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