Dolbeault closedness of Chern curvature
= Dolbeault closedness of Chern curvature
{c}
{title2=$\Theta\in A^{1,1}(\operatorname{End}E),\quad\bar\partial\Theta=0$}
In a <holomorphic local frame> the Chern matrix is $A=H^{-1}\partial H$. The identity $\partial A+A\wedge A=0$ gives <vector-bundle curvature> $\Theta=\bar\partial A$ of type $(1,1)$. Therefore $\bar\partial_{\operatorname{End}E}\Theta=0$, allowing the <vector-bundle curvature> to represent a bundle-valued Dolbeault class.