The Multiplicity-polar Theorem

The heart of intersection theory resides in the following situation. Given a purely d-dimensional analytic set X, and subsets V and W intersecting at a point x ∈ X, V and W given by i and j equations respectively where i + j = d, then the correct number of times to count x is the multiplicity of the ideal I in the local ring of X at x, where I is the ideal generated by the equations of V and W. The multiplicity is the correct number, because when we deform V so that it intersects W transversely at smooth points of W , the number of points is exactly the multiplicity of I. In many situations it is desirable to solve the following intersection problem. Given a vector bundle of rank e, on a purely d-dimensional analytic set X, and two collections of sections of the bundle, such the total number of sections is d + e − 1 and an isolated point x ∈ X, where the sections fail to have rank e, then how many times should we count x, if we are counting the number of points where our two collections fail to have maximal rank? This problem occurs in calculating the contribution of a point to the top Chern class of a bundle on X or the index of a differential 1-form with an isolated singularity on X, X with an isolated singularity. We may ask for more. First, just as in the case of intersecting sets we replace functions by ideals, we replace collections of vector fields by the module M they generate. Suppose the set where the rank of M is less than maximal is non-isolated, but in a natural way from the problem being considered, the module M is a submodule of a module N where M is of finite colength inside N. An example of this situation is given by a 1-form on a space with isolated singularities, which is not a complete intersection, in which the 1-form still has an isolated singularity in some sense. We would like to calculate the index of the 1-form, in a way which fits well with deformations of the original 1-form. (In this case, the modules M and N are described in section 2 in the material around Theorem 2.8.) In this paper we show that the correct solution to this problem is …