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 . (more…)