Hi, 
as the title says i'm wondering if there's a "simple" and known commutation relation between the following two differential operators. Let $E$ be a holomorphic vector bundle over a compact kahler manifold $X$, fix a Hermitian metric $h$ on $E$ and a kahler metric $g$ on $X$. Denote with $D=D^{'}+D^{''}$ the associated Chern connection and its (1,0) and (0,1) parts. Denote with $\delta=\delta^{'}+\delta^{''}$ the adjoint respect to $h$ of $D$.
So my question is what is $[D^{''},\delta^{'}]$? Is there a relation like kahler relations?

Thank you in advance.