Normalized curvature form of a Hermitian holomorphic line bundle (${\omega }_{(L,h)}=i{F}_h/(2\pi )$)

For a holomorphic line bundle with Hermitian metric $h$, its normalized Chern curvature iswhere $e$ is a holomorphic local frame and $h_e=h(e,e)>0$. A change of frame multiplies $h_e$ by the squared absolute value of a holomorphic unit; its logarithm has zero mixed derivative. The local expressions therefore define a closed real (1, 1)-form representing the First Chern class.

Positive curvature is an additional condition on the metric, distinct from positivity of each fibre norm. On the trivial line over $\mathbb{C}$, the metric $h_e=1$ is positive definite but has zero curvature. The metric $h_e=e^{-|z{|}^2}$ instead gives ${\omega }_{(L,h)}=idz\wedge d\bar{z}/(2\pi )$, a positive real (1, 1)-form. Globally, admitting such a positively curved metric is the definition of a positive holomorphic line bundle.

Replacing $h$ by $e^{-u}h$ adds $(i/(2\pi ))\partial \bar{\partial }u$. Thus the normalized Chern curvature changes by an exact differential form, while its First Chern class stays fixed.