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. 



Examples of compact complex manifolds for which the $dd^c$ lemma does not hold
Gauduchon proved that a compact complex manifold satisfies the $dd^c$ lemma for $(1, 1)$-forms if and only if $b\_1 = 2h^{0,1}$. As a compact complex surface is Kähler if and only if $b\_1$ is even, a compact complex non-Kähler surface does not satisfy the $dd^c$ lemma.


The reference for the above result of Gauduchon is the paper 


Gauduchon, P. *La classe de Chern pluriharmonique d’un fibré en droites*, C. R. Acad. Sci. Paris Sér. A-B **282** (1976), no. 9, Aii, A479–A482


a link to which can be found in the comments below. Better still, an elementary proof of this result is given in the exercises from a course taught by Valentino Tosatti. A link to the exercises, complete with solutions, also appears in the comments. Thanks to YangMills for both links.


