= Flatness criterion from Lefschetz commutation
{title2=$[L,\Delta_{\bar\partial}]=0\Longleftrightarrow\Theta=0$}
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.
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