Consideration of the curvature allows to define positivity notions for metrized line bundles. Namely, one says that a smooth metrized line bundle $\overline{L}$ is positive (resp. semi-positive) if its curvature form is a positive (resp. a non-negative) $(1,1)$-form. This means that for any point $x\in\mathrm{X}$, the hermitian form $c_{1}(\overline{L})_{x}$ on the complex tangent space $\mathrm{T}_{x}\mathrm{X}$ is positive definite (resp. non-negative). As a crucial example, the line bundle $\mathscr{O}(1)$ with its Fubini-Study metric is positive. The pull-back of a positive metrized line bundle by an immersion is positive. In particular, ample line bundles can be endowed with a positive smooth metric ; Kodaira’s embedding theorem asserts the converse : if a line bundle possesses a positive smooth metric, then it is ample.

The pull-back of a semi-positive metrized line bundle by any morphism is still semi-positive. If $\overline{L}$ is semi-positive, then the measure $c_{1}(\overline{L})^{n}$ is a positive measure.