Let be a genus. We might ask when satisfies the following multiplicative property:

**Property:** For any appropriate fiber bundle of manifolds, we have

When is simply connected, this is true for the signature by an old theorem of Chern, Hirzebruch, and Serre.

A special case of the property (1) is that whenever is an even-dimensional complex vector bundle, then we have

for the projectivization: this is because is a fiber bundle whose fibers are odd-dimensional complex projective spaces, which vanish in the cobordism ring.

Ochanine has given a complete characterization of the genera which satisfy this property.

Theorem 1 (Ochanine)A genus annihilates the projectivizations of even-dimensional complex vector bundles if and only if the associated log series is given by an elliptic integral

for for constants .

Such genera are called **elliptic genera.** Observe for instance that in the case , then

so that we get the signature as an example of an elliptic genus (the signature has as logarithm, as we saw in the previous post).

I’d like to try to understand the proof of Ochanine’s theorem in the next couple of posts. In this one, I’ll describe the proof that an elliptic genus in fact annihilates projectivizations of even-dimensional bundles .

**1. Calculations in the projectivization**

Suppose is the genera associated to an elliptic integral: that is, suppose the logarithm is given by the formal power series

where are “generic.”

Let be a manifold, and let be an even-dimensional complex vector bundle. We would like to show that . Let be the natural fiber bundle. First, we note that we have a splitting

where consists of the vectors “tangent to the fiber.” We have

where is the dual to the tautological bundle on . This follows as in the case of complex projective space.

Let be the characteristic power series of the genus , as in the previous post; then

and defines a stable characteristic class for real vector bundles. Then . This is what we want to compute.

Fortunately, we have a pretty good hand on . In fact,

if , and where is in degree . Consequently,

The fundamental class of is given by the fundamental class of times . So, in other words, we have to show:

Lemma 2If is even, then the coefficient of in vanishes.

**2. Reduction to a fact about elliptic functions**

So, we’ve reduced to a lemma on computing the characteristic class of . Let be formal Chern roots of ; then the formal Chern roots for are because the Chern classes in ordinary cohomology are additive under the tensor product of line bundles.

In particular, we get

We have to show that there is no coefficient of in this.

Here Ochanine’s clever idea is to use the fact that (this is the relation that satisfies in terms of the Chern classes) and then to use facts about elliptic functions. Namely, let be a variable, and let be formal variables. We can write

where is of degree in . Substituting in and , we find

So we want to show that the coefficient of in is zero. We will do this by doing so in the universal case, where we can introduce denominators.

Lemma 3If is even, the coefficient of in is zero.

So the strategy will be to work out what looks like. We can do this in the case of indeterminates, as before. Since is a polynomial of degree , we can do this using Lagrange interpolation. Namely, we have for each ,

Since we have obtained values of , we can thus write

Note that it is important that we are working in the universal case (with indeterminates) here, because we have introduced denominators. The coefficient of in this expression is given by

**3. Some facts about elliptic integrals**

So, we need to prove that this is zero. We have used essentially nothing so far; it is here that where we will need the specific facts about elliptic integrals. It follows that we need to prove:

Lemma 4If is an elliptic integral , then

.

We will prove this by treating as a function and applying to complex numbers .

The functions (for the Weierstrass function for an appropriate lattice) can be taken as the inverses to elliptic integrals of the given form, and in particular for some choice of lattice (at least if are suitably nondegenerate). It follows that we need to prove

The idea is that the function

is, for even, an honest elliptic function, with simple poles at the . The function (as a square root of an even function) was not necessarily elliptic; it was only elliptic up to factors of . But since is even, the product of all these is elliptic. Now, for any elliptic function , the sum of the residues is zero: this follows by integrating the differential . But the sum of the residues of is given by the sum in question.

Anyway, this proves one half of Ochaine’s theorem. It turns out that much more is true (namely, one has multiplicative in suitable fiber bundles where the fiber is a spin manifold); this follows from work of Bott and Taubes.

## Leave a Reply