Bundle-valued Kähler identities
= Bundle-valued Kähler identities
{title2=$[\Lambda,\bar\partial_E]=-i(D'_E)^*$}
On a <Kähler manifold> the <Chern connection> components satisfy $[\Lambda,\bar\partial_E]=-i(D'_E)^*$ and $[\Lambda,D'_E]=i\bar\partial_E^*$. Holomorphic normal coordinates and a <Chern connection in a normal holomorphic frame> reduce them to the scalar identities pointwise. Closedness of the Kähler form also gives $[\Lambda,\bar\partial_E^*]=0$.