Weitzenböck Identities

Question

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).

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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?

Answer 1

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.

Answer 2

The most general version of Weitzenbock identities (with coefficients in appropriate universal enveloping algebras) is due to Uwe Semmelmann and Gregor Weingart: http://arxiv.org/abs/math/0702031 "The Weitzenböck Machine".

Answer 3

As far as references are concerned, you can read about the Bochner technique in H.Wu, "The Bochner technique in differential geometry". For some applications of the Lichnerowicz formula you can check Ch.3 of Berline, Getzler, Vergne, "Heat Kernels and Dirac operators", and Ch.3, 5 of T.Friedrich's "Dirac Operators in Riemannian geometry".

Answer 4

I don't know of any precise definition of Weitzenbock identities, which is closely related to or also known as the Bochner technique. It is basically a way to write some invariantly defined second order linear differential operator $D$ on a vector bundle over a manifold (often a complex one) in two different ways. One way, usually written in terms of a coboundary operator and its adjoint, shows that the kernel of the operator is an interesting topological or holomorphic invariant (usually some kind of cohomology). The other way is in the form $D = P^*P + R$, where $P$ is a linear first order operator and $R$ is a naturally defined geometric tensor, usually called some kind of curvature. This allows the kernel of $D$ to be studied under suitable assumptions on $R$ (usually that it is positive or nonnegative definite) by studying the kernel of $P$.

Also, you get from one form of the operator to the other by commuting covariant derivatives, which is how the $0$-th order curvature term $R$ appears.

Answer 5

You may also be amused by Bérard's article "From vanishing theorems to estimating theorems: the Bochner technique revisited", Link