In this book Geometric and Algebraic Topological Methods in Quantum
Mechanics, G. Giachetta,L. Mangiarotti,Gennadiĭ Aleksandrovich
Sardanashvili page 181

in remark 2.6.2, it has been written that if $X $ be a simply
connected non-compact manifold then then local Kahler potential can be
glued into global one, and if $X$ is not simply connected then local
potential still exists on an open subset $U$ obtained from $X$ by
deleting a real submanifold of lower dimension

Recently A. Loi showed that

Let$ (M, ω) $ be a homogeneous Kahler manifold. Then the following
are equivalent:

(a)$ M$ is contractible.

(b)$ (M, ω)$ admits a global Kahler potential.

(c) $(M, ω) $admits a Berezin quantization.

If $\omega$ is a real $d$-exact $(1,1)$-form, then $\omega$ is $\sqrt{-1}∂∂$-exact , i.e. there exits a real
function $u$ such that locally $\omega = i∂∂u$

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

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

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.

**In Sasakian cone we have global Kahler potential:**

A compact Riemannian manifold $(S,g)$ is Sasakian if and only if its metric cone $(C(S)=R_{>0}\times S, \bar g=dr^2+r^2g)$ is Kahler. which Sasakian manifolds are the odd dimentional view of Kahler manifolds

Then if $S$ be Sasakian, then the cone $C(S)$ has

$$\omega=\frac{1}{2}\sqrt {-1}\partial\bar\partial r^2$$

the function $\frac{1}{2}r^2$ is hence a **global Kahler potential** for the cone metric