Assume $X$ is smooth. A class $[\omega]$ in $H^{1}(X,\mathcal{PH}_{X})$ will be called numerically base point free iff there exists a proper surjective holomorphic mapping $X\to Y$, $Y$ normal, such that $[\omega]$ is the pull back of a Kähler class on $Y$. This is a stronger condition than being semi-Kähler.

In the non-big case (i.e.: $\int_{X}\omega^{n}=0$), it is straightforward to construct semi-Kähler classes that are not numerically base point free (e.g. on complex tori). On the other hand, it is still unknown whether there exists a smooth projective variety $X$ and a semi-Kähler form $\omega$ which is big without being numerically base point free.