TITLE: Examples of compact complex manifolds for which the $dd^c$ lemma does not hold
QUESTION [8 upvotes]: The well known $dd^{c}$ lemma in complex geometry claimed that 

Let $X$ be a compact Kähler manifold. Let $p,q\ge 1$. Let $\eta$ be a
  $(p,q)$ form on $X$ and assume $\eta$ is $d$-exact. Then there exists
  a $(p-1,q-1)$ form $\beta$ such that $$ \eta=dd^{c}\beta $$ If $p=q$
  and $\eta$ is real, then we may take $\beta$ to be real.

(Source, Griffths and Harris, page 149, sign of $\beta$ corrected). The traditional proof breaks down for non-Kähler complex manifolds because in this case $\Delta_{\overline{\partial}}=\Delta_{\partial}=\frac{1}{2}\Delta_{d}$ may no longer holds. And then the lemma seems too good to be true in general. 
I am wondering if we have examples of compact complex manifolds for which we know the $dd^{c}$ lemma does not hold. Here is a counter-example for an open domain. The Hodge decomposition still holds for any elliptic pseudodifferential operator, and we can still take its complex conjugate. But I am kind of lost how to construct a specific counter-example. The topic is classical, so I am not sure if this is common sense among expert circle already. 
If the statement is true for certain non-Kähler complex manifolds, then I suspect we need to show that
$$
(\eta, (\partial\overline{\partial}^{*}+\overline{\partial}^{*}\partial)\eta)\le 0, \forall \eta\in \mathcal{A}^{p,q}(M)
$$
in these cases. I am wondering what does this imply for $M$ from an analysis perspective.

REPLY [2 votes]: Below are just some more examples illustrating Donu's answer.
In the paper A new geometric construction of compact complex manifolds in any dimension Laurent Meersseman constructed a large family of complex non-Kahler manifolds. Each of this manifolds is (a deformation of) a principle torus bundle over toric variety.
The same manifolds are known in the toric topology under the name moment-angle-manifolds (see, e.g., Real quadrics in ℂn, complex manifolds and convex polytopes and Complex-analytic structures on moment-angle manifolds) and it is has been demonstrated that many of them have non-trivial triple Massey products.