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 $b^1 = 2h^{1,0}$.