# Examples of smooth compact Kähler manifolds with semipositive canonical class

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.

• Any variety of general type for which $K$ is not ample, there are zillions of these. For a random example, take a surface in $\Bbb{P}^3$ of degree $\geq 5$ with some nodes and blow up the nodes.
– abx
Commented Oct 21, 2022 at 5:51

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 $$C_i$$.