I asked this question at Maths Stack Exchange, 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).
The Wikipedia page for Weitzenböck identities 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?
Best Answer
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.