I am looking for a reference that details in precision what happens when dealing with subbundles of Hermitian holomorphic vector bundles. This topic is covered in Kobayashi's Differential geometry of complex vector bundles (Chapter I, section 6) but I find it extremely hard to understand in detail as the text skips many steps.
I have searched in the literature that I have available (Wells' Differential Analysis on Complex Manifolds, Huybrechts' Complex geometry) but I cannot find a good exposition on the second fundamental form of a subbundle of a Hermitian vector bundle, and furthermore how to express the connection and curvatures of the subbundles in terms of the ambient one.
For a brief exposition of what I am looking for: suppose $E\rightarrow M$ is a Hermitian holomorphic subbundle with the Hermitian connection $D$ (metric compatible and $D^{(0,1)}=0$). I want to express the curvature of the Hermitian bundle $E$ in terms on the curvatures of the subbundles $S,S^\perp \subset E$, where $S$ is a holomorphic subbundle.
My aim is to understand the following formula in detail:
$$ R = \left(\begin{array}{cc} R_S - B\wedge B^* & D^{(1,0)}B\\ -D^{(0,1)}B^* & R_{S^\perp} - B^*\wedge B \end{array}\right) $$ where $B$ is the second fundamental form of $S$ within $E$.