Lefschetz operator preserves harmonic forms
= Lefschetz operator preserves harmonic forms
On a <Kähler manifold>, the <Lefschetz operator of a Kähler manifold> $L=\omega\wedge-$ commutes with the <Dolbeault Laplacian>:
$$
[L,\Delta_{\bar\partial}]=0.
$$
Consequently, if $\alpha$ is $\Delta_{\bar\partial}$-harmonic, then every $\alpha\wedge\omega^k=L^k\alpha$ is also harmonic.