Chern connection component adjoint
= Chern connection component adjoint
{c}
{title2=$(D'_E)^*=-*_{E^*}D'_{E^*}*_E$}
For the $(1,0)$ component of a <Chern connection>, $(D'_E)^*=-*_{E^*}D'_{E^*}*_E$, where the middle operator is the corresponding component of the dual <Chern connection>. This is the holomorphic-degree counterpart of the <bundle-valued Dolbeault adjoint>.