 Local formula for the Chern connection on a line bundle

Let $e$ be a holomorphic local frame of a Hermitian holomorphic line bundle and put $h=h(e,e)$. The Chern connection and its curvature are locallyThus $F_{\nabla }$ has type $(1,1)$. Because the connection is unitary, ${\bar{F}}_{\nabla }=-F_{\nabla }$, so $i{F}_{\nabla }$ is a real (1, 1)-form.