= Curvature difference of two Chern connections
{title2=$F_{D_1}-F_{D_2}=\bar\partial a$}
Let $D_1$ and $D_2$ be <Chern connections> on the same <holomorphic vector bundle>, possibly for different <Hermitian metrics>, and put $a=D_1-D_2$. Then $a$ has type $(1,0)$ and
$$
F_{D_1}-F_{D_2}=\bar\partial_{\operatorname{End}E}a.
$$
Indeed, the <curvature difference formula> has only types $(2,0)$ and $(1,1)$; both Chern curvatures have type $(1,1)$, so the $(2,0)$ part vanishes and the remaining part is $D_2^{0,1}a=\bar\partial_{\operatorname{End}E}a$.
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