In a previous post, I began discussing a theorem of Ochanine:

Theorem 1 (Ochanine) A genus ${\phi: \Omega_{SO} \rightarrow \Lambda}$ annihilates the projectivization ${\mathbb{P}(E)}$ of every even-dimensional complex bundle ${E \rightarrow M}$ if and only if the logarithm of ${\phi}$ is an elliptic integral

$\displaystyle g(x) = \int_0^x (1 - 2\delta u^2 + \epsilon u^4)^{-1/2} du.$

In the previous post, we described Ochanine’s proof that a genus whose logarithm is an elliptic integral (a so-called elliptic genus) annihilated any such projectivization. The proof relied on some computations in the projectivization and then some trickery with elliptic functions. The purpose of this post is to prove the converse: a genus with a suitably large kernel comes from an elliptic integral. (more…)

Let ${\phi: \Omega_{SO} \rightarrow \Lambda}$ be a genus. We might ask when ${\phi}$ satisfies the following multiplicative property:

Property: For any appropriate fiber bundle ${F \rightarrow E \rightarrow B}$ of manifolds, we have

$\displaystyle \phi(E) = \phi(B) \phi(F). \ \ \ \ \ (1)$

When ${B}$ 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 ${E \rightarrow B}$ is an even-dimensional complex vector bundle, then we have

$\displaystyle \phi(\mathbb{P}(E)) = 0,$

for ${\mathbb{P}(E)}$ the projectivization: this is because ${\mathbb{P}(E) \rightarrow B}$ 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 ${\phi}$ annihilates the projectivizations ${\mathbb{P}(E)}$ of even-dimensional complex vector bundles if and only if the associated log series ${g(x) = \sum \frac{\phi(\mathbb{CP}^{2i})}{2i+1} x^{2i+1}}$ is given by an elliptic integral

$\displaystyle g(x) = \int_0^x Q(u)^{-1/2} du,$

for ${Q(u) = 1 - 2\delta u^2 + \epsilon u^4}$ for constants ${\delta, \epsilon}$.

Such genera are called elliptic genera. Observe for instance that in the case ${\epsilon = 1, \delta = 1}$, then

$\displaystyle g(x) = \int_0^x \frac{du}{1 - u^2} = \tanh^{-1}(u),$

so that we get the signature as an example of an elliptic genus (the signature has ${\tanh^{-1}}$ 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 ${\mathbb{P}(E)}$ of even-dimensional bundles ${E}$. (more…)

Let ${\Lambda}$ be a ${\mathbb{Q}}$-algebra. A genus is a homomorphism

$\displaystyle \phi: \Omega_{SO} \rightarrow \Lambda,$

where ${\Omega_{SO} }$ is the oriented cobordism ring. In other words, a genus ${\phi}$ assigns to every compact, oriented manifold ${M}$ an element ${\phi(M) \in \Lambda}$. This satisfies the conditions:

1. ${\phi(M \sqcup M') = \phi(M) + \phi(M')}$.
2. ${\phi(M \times M') = \phi(M) \phi(M')}$.
3. ${\phi(\partial N) =0 }$ for any manifold-with-boundary ${N}$.

A fundamental example of a genus is the signature ${\sigma}$, which assigns to every manifold ${M}$ of dimension ${4k}$ the signature of the quadratic form on ${H^{2k}(M; \mathbb{R})}$. (Also, ${\sigma}$ is zero on manifolds whose dimension is not divisible by four.)