Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 118 / 4 / b

Represent the middle-degree class by its unique harmonic differential form $h$. The Lefschetz commutator on degree $k$ is $[\Lambda ,L]=(n-k)\mathrm{id}$, so at $k=n$ the operators commute. Since the adjoint Lefschetz operator is the adjoint of the Lefschetz operator of a Kähler manifold,Both $Lh$ and $\Lambda h$ are harmonic. A harmonic form represents zero in de Rham cohomology exactly when the form itself is zero. Consequently $L[\alpha ]=0$ if and only if $Lh=0$, which by the norm identity is equivalent to $\Lambda h=0$, hence to $\Lambda [\alpha ]=0$. ThereforeThis identifies the two middle-degree descriptions of a primitive differential form on a Kähler manifold at the cohomology level.