The Multiplicity Polar Theorem and isolated singularities

Abstract

We show how the Multiplicity Polar Theorem can be used to calculate invariants which describe an “isolated singularity”. Examples include the defect of a function, which is related to the Euler obstruction, the index of a differential form, the dimension of the space of vanishing cycles of a sheaf of D-modules M M relative to a function f f at 0 0 , and a formula for the relative cohomology of the Milnor fiber of f f where f f has an isolated singularity on a complex analytic set with possibly non-isolated singularities. We apply the result on the defect to refine previous work on the A f \mathrm {A}_f condition.