_1 = 2h^{0,1}$.
There are manifolds which support the $dd^c$-lemma but are not in the Fujiki class $\mathcal{C}$ (see, for instance, Remark 1.17 of Angella's thesis).
Question: Are there compact balanced manifolds that do not satisfy the $dd^c$-lemma on $(1,1)$-forms?
Note that if $X$ is the Iwasawa threefold, page 44 of Angella's thesis shows that $b_1 = 4 = 2 h^{0,1} = 2 \cdot 2$, and so the Iwasawa threefold satisfies the $dd^c$-lemma on $(1,1)$-forms.
Another non-example to bear in mind: The Calabi-Eckman manifold $\mathbf{S}^3 \times \mathbf{S}^3$ does not satisfy the $dd^c$-lemma on $(1,1)$-forms, since $b_1 \neq 2 h^{0,1}$. However, Michelsohn showed that Calabi-Eckman manifolds never support balanced metrics.
