I asked this question at [Maths Stack Exchange](https://math.stackexchange.com/q/194749/39599), but I haven't received any replies yet (I'm not sure how long I should wait before it is acceptable to ask here, assuming there is such a period of time).




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The Wikipedia page for [Weitzenböck identities](http://en.wikipedia.org/wiki/Weitzenb%C3%B6ck_identity) is explicitly example based. I am looking for a reference which takes a more rigorous approach (as well as a discussion of the Bochner technique). In particular, I am interested in references which focus on these identities in complex geometry.


I have already consulted Griffiths & Harris which is mentioned in the article, but it only contains one example. Berger's *A Panoramic View of Riemannian Geometry* doesn't have much more.


My interest in Weitzenböck identities has been motivated by a question arising from the following theorem:



> 
> Let $X$ be a Kähler manifold and $E$ a hermitian holomorphic vector bundle with Chern connection $\nabla$. Then for the Laplacians $\Delta\_{\bar{\partial}} = \bar{\partial}\bar{\partial}^\* + \bar{\partial}^\*\bar{\partial}$, $\Delta\_{\partial} = \partial\partial^\* + \partial^\*\partial$, we have $\Delta\_{\bar{\partial}} = \Delta\_{\partial} + [iF\_{\nabla}, \Lambda]$.
> 
> 
> 


Is $\Delta\_{\bar{\partial}} = \Delta\_{\partial} + [iF\_{\nabla}, \Lambda]$ an example of a Weitzenböck identity?



Weitzenböck Identities
Here is roughly the philosophy of the Weitzenbock technique. (Most of what follows is taken from Berline-Getzler-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](http://www.nd.edu/~lnicolae/Lectures.pdf).


