It is well-known that the Weitzenböck formula for the real Laplacian is $$\frac12 Δ|∇f|2=|Hessf|2+⟨∇f,∇Δf⟩+Ricci(∇f,∇f)$$ where $Hess$ denotes the Hessian tensor of $f$. and $\nabla f$ denotes the gradient vector of $f$, $Ricci$ denotes the Ricci curvature of the manifold $M$.

If $\Delta_{\bar\partial}$ denotes the $\bar\partial$-Laplacian, it is well-known that it is half of the real Laplacian. So I am wondering is there any formula of the Weitzenböck formula in complex coordinates. (Assume the manifold is Kähler). Apparently one can devided the above formula by 2 to the one, but the expression I want should be expressed by $f_{i\bar j}$ and etc.

ps. The Comparison Geometry of Ricci Curvature, by Shunhui Zhu, 221-262 had a very nice introduction to this formula in real case. http://library.msri.org/books/Book30/contents.html

However I am not familiar with Kaehler case, for example, I dont know the such a formula can be derived in the same fashion as in Zhu's paper? Any book or paper with detailed calculation would be helpful.