Examples of excess intersection theory? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T21:06:47Z http://mathoverflow.net/feeds/question/107292 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107292/examples-of-excess-intersection-theory Examples of excess intersection theory? Koopa 2012-09-15T22:53:50Z 2012-09-17T00:36:57Z <p>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. </p> <p>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. </p> <p>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?</p> <p><strong>Edit</strong> 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. </p> http://mathoverflow.net/questions/107292/examples-of-excess-intersection-theory/107311#107311 Answer by Mark Grant for Examples of excess intersection theory? Mark Grant 2012-09-16T10:39:49Z 2012-09-16T10:39:49Z <p>A good example might be the <em>self-intersection</em> 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 <em>cleanly</em> (in the terminology of Bott and Quillen) and use the excess intersection formula.</p> <p>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, $i_!e(\nu_i)$.</p> <p>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)). </p> http://mathoverflow.net/questions/107292/examples-of-excess-intersection-theory/107320#107320 Answer by unknown (google) for Examples of excess intersection theory? unknown (google) 2012-09-16T14:06:08Z 2012-09-16T14:06:08Z <p>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.</p> <p>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).</p> <p>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.)</p>