I'm looking at Kahler geometry at the moment and admiring how it manages to do so much with clean global algebraic arguments. One of the big exceptions to all this, however, is the proof of the Kahler identities $$ [\Lambda,\overline{\partial}]=-i \partial^\ast, ~~~~~~ [\Lambda,\partial]=-i \overline{\partial}^\ast. $$ In the two standard references, Voisin, and Griff + Harris, the identities are proved using arguments that are local and somewhat analaytic. Does there exists anywhere a nice global algebraic proof?