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?

As I have not received any comments or answers, I have posted this question on MathOverflow. Those interested may be able to find some helpful answers there.

  • 1
    $\begingroup$ 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? $\endgroup$ – t.b. Sep 14 '12 at 13:57
  • $\begingroup$ @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. $\endgroup$ – 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.


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