A closed $(1,1)$-form $\theta$ is said to be:

• (i)

  semipositive if $\theta_{\mathcal{X}}\in N^{1}(\mathcal{X}/S)$ is nef for some (or, equivalently, any) determination $\mathcal{X}$ of $\theta$;

• (ii)

  $\mathcal{X}$-positive if $\mathcal{X}\in\mathcal{M}_{X}$ is a determination of $\theta$ and $\theta_{\mathcal{X}}\in N^{1}(\mathcal{X}/S)$ is ample.

A model metric $\|\cdot\|$ on a line bundle $L$ is said to be semipositive if the curvature form $c_{1}(L,\|\cdot\|)$ is semipositive.