# Variation of the Chern connection according to the variation of hermitian metric

Whats is the relation between the Chern connections of tow Hermitian metrics in a holomorphic vector bundle?

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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 $$\Theta(E)=\bar\partial\bigl(\overline H^{-1}\partial\overline H\bigr).$$ – diverietti Feb 25 '12 at 15:36
I am looking for a comparison formula,not the local frame representation of Chern connection or its curvature:) – Hamed Feb 25 '12 at 16:35
Which kind of comparison? Infinitesimal variation? Or arbitrary comparison between two random metrics? – diverietti Feb 25 '12 at 17:59
I believe Hamed is asking for a precise description of the difference between the two connections. – Deane Yang Feb 25 '12 at 20:36
Deane Yang is right. – Hamed Feb 26 '12 at 5:15