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 2 If
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 3 If
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 4 If
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.
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