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.