# Weitzenböck Identities

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?

• It is usually recommended that one waits a bit longer than a day, say a week or so, but I think in this case it is fine and you already received quite a bit of input. See this thread on meta.MO. May I suggest that you post and accept a short answer in a few days, indicating that you received a satisfactory answer on MO so that the question here is considered answered here? – t.b. Sep 14 '12 at 13:57
• @t.b.: I wasn't sure whether this question was appropriate for MO or not so I asked it here first. If ever I am in this situation again, I will wait a bit longer. Once I accept an answer on MO I will make sure I post something about it here in an answer. – Michael Albanese Sep 14 '12 at 16:04

This question has been asked and answered on MathOverflow. I have replicated the accepted answer by Liviu Nicolaescu below.

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.