What is the Weitzenböck formula for the $\bar\partial$-Laplacian

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 $g$.

The formula is called the Bochner-Kodaira formula. It involves as all other Weitzenböck Formulars the curvature of the bundle, this time a holomorphic twisting bundle. Striking consequences are for example the Kodaira-Vanishing theorem.

You may take a look for example in Berline Getzler Vergne "Heat kernels and the dirac operator", page 135, Proposition 3.71 or in Lawson, Michelson "Spin geometry" Thm D.12

The proofs are always quite similar, you use that the symmetries of the curvature tensors and the clifford multiplication cancel each other to reduce the term which is not the connection laplace.

Weitzenböck Identities for Kahler manifolds is as follows

Let $\Theta $ be the Chern curvature form of $h$ for the Hermitian line bundle over Kahler manifold $(X,\omega)$

For any smooth section $\xi ∈ Γ(T^{0,1}X \otimes L)$ we have,

$$(\bar{\partial^*}\bar\partial+\bar\partial\bar{\partial^*})\xi=\bar{\nabla^*}\bar\nabla\xi+(\Theta+Ric(\omega))(\xi,.)$$

and

$$(\bar{\partial^*}\bar\partial+\bar\partial\bar{\partial^*})\xi={\nabla^*}\nabla\xi+(\Theta(\xi,.)-(tr_\omega\Theta)\xi$$

Note that here Ricci make sense as follows

If we take a section $s\in H^0(X,L)$ and $\xi=\alpha_{\bar i}d\bar z^i\otimes s$ then Ricci can be defined as follows

$$Ric(\xi,.)=R_{i\bar j}\alpha_{\bar i}d\bar z^j\otimes s$$

I learn it from paper of Tian