> 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][1]

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][2]


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][3]

=========

> 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]: http://arxiv.org/pdf/1502.00011
  [2]: https://www.ma.utexas.edu/users/rjain/CalabiConjecture.pdf
  [3]: http://arxiv.org/pdf/math/0501406.pdf