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.
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
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.