If $\omega$ is a real  $d$-exact $(1,1)$-form, then $α$ is $\sqrt{-1}∂∂$-exact , i.e. there exits a real
function $u \in C^{\infty}(X,\mathbb{R})$ such that $α = i∂∂u$

If $ M$ is a compact Kahler manifold then $M$ cannot have a global Kahler potential.see page 6 [here][1] 

for non-compact case we don't have in general $dd^c $-lemma

In general Gauduchon showed that for a compact complex manifold $M$, $dd^c$
-lemma for $(1, 1)$-forms
is equivalent to the equality $b^1 = 2h^{1,0}$.

 Deligne  showed that Complex manifolds satisfying
the $dd^c$-lemma are formal [see New aspects of the ddc
-lemma][2]

If $\omega$ be a Kahler current instead of Kahler form, then still $dd^c $
-lemma holds

Let $X$ be a compact complex manifold. The equality

$$\sum_{p+q=k}(dim_\mathbb CH_{BC}^{p,q}(X)+dim_A^{p,q}(X))=2dim_\mathbb CH_{dR}^k(X,\mathbb C)$$


holds for every $k \in N$ if and only if $X$ satisfies the $∂\bar∂$-Lemma.
here , $H_{BC}^{•,•} (X)$, is Bott-Chern  and $H^{•,•}_A(X) $ is Aeppli cohomologies.


  [1]: https://www.ma.utexas.edu/users/rjain/CalabiConjecture.pdf
  [2]: http://arxiv.org/pdf/math/0501406.pdf