Here is roughly the  philosophy of  the Weitzenbock technique. (Most of what follows is taken from  Berline Getzelr Vergne book.)

Suppose that  $E_0,E_1\to M$ are  vector bundles on an oriented Riemann manifolds  $M$ equipped with  hermitian metrics.  Denote by $C^\infty(E_i)$ the space of  smooth sections of $E_i$. 

A symmetric 2nd order  differential operator  $L: C^\infty(E_0)\to C^\infty(E_0)$ is called a  *generalized Laplacian*  on $E_0$ if its principal symbol  $\sigma_L$ coincides with the principal symbol of a Laplacian. Concretely this means  the following. 

 For a smooth function $f\in C^\infty(M)$ denote by $M_f$ the linear operator $C^\infty(E_0)\to C^\infty( E_0)$ defined by the multiplication with $f$. Then $L$ is a generalized Laplacian if for any $f_0,f_1\in C^\infty(M)$  and any  $u\in C^\infty(E_0)$ we have

$$ [\; [\; L,M_{f_0}\; ], M_{f_1}\; ]u = -2g( df_0,df_1)\cdot u $$

where $[-,-]$ denotes the commutator  of two operators.   Equivalently, this means

$$[[L,M_{f_0}],M_{f_1}]=-2M_{g(df_0,df_1)}. $$

One can show that if $L$ is a generalized Laplacian on $E_0$, then there exists a connection   $\nabla$ on $E_0$, compatible with the metric on $E_0$, and a symmetric  endomorphism $W$ of $E_0$ such that

$$ L =\nabla^*\nabla +W. $$

The classical Weitzenbock formulas give explicit descriptions to the Weitzenbock remainder  $W$ and the connection $\nabla$.   



Usually  the  generalized Laplacians are  obtained through  *Dirac type* operators which  are first order differential operators $D: C^\infty(E_0)\to C^\infty(E_1)$ such that both operators $D^\ast D$ and  $D D^\ast$ are generalized Laplacians on $E_0$ and respectively $E_1$.  We can rewrite this in a compact form by using the operator

$$\mathscr{D}: C^\infty(E_0)\oplus C^\infty(E_1)\to C^\infty(E_0)\oplus C^\infty(E_1), $$

$$\mathscr{D}(u_0\oplus u_1)= (D^*u_1)\oplus (D u_0). $$

Then $D$ is Dirac type  iff $\mathscr{D}^2$ is a generalized Laplacian.



The Weitzenbock remainders of $D^\ast D$ and $D D^\ast$ involve curvature terms. If the Weitzenbock remainder of $D^*D$ happens to be a positive endomorphism of $E_0$, then one can conclude that

$$\ker D=\ker D^\ast D=0. $$

The  Hodge-Dolbeault operator 

$$\frac{1}{\sqrt{2}}(\bar{\partial}+\bar{\partial}^*): \Omega^{0,even}(M)\to \Omega^{0,odd}(M) $$

on a Kahler manifold $M$ is a Dirac type operator. For more details and examples  you can check Sec. 10.1 and Chap 11 of [my lecture notes][1].
 


  [1]: http://www.nd.edu/~lnicolae/Lectures.pdf