Is there an example of compact complex manifold, which is formal, but does not admit complex structure satisfying $dd^c$-lemma? 



Formal complex manifold without dd^c
The Hopf surface $S^1 \times S^3$ is formal but does not admit a complex structure satisfying the $dd^c$ lemma. Indeed, a closed four-manifold admitting a complex structure satisfying the $dd^c$ lemma (equivalently the $\partial \bar \partial$ lemma) admits a Kähler structure and thus has even first Betti number.


