It may be difficult to give some special and non-trivial examples of Kähler cones.The examples I know are the following:

- for complex tori, the Kähler cone is just the set of positive hermitian matrices;
- for a Kähler manifold $M$ with $h^{1,1}(M)=1$, then the Kähler cone is just $\mathbb{R}_{+}$.

Thus in order to give a non-trivial example, firstly we must require $h^{1,1}(M)>1$. I would like to see an example such that:

- the cone equation or the boundary of the cone is clear;
- $c_{1}(M)>0$, or the sign of $c_{1}(M)$ is not definite (are there some intresting manifolds of this type?).

Of course, any explicit shape will be welcome.