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](https://mathoverflow.net/questions/59479/partial-bar-partial-lemma-for-contractible-domains). 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.