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 well-known, 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, anti-holomorphic, and de Rham cohomologies? For example, in Voisin and Huybrecht's books they do not even appear.



Are the Kahler Identities for a Holomorphic Vector Bundle Actually Interesting?
They can be used to prove the vanishing theorems (and more). Kodaira-Nakano 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.


