 Lefschetz-Dolbeault curvature commutator

For a Hermitian holomorphic bundle over a Kähler manifold, let $\Theta$ act by exterior multiplication. The bundle-valued Kähler identities give $[\Lambda ,{\Delta }_{\bar{\partial }}]=-i(\Theta \wedge -{)}^{\ast }$, and taking adjoints yields $[{\Delta }_{\bar{\partial }},L]=i\Theta \wedge -$. This measures the failure of the bundle-valued Dolbeault Laplacian to commute with the Lefschetz operator.