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.
Best Answer
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)).