I asked this question at <a href="http://math.stackexchange.com/q/194749/39599">Maths Stack Exchange</a>, 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 <a href="http://en.wikipedia.org/wiki/Weitzenb%C3%B6ck_identity">Weitzenböck identities</a> 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?