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

For a holomorphic local frame $e$ with $h_e=h(e,e)>0$, this form isIt is a closed real (1, 1)-form representing the First Chern class. A positive Hermitian metric makes it a positive real (1, 1)-form. Replacing $h$ by $e^{-u}h$ adds $(i/(2\pi ))\partial \bar{\partial }u$ to the form.