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.
Best Answer
You can just prove it yourself directly in local holomorphic coordinates. Indeed, the $\overline{\partial}$ Laplacian on functions is equal to $\Delta_{\overline{\partial}}f=g^{i\overline{j}}\partial_i \partial_{\overline{j}}f$. Apply this to $|\partial f|^2=g^{k\overline{\ell}}\partial_k f \partial_{\overline{\ell}}f$ (length squared of $\partial f=(df)^{(1,0)}$, which equals $1/2$ of the usual $|\nabla f|^2$), using if you want local holomorphic normal coordinates for $g$ at a point, and you will immediately get
$$\Delta_{\overline{\partial}}|\partial f|^2=|\nabla_i \nabla_j f|^2+|\nabla_i \nabla_{\overline{j}} f|^2+2\mathrm{Re}\langle \partial f, \partial\Delta_{\overline{\partial}}f\rangle+R^{i\overline{j}}\partial_i f\partial_{\overline{j}}f,$$ where $R^{i\overline{j}}$ is the Ricci curvature of $g$ with the indices raised.
If $g$ is not Kähler, and you define the complex Laplacian by the same formula $g^{i\overline{j}}\partial_i \partial_{\overline{j}}f,$ then a similar result holds, with the Ricci curvature now being one of the several Ricci curvatures of the Chern connection of $g$, and with several new terms involving the torsion of $g$ and its covariant derivative. The calculation is again completely strightforward, using local holomorphic coordinates (not normal anymore!), and using the definitions of covariant derivative and curvature of the Chern connection of $g$.