The Multiplicity Polar Theorem and Isolated Singularities
  Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse intersection. A problem with the multiplicity of an ideal or module is that it is only defined for modules and ideals of finite colength.   In this paper we use pairs of modules and their multiplicities as a way around this difficulty. A key tool is the multiplicity-polar theorem which applies to families of pairs of modules, linking the multiplicity at a general pair of the family to the multiplicity of the pair at the special fiber.   The specific problem we work on is:   Suppose X,0 is an equidimensional complex analytic set in C^n, and T^*_X(C^n) is the closure of the conormal vectors to the smooth part of X, s, a section of T^*(C^n), the cotangent bundle of C^n restricted to X. Suppose the intersection of the image of s, denoted im(s), with T^*_X(C^n) is isolated. Then we calculate the intersection multiplicity im(s)\cdot T^*_X(C^n) using the multiplicity of pairs. It is easy to see that if X=C^n and s(x)=df(x), where f has an isolated singularity at the origin, then this number is the Milnor number of f.   Variations of this number appear in several different situations, such as, the theory of differential forms on singular spaces, the theory of D-modules, the Milnor fiber of a function with an isolated singularity, the A_f stratification condition of Thom, and the theory of the Euler invariant. We discuss the application of our result to all of these situations. 
Terence Gaffney