Let $(M,\omega)$ be a Quantized closed Kaehler manifold then by Koderia embedding theorem , $M$ must be algebraicly projective i.e, we have the embedding


$$\phi: (M,\omega)\to  (\mathbb CP^N, \omega\_{FS})$$ So $$\phi^\*\omega\_{FS}=\omega+\frac{i}{2\pi}\partial\bar \partial \epsilon $$


where $\epsilon$ is a smooth function and is defined as follows: 


**Definition of $\epsilon$ function**: Let $\pi:(L,h)\to (M,\omega)$ be a prequantum line bundle and let $x\in M$ and $q\in L^+$ such that $\pi(q)=x$ and $H$ is the Hilbert space of global holomorphic sections ($h$ is hermitian metric). Then we can write $s(x)=\delta\_q(s)q$ where $\delta\_q:H\to \mathbb C$ is a linear continous functional of $s$ and by Riesz theorem $\delta\_q(s)=\langle s,e\_q \rangle\_h$ where $e\_q\in H$ and thus $s(x)= \langle s,e\_q\rangle\_hq$ and we can define the real valued function on $M$ by the formula 


$$\epsilon(x)=h(q,q)\left \| e\_q \right \|\_h^2$$



> 
> Now the **conjecture** is that, if $\epsilon$ be constant then $M$ is
>  homogeneous space? Is there any counterexample or proof for it?
> 
> 
> 


***This question is known as Andrea Loi's conjecture in his doctoral thesis***


Peter Crooks gave a counterexample and I removed the part *simply connected*, I want to see this conjecture still is conjecture :)



A question about Quantized closed Kaehler manifolds
I do not believe this is the case. If you have any smooth complex submanifold $X$ of $\mathbb{CP}^n$, then the Kahler form on $\mathbb{CP}^n$ pulls-back to a Kahler form $\omega$ on $X$ (so $\phi^\*\omega\_{FS}=\omega$). You can take $X$ to be any smooth projective variety. These are not all simply-connected. For instance, let $X$ be an elliptic curve.


