Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 15 / 4 / a / Solution

On the Kähler manifold define $L\alpha =\omega \wedge \alpha$ using the Kähler form; the operator $\Lambda =L^{\ast }$ is the adjoint Lefschetz operator, the pointwise contraction adjoint to wedging with $\omega$. It lowers bidegree by $(1,1)$. Setand 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 $[\Lambda ,\partial ]=i{\bar{\partial }}^{\ast }$ gives ${\bar{\partial }}^{\ast }=-i[\Lambda ,\partial ]$. Expanding rather than assuming the desired anticommutation,because ${\partial }^2=0$. Hence $\boxed{\partial {\bar{\partial }}^{\ast }+{\bar{\partial }}^{\ast }\partial =0}$.