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

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.

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

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

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

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.

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

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.

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

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

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

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.

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

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.