There are a number of results in geometry which allow to conclude that a certain group vanishes or is bounded under hypotheses on the curvature. For instance, we have:

Theorem 1If is a compact manifold of positive curvature, then .

Another such result is the Kodaira vanishing theorem, which enables one to show that certain cohomology groups of an ample line bundle on a smooth projective variety vanish in characteristic zero.

I’ve been trying to gain an understanding of such results, and it seems that there is a common technique in such arguments. The first strategy is to identify the desired cohomology group (e.g. ) with the kernel of a Laplacian-type operator, by Hodge theory. The second step is to bound below the relevant Laplacian-type operator. In this post, I’d like to try to explain what’s going on, in a special case.

**1. Hodge theory**

It is a classical fact that on a smooth manifold , one can write

In other words, one can use differential forms and the exterior derivative as a replacement for cochains, and one gets a representation of in terms of cocycles (closed forms) modulo coboundaries (exact forms).

Hodge theory enables one to pick out a subspace of closed -forms such that the natural map

is an isomorphism. The idea is that, if one puts a norm on the space of all -forms, then each element in the quotient can be represented by an element of *smallest* norm. In other words, to a cohomology class , we can choose a representative closed form which is *orthogonal* to the subspace of exact forms (by which we are quotienting). That is, we require

with respect to some inner product on the space of -forms. Ideally, for every cohomology class , we should be able to find a *unique* representative which is in the cohomology class and which satisfies the additional condition (1). As a result, we will want to take as the subspace of closed forms satisfying (1); it turns out that this is in fact isomorphic to , as desired.

If one has an *oriented* Riemannian manifold , then one can introduce a metric on the space of all -forms: namely, one introduces a metric in for each , and then a metric on global sections by integrating . In this case, will have an *adjoint* operator that sends a -form to a -form. Then (1) can be rephrased as

This is equivalent to saying that

The main result is:

Theorem 2 (Hodge theorem)Every cohomology class has a unique representative such that . As a result, is isomorphic to the space of forms such that .

In other words, every cohomology class has a representative of smallest length. In fact, it can be stated via a direct sum decomposition of the space of all -forms on :

where is the space of forms in the kernel of both and .

Definition 3A form isharmonicif .

Thus, the Hodge theorem states that the space of harmonic forms is isomorphic to the cohomology . We can give a simpler characterization of harmonic forms.

Definition 4TheHodge Laplacianis given by on forms.

The elements of are precisely the harmonic forms (this is a general fact, easily proved, about inner product spaces), and the Laplacian is a nonnegative, self-adjoint operator on the space of all forms.

Let’s try to see why the Hodge theorem should be believable. As I said, is a self-adjoint, nonnegative operator on forms. It is, moreover, elliptic, and so we should think of it as something like a Fredholm operator. For a Fredholm operator, we get a decomposition

but since is self-adjoint, we have

One part of the direct sum decomposition is the space of harmonic forms, as desired. Since is a subset of (note that the two spaces are orthogonal as ), we can conclude the decomposition (2). From this, the Hodge theorem follows.

The problem, of course, is that the space of smooth -forms on is not a Hilbert space, and the talk of Fredholm operators was a bit imprecise. Fortunately, though, there is a detailed theory of Sobolev spaces which allows one to make the above ideas precise.

**2. The exterior derivative from the connection**

The idea of the Bochner technique is, in effect, “completion of squares.” Namely, we have seen that we can represent cohomology classes on a smooth, oriented Riemannian manifold via elements in . The strategy is now to bound below the nonnegative operator .

In order to do so, we’ll need to construct more second-order, nonnegative operators on . Namely, the theme is to show that

where is a first-order operator (so is nonnegative), and is a lower-order term that can be computed explicitly. If we can bound from below and show that it is positive, it will follow is positive, and we can thus show that the space of harmonic forms vanishes. This is the technique of completion of squares.

The relevant operator in this case is the covariant derivative : this sends a -form to a -valued -form. That is, if is a local orthonormal frame for the tangent bundle, with dual basis , then we have

where is differentiation associated to the Levi-Civita connection on . Here the Einstein summation convention is in effect.

One of the consequences of ‘s being the Levi-Civita connection (in particular, torsion-free), is that is the same as the usual exterior derivative : that is, covariant differentiation becomes exterior differentiation after antisymmetrizing. We can state this formally:

Proposition 5Notation as above,

thus the exterior derivative can be obtained from the covariant derivative.

*Proof:* The strategy is to note that are tensorial, so we may prove the equality (3) desired by working in a local coordinate system that is particularly convenient. Fix a point . To prove (3), we may assume that we have chosen local coordinates around such that are orthonormal at and such that

That is, we are choosing *normal coordinates* such that the metric is to order one the same as the euclidean metric. Equivalently, we are assuming that the Christoffel symbols vanish at .

At the point , we can thus prove (3) by taking . If is, say, a form , then we have

while

So, , and the two are equal as desired.

The above technique, though elementary, is pretty powerful in differential geometry: when one wants to prove that two invariantly defined objects are equal, it may be useful to prove it a point by working in a particularly convenient system of local coordinates *at that point.* This is the basis for the so-called “Kähler identities” that lead to the high degree of structure in the cohomology of a Kähler manifold.

By a similar technique, we can get a clean description of the adjoint :

Proposition 6Notation as above, for the interior product,

This proposition is a corollary of the previous one if we use the Hodge star operator.

Anyway, so is pretty close to , except for the lack of antisymmetrizing. This suggests we might look for a relation between

It will turn out that there is one; however, there is an additional term that comes up because the covariant derivative is not symmetric. This is related to the curvature.

**3. The Weitzenbock formula**

Let us start by studying the Laplacian . Let’s choose any orthonormal frame of vector fields, locally, with dual coframe field . For a -form , we can use the formulas of the previous section to see that

Similarly,

Let us now suppose the are *normal* at a point , i.e. at . If we work only at , then we can simplify some of the terms. Namely, from the first equation, we get (at ):

From the second equation, we get

These two look almost the same, and there is indeed a fair bit of cancellation.

We shall use the following formula. Consider the operators on the exterior algebra given by wedging with and . Then the satisfy the *Clifford relations*

In fact, the operators give an action of an appropriate Clifford algebra on the exterior algebra.

So, anyway, we have proved that, at ,

We can simplify this. When , we get a term . When , we have the sum

at least if we make the further simplifying assumption that at . By the commutator relations, this becomes

However, let’s note that since we are in a good frame which has no covariant derivatives at , we have that the adjoint to is (since it just looks like differentiation in this frame), and consequently, we have proved that .

Definition 7Thecurvature operatoris the operator on the space of forms. Note that this is invariantly defined for an orthonormal frame.

As a result, we can state:

Theorem 8 (Weitzenbock formula)On an oriented Riemannian manifold, we have:

Again, we have proved the Weitzenbock formula purely in a very specific choice of local coordinates, but since the two operators are invariantly defined and global, that is not a problem.

**4. Applications to geometry**

With the “completion of squares” identity established in (7), we can make an observation.

**Meta-observation:** If is a nonnegative operator, then any harmonic form on the compact Riemannian manifold is parallel. If is positive, then there are no nontrivial harmonic forms.

In fact, we notice that the operator is always nonnegative, and the elements in the kernel are precisely the parallel forms. So if is harmonic, then we have

and consequently, taking inner products with , we get:

As a result, if we can bound below (by zero), then we will get very strong results in the cohomology of the manifold.

Theorem 9 (Bochner)If has nonnegative (resp. positive) Ricci curvature, then the curvature operator is nonnegative (resp. positive) on the space of 1-forms. Consequently, under these hypotheses we have

resp.

To prove this result, we will need to study the curvature operator, which was defined in local coordinates as , on one-forms. (The notation is as in the previous section; is a local frame, not necessarily normal.) The key point is the following:

Proposition 10The curvature operator on one-forms can be identified with the Ricci tensor.

*Proof:* The Riemann tensor is given in a local orthonormal frame via four indices (which is defined as the component of in ), and the Ricci tensor is a contraction of it,

The curvature operator, as defined, takes in two tangent vectors and a form and spits out a form. We can thus get it by raising and lowering indices in the Riemann tensor (which does nothing since we are in an orthonormal frame)

The operator is wedging by . Consequently, we can work out the action of the curvature operator:

But acting by multiplies by , while acting by lops off a factor of if it exists and otherwise does nothing. So the only terms occur for , and we have

and this is the Ricci tensor acting on . In fact, we have .

Now it is easy to prove Bochner’s theorem. We have shown that if the Ricci curvature is nonnegative (resp. positive), then the operator on 1-forms is just the Ricci tensor, and consequently nonnegative (resp. positive). This means that a harmonic form is in the kernel of (resp. is zero). So we’ve established the second part of the theorem. In the first case, we’ve shown that harmonic forms are parallel, and since a parallel form is determined by its value at a point, we get the upper bound on .

Incidentally, the result in the case of positive curvature can be strenghtened: by a theorem of Myers and Bonnet, the fundamental group of such a manifold (compact, with positive Ricci curvature) is in fact finite.

February 18, 2012 at 1:54 pm

So what part exactly is the Bochner technique -are there various meanings?

I quickly looked at the wiki link you give on the Kodaira vanishing theorem. They mention Deligne-Illusie. Did you have that in mind when giving the link? I mean, what is the Bochner technique in algebraic geometry, for the Kodaira vanishing theorem in particular? And are there other applications? Is the Deligne-Illusie spectral sequence argument to be understood as a Bochner technique?

I guess another post is coming so you may keep replies for it, but otherwise I’d appreciate any insight.

February 18, 2012 at 2:59 pm

I think the Bochner technique refers to this method of showing that certain cohomology groups vanish by a) representing them as kernels of some Laplacian-like operator and b) writing said Laplacian-like operator as the sum of another nonnegative Laplacian-like operator and some positive term (usually involving the curvature). For instance, that’s how the analytic proof of Kodaira vanishing goes: you use the Kahler identities to write the -operator (the -Laplacian) in terms of the usual Laplacian plus the curvature. I’ll try to explain this next time. I’m pretty sure Deligne-Illusie gave an algebraic proof of Kodaira vanishing for smooth projective varieties (a special case, but not the only case, of compact Kahler manifolds), but I’m not familiar with their methods.

February 20, 2012 at 2:56 pm

I should just add a few comments and references:

http://www.ams.org/journals/bull/1988-19-02/S0273-0979-1988-15679-0/S0273-0979-1988-15679-0.pdf

Bérard gives a nice overview and references of the Bochner technique.

Several riemannian geometry textbooks treat the topic, each with its own interpretation, Jost, Gallot-Hulain-Lafontaine, Petersen.

Gromov’s “Metric Structures for Riemannian and Non-riemannian Spaces” has a chapter on riemannian manifolds of bounded (mostly below) Ricci curvature. It seems that Bishop’s inequality, based on the Bochner formula and method, is basic to much of that study.

Esnault-Viehweg have detailed lectures on Deligne-Illusie. I gather from looking at those and the original paper that the idea of positivity of curvature is present in some way in those arguments, but I have not understood the details.

In any case thanks for the post.

April 13, 2014 at 8:18 pm

Small typo: the adjoint of sends a -form to a -form, not to a -form.