A smooth metric is said to be ample if it is defined by a model $(\mathfrak{X},\mathfrak{L},e)$ such that the restriction $\mathfrak{L}_{\tilde{K}}$ of $\mathfrak{L}$ to the closed fiber $\mathfrak{X}_{\tilde{K}}$ is ample. The Weil metric on the line bundle $\mathscr{O}(1)$ on the projective space is ample. The proof given above of the existence of smooth metrics shows, more precisely, that ample line bundles admit ample metrics, and that the pull-back of a smooth ample metric by an immersion is a smooth ample metric.

A smooth metric is said to be semi-positive if it can be defined on a model $(\mathfrak{X},\mathfrak{L},e)$ such that the restriction $\mathfrak{L}_{\tilde{K}}$ of $\mathfrak{L}$ to the closed fiber $\mathfrak{X}_{\tilde{K}}$ is numerically effective : for any projective curve $C\subset\mathfrak{X}_{\tilde{K}}$, the degree of $\mathfrak{L}_{\tilde{K}}$ is non-negative. Ample metrics are semi-positive.

The pull-back of a smooth semi-positive metric by any morphism is semi-positive.