Ochanine’s theorem on elliptic genera

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 .

 

The classical zero-free region for L-functions

Apologies for the lack of posts lately; it’s been a busy semester. This post is essentially my notes for a talk I gave in my analytic number theory class.

Our goal is to obtain bounds on the distribution of prime numbers, that is, on functions of the form . The closely related function

turns out to be amenable to study by analytic means; here is the von Mangolt function,

Bounds on will imply corresponding bounds on by fairly straightforward arguments. For instance, the prime number theorem is equivalent to .

The function is naturally connected to the -function in view of the formula

In other words, is the Dirichlet series associated to the function . Using the theory of Mellin inversion, we can recover partial sums by integration of along a vertical line. That is, we have

at least for , in which case the integral converges. Under hypotheses on the poles of (equivalently, on the zeros of ), we can shift the contour appropriately, and estimate the integral to derive the prime number theorem.

 

The index theorem II

Let be a compact manifold, vector bundles over . Last time, I sketched the definition of what it means for a differential operator

to be elliptic: the associated symbol

was required to be an isomorphism outside the zero section. The goal of the index theorem is to use this symbol to compute the index of , which we saw last time was a well-defined number

invariant under continuous perturbations of  through elliptic operators (by general facts about Fredholm operators).

The main observation is that , in virtue of its symbol, determines an element of . (Henceforth, we shall identify the tangent bundle with the cotangent bundle , by choice of a Riemannian metric; the specific metric is not really important since -theory is a homotopy invariant.) In fact, we have that is the (reduced) -theory of the Thom space, so it is equivalently for the unit ball bundle and the unit sphere bundle. But we have seen that to give an element of is the same as giving a pair of vector bundles on together with an isomorphism on , modulo certain relations.

Observation: The symbol of an elliptic operator determines an element in .

 

The index theorem I

This is the first in a series of posts about the Atiyah-Singer index theorem.

Let be finite-dimensional vector spaces (over , say), and consider the space of linear maps . To each , we can assign two numbers: the dimension of the kernel and the dimension of the cokernel . These are obviously nonconstant, and not even locally constant. However, the difference is constant in .

This was a trivial observation, but it leads to something deeper. More generally, let’s consider an operator (such as, eventually, a differential operator), on an infinite-dimensional Hilbert space. Choose separable, infinite-dimensional Hilbert spaces ; while they are abstractly isomorphic, we don’t necessarily want to choose an isomorphism between them. Consider a bounded linear operator .

Definition 1  is Fredholm if is “invertible up to compact operators,” i.e. there is a bounded operator such that and are compact.

In other words, if one forms the category of Hilbert spaces and bounded operators, and quotients by the ideal (in this category) of compact operators, then is invertible in the quotient category. It thus follows that adding a compact operator does not change Fredholmness: in particular, is Fredholm if and is compact.

Fredholm operators are the appropriate setting for generalizing the small bit of linear algebra I mentioned earlier. In fact,

Proposition 2 A Fredholm operator has a finite-dimensional kernel and cokernel.

Proof: In fact, let be the kernel. Then if , we have

where is a “pseudoinverse” to as above. If we let range over the elements of of norm one, then the right-hand-side ranges over a compact set by assumption. But a locally compact Banach space is finite-dimensional, so is finite-dimensional. Taking adjoints, we can similarly see that the cokernel is finite-dimensional (because the adjoint is also Fredholm).

The space of Fredholm operators between a pair of separable, infinite-dimensional Hilbert spaces is interesting. For instance, it has the homotopy type of , so it is a representing space for K-theory. In particular, the space of its connected components is just . The stratification of the space of Fredholm operators is given by the index.

Definition 3 Given a Fredholm operator , we define the index of to be .

 

Analytic spaces

I’ve been trying to understand some complex analytic geometry as of late; here is an overview of Oka’s theorem.

Consider the space and the sheaf of holomorphic functions on it. One should think of this as the analog of complex affine space , with the Zariski topology, and with the sheaf of regular functions.

In algebraic geometry, if is an ideal, or if is a coherent sheaf of ideals, then we can define a closed subset of corresponding to the roots of the polynomials in . This construction gives the notion of an affine variety, and by gluing these one gets general varieties.

More precisely, here is what an affine variety is. If is a coherent sheaf of ideals, then we define a ringed space ; this gives the associated affine variety. Here the “support” corresponds to taking the common zero locus of the functions in . In this way an affine variety is not just a subset of , but a locally ringed space.

Now we want to repeat this construction in the holomorphic category. If is a finitely generated ideal—that is, an ideal which is locally finitely generated—in the sheaf of holomorphic functions on , then we define the space cut out by to be . We can think of these as “affine analytic spaces.”

Definition 1 An analytic space is a locally ringed space which is locally isomorphic to an “affine analytic space.”

 

The Weierstrass preparation theorem

First, thanks to all who kindly contributed advice on the previous post.

A long on-and-off project of mine has been to learn several complex variables. My latest attempt started a few days back, though the commencement of the fall semester may derail it. As is now customary for myself, I have started writing a set of notes that I intend to make grow reasonably large, though right now it is less than thirty pages (even with parts of my AG notes on coherence copied and pasted in). I shall post here an excerpt on the Weierstrass preparation theorem, which will assume only the definition of a holomorphic function in several variables).

 

Dirichlet’s theorem on primes in arithmetic progressions (digression)

So.  First off, surely the five remaining readers of this increasingly erratic blog have noticed the change of theme.

I want to next discuss the second inequality in class field theory, which is an upper bound on the norm index of the idele group.  There are two ways I know of to prove this: one analytic, one algebraic.  I will first sketch the analytic one. I say sketch because to do a full proof would get into the details of Dirichlet series, lattice points in homogeneously expanding domains, the construction of a certain fundamental domain for the action of the units, etc., etc., and I’d rather outline those ideas rather than do all the details because this is a series on class field theory. What I do plan on doing properly, however, is the algebraic (due to Chevalley in 1940) proof of the second inequality, which heavily uses results of field theory (e.g. Kummer theory) and local fields (e.g. power index computations).  I still thought it worthwhile to sketch the analytic approach, though. Rather than jumping right into it (I have to first say something about how the ideal and idele groups are connected), I decided to give an expository post on L-functions and Dirichlet’s theorem—in the case of the rational numbers.

1. Ramblings on the Riemann-zeta function

Recall that the Riemann-zeta function is defined by , and that it is intimately connected with the distribution of the prime numbers because of the product formula

valid for , and which is a simple example of unique factorization. In particular, we have

It is known that has an analytic continuation to the whole plane with a simple pole with residue one at . The easiest way to see this is to construct the analytic continuation for . For instance, can be represented as a certain integral for that actually converges for though. (The functional equation is then used for the rest of the analytic continuation.) The details are here for instance. As a corollary, it follows that

This fact can be used in deducing properties about the prime numbers. (Maybe sometime I’ll discuss the proof of the prime number theorem on this blog.) Much simpler than that, however, is the proof of Dirichlet’s theorem on the infinitude of primes in arithmetic progressions. I will briefly outline the proof of this theorem, since it will motivate the idea of L-functions.

Theorem 1 (Dirichlet) Let be an arithmetic progression with relatively prime. Then it contains infinitely many primes.

The idea of this proof is to note that the elements of the arithmetic progression can be characterized by so-called “Dirichlet characters.” This is actually a general and very useful (though technically trivial) fact about abelian groups, which I will describe now.

 

A funky lemma in complex analysis

I fell a bit behind on the continuation of the class field theory series because I was setting up a new laptop. Before I resume that, I want to talk about something very weird that I learned today.

Let be a set that omits at least two points. If holomorphic and is such that at one , then is the identity.

This is a striking rigidity phenomenon!

But how do we prove it? The idea is to consider the sequence of iterates . Suppose for simplicity . Then in a neighborhood of , we can write , where the are omitted higher terms. If is not identically the identity, then .

So, similarly, by direct computation, in some neighborhood of , we have . Similarly, if we define for notational convenience, we have

But the are all holomorphic maps into . Since omits at least two points, the family is normal by Montel’s theorem and consequently has a subsequence that converges uniformly on compact sets.

Thus the derivatives converge, which is impossible unless .

Huh? I didn’t exactly see that coming. If is the unit disk, then at least it looks familiar. A holomorphic map of the unit disk into itself sending zero to zero must satisfy , and if equality holds is a rotation. So perhaps this result should be thought of as a generalization of Schwarz’s lemma? (Nevertheless, the use of Montel’s theorem is quite a sledgehammer to prove something as elementary as Schwarz.)

I should say where I got this from: Krant’z Complex Analysis: The Geometric Viewpoint. Krantz didn’t prove exactly this, but the argument is the same.  Either this is standard fare that I missed when learning basic complex analysis, or I’m turning Climbing Mount Bourbaki into a comedy routine.

 

Applications of ergodic theory to equidistribution

We will now apply the machinery already developed to a few concrete problems.

Proposition 1 Let be a compact abelian group and the rotation by . Then is uniquely ergodic (with the Haar measure invariant) if is dense in .

 

The proof is straightforward. Suppose is invariant with respect to rotations by . Then for , we have

and hence

for any , which means that must be Haar measure (which is unique).

Corollary 2 An irrational rotation of the unit circle is uniquely ergodic.

 

Application: Equidistribution

 

Theorem 3 Let be irrational and let be continuous and -periodic. Then

 

Unique ergodicity

So, let’s fix a compact metric space and a transformation which is continuous. We defined the space of probability Borel measures which are -invariant, showed it was nonempty, and proved that the extreme points correspond to ergodic measures (i.e. measures with respect to which is ergodic). We are interested in knowing what looks like, based solely on the topological properties of . Here are some techniques we can use:

1) If has no fixed points, then cannot have any atoms (i.e. ). Otherwise would have infinite measure.

2) The set of recurrent points in (i.e. such that there exists a sequence with ) has -measure one. We proved this earlier.

3) The set of non-wandering points has measure one. We define this notion now. Say that is wandering if there is a neighborhood of such that . In other words, the family of sets is disjoint. If not, say that is non-wandering. Any recurrent point, for instance, is non-wandering, which implies that the set of non-wandering points has measure one.

Here is an example.