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?

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 kform $\alpha \in P^k$ we have: $\ast L^j (\alpha) = (1)^\frac{k(k+1)}{2} \frac{j!}{(nkj)!} L^{nkj}I(\alpha)$ 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 121122. 


Since the Kähler identities do not involve coordinates, it is natural to expect that there is a coordinatefree proof. However, it is completely unreasonable to expect that the coordinatefree proof is in any way more elegant. Here is my rationale: if you read a proof using coordinates, and \it{this proof does not use any special property of the chosen coordinates}, then you can translate the argument into a coordinatefree one. But the argument for the Kaehler identities is NOT: choose an arbitrary coordinate system and do a stupid calculation. If this were the case, Deligne, Sullivan, Griffiths and Morgan would surely have avoided coordinates. Instead, the argument is (in my opinion this argument is marvelous):
The standard proof of point 1 is a bit messy, and my suggestion is to make it slicker by using the symmetries of $C^n$. Observe that all expressions on $C^n$ are translationinvariant and invariant under $U(n)$. Moreover, the effect of scaling $C^n$ and conjugation is easy to figure out. These properties should be enough to force the Kähler identities. In a completely coordinatefree proof, you do no longer have these special coordinates at hand, and thus I expect the proof, though possible, to be much more complicated. EDIT: as RdN points out, Huybrechts gives a coordinatefree proof in his book. Huybrecht's proof is about as elegant and complicated as the standard proof. 


Actually, there is an argument that doesn't use neither Weil identities nor flat up to a second order coordinates, instead, it uses the symplectic Hodge star, invented by Brylinski. When you have a 2ndimensional manifold $M$ with the nondegenerate 2form $\omega$, you can define an operator $*_s:\Lambda^k(M) \mapsto \Lambda^{2nk}(M)$ by the standard identity $\alpha \wedge *_s\beta = \omega(\alpha,\beta)\omega^n$. Then, for a closed $\omega$ a really simple computation (you can do it only for the case of two variables, and then use induction on dimension) in Darboux coordinates shows that, up to some sign ($(1)^{k+1}$, i guess), $*_sd*_s=[\Lambda, d].$ It is written in the Brylinski's article "A differential complex for Poisson manifolds". And then you need to observe that for a Kaehler $M$ symplectic and Riemannian Hodge stars differ by an action of $I$. The Kaehler identity $d^*=[\Lambda, d^c]$ follows from that. 

