Our convention is $d=\partial+\bar{\partial}$, $d^{c}=\frac{\sqrt{-1}}{2\pi}(-\partial+\bar{\partial})$, so $dd^{c}=\frac{\sqrt{-1}}{\pi}\partial\bar{\partial}$. Given a Hermitian metric $h$ on a line bundle $L$, its curvature form is $-dd^{c}\log h^{1/2}$ in the class $c_{1}(L)$.