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

conjectureis 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 :)

exactlywhat you mean by «homogeneous space». – Mariano Suárez-Alvarez♦ May 25 at 19:33