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?



Global Algebraic Proof of the Kahler Identities?
I don't know of a proof that would really be characterized as global, and if I saw one I would immediately try to figure out how it's really local. There is, however a proof along different lines than that in GH or Voison, and seems more enlightening to me. Huybrechts in his Complex Geometry book gives one that is more representation theoretic/linear algebraic. First he proves a formula (due to Weil?): for a primitive k-form $\alpha \in P^k$ we have:


$\ast L^j (\alpha) = (-1)^\frac{k(k+1)}{2} \frac{j!}{(n-k-j)!} L^{n-k-j}I(\alpha)$


Here n is of course the complex dimension of the manifold, and $I$ is the operator induced on forms by the almost complex structure (page 37 of Huybrechts). This underappreciated formula seems to only be found in this book.


He then uses this together with the purely linear algebraic Lefschetz decomposition on $\alpha$ and $d\alpha$ to prove the commutation relation in the form $[\Lambda, d] = -d^c$. The proof is a bit calculational though. See page 121-122.


