Question

Let $M$ be a smooth manifold of dimension $m$ and $\pi:E\rightarrow M$ a vector bundle of rank $e$. Given a section $s$ of the bundle $\pi:E\rightarrow M$, we expect that the zero locus $Z(s)$ of $s$ is a submanifold $N\subset M$ of dimension $m-e$. This is true if $s$ can be perturbed into a general position so that $s(M)$ and the zero section intersect transversally.

Perturbation is not always possible (for example in holomorphic category category). In this case we need "excess intersection theory"; if the section $s$ lies in a subbundle $F\subset E$ and is a transversal section of $F$, the correct $(m-e)$-cycle we should take is the Euler class of the quotient bundle $E/F$, which is homologous to $Z(s)$ if transverse perturbation of $s$ exists.

My problem is that I don't really know good explicit examples with which I can compute things. Could anyone give me an example or reference, which shows how useful excess intersection theory is?

Edit My motivation to study excess intersection theory is virtual cycles of moduli spaces, which of course are very good examples of excess intersection theory. But I am looking for some elementary examples on which I can compute things. I want to convince myself that the theory is really reasonable by computing a few simple examples.

Answer 1

A good example might be the self-intersection of a submanifold $A\subset M$. We would like this to be the intersection of $A$ with a perturbation $A'\subset M$ such that the intersection is transverse. However once we start perturbing things, we lose control, so its better to notice that $A$ intersects itself cleanly (in the terminology of Bott and Quillen) and use the excess intersection formula.

In the setup you give, the bundle $\pi\colon E\to M$ is the tangent bundle of $M$, whose total space we can identify with an open neighbourhood of the diagonal $M\subset M\times M$. The section $s\colon M\to E$ is given by the diagonal embedding. The excess bundle $E/F$ is therefore identified with the normal bundle of $A$ in $M$. In this case, then, the excess intersection formula gives that the self-intersection of $i\colon A\subset M$ is given by the push-forward of the Euler class of its normal bundle, $i_!e(\nu_i)$.

This is of course a very basic example of the excess intersection formula. You'll find more in-depth Algebraic Geometry applications in the book "Intersection Theory" by William Fulton (see in particular chapters 6 and 9). In the topological setting, Quillen used excess intersections in his seminal work on cobordism theory ("Elementary proofs of some results in cobordism theory using Steenrod operations", Adv. Math. 7 1971 29–56 (1971)).

Answer 2

Surely not the simplest example but certainly one of the reason why excess intersection theory is useful is the theory of virtual fundamental classes. Suppose that A is some moduli space and that you can find M,E, s as in the question such that A = Z(s). E/F is called the obstruction bundle (it is the obstruction to the transversality of s). Then the pullback to A of the Poincare dual of the top chern class of the obstruction bundle is an homology class of the "expected " dimension m-e. In order to simplify suppose that this dimension is 0 then the integral of 1 over the virtual fundamental class has to be interpreted as the number of elements in A for a generic perturbation of the parameters of which M can depend, even if such a deformation actually does not exist.

In general, there does not exist M,E and s but sometimes it exists locally and one can still use excess intersection theory in order to define a virtual fundamental class. Most of the counting theories of curves uses this idea (Gromov-Witten, Donaldaon-Thoms, stables pairs ...)(see for example for a survey "13/2 ways to count curves" of R.Pandharipande and R.P.Thomas).

In this way, one obtains a lot of examples which can be seen as application of excess intersection theory : when A is smooth, the obstruction is really a bundle and you just have to calculate a top chern class (just to cite a real "concrete" simple example : Gromov-Witten invariants for homology class 0 can be written has an integral of caracteristic classes over the moduli space of curves. There are surely thousands of such examples.)