Let $E\to X$ a holomorphic hermitian vector bundle on a complex manifold $X$. Let $h$ the hermitian metric on $X$ and choose a local holomorphic trivialization for $E$. Call $H$ the hermitian matrix with smooth coefficients representing the metric along the fibers of $E$ over the given trivialization. Then, the Chern curvature tensor $\Theta(E,h)$ is given by $$ \Theta(E)=\bar\partial\bigl(\overline H^{-1}\partial\overline H\bigr). $$
– diveriettiFeb 25 '12 at 15:36

I am looking for a comparison formula,not the local frame representation of Chern connection or its curvature:)
– HamedFeb 25 '12 at 16:35

Which kind of comparison? Infinitesimal variation? Or arbitrary comparison between two random metrics?
– diveriettiFeb 25 '12 at 17:59

I believe Hamed is asking for a precise description of the difference between the two connections.
– Deane YangFeb 25 '12 at 20:36