Let $M$ be an even dimensional smooth manifold.
I want to find an example $M$ satisfying the following conditions,

1. $M$ admits a Kahler structure.
2. $\omega$ is a symplectic form on $M$.
3. There is no Kahler structure $(M,\omega',J)$ such that $[\omega']=[\omega] \in H^2(M;\mathbb{R})$

(I mean, want to find an example $M$ such that "Kahler cone $\neq$ symplectic cone" with non-empty Kahler cone.)

Thank you in advance.