Let $M$ denote a Kähler manifold with Kähler form $\omega$. Let $L$ denote the Lefschetz operator acting on on $A^k(M,\mathbb{C}) \to A^{k+2}(M,\mathbb{C})$ such that $L(\eta)=\omega \wedge \eta$.
Now let $\Lambda = L^{*}$, that is $\Lambda$ defines the adjoint of the Lefschetz operator.
I am looking at the proof that $[\Lambda , \overline{\partial}]=- i \partial^{*}$, which is given by Proposition 6.5 in 'Hodge Theory and Complex Algebraic Geometry' by Claire Voisin. It uses the fact that locally $\omega$ is equal to the standard Kähler form on $\mathbb{C}^n$ up to order two in $z$, and that $\Lambda$ and $\overline{\partial}$ are zero and first order differential operators respectively. It then says that it suffices to prove the proposition for the standard Kähler form on $\mathbb{C}^n$ as .
I don't understand this, can somebody explan this logic. The second order terms don't disappear after applying a first order differential onto them, so I don't understand why we can neglect them in the proof?