Local formula for the Chern connection on a line bundle
= 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 locally
$$
\nabla e=(\partial\log h)e,
\qquad
F_\nabla=\bar\partial\partial\log h=-\partial\bar\partial\log h.
$$
Thus $F_\nabla$ has type $(1,1)$. Because the connection is <unitary>, $\bar F_\nabla=-F_\nabla$, so $iF_\nabla$ is a <real (1, 1)-form>.