Suppose $(M, \omega)$ is a Kähler manifold, and I am looking for examples of compact Kähler manifolds with $c_1(K_{M}) \geq 0$. A $(1,1)$ form $\eta$ is semi-positive if in local coordinates its metric tensor $(g_{i \bar j})$ is positive semi-definite.

Apparently Calabi-Yau manifolds serve as abundant examples. However, I wonder are there any easy examples where it is not equal to or $> 0$ everywhere. Also wondering is there a name for compact manifolds with its canonical class $>0$ as opposed to Fano manifolds.