= Lefschetz-Dolbeault curvature commutator
{c}
{title2=$[\Delta_{\bar\partial},L]=i\Theta\wedge-$}
For a Hermitian holomorphic bundle over a <Kähler manifold>, let $\Theta$ act by exterior multiplication. The <bundle-valued Kähler identities> give $[\Lambda,\Delta_{\bar\partial}]=-i(\Theta\wedge-)^*$, 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.
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