 Bundle-valued Kähler identities

On a Kähler manifold the Chern connection components satisfy $[\Lambda ,{\bar{\partial }}_E]=-i(D_E^{\prime }{)}^{\ast }$ and $[\Lambda ,D_E^{\prime }]=i{\bar{\partial }}_E^{\ast }$. 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 $[\Lambda ,{\bar{\partial }}_E^{\ast }]=0$.