Our convention is $d=\partial+\bar{\partial}$, $d^{c}=\frac{\sqrt{-1}}{2\pi}(-\partial+\bar{\partial})$, so $dd^{c}=\frac{\sqrt{-1}}{\pi}\partial\bar{\partial}$. The relation between Kähler potentials and Kähler metrics is $\omega_{\phi}=\omega+dd^{c}\phi$. Alternatively, we think of a Kähler metric in terms of local absolute potentials, meaning $\omega=dd^{c}\varphi$ for locally defined psh functions $\varphi$. Given a Hermitian metric $h$ on a line bundle $L$, its curvature form is $-dd^{c}\log h^{1/2}$ in the class $c_{1}(L)$.