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I read a local formula like this: Under some sort of Kahler condition, $$Ric(h)=-i\partial\bar\partial \log \det(h_{\alpha\bar\beta})$$

where $h_{\alpha\bar\beta}$ is the matrix of the Hermitian metric $H$ on a local trivialization of a holomorphic vector bundle $E$ over a complex manifold $M$. If $E$ is the holomorphic tangent bundle, then $Ric(h)$ is the Ricci curvature of the manifold $M$ with Hermitian metric $H$, and we know this formula works if $M$ is Kahler. My question is:

What is the exact expression of Kahler condition here if we replace $T^{(1,0)}M$ by a general Hermitian and holomorphic bundle $E$?

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