= Solution
On the <Kähler manifold> define $L\alpha=\omega\wedge\alpha$ using the <Kähler form>; the operator $\Lambda=L^*$ is the <adjoint Lefschetz operator>, the pointwise contraction adjoint to wedging with $\omega$. It lowers bidegree by $(1,1)$. Set
$$
\Delta_d=dd^*+d^*d,\qquad \Delta_\partial=\partial\partial^*+\partial^*\partial,
$$
and keep $\Delta_{\bar\partial}$ as above. These are the <Hodge Laplacian> and the holomorphic and antiholomorphic Laplacians respectively. The permitted identity among the <Kähler identities> $[\Lambda,\partial]=i\bar\partial^*$ gives $\bar\partial^*=-i[\Lambda,\partial]$. Expanding rather than assuming the desired anticommutation,
$$
\partial\bar\partial^*+\bar\partial^*\partial
=-i\bigl(\partial\Lambda\partial-\partial^2\Lambda+\Lambda\partial^2-\partial\Lambda\partial\bigr)=0,
$$
because $\partial^2=0$. Hence \b[$\boxed{\partial\bar\partial^*+\bar\partial^*\partial=0}$].
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