For a Kahler manifold $M$, and a smooth vector bundle $E$ over $M$, let us denote by $A^{(p,q)} := \Omega^{(p,q)} \otimes E$ the bundle of forms with values in $E$. Now with respect to a choice of connection on $E$, we can extend $d$ to a mapping d$ _E: A^{(p,q)} \to A^{(p+1,q+1)}$. Moreover, we can extend $\partial$, and $\overline{\partial}$, to mappings $\partial_E: A^{(p,q)} \to A^{(p+1,q)}$, and $\overline{\partial}_E: A^{(p,q)} \to A^{(p,q+1)}$ respectively. As is wellknown, if we also assume that $E$ is holomorphic, then we can choose a connection such that $\overline{\partial}_E: A^{(0,\bullet)} \to A^{(0,\bullet)}$ is a chain complex. However, the same cannot be said for the sequences ${\partial}_E: A^{(\bullet,0)} \to A^{(\bullet,0)}$, and d$ _E: A^{(\bullet,\bullet)} \to A^{(\bullet,\bullet)}$, meaning that in general we have no analogue for holomorphic Dolbeault, or de Rham, cohomologies. We do have, however, natural analogues of the Kahler identities. What I would like to know is, are these still of any real interest, given that they can no longer be used to prove equality of holomorphic, antiholomorphic, and de Rham cohomologies? For example, in Voisin and Huybrecht's books they do not even appear.
They can be used to prove the vanishing theorems (and more). KodairaNakano vanishing theorem and Kodaira embedding theorem follow from these identities. One proves that the difference of the $\partial$ and $\bar\partial$Laplacians is a commutator of Hodge $\Lambda$ operator and the curvature. This commutator happens to be positive or negative when the bundle is positive or negative; vanishing of cohomology follows immediately, because the Laplacians themselves are positive operators.

2$\begingroup$ Demailly goes through all the details of this in Chapter VII of the good book (wwwfourier.ujfgrenoble.fr/~demailly/manuscripts/agbook.pdf), including what look like the commutation relations the OP asked about. $\endgroup$ – Gunnar Þór Magnússon Nov 12 '13 at 17:10