Gayana Jayasinghe

Lefschetz fixed point theorems

"Sum of the traces over the local holomorphic functions at each fixed point equals the trace over the global holomorphic function". That's a good first introduction to the holomorphic Lefschetz fixed point theorem. Let's look at some singular examples. These examples are worked out in more detail in this article.

Fig.1: Rotation of a sphere

Here's a picture of a 2 dimensional sphere, which has been rotated. There is a circle's worth of maps given by rotating the sphere around a fixed axis, parametrized (labelled) by the angle of rotation.

What is the equation? The two terms on the left hand side are "local holomorphic Lefschetz numbers" and correspond to the two fixed points on the sphere.

Consider the first of them. Taking a Laurent expansion, one gets sums of powers of lambda. This is the trace of the map induced by the rotation on the space of local holomorphic functions near the fixed point (with a basis 1, Z , Z^2,...). Since Z goes to lambda Z by the map, the trace is (1+ lambda + lambda^2 +...).

The term on the right hand side is the "global holomorphic Lefschetz number". Since the only global holomorphic functions are the constants, the map induced on that one dimensional vector space by the map is the identity, the trace of which is 1.

Fig. 2: Conic singularity of the "Cusp" curve

We are all used to seeing eyes scream cones. But what if the cross section was a knotted circle, as opposed to the standard circle? The knot in blue appearing here is called the Trefoil knot.

Why do I bring this up?

Consider deforming a torus (a doughnut) as follows. First pinch the loop in green to a point. Then make the hole smaller. What would we get in the limit?

Fig. 3: Deforming a torus (elliptic curve)

Well if we do this is a certain way, we get what is known as a cusp curve. If we look closely near the singular point, this is the shape we will see. What we saw in Fig. 2

Fig. 4: Cusp curve

In fact, this is the vanishing set of the polynomial that I've written here. If we just take the polynomial to be in two real variables, the curve in black will be the curve you see (all the points that satisfy the equation). If we allow them to be complex variables, then we see this object.

Fig. 5: Circle action ("Rotation") of a cusp curve

Rotating the knotted circle, we can see that there is a fixed point at the singularity. In fact we can find a space which is smooth away from one point where the singularity looks exactly like this cone over a trefoil, and there is a "rotation" which fixes two points, one of which is the singularity.

This "cusp curve" is called as such since the restriction of y^2=x^3 with complex valued x,y to real valued x,y, looks like a cusp, with the two black lines that meet at the point. I show that the metric is "wedge" (roughly asymptotically conic) when you consider the complex valued x,y values. Thus one can compute local and global Lefschetz numbers.

A pinched solid torus, before and after I performed surgery on it. It was delicious!

Now lets look at a stratified space with a depth 2 singularity. This is a four dimensional space! Or a caricature of one.

Fig. 6: Stratified space of depth 2

Here is a space with a singularity of depth 2. Here's how to construct it. Take a cone over a two dimensional torus, like those in blue drawn here. The cone point is then a depth 1 singularity. Then take a cone over that. The cone point of the new cone is a depth 2 singularity, while there is a depth 1 singularity which corresponds to the set of all cone points of the cones over the tori.

Do these arise naturally? For instance as a vanishing set of a polynomial? You bet!

Fig. 7: L^2 Holomorphic Lefschetz numbers on a singular toric variety.

Another 4 dimensional space. The equation in blue describes the entire space. Within it, there is a singular set when X and Z both vanish as well. This singular set can be thought of as a two dimensional sphere, roughly. And one has to imagine some other objects with cone angle (6pi) at every point of that green set.

The depth 2 singularity is when Y also vanishes.

More cryptic sums of rational functions.

Four dimensional spaces are hard (impossible!) to draw. Here's a more accurate figure of the same space we discussed above. The cone angles are (6pi) with a cone over a T(4,3) knot.

Fig. 8: Baum-Fulton-Quart Lefschetz numbers for the space in Fig. 8

Here we compute what is known as the Baum-Fulton-Quart Lefschetz number, on the same space for which we computed the L^2 version in Fig. 7.

So what are these weird rational functions that add up to polynomials?

You will find more rigorous descriptions in this article. They are holomorphic Lefschetz numbers in various cohomology theories. The polynomials appearing are global Lefschetz numbers and are well known to be supertraces on global cohomology. Roughly this means that it is counting something global.

The simplest phrasing of the Atiyah-Bott-Lefschetz fixed point theorem is the miraculous fact that the global Lefschetz number is equal to the sum of local Lefschetz numbers over fixed points of a "generalized rotation".

Basically when you have some "generalized rotation", you can find some local rational functions at the fixed points which are called local Lefschetz numbers, which reflect the local geometry of the singularities (or smoothness) at fixed points. I show that these are supertraces on local cohomology, which counts something local near fixed points, smooth or singular.

For instance in the figure above, there are two terms in black, reflecting that there is a 2 dimensional space of global functions, in a generalized sense. The three local terms are from the three fixed points of a "generalized rotation", one on the sets of each depth (the nonsingular set is depth 0). The rational polynomials at each point correspond to supertraces over infinite dimensional spaces of functions at the fixed points. That summing up infinitely many of them makes sense can be seen via carefully analyzing the supersymmetry of Dirac type operators and by an idea that is sometimes called renormalization.

Roughly, if you do power series expansions (any choice of Laurent series), that is the supertrace over infinitely many terms. That the sum of local rational functions add up to a simple polynomial can be explained in terms of what is sometimes called Supersymmetric cancellation by physicists.

In the rougher figure above, we have a different formula. That's relating a different class of local and global functions. I explain what is happening more carefully in the last example of the paper, including why there is an extra term in the right hand side of the equation on the last figure, as opposed to the right hand side equation on the preceding figure, explaining how one can write that function explicitly.

There are various such formulas for various Dirac type operators and various cohomology theories, even on the same space with the same generalized rotation.

This needs to be updated now. The Lefschetz formulas have been upgraded to Morse polynomials in this article, and there's some work for non-isolated fixed/critical points in this article. Will do as I have time.

These are some Vesak Lanterns from Sri Lanka.

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