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It is well-known that the Weitzenböck formula for the real Laplacian is $$ \Delta |\nabla f|^2 =|\operatorname{Hess} f|^2 + \langle \nabla f, \nabla \Delta f\rangle + \operatorname{Ricci}(\nabla f, \nabla f) $$

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). The expression I want should be expressed by $f_{i\bar j}$ etc. Any book or paper with explicit proof would be helpful!

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Also on MathOverflow –  Martin Jan 31 '13 at 0:27

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