Lefschetz-Dolbeault curvature commutator ($[{\Delta }_{\bar{\partial }},L]=i\Theta \wedge -$)

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.

The Lefschetz operator commutes with the bundle-valued Dolbeault Laplacian on the entire smooth form space exactly when the Chern connection is flat. The Lefschetz-Dolbeault curvature commutator gives one direction. Conversely test its curvature-wedge operator on degree-zero sections with arbitrary prescribed values at any point to force every vector-bundle curvature endomorphism to vanish.