Examples of smooth compact Kähler manifolds with semipositive canonical class

Question

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.

Answer 1

As explained by abx in his comment, any variety of general type whose canonical class $K = \det \Omega^1$ is not ample provides an example of what you are looking for.

If you need an explicit series of examples, take $2n$ lines $L_1, \ldots L_{2n}$ in general position in the complex projective plane $\mathbb{P}^2$ and consider the double cover $f \colon X \to \mathbb{P}^2$ branched over the union of these lines.

The surface $X$ contains $n(2n-1)$ nodes, corresponding to the pairwise intersection points of the lines $L_i$. Blowing up these nodes, one obtains a smooth surface $\bar{X}$ containing $n(2n-1)$ smooth rational curves $C_1, \ldots, C_{n(2n-1)}$, over which the restriction of $K_{\bar{X}}$ is trivial.

On the other hand, when $n \geq 4$ the canonical class of the singular model $X$ is ample, hence $c_1(K_{\bar{X}})$ is stricly positive on the complement of the $C_i$.